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UNIV. or
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LIBRARY
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THE MONIST
A QUARTERLY MAGAZINE
DEVOTED TO THE PHILOSOPHY OF SCIENCE
VOLUME XXVII
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CHICAGO
THE OPEN COURT PUBLISHING COMPANY
1917
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COPYRIGHT BY
THE OPEN COURT PUBLISHING COMPANY
1916-1917
CONTENTS OF VOLUME XXVII.
ARTICLES AND AUTHORS.
PACE
Bacon, Ernst Lecher. Our Musical Idiom 560
Bergsonism in England. By J. W. Scott 179
Bolzano, Bernard (1781-1848). By Dorothy Maud Wrinch 83
Burns, C. Delisle. A Medieval Internationalist (Pierre Dubois) 105
Bussey, Gertrude Carman. Mechanism and the Problem of Freedom 295
Cal-Dif-Fluk Saga (Poem). By J. M. Child 467
Carus, Paul. Belief in God and Immortality, 311; A Chinese Poet's Con-
templation of Life (translated poem), 128; Determinism of Free
Will, 306;£Leibniz and Locke J137; Nirvana (poem), 233; Sir
Oliver Lodge on Life After Death, 316.
Chatley, Herbert. Idealism as a Force : A Mechanical Analogy 151
Child, J. M. Cal-Dif-Fluk Saga (Poem), 467; The Manuscripts of Leib-
niz on his Discovery of the Differential Calculus, Part II, 238, 411.
Chinese Poet's Contemplation of Life, A (Su Tung P'o). Tr. by Paul Carus 128
Class, Function, Concept, Relation. By Gottlob Frege 114
Confucianism, Classical. By Suh Hu 157
De Vries, Hugo. The Origin of the Mutation Theory 403
Determinism of Free Will. By Paul Carus 306
Dogma? What is a. By Edouard Le Roy 481
Dubois, Pierre, a Medieval Internationalist. By C. Delisle Burns 105
Edmunds, Albert J. The Text of the Resurrection in Mark and its Testi-
mony to the Apparitional Theory 161
Electronic Theory of Matter, The. By William Benjamin Smith 321
Existents and Entities. By Philip E. B. Jourdain 142
Feingold, Gustave A. The Present Status of the Unconscious 205
Free Will, Determinism of. By Paul Carus 306
Freedom, Mechanism and the Problem of. By Gertrude Carman Bussey. . 295
Frege, Gottlob. Class, Function, Concept, Relation 114
Gerhardt, Karl Immanuel. ^Leibniz in London^ 524
Grassmann, Hermann (1809'-1877). By A. E. Heath 1
Grassmann, The Geometrical Analysis of, and its Connection with Leibniz's
Characteristic. By A. E. Heath 36
Grassmann, The Neglect of the Work of. By A. E. Heath 22
iv THK MONIST.
PAGE
k l.lca* <>i .4n Aiu-i \\..rld. By Orland O. Norris 57
I lermann Grassmann, 1 ; The Neglect of the Work of Grass-
niann. 22; The Geometrical Analysis of Grassmann and its Con-
nection with Leibniz's Characteristic, 36.
1 lu, Suh. Classical Confucianism 157
Idealism as a Force: A Mechanical Analogy. By Herbert Chatley 151
Immortality, Primitive and Modern Conceptions of. By J. II. Leuba 608
Internationalist, A Medieval. By C Delisle Burns 105
Jourdain, Philip E. B. Existents and Entities, 142; Logic and Psychology, 460
Koopman, Harry Lyman. Libra: The Eternal Balance of Good and 111
(Poem) 455
rLeibniz and LockcJ By Paul Carus 137
VLeibniz's Characteristic The Geometrical Analysis of Grassmann and its
Connection with. By A. E. Heath 36
/ Leipniz in London. By Karl Immanuel Gerhardt 524
\Leibniz, The Manuscripts of, on his Discovery of the Differential Cal-
culus, Part II. By J. M. Child 238, 411
Le Roy, Edouard. What is a Dogma ? 481
Leuba, J. H. The Primitive and the Modern Conceptions of Personal Im-
. mortality 608
Libra: The Eternal Balance of Good and 111 (Poem). By Harry Lyman
Koopman 455
Liu, King Shu. The Origin of Taoism 376
Locke, Leibniz and. By Paul Carus 137
Logic and Psychology. By Philip E. B. Jourdain 460
Mark, The Text of the Resurrection in, and Its Testimony to the Appa-
ritional Theory. By Albert J. Edmunds 161
Mechanism and the Problem of Freedom. By Gertrude Carman Bussey. . 295
Medical Science and Practice, The Contributions of Paracelsus to. By
J. M. Stillman 390
Musical Idiom, Our. By Ernst Lecher Bacon 560
Mutation Theory, The Origin of. By Hugo De Vries 403
Nirvana ( Poem) . By Paul Carus 233
Norris, Orland O. Greek Ideas of an Afterworld 57
Paracelsus, The Contributions of, to Medical Science and Practice. By
J. M. Stillman 390
Perry, Ralph Barton. Purpose as Systematic Unity 352
Philosophy of Science, Notes on Recent Work in 618
Psychology, Logic and. By Philip E. B. Jourdain 460
Purpose as Systematic Unity. By Ralph Barton Perry 352
Scarlet Cliff, The (Chinese Poem). Tr. by Paul Carus 128
Scott, J. W. Bergsonism in England 179
Smith, David Eugene. Notes on De Morgan's Budget of Paradoxes 474
Smith, William Benjamin. The Electronic Theory of Matter 321
Stillman, J. M. The Contributions of Paracelsus to Medical Science and
Practice 390
Taoism, The Origin of. King Shu Liu 376
Unconscious, The Present Status of the. By Gustave A. Feingold 205
Wrinch, Dorothy Maud Bernard Bolzano (1781-1848) . ... 83
CONTENTS OF VOLUME XXVII.
BOOK REVIEWS AND NOTES.
PAGE
American Mathematical Monthly 625, 628, 629
Bollettino di bibliografia e storia delle sciense matcinatichc 320
Bulletin of the American Mathematical Society 319, 630, 631
Cook, Stanley A. The Study of Religions 480
Dawson, Miles Menander. The Ethics of Confucius 157
De Morgan, A. Budget of Paradoxes 474
Franceschi, Pietro. De corporibus regularibus 319
Jourdain, Margaret (Tr.) Diderot's Early Philosophical Works 639
Keyser, Cassius J. The New Infinite and the Old Theology 479
Le Roy, Edouard. The New Philosophy of Henri Bergson 640
Leuba, James H.. The Belief in God and Immortality 311, 608
Lodge, Sir Oliver. Raymond, or Life After Death 316
Mind 634
Rendiconti delta R. Accademia del Lined 319, 621
Revue de mctaphysique et dc morale 320, 618, 620
Science 623
Scientia 624, 637, 638
Sorel, Georges. Reflections on Violence 478
Transactions of the American Mathematical Society 319
VOL. XXVII. JANUARY, 1917 NO. i
THE MONIST
HERMANN GRASSMANN.
1809-1877.
WE like to believe that the final significance of any
thinker's work is independent of his time and place
and is fixed by reference to some absolute standard. How-
ever that may be, it seems quite clear that his importance
in his own age, and hence his effect on the next succeeding
generations, depends to some extent on other factors than
his intrinsic value. And so in judging that value we must
distinguish plainly between it and what we might call the
relative or historical importance of the man's work. This
latter may well be compared to the potential of a body in
electrostatics. For just as that potential depends not only
on the actual charge on the body, but also on the charges
on neighboring bodies; so also the relative importance of
a man is not determined alone by the content of his life
and work, but is affected also by his milieu and by the
reactions of that milieu to it.
This is the reason why the contemporary estimate of a
thinker is often so utterly wrong. At the time, the external
man and his work are more easily seen; but the subtle
tendencies of the age are not so readily understood, nor
can the observer escape the distortion of vision wrought
by prevailing influences on himself. So it comes about
that he who is written down a failure in one age may stand
out a very genius in the next.
These reflections are quite pertinent to any inquiry into
2 THE MONIST.
the life and work of the author of the Ausdehnungslehre —
Hermann Giinther Grassmann, the distinguished mathe-
matician whose own generation passed him by. Although
he reached eminence in other branches of human activity,
we speak of him as a mathematician because that was cer-
tainly the subject he loved most and in which his influence
will be most felt in the future. Of him, on the occasion
of his centenary (1909), F. Engel1 could say: "To-day he
is known by name to mathematicians, but few have read his
writings. Even where his ideas and methods have been dif-
fused in mathematical physics people learn them second-
hand, sometimes not even under his name." So in Grass-
mann we have a straightforward example of a man be-
tween whose work and whose influence on his own and
immediately succeeding generations we must sharply dis-
tinguish if we are to avoid underrating his significance.
He was born on April 15, 1809, m Stettin.2 His father,
Justus Gunther Grassmann, was a teacher in the Gym-
nasium there, and was himself a good mathematician and
physicist.8 His school days passed without his showing
any inclination or aptitude in special studies. He had
however great skill in and fondness for music, and received
a good foundation in piano and counterpoint from the
famous composer Loewe. The latter was appointed teacher
in the Stettin Gymnasium in 1820 and lived for the first
year in the house of the Grassmanns, where he found very
congenial society in Hermann and his brothers and sisters,
all of whom were musical. With them Loewe often used
to try over his new quartettes.
Of Grassman's inner development during these out-
1 F. Engel, Speech on "Grassmann in Berlin," to the Berliner Mathema-
tische Gesellschaft (1909). To this I owe most of the information about
Grassmann's early life given in what follows.
2 The same date as Euler.
8 He invented an air-pump cock which was given his name, and also con-
structed a useful index notation of crystals.
HERMANN GRASSMANN. 3
wardly calm and uneventful years we can form a clear
picture from his own writings. For in 1831 he wrote an
account of his life in Latin in connection with the examina-
tion for his teacher's certificate; and later, in 1834, he
handed in an autobiography to the Konsistorium in Stet-
tin when he was passing his first theological examina-
tion. He refers to those earlier years as a period of slum-
ber, his life being filled for the most part with idle reveries
in which he himself occupied the central place. He says
that he seemed incapable of mental application, and men-
tions especially his weakness of memory. He relates that
his father used to say he would be contented if his son
Hermann would be a gardener or artisan of some kind,
provided he took up work he was fitted for and that he
pursued it with honor and advantage to his fellow men.
As he usually spent his holidays in the country among
relatives, and nearly always in the families of clergymen,
he conceived the desire to prepare himself for the ministry.
But he soon came, partly from the ridicule of his compan-
ions and partly from the warnings of his parents, to doubt
his capacity. He says however that, after his course of
instruction for confirmation, a light came into his dreams.
Suddenly he determined to exercise all his intellectual
powers and to overcome as far as possible the phlegmatic
character of his temperament. And this resolution he
carried out with resistless energy.
F. Engel4 sums up these early years in the following
words : "He does not belong to those early ripening geniuses
who, even in childhood's years, know whither their gifts
will lead them, and turn without doubt or hesitation to that
branch of knowledge to which they are called. He was
exceptionally gifted on too many sides for that. But even
these many-sided gifts by no means showed themselves at
the beginning; and that they developed themselves richly
4 THE MONIST.
later came by no means without effort, but was the direct
result of many years of concentrated work which he did
in order to develop his character and to solidify his moral
outlook and grasp of life."
In the August of 1827 Grassmann and his elder brother
Gustav entered the University of Berlin with the intention
of studying theology. Two days after their arrival Her-
mann wrote a droll letter to his mother vividly describing
how they had settled in. He tells how they had to climb
seventy-two steps to their attic dwelling at 53 Dorotheen-
strasse at the corner of Friedrichstrasse. They had only
room for their beds and two chairs, but he comments hu-
morously on their extra fine look-out over the gardens and
houses of the city, and adds that though the rooms were
small they could be the more easily heated. His landlady
will be recognized by students all the world over in his pen
picture, "If she does talk too much, she is very pleasant
and industrious." Particularly amusing is the manner in
which he tells how they had spent practically all their
money in two days. He enumerates all the possible and
impossible things on which they had not spent the money,
and finally confesses that their sudden impecuniosity was
due to the piano which they luckily bought for 50 Taler.
Grassmann admitted later that when he first came to
the university he was quite dependent on the guidance of
the professors. He was easily impressed by the lectures
he heard and tended to fit in his studies with the lectures he
chanced upon rather than to take those corresponding to a
course of study. At first he came specially under the in-
fluence of the well-known church historian Neander. Grad-
ually, however, he became attracted more and more to
Schleiermacher to whom he acknowledges great indebted-
ness. He wrote: "Early in my second year I attended
Schleiermacher's lectures, which of course I did not under-
stand ; but his sermons began to exercise an influence upon
HERMANN GRASSMANN. 5
me. However it was not until my third, and last, year that
Schleiermacher entirely engaged my thought, and although
at that time I was more occupied with philology, yet I then
for the first time recognized how one could learn something
from him for every branch of knowledge, because he aimed
less at giving positive information than at making us ca-
pable of attacking each investigation in the right way and
of carrying it on independently." From this we can see
how Grassmann was coming to feel the joy of original
creative work.
Though he had studied theology with his heart in his
subject, he had by this time reached the decision to lay it
aside. He says that he had noticed that clergymen who
lived in country parishes, shut off from intercourse with
scholars, lost grasp of their studies, however enthusiastic
they had previously been, and ceased to pursue any investi-
gation on their own account. To escape such a fate he
decided to prepare himself as broadly as possible. For this
reason he began the study of philology, but he continued
it from sheer love of the subject. He had also made the
discovery by this time that academic lectures are only of
profit if taken in moderation ; so he confined himself to two
courses under Professor Boeckh, on the history of Greek
literature and on Greek antiquities respectively. But he
planned out a tremendous course of study, intending to
begin with Greek grammar, then to read the Attic authors
chiefly the historians, with the study of whom he would
combine Greek history and antiquities — next the trage-
dians with mythology and poetic forms, and afterward
Homer and Herodotus. Meanwhile he would seek variety
by reading Roman authors. Finally, as he intended to
follow his linguistic studies with mathematics, he meant
to save Plato and Demosthenes until he began that study.
This exhaustive program he was not able to complete
in Berlin. When he had reached the Attic authors he was
6 THE MONIST.
taken ill in consequence of over-work. He describes his
illness as neither severe nor dangerous, but it compelled
him to slow down and to introduce more variety in order
to avoid mental strain.
In this way he was led to the study of the sciences, but
he showed his growing independence by working free of
the schools. He did not attend a single mathematical lec-
ture while a student in Berlin.
We may now see how wide his range of interests was
throughout his university career. He seems to have been
striving for as broad a foundation as possible, while at
the same time he was building up a truly scientific attitude
of mind which would enable him successfully to attack any
subject he might turn his attention to. It is as though, as
Engel says, he knew from the first that it would be neces-
sary in his life to have more than one iron in the fire.
In the autumn of 1830 he returned to Stettin, and late
in the following year took an examination for a teacher's
certificate before the Scientific Examination Commission
in Berlin. It was at this examination that he handed in
the Latin autobiography we have previously referred to,
and concerning which Kopke, rector of the monastary
school of the Grey Friars in Berlin, comments "Specimen
turn propter rerum ubertatem turn propter stili venustatem
et elegantiam laude dignum." He was given permission
to teach philology, history, mathematics, German and re-
ligious knowledge in lower and middle classes; but the
commission at the same time expressed their expectation
that he might easily perfect himself for teaching ancient
languages and mathematics in all classes. This may have
stimulated Grassmann to further mathematical studies,
though he had already thrown himself with energy into
them under the influence of his father, whose text-books
he would naturally use.
HERMANN GRASSMANN. 7
He became assistant teacher (Hilfslehrer) in the Stet-
tin gymnasium, and in 1832 began to lay the foundations
of his great work, the "Theory of Extension" (Ausdeh-
nungslehre). He began by working at the geometrical
addition of straight lines, or what we now call vector addi-
tion. From this he was led to the notion of the geometrical
product of straight lines. The direct influence of his father
can best be shown by his own words :5 "But I had not the
slightest idea into what a rich and fruitful province I had
here arrived; rather did this result6 appear to me to be little
worthy of notice until I combined it with a closely related
idea. Namely, by following it up with the same idea of the
product in geometry as my father had held,7 it became
evident to me that not only the rectangle but also the
parallelogram in general may be considered as the product
of two adjoining sides."
He goes on to add that he was surprised to find that
he had thus reached a product which changed in sign if
its factors were interchanged. And this, together with
the fact that he was drawn into other spheres of work —
one of which was the passing of the first theological exam-
ination at Stettin — caused this seed-idea to remain dormant
for some considerable time.
In October, 1834, Grassmann returned to Berlin, this
time as mathematics master in a trade school. Soon after-
ward he applied for a better position than the one he held
and his principal gave the following characterization of
him: "Mr. Grassmann is a young man not lacking in at-
tainments. It is also apparent that he has given particular
attention to the elements of mathematics, and thinks with
especial clearness along that line, but he seems to have had
6 Preface to the first edition of the 1844 Ausdehnungslehre.
6 The notion of writing AB -f- BC = AC whether the three points A, B, C
are in the same straight line or not.
7 Cf. J. G. Grassmann, Raumlehre, Part II, p. 164, and his Trigonometrie,
p. 10.
g THE MONIST.
little intercourse with people and is therefore backward
in the usual forms of social life, shy, easily embarrassed
and then very awkward. In the classroom all this vanishes
when he does not know that he is observed. He then
moves with ease, control, and certainty. In my presence,
in spite of the fact that I have done all I could to give him
confidence, he has not been able to become fully master of
his embarrassment, which caused him much concern. My
judgment of him is therefore as yet uncertain, and I cannot
say whether he will be able suitably to fill the present
vacancy."
As a matter of fact the vacancy was not an easy one to
fill, since it had previously been held by no less a person
than Jacob Steiner, the geometrician, who had been ap-
pointed to the university but retained some of the higher
classes in geometry. Grassmann obtained the appoint-
ment ; and as Steiner had bound himself to initiate his suc-
cessor as far as possible into his own method of geometrical
instruction, one would have expected interesting develop-
ments from the contact between the two men. There ap-
pears however to have been very little intimacy between
them. There was a difference of thirteen years in their
ages, and a wide contrast in temperament — the one self-
reliant but thoroughly one-sided, the other diffident and
many-sided. To these differences in personal character-
istics Carl Musebeck8 is inclined to attribute their small
effect on each other. Victor Schlegel's9 view was that it
was caused by the great difference in the methods em-
ployed by the two mathematicians. Whatever may have
8 Carl Musebeck, article on Hermann Grassmann, No. 3, Jahrgang 6 of the
Mathematisch-Natururissenschaftliche Blatter, p. 1, note.
9 It is curious to note that V. Schlegel, who, as we shall see, was one of the
first appreciators of Grassmann's work, long afterward used the methods of
Grassmann's "Geometrical Analysis" to attack the problem of the minimum
sum of the distances of a point from given points (Bull. Amer. Math. Soc.,
Vol. I, 1894, p. 33) and reached a general result which reduces to Steiner's
form of solution as a special case ; thus illustrating the power of the method.
HERMANN GRASSMANN. 9
been the cause it is at any rate clear that Steiner's method
of handling geometry had no influence whatever upon
Grassmann's manner of thinking.
Several things combined to make Grassmann's stay in
Berlin short. He was greatly distressed by the loss of his
youngest sister, who was scarcely four years old, and this
increased his inclination to religious brooding — to which
he was the more inclined as he lacked suitable companion-
ship. His eyesight also gave him some trouble, so that
after a year and a quarter he gladly returned to Stettin on
January I, 1836, and became teacher in the Ottoschule.
He had, however, pleasant memories of these months
in Berlin, as we can see from a letter written to his brother
Robert, in which after speaking with pleasure of his return
to Stettin he acknowledges the freedom and mental stimu-
lation afforded by Berlin. At first glance this move from
the capital seems a pity, since recognition of his talents
might have come to him if he had stayed on. But we must
remember to set against this, that he was very high-strung
and energetic in mind and could be easily over-stimulated
— an effect helped by the quiet life he lived — and also that
a calmer atmosphere was more suitable to the long and
careful development of his very original way of thought.
While still at the Ottoschule Grassmann entered for
and passed the second theological examination in Stettin
in July, 1839. We may note here that he was deeply at-
tached to the study of positive theology throughout his life.
After passing his theological examinations he became sec-
retary and then president of the "Pomeranian Central So-
ciety for the Evangelization of China." And it is note-
worthy in this respect that his last work was on "The
Falling Away from Belief."
A few months before he submitted his essay for this
last theological test, he was examined by the Berlin Scien-
IO THE MONIST.
tific Examination Commission in mathematics and physics.
It was in connection with this that an event fraught with
great consequences to his lifework happened to Grassmann ;
for he was set the task, by Professor Conrad of the Joa-
chimsthal Gymnasium, of developing the theory of tides.
It is uncertain whether the subject was chosen by Conrad
on his own initiative or was suggested by Grassmann him-
self. In any case it was precisely the practical need which
was best calculated to spur him on to the development of
his dormant mathematical ideas. Later on he spoke10 of the
necessity, in expounding the claims of a new mathematical
discipline, of showing its application. And it seems clear
that, faced with the difficulties and complications of La-
place's tidal theory, he was led at once to the idea of trans-
forming analytical mechanics by the introduction of his
own rudimentary analytical notions. He found to his de-
light that the new analysis proved a powerful simplifying
tool when applied to the equations of Lagrange's Mecanique
analytique. This initial success encouraged him to extend
his method and to clothe many other conceptions such as
exponentials, the angle, and the trigonometrical functions,
in the form of that analysis. He was then able to simplify
and render symmetrical the intricate formulas of the tidal
theory. Furthermore he found that the elimination of ar-
bitrary coordinates so effected left the ideas, their develop-
ment, and their interrelations much less obscured by ana-
lytical machinery.
The thesis Grassmann sent to Berlin in April 1840 was
of an unusual size ;" and, in the opinion of Engel,12 "judged
by the number of new thoughts and methods contained in
it, there is only one other to be compared with it — the thesis
which Weierstrass submitted a year later to the Commis-
10 In the Preface to the first edition of the Ausdehnungslehre of 1844.
11 It fills 190 pages of royal octavo in the third volume of his Werke.
12 F. Engel, he. cit.
HERMANN GRASSMANN. II
sion at Munster." The two works were, however, ac-
corded very different receptions; and it is evident that
Professor Conrad had no idea of the remarkable work he
had called into being. His report runs: "The test treats
the theory of the tides with thoroughness and strength
throughout; and he has chosen, not unhappily, a peculiar
method which departs in many particulars from the theory
of Laplace." It remains an evil omen for the fate of
Grassmann's later work that his examination thesis should
thus have failed to find recognition. It must be added
that Conrad could scarcely have read the work and still
less have been able to estimate it at its true value. For
he received it on May 26 and returned it five days later at
the oral examination — in which Grassmann fared better,
being granted full recognition of his mathematical ability.
Grassmann probably realized that this thesis on tidal
theory was but a first fruit of his methods and that those
methods themselves were much more general and capable
of immense development. This work he threw himself
into with characteristic energy in the next few years. He
left the Ottoschule at Michaelmas, 1842, and spent six
months teaching at the Stettin Gymnasium; after which
he entered the Friedrich-Wilhelm-Schule which had been
founded a few years before, and of which his eldest son
Justus Grassmann is now the principal.
By 1842 Grassmann had completed the main outlines
of his new analytical method. He tried to make the ideas
known to his own circle by lectures, in which he showed
the power of the new "science of extended magnitudes"
by further application to mechanics and crystallography.
Desiring to expound his method by reference to well-known
results he was led to the barycentric calculus of Mobius
and to Poncelet. The first of these illustrations was the
"Theorie der Zentralen" (Crelle's Journal, Vol. XXIV,
12 THE MONIST.
1842) in which, without using his own analysis, he made
a general statement in which not only all Poncelet's results
but also further important general properties of curves
and surfaces are contained as special cases. Such wide
generalization is characteristic of his method. In 1844 h*3
Ausdehnungslehre was published, being designed as the
first part of the complete work. This part, which he pro-
posed to follow up with a second later, he called" Die lineale
Ausdehnungslehre, a new branch of mathematics."
The fate of this book was a tragic one. It remained
unread and unsold until the publisher had to get rid of the
whole edition as waste paper. Not even a review was
granted to it; and what criticism there was had so little
basis of understanding that it led to no deeper study of the
work. Gauss wrote of it, in 1844, that its tendencies partly
went in the same direction in which he himself for almost
half a century had wandered; but there seemed to him to
be only a partial and distant resemblance in the tendency.
He thought it would be necessary to familiarize oneself
with the special terminology to get at the real kernel of
the book. Grunert declared that he had not completely
succeeded in forming a definite and clear opinion about the
work. Mobius, whom Grassmann had asked for a review
in some critical journal because he stood nearest to the
ideas in the book, answered that this mental relationship
only existed in regard to mathematics, not with reference
to philosophy; and that he considered himself incapable of
estimating and appreciating the philosophical element of
the excellent work — which lies at the base of all mathe-
matics. But he added that he recognized that, next to the
great simplification of method, the principal gain consisted
in the fact that by a more general comprehension of funda-
mental mathematical operations the difficulties of many
analytical concepts are removed.
HERMANN GRASSMANN. 13
Without entering in detail into a discussion of the
causes of this neglect of Grassmann's work13 we may note
that its great generality, its philosophical form, and its
original and technical symbolism were contributing factors
which also make it very difficult to give any account of the
work for the general reader.14 But the importance of the
ideas hidden away in this forbidding volume may be gath-
ered from the words written of it by Carl Miisebeck many
years later: "Earlier than Riemann, Grassmann evolved
manifolds of n dimensions in mathematical analysis. In
a lighter and less constrained manner Grassmann arrives
by his combinatory multiplication at the fundamental prin-
ciples of determinant-theory, and the elementary solution
of various problems of elimination. In him one finds indi-
cated both Bellavitis's Equipollences and Hamilton's Qua-
ternions." And yet the only recognition given by mathe-
maticians to the ideas of Grassmann was the award to him
by the Jablonowski Society at Leipsic for a prize essay15 on
the "Geometrical Calculus of Leibniz" in 1846.
It must not be supposed, however, that Grassmann sat
quietly down to neglect. He brought out the importance
and applicability of his investigation by numerous valuable
articles in Crelle's Journal, and later in Mathematische
Annalen and the Nachrichten of the Royal Society of Sci-
ence of Gottingen. Furthermore, in 1845 ne published in
Grunert's Archiv, Vol. VI, a detailed abstract16 of the Aus-
dehnungslehre, intended for mathematicians. Thirty years
later Grassmann spoke to Delbriick17 with youthful ardor
18 See the article below on "The Neglect of the work of H. Grassmann."
14 An attempt was made to do this by Justus Grassmann in an address
delivered at the opening of his school year on April 16, 1909, when the cen-
tenary of his father was being celebrated.
15 Geometrische Analyse, published 1847. This treatise is to some extent
a substitute for the second part of the Ausdehnungslehre of 1844, anticipated
in the preface to that work but never written.
16 Reprinted in the Werke, Vol. I, Part I, p. 297.
17 B. Delbriick, "Hermann Grassmann," Supplement to the Allgemeine Zei-
tung, Oct. 18, 1877.
14 THE MONIST.
of this period as one of happy restlessness and joy in dis-
covery. Such joy in original work and faith in the power
of his mathematical methods he always retained in spite
of a succession of disappointments which would have
quenched a less ardent spirit.
It is an extraordinary thing that it was not only in his
mathematical work that he failed to find recognition, but
also in his contributions to physics. In 1845 ne published
in Poggendorff's Annalen a statement of the mutual inter-
action of two electric stream lines which was re-discovered
thirty-one years later by Clausius. In a school syllabus
in 1854 Grassmann stated that the vowels of the human
voice owe their character to the presence of certain partial
tones of the mouth cavity, a view of the nature of vowel
sounds which is usually ascribed to Willis and Helmholtz.
Of his other purely physical work we may mention his
notes on the mixing of colors and his design of a very
simple but practical heliostat.18 Still he continued to hope
that the value of his work would be appreciated. He had
himself foreseen19 that the dislike of mathematicians for a
philosophical form might deter them from considering his
work, and the comments of Mobius and Grunert on this
had shown his fears to be well founded. So he yielded to
the often expressed wish of Mobius that he should rewrite
the Ausdehnungslehre. in a form more attractive to mathe-
maticians. In the new work, published in 1862, he chose
a more deductive method — one moreover which is not alto-
gether suited to the subject matter, but it did succeed in
bringing forward more clearly the original operations and
characteristics of the Ausdehnungslehre. All was in vain.
Neither genius nor indomitable energy could contend
against so unresponsive an environment.
We must remember that Grassmann's continued output
18 A model was constructed by the Stettin Physical Society.
19 Preface to the first edition of the 1844 Ausdehnungslehre.
HERMANN GRASSMANN. 1 5
of virile original work was done in the scanty leisure of
an energetic schoolmaster. He had been nominated head-
teacher at the Friedrich-Wilhelm-Schule in 1847, and five
years later he was appointed successor to his father at the
gymnasium. There he remained for a quarter of a cen-
tury. He had hoped that his mathematical writings would
win for him some position in which he would have more
leisure for research and be in closer contact with other
scientific workers. But it must not be supposed for an
instant that this lessened his intense interest in the work
at hand. He wrote articles on educational subjects as
well as a number of text-books for school use. Of these
his Arithmetik, written in collaboration with his brother
Robert showed a strictness in its proofs which made it a
good introduction to the theory of numbers. His Trigono-
metrie has a richness of content in small space and an
originality of plan not often then found in elementary hand-
books.
Miisebeck has questioned some of Grassmann's pupils
on his methods of teaching. They appear in the main to
agree with Wandel, who says in his "Studies and Char-
acters from Ancient and Modern Pomerania" that he was
a lovable and painstaking master whose kindly instruction
was sometimes too difficult for them. The lively interest
he took in the independence of those he taught is shown by
the fact that, according to Schlegel, he formed a society
out of every three scholars in his chemistry class, the
members of which had to demonstrate and lecture to the
others on some substance and its combinations. The pleas-
ant footing he established between himself and his classes
may be judged from the fact that they were willing to co
operate in classwork with him when in later years he had
to be taken to school in a wheeled chair. Whenever any of
his old pupils speak of him they do so with the greatest
admiration and respect.
l6 THE MONIST.
It is difficult in thus giving an account of Grassmann's
educational and scientific activity to avoid at the same
time conveying the impression of a mere enthusiastic ped-
ant. It does not seem that there could have been time for
anything else. And yet such a view would be widely re-
moved from the truth. For in the midst of all these exact-
ing duties he had many social and general interests. In
1848 he took an active part in politics, expressing anti-
revolutionary sympathies; he attempted to introduce a
German plant-terminology into botany; and his early de-
veloped love for music found expression in organizing an
orchestra of his scholars and in collecting numerous folk-
songs, which he set for three voices, to be sung in his
family.
We have been led, by the necessity of obtaining some
idea of the actual conditions under which Grassmann
worked, to speak of his later life. We must now return to the
time when he first began to realize how slight a recognition
was to be accorded to his mathematical writings — that is
to say about the year 1852. Great as was his inner sure-
ness of the value of the work, yet his was not the type of
mind to be satisfied with a partial success. And so he took
the astonishing (and almost unprecedented)20 step of turn-
ing his attention to another field of knowledge altogether
and quickly winning the recognition of experts. The plia-
bility of his genius enabled him to force his way into a new
subject, philology, and to produce results of outstanding
merit in it.
B. Delbriick21 gives an interesting account of how
Grassmann turned to philology. The rules of the tradi-
tional school grammar with its mass of exceptions must
have been painful to his mathematical understanding, and
20 The equally neglected English genius Thomas Young combined mathe-
matical and philological ability.
21 B. Delbruck, loc. cit.
HERMANN GRASSMANN. I?
so he first planned a grammar and reading book in which
scientific laws replaced the old rule-of-thumb methods
wherever possible. It is natural therefore that he should
next turn his attention to that sphere of language in which
such laws are most easily recognizable, namely phonetics.
His first attempt in the realm of comparative philology
was on this subject. It was an article, published in 1859,
on the influence of v and j on neighboring consonants, and
on certain phenomena in connection with aspiration. Del-
briick expresses the opinion that his work in this field is
not distinguished either for breadth of scholarship, since
he worked with few books, or for etymological depth.
"But," he says, "it is the clearness of reflection which pene-
trates into all corners of the subject, the persistence with
which the material has been so long accumulated until it
became possible to reach the simplest formulation of the
governing law, and the untiring nature of the mathematical
abstraction which in these undertakings so clearly comes
to light."
Grassmann must have quickly recognized how valuable
in all researches into comparative philology a deep acquain-
tance with the oldest Indian languages would be, and he
determined with his usual persistency to make himself at
home in the hymns of the Vedas. These Sanskrit studies
led him to the production of works which rendered his name
famous. In 1861 he had only the first volume of the up-
right text and scarcely half of the Bohtlingk-Roth diction-
ary. Yet with these means he succeeded in mastering the
extraordinary difficulties of the texts, and began his dic-
tionary and translation of the Rig- Veda. He arranged his
dictionary in an original manner so as to be able to give
the meaning of each form according to the place in which
it occurred. Although Delbriick credits the first volume
of the dictionary with etymological value for its grammat-
ical subdivision of the roots, yet he regards the arrange-
l8 THE MONIST.
ment just mentioned as unphilological. It aims less at
giving definite historical and philological information than
at making successive attempts at explanation. As, how-
ever, the work progressed, aided by the stream of material
reaching the author from the growing Roth dictionary
and elsewhere, it became more philological. Still the
method pursued was the same, and Grassmann completed
the translation side by side with the dictionary. For long
these works formed a useful tool in attacking the difficul-
ties of the Vedas. The recognition of experts was worthily
expressed by Rudolph Roth, on whose word the University
of Tubingen conferred upon Grassmann the honorary de-
gree of Doctor of Philosophy. He spoke of him as a man
qui acutissima vedicorum carminum interpretation nomen
suum reddidit illustrissimum.
During this period of his life when he was winning
fame in another sphere of work, Grassmann's mathemat-
ical writings were gradually obtaining the recognition
which was their due. Toward the end of the sixties con-
siderable attention was paid by mathematicians to higher
algebra, and the quickening of thought along those lines
made recognition much more likely. Hermann Hankel in
his Theorie der complexen Zahlensysteme of 1867 was the
first to call attention to Grassmann's work. Clebsch22 also
shortly afterward accorded him a full measure of admira-
tion. Grassmann23 believed that Clebsch would have fer-
tilized the theory of extension with far-reaching new ideas
of his own if death had not cut short his promising career.
Some of the younger teachers at the Stettin Gym-
nasium had become pupils of Grassmann ; and one of these,
the mathematician Victor Schlegel, in his System der
22 Clebsch, Zum Ged'dchtniss an Julius Pliicker, 1872.
28 See preface to second edition of the Ausdehnungslehre of 1844, pub-
lished in 1878.
HERMANN GRASSMANN. 19
Raumlehre (ist part 1872, 2d part 1875) made his works
more accessible by a clear exposition and application of
them. The best kind of approval from authorities came to
him in their use of his methods in various fields ; and Grass-
mann himself, after a long interval, again took up his
mathematical labors. Of the many articles from his pen,
we may mention especially that on the application of his
work to mechanics,24 because it was in this domain that he
considered the theory of extension to be particularly suc-
cessful. He expressed the desire that it might be granted
to him to write a treatise on mechanics based on his prin-
ciples. This was denied him. He lived, however, to see
a second edition of his ill-fated Ausdehnungslehre of 1844
called for ; and died, while it was passing through the press,
on September 26, 1877, in his sixty-ninth year. To the
last, in spite of great bodily suffering, he retained his vigor
and enthusiasm. Five essays published in the year of his
death testify to this.
It is a pleasant thing to think that he received such
rich recognition before he died; though it must always
remain a source of regret that he never succeeded in ob-
taining the position he hoped for, which would have enabled
his powers to be more fully developed and his influence
more widely expressed. And yet, there can be no cause
for sorrow if we think of the fortitude of this strong soul,
and remember the firm conviction expressed in the closing
words of the introduction to the Ausdehnungslehre of 1862,
that his mathematical ideas would some day arise again,
though perhaps in a new form, and become part of living
thought. To some extent that conviction has proved a
justifiable one. The publication of his Collected Works
was suggested by Professor Klein of Gottingen. After ob-
24 "Die Mechanik nach den Principien der Ausdehnungslehre," Math.
Annalen, Vol. XII, 1877.
2O THE MONIST.
taining the consent of Grassmann's relatives he laid the
matter before the Royal Saxon Academy of Sciences in
October, 1892. A committee was formed and F. Engel
made chief editor. The first part of the first volume ap-
peared in 1894.
Since then there have been many works on the calculus
of extension, but it can scarcely be held that they have
done more than make a beginning of the development of the
suggestions in Grassmann's work. What has been done
has been mainly in the domains of spatial theory and higher
algebra ; mechanics remains still burdened with traditional
coordinate systems. This is the more remarkable since the
principle of relativity, with its demand for a generalized dy-
namics of which ordinary dynamics is a special case, offers
such a promising field of application.
There is usually, in the sphere of thought, a rational
explanation of apparently irrational facts. A minute in-
fluence translated into action by the mass of thinking men
may give rise to the spirit of their age ; and thus its effects,
and the negative effects may be just as great as the positive,
carried forward in ever-increasing circles to distant gen-
erations. So it has been with whatever lies at the base of
the neglect of Hermann Grassmann. There has been be-
queathed to us something like an unreasoning distaste for
his and similar analytical methods, from which has arisen
the need for a definite effort to break the spell of the past.
The formation of an "International Association for Pro-
moting the Study of Quaternions and Allied Systems of
Mathematics" took its origin from such a need.25 It may
therefore be that a just estimate both of the value and limi-
tations of Grassmann's work will only come by the appli-
cation of a critical method of wider scope than those of his
28 P. Molenbrock and Shunkichi Kimura, letter to Nature, Oct. 3, 1895.
HERMANN GRASSMANN. 21
own period. Indications are indeed not wanting that in
the modern theory of transformation-groups26 lies the cri-
terion for a final judgment.
A. E. HEATH."
BEDALES, PETERSFIELD, ENGLAND.
26 Lie and Engel, Theorie der Transformationsgruppen, Vol. II, p. 748 ;
M. Abraham and P. Langevin, "Notions geometriques fondamentales," Encyc.
des sciences mathematiques, Tome IV, Vol. 5, p. 2.
27 I wish to thank Miss Vinvela Cummin and Mr. R. E. Roper for help
in the translation of materials for this sketch.
THE NEGLECT OF THE WORK OF H. GRASS-
MANN.
IT must not be supposed that the neglect of Hermann
Grassmann's mathematical work by his contemporaries
is merely an incident of his biography. Its consideration
involves a much larger question, because Grassmann's fate
was shared by other mathematicians of the period in whose
work stress was laid on form rather than content. The
distinction between the two may be illustrated by reference
to the mathematical treatment of quantity. As soon as
analysis had generalized that idea so as to include complex
quantities, a mathematics based on formal definitions and
of a general character could be developed to include them.
The meaning of the propositions of such a calculus need
not enter into this study. The propositions would consti-
tute a formal deductive series which could be developed
without any reference to content. That Grassmann was
a pioneer in the movement which made magnitude sub-
ordinate and posterior to a science of form was recognized
by Hankel,1 who says, "It was Grassmann who took up this
idea for the first time in a truly philosophical spirit and
treated it from a comprehensive point of view." In the
Introduction (A) to the Ausdehnungslehre of 1844 Grass-
mann puts the matter thus : "The chief division of all sci-
ences is that into real and formal. The former sciences
1 Theorie der complexen Zahlensysteme, p. 16.
THE NEGLECT OF THE WORK OF H. GRASSMANN. 2$
image in thought the existent as independent of thinking,
and their truth consists in the agreement of the thought
with the existent; the latter sciences on the contrary have
for their subject-matter that which has been determined
by thought itself, and their truth is shown in the mutual
agreement between processes of thought." He goes on to
consider mathematics and formal logic as branches of a
general science of form, and seeks to dissociate this science
from such real sciences as the geometry of actual space,
although it must form the basis on which all such are built.
That the neglect accorded to Grassmann had nothing
to do with any accident of birth or position is shown by
the fact that Leibniz, whose name was famous in both
mathematical and philosophical circles, shared the same
fate in regard to his Dissertatio de Arte Combinatoria
and later writings of the same kind, in which he sought
to set up a formal symbolical calculus with similar aims.
Of Grassmann's contemporaries who worked in the same
field, we need mention only George Boole (1815-1864)
who failed to obtain anything like a due recognition of his
genius; and Sir. W. R. Hamilton whose early papers on
quaternions were regarded as mere curiosities. Even when
the applications of these generalized formal methods to the
founding of a calculus of directed quantities of immediate
value to physics had been made, we find the important
work of Willard Gibbs waiting for years before it became
known and made full use of. If, then, we are to explain
the neglect of Grassmann's work we shall have to analyze
the causes of the apathy and mistrust with which all such
work has been received.
The view held by Carl Miisebeck is that in the almost
exclusively philosophical form of representation, which
however was grounded in the whole system, we have to
seek the reason why the contemporaries of Grassmann
24 THE MONIST.
drew back in terror from deeper study of his early work.
He says2: "Such a height of mathematical abstraction in
which, with the help of a new calculus, laws are inferred
in abstract regions about the mutual dependence of abstract
constructions in which not even the character of the spatial
is maintained, although at the conclusion of almost every
section it is shown how the new method could be used with
advantage, was never before known." That this has been
a very important factor cannot be doubted. Dislike of the
philosophical form of his work was expressed to Grass-
mann by the few mathematicians who noticed his first Aus-
dehnungslehre. He himself says in the preface to the
second edition of this book that he expected the work to
find fullest recognition from the more philosophically in-
clined reader. It is only necessary to refer to the appli-
cation and extension of his ideas which have come from
A. N. Whitehead3 in England and from G. Peano4 and C.
Burali-Forti5 in Italy to show how well-founded this fore-
cast was. But the analysis cannot rest there. We must
inquire further how this dislike arose.
J. T. Merz6 in his chapter on "The Development of
Mathematical Thought in the iQth Century," inclines to
the view that a definite distaste for a philosophical form
had set in among German mathematicians as a part of the
reaction against the exaggerations of the metaphysical uni-
fication of knowledge in the schools of Schelling and Hegel.
But mathematicians in modern times have, on the whole,
been singularly unaffected by philosophical movements.
Furthermore the calculus of extension and allied systems
have not fully come into their own even in our own day,
2 In his memoir of Hermann Grassmann, Stettin, 1877.
3 Universal Algebra, Cambridge, 1898.
4 Calcolo Geometrico secondo I'Ausdehnungslehre di H. Grassmann, Turin,
1888.
e Introduction a la geometric differ entielle, suivant la methode de H. Grass-
mann, Paris, 1897.
6 History of European Thought in the loth Century, Vol. I, p. 243.
THE NEGLECT OF THE WORK OF H. GRASSMANN. 25
when wide syntheses are eagerly sought. It seems to the
present writer that it is in the attitude of the plain anti-
metaphysical mathematician that we must seek for the
explanation of the want of understanding which leads to
mistrust of philosophical form. An immense amount of
prejudice barred the way to the full development of a gen-
eral science of form — prejudice due to non-realization of
the purely formal claims of such a calculus.7 And if we
could get at the bottom of this not altogether unreasoning
mistrust it might be possible to clear away some of the
hindrances to a proper understanding of the fundamental
importance of Grassmann's work.
To do this we must push our analysis a step further.
What steady cause can have been operating over such a
long period which could so affect the attitude of the indi-
vidual as to create what amounts almost to a general
blindness to the importance of a whole body of contribu-
tions to thought ? I believe that the root of the matter lies
in wrong principles of instruction. It may be that this at
first sight appears too small an influence to have such con-
sequences; but so did the minute geological influences of
the uniformitarians to those who sought for explanations
in more dramatic cataclysms. It is as unscientific to neglect
the unobtrusive but persistent influences of educational
methods on pure thought as it would be to treat of the
social conditions of a people without taking into account
their mind-development.
We will only give one well-recognized example of the
importance of methods of exposition on mathematical his-
tory. Merz places Gauss at the head of the critical move-
ment which began the nineteenth century. He adds,8 how-
ever, that it was not to him primarily that the great change
7 Cf. the article below on "The Geometrical Analysis of Grassmann and its
Connection with Leibniz's Characteristic," § 2.
8 Op. cit.f Vol. II, p. 636.
26 THE MONIST.
which came over mathematics was due, but to Cauchy.
Gauss, while issuing finished and perfect though some-
times irritatingly unintelligible tracts, hated lecturing; in
contrast to this Cauchy gained the merit, through his en-
thusiasm and patience as a teacher, of creating a new
school of thought — and earned the gratitude of the greatest
intellects, such as Abel, for having pointed out the right
road of progress. But it is not so much upon the manner
of exposition of original mathematicians themselves that
stress must be laid. It has without doubt often happened
that writers of great analytical insight have failed to see
that it is no more a descent to a common level to seek out
and use the best methods of enforcing consideration of
their work, than it is to use a printing-press instead of a
town crier the more effectively to reach their audience.
Grassmann himself, however, did all that was humanly
possible in this way, although Jahnke is of the opinion
that he was inclined to the belief that even first instruction
should be rigorous; and kept back applications until too
late. It is rather that teaching methods in general dur-
ing the nineteenth century have always lagged too far be-
hind discovery. And so they have left the students of
one generation, who are the potential original workers
of the next, with minds unreceptive to newer and more
delicate methods. It might be urged that this would
affect equally all branches of mathematics, but I think it
can be shown that it is on the reception of such funda-
mental analytical methods as Grassmann's that its evil in-
fluence more particularly falls.
It is quite obvious that the subject must be limited if
we are to deal in detail with the suggested effects of in-
adequate educational methods. So I shall confine myself
in what follows to the consideration of the difficulties which
beset the path of the teacher who has to explain the ordi-
THE NEGLECT OF THE WORK OF H. GRASSMANN. 2?
nary concepts of mechanics ; and attempt to show how fail-
ure to realize the nature of those difficulties tends to pro-
duce an unreceptive attitude to modern analysis. I have
chosen this subject for two reasons. Firstly, it seems to
me that if the concepts of mechanics were properly treated
they would finally appear to the pupil as useful construc-
tions instead of as the dogmatically asserted existents they
are still commonly held to be; and so the formal science
underlying the real science of mechanics would naturally
arise for him as the final result of analysis, and not as the
unreal fabric of a philosopher's dream. And secondly, it
is the domain to which the various "extensive algebras"
have peculiar applicability, as Grassmann himself felt
strongly. It is highly significant therefore that it is pre-
cisely Grassmann's suggestive applications to mechanics
whose neglect is the most noticeable. That this is so is.
on my view, because sounder and more philosophical no-
tions of geometrical as opposed to mechanical concepts
were already coming into exchange in Grassmann's own
day so that geometrical applications were thereby rendered
more understandable.
At the very outset of our discussion we are faced with
the difficulty that so much difference of opinion exists be-
tween teachers of mechanics that many have been forced
into the conclusion that, since the enthusiast with an un-
philosophical method of his own can yet reap good results,
method is unimportant. This, of course, is only partially
true. If it were wholly true it would mean an end to all
possibility of coordination — an end, in fact, to the claims
of education to be a science. To grant that education is
an art is not to forego all its claims to be a science. For
we must regard all art as applied science "unless we are
willing, with the multitude, to consider art as guessing
28 THE MONIST.
and aiming well."1 Beneath the apparent chaos of opinion
on the teaching of mechanics there is however some order
if one can avoid certain sources of confusion which have
led to superficial differences of opinion where nothing
deeper exists.
One source of confusion is the absence of a clear idea
of the difference in educational theory between an imper-
sonal principle and the more personal element — the method
of applying the principle. This distinction is insisted on
by Mr. E. G. A. Holmes,10 and seems a real one. If once
we realize it we can see how it is possible for there to be
fairly well accepted scientific principles of teaching at the
same time as a wide divergence of method in use by dif-
ferent teachers under differing conditions. And indeed if
one looks carefully into much of the polemical writing on
mechanics teaching it is seen to be caused less by funda-
mental differences of principle than by differences of method.
It is still more necessary to clear away a second source of
unsatisfactory discussion. A superficial glance through the
mass of controversial writing on science teaching in recent
years would lead one to suppose there was a sharp division
of principle between those who believe in a logically ordered
course with emphasis on what one may call the instruc-
tional method, and those who prefer a looser, more empir-
ical, treatment usually embodying heuristic methods. It
would be possible, however, to reconcile many of the com-
batants if they could be persuaded to see that so direct an
opposition is far too simple a statement of the problem, and
that each may be partial statements of the real solution.
And this becomes possible, I think, if once the disputants
grant the importance of the biogenetic or embryonic prin-
ciple as applied to education — the principle, that is to say,
8 Reference to Plato, Philebus : G. Boole, The Mathematical Analysis of
Logic, note p. 7.
10 E. G. A. Holmes, The Montessori System of Education, English Board
of Education Pamphlet, No. 24, p. 3.
THE NEGLECT OF THE WORK OF H. GRASSMANN. 2Q
that the development of the individual is a recapitulation
of the development of the race. It seems strange that it
should be necessary at this stage to call attention to a
principle so well known11 and so much applied, and yet
one often has the spectacle of a successful teacher of higher
classes urging the claims of logical order against an equally
successful empiricist whose experience has been with
younger pupils. The truth is, of course, that no one
method is applicable to all ages. If the biogenetic law
holds, then the natural principle would be to use, in general,
modes of teaching a subject similar at each stage to those
by which the race has gathered its knowledge of that sub-
ject. In mechanics this would mean that a more rigidly
logical course would follow empirical experiments and the
handling of simple machines.
We will now pass on to our main investigation of the
factors which must be taken into account in avoiding the
creation of an atmosphere uncongenial to a final abstract
analysis. In doing so I will indicate what appear to
be the general principles by which one must work in
giving to beginners living ideas of the entities of mechan-
ics, and failure to comply with which leads to the produc-
tion of passively instructed, rather than of irritable and
responsive, organisms. The concepts of mechanics are
produced from the raw material of experience by the proc-
ess of abstraction, and a beginner must therefore pass
through an experimental stage before he is introduced to
the logically defined concepts themselves. In fact he must
first use and handle rough ideas and thence be led to build
up the more rigidly exact definitions of them for himself.
It follows from this that any information we can glean
11 It is a very remarkable thing that De Morgan in his Study and Diffi-
culties of Mathematics, first published in 1831, or 28 years before the Origin
of Species, should have stated this principle so concisely in the words (p. 186)
referring to discussions of first principles : "the progress of nations has ex-
hibited throughout a strong resemblance to that of individuals."
3O THE MONIST.
about the actual historical process by which man came to
form and use concepts may be of vital importance to a
teacher. In mechanics particularly, where the concepts
are less obvious than in geometry (the first ideas of force,
mass, acceleration and energy, regarded however not as
constructions but as real entities, were only developed to
any clearness after Galileo — that is at quite a late stage
in man's history) any fogginess about their nature and
use means endless confusion; and that accounts for most
of the difficulties commonly experienced.
It was Locke who first plainly showed how concepts
arise from the material of immediate perception. If we
think of the flux and confusion of our perceptions — the
colors, sounds, smells, sensations of touch, at any instant
we find our attention drawn to some more insistent parts
of that flux. When these continually recur we use nouns,
adjectives and verbs to identify them. Such is the begin-
ning of the formation of concepts. These are regrouped to
form other concepts. Thus a wide experience of animals
would lead us to group them and to speak, for example,
of a class "dog." Once classed we can treat all instances
as having the general properties of the class. The prac-
tical advantages are obvious. "The intellectual life of man
consists almost wholly in his substitution of a conceptual
order for the perceptual order in which his experience
originally comes," says William James.12 Once concepts
are formed they enable us to handle our immediate ex-
perience with greater ease. And by building up more and
more complex concepts and tracing the connections be-
tween them we create our mathematics and our sciences.
Even animals may form rough concepts.18 A dog by
experience comes to know the difference between "man"
12 Some Problems of Philosophy, p. 51.
18 This treatment of the origination of concepts is founded largely on that
of E. Mach in his chapter on Concepts in the volume Erkenntnis und Irrtum.
THE NEGLECT OF THE WORK OF H. GRASSMANN. 3!
and other animals. Furthermore if he met a dummy man
he would soon find out that the reactions he ordinarily
associated with "man" failed to be reproduced, and so
would reject that experience for his man-class. In a
similar way man must have formed concepts becoming
more and more complicated but more firm in outline as
his experience became richer. But it is to be noticed that
the growth of concepts in a body of experience depends
on the number and interest of our observations in the
region concerned. For this reason interest in, and con-
sequent familiarization with, simple machines and mechan-
ical toys may well be the child's best introduction to me-
chanics. Model monoplanes, an old petrol engine from a
motor cycle, pumps, a screw, levers, a jack, Hero's turbine
model — all these can easily be got at; few young children
will show no interest, while many of them will possess
already in these days of mechanical toys a considerable
knowledge of manipulation. Simple explanations of the
working of such apparatus are absorbed with astonishing
readiness. In larger schools where there is an engineering
workshop this method of introducing young boys to me-
chanics by way of machinery has been tried with con-
siderable success. Knowledge gets picked up as it were
"by contact." The concepts which arise at this stage are
necessarily crude — general ideas of force, speed, work and
friction; this latter is, of course, one of the first things
to notice — not the last to be dealt with as is usually the
case. Simple as these considerations are, they are not yet
fully appreciated. The London Mathematical Association's
Report on the Teaching of Elementary Mechanics sug-
gested some time ago that the phrase "Mechanical Advan-
tage" be replaced by "Force-Ratio." For beginners neither
of these is intelligible ; but they very soon know "how much
stronger" a machine makes you. And that conception is
quite good enough for them to use.
32 THE MONIST.
In introducing simple mechanical concepts to beginners,
therefore, the principle to use is that the concepts must
arise naturally from experience and not be handed out as
definitions. Dictated definitions not founded on sufficient
knowledge of facts are flimsy constructions ready to fall
at the first breath of difficulty. They do not perform that
primary function of concepts of helping one to classify and
handle facts, because the facts to be handled are not in the
mind when the concept is formulated. "How much stronger
a machine makes you" is a phrase which reminds the
hearer at once of the assistance it gives him in grouping
machines and using them intelligently for different pur-
poses. A note-book definition of "mechanical advantage"
is likely to present another arithmetical puzzle instead
of serving to remind the learner of the solution of old
ones. The principle here advocated was well expressed
in the discussion on mechanics teaching at the British
Association in 1905 by the president of the section, Pro-
fessor Forsyth. He said, "What you want to do in the
first instance is to accustom the boys to the ordinary rela-
tions of bodies and of their properties, and afterward you
can attempt to give some definitions which will be more or
less accurate; but do not begin with the definitions, begin
with the things themselves." And the philosophical basis
for the principle is, that the significance of concepts is
always learned from their relations to perceptual particu-
lars, their utility depending on the power they give us of
coordinating perceptual facts. From this it follows further
that concepts and names should never be introduced where
there is no direct and immediate gain in so doing. Such
terms as "centrifugal" and "centripetal" forces, and the
endless discussion to which they lead, are thus beside the
mark. "Force toward, or away from, the center" does
all that is necessary without introducing new words of
really less precision,
THE NEGLECT OF THE WORK OF H. GRASSMANN. 33
It should be noted that some of the crude concepts
arrived at in the early stages are really, when one comes
to analyze them, very complex, and Ostwald's warning14
against the error of supposing that the less simple concepts
have always been reached by compounding simple ones
has application here. As he says, complex concepts often
in origin have existed first. We can now see more clearly
why the teacher of mechanics so often complains of the
difficulty of giving the average child a satisfactory notion
of force.15 The difficulty is largely due to the teacher who
knows the concept to be complicated, and seeks to define it
in terms of mass-acceleration — thus involving two more
concepts, one of which (mass) is at least as difficult to
understand as force. A rough idea of force, considered
simply as a "push" or a "pull," can be assimilated at a very
early stage ; that of mass-acceleration must come very much
later.
The bearing of this preliminary stage in the formation
of concepts on our main thesis may now be traced. It is
quite evident that the individual has very limited powers
of absorbing the logically ordered account of a science in
which stress is laid on abstract notions before such notions
have grown up naturally by use. Now this difficult step
for the beginner from the perceptual to the conceptual is
very similar to that which leads from ordinary mechanics
to such a treatment of the subject as that of Grassmann.
Both lead into regions of greater abstraction. In the latter
case we can get rid of concepts in so far as they relate to
the existent, and reach a statement of mechanical principles
in terms of a generalized form-theory. We may illustrate,
roughly, the meaning of this by the following analogue.
At different stages in the history of physics various the-
14 Ostwald, Natural Philosophy, p. 20.
15 Cf. C. Godfrey, Brit. Association Report on Mechnics Teaching, p. 41.
34 THE MONIST.
ories of light have been held. The concepts used in these
theories (corpuscle, elastic-solid ether, electro-magnetic
medium) have possessed widely different "qualities"; but
the equations expressing the relation between the concep-
tual elements have throughout possessed similarity of form.
A science of form would hence lay emphasis on the in-
variant relations, refine away the particular concepts, and
leave a much more abstract and generalized science.
But if racial development is in the main similar to the
progress of the individual this will explain the great diffi-
culty experienced by whole generations of mathematicians
in understanding work of the type of Grassmann's.
Furthermore, it is at this point that defective scientific
training looms into importance. For unless great care
has been taken in avoiding the too early definition of con-
cepts, a rigid view of them is promulgated. The older
dogmatic and orderly methods of teaching tended inevi-
tably to this. The consequence was that when the time
came for polishing and development of the concepts ob-
tained, and for the deliberate building up of more com-
plex ones — it was found that the capacity for subtle gen-
eralized views had been destroyed. A mind forced into
passivity and filled with inert knowledge cannot suddenly
be brought to discard it in response to the stimulus of a
tentative generalization. To take a simple example, the
idea of a new kind of addition, applicable to vectors, shocks
and confuses a pupil who has been dogmatically instructed
in algebra as though it were a sacred rite. As with the
child under such a system, so with the generation of which
he forms a part. Jahnke states that many mathematicians
were put off by meeting in Grassmann's work a product
which equals zero without either factor doing so. Formal
logical development often leads to conclusions which are
not capable of any mental image.16 Such abstractions are
" Cf. F. Klein, The Evanston Colloquium, Lecture 6.
THE NEGLECT OF THE WORK OF H. GRASSMANN. 35
out of reach of those who have never been freed from the
confines of the existent world.
Cajori17 in a notice in 1874 of the publication called
The Analyst, Des Moines, Iowa, said that it bore evidence
of an approaching departure from antiquated views and
methods, of a tendency among teachers to look into the
history and philosophy of mathematics. My thesis is that
such a movement, which certainly has not yet been realized,
would remove the main cause of the neglect of Hermann
Grassmann's work, which even in these days is often
granted the kind of recognition accorded to certain literary
classics, which are famous but never read. Perhaps it is
an earnest of the future that the copy of The Analyst re-
ferred to by Cajori contained a brief account18 of the essen-
tial features of Grassmann's Ausdehnungslehre.
A. E. HEATH.
BEDALES, PETERSFIELD, ENGLAND.
17 Teaching and History of Mathematics in the United States.
18 Translated by W. W. Beman of the University of Michigan.
THE GEOMETRICAL ANALYSIS OF GRASSMANN
AND ITS CONNECTION WITH LEIB-
NIZ'S CHARACTERISTIC.
§ I-
BY a curious turn of fate Grassmann wrote, in the
introduction to his "Geometrical Analysis," concern-
ing Leibniz's early work on the same subject, words which
were to apply with prophetic force to his own Ausdeh-
nungslehre. "When the special power of a genius .... is so
revealed that he is able to grasp and extend the ideas
toward which the development of his time is directed, and
so appears representative of his period, then that power
shows itself still more remarkable when it can seize ideas
in those realms of thought in advance of their day and
forecast for hundreds of years the line of their develop-
ment. While ideas of the first kind are often developed
simultaneously by the outstanding spirits of the age, as
when both Leibniz and Newton founded the differential
calculus — a certain stage of fruition being reached — ideas
of the latter kind appeared as the special characteristic
of the individual, the innermost revelations of his mind
into which only a few elect contemporaries could enter
and have a foreshadowing of the developments which were
to spread from them in the future. While the first received
great applause and aroused movement in their own day,
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 37
because they represent the summit of the epoch, the others
for the most part fall without effect in the contemporary
period since they are only understood by a few, and then
only partially. Often afterward does such thought be-
come the seed of a rich harvest. That this great idea of
Leibniz — namely, the idea of a true geometrical analysis —
belongs to this preparatory and, as it were, prophetic class
cannot be doubted for a moment. It has also shared the
fate of such. Indeed by a special ill favor of circumstances
it has remained hidden far beyond the time when it might
have had a powerful influence. For even before it was
brought out of its hiding place by Uylenbroek, paths
toward a similar analysis had been made in other ways."
At the time when these words were written Grassmann
could have had no idea of the disappointment which was
to come to him in the neglect of his own work. The first
edition of the Ausdehnungslehre, or theory of extended
magnitudes, had been published in 1844 and had received
no attention from mathematicians with the exception of a
few individuals. Grassmann, however, believed that rec-
ognition was only a matter of time and sought to bring
out the importance and applicability of the new analysis.
For the year 1845 (but extended to 1846 to coincide with
the two hundredth anniversary of Leibniz's birth) the
Jablonowski Society of Leipsic set a prize essay demanding
the restatement or further development of the geometrical
calculus discovered by Leibniz or the setting up of a simi-
lar calculus; and the award was made to Grassmann for
the essay, printed in 1847, from which I made the above
quotation. This was the first and the only acknowledg-
ment of the value of his work which he received from
mathematicians until long after many of the ideas he
formulated had been reached and applied by other methods
and other thinkers.
I have laid stress on the similarity of treatment meted
38 THE MONIST.
out to the fundamentally important work of the two men
because I believe that in some elements of its explanation
lies the clue to unravel the difficulties of their subject
matter and connection with each other. The more general
aims of both Leibniz and Grassmann were the same — the
setting up of a convenient calculus or art of manipulating
signs by fixed rules, and of deducing therefrom true propo-
sitions for the things represented by the signs, for use as
a generalized mathematics. In each case their geometrical
calculus was a particular application to geometry of a
wider calculus for which each desired more than mere
applicability to mathematics.
In a letter to Arnauld, dated January 14, 1688, Leibniz
writes1: "Some day, if I find leisure, I hope to write out
my meditations upon the general characteristic or method
of universal calculus, which should be of service in the
other sciences as well as in mathematics. I have defini-
tions, axioms, and very remarkable theorems and prob-
lems in regard to coincidence, determination, similitude,
relation in general, power or cause, and substance, and
everywhere I advance with symbols in a precise and strict
manner as in algebra. I have made some applications of
it in jurisprudence." Similarly Grassmann2 says: "By a
general science of symbols (Formenlehre) we understand
that body of truths which apply alike to every branch of
mathematics, and which presuppose only the universal con-
cepts of similarity and difference, connection and disjunc-
tion." The symbols are made so general as to be applicable
to both logic3 and mathematics, although in the Ausdeh-
1 George R. Montgomery (trans.), Leibniz: Discourse on Metaphysics,
Correspondence with Arnauld, and Monadology, p. 241.
2 Ausdehnungslehre of 1844, p. 2.
8 The application of such a general science of symbols to formal logic
was made by both H. Grassmann and his brother Robert.
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 39
nungslehre they are only applied to the domain of mathe-
matics.4
It is clear that both Leibniz and Grassmann, but espe-
cially the former, claimed great scope for their calculus,
a fact which tended to make their writings generalized
and difficult to understand. In the preface to his Universal
Algebra (1898) Professor Whitehead expresses his belief
that lack of unity in presentation (which of course would
be the tendency in dealing with a method applicable to
many fields) discourages attention to such a subject. But
that is not all. A new mathematical method, to make
itself known, has to appeal in the main to mathematicians
and not to philosophers. So that a wide and philosophical
treatment is apt to be discounted by the ordinary man who
thinks logic can be made to prove anything.
§2.
Before we condemn this attitude we must first of all
inquire as to what exactly the common man means by the
dangers of logic. What he really fears is not logic but
fallacy. Without realizing it he distrusts a mechanical
dexterity in reasoning because the attainment of truth de-
pends not only on a facility in manipulating logical proc-
esses but also on the sifting of first principles. When
Leibniz claims for his char act eristic a universalis or "uni-
versal mathematics," the germ of which he produced in
his De arte combinatoria published when he was twenty,
that " . . . . there would be no more need of disputation
between two philosophers than between two accountants.
For it would suffice to take their pencils in their hands, to
sit down to their slates, and to say to each other (with a
*In the Ausdehnungslehre , however, are expressions directly applicable
to logic, e. g., there is the generalized expression for the result of division
C+ O/B where O/B is an indefinite form (p. 213) — an anticipation of Boole's
use of p/O to symbolize perfect indefiniteness, as pointed out by Venn in his
Symbolic Logic, note p. 268 (2d ed.).
40 THE MONIST.
friend as witness, if they liked) : Let us calculate" — he is
running counter to the plain man's knowledge that there
are two parts of a logical process, the first the choosing of
an assumption, the second the arguing upon it.
Now Leibniz realized of course that premises are re--
quired first, but he thought they could be obtained very sim-
ply. By analyzing any notion until it was simple he thought
that all axioms or assumptions followed as identical propo-
sitions. Thus he was led, by his view of ideas, to believe
that even the axioms of Euclid could be proved. So in his
New Essays, "I would have people seek even the demon-
stration of the axioms of Euclid .... And when I am asked
the means of knowing and examining innate principles, I
reply. . . .we must try to reduce them to first principles,
i. e., to axioms which are identical, or immediate by means
of definitions which are nothing but a distinct exposition
of ideas." This is connected with his view that all our
ideas are composed of a very small number of simple ideas,
which together form an alphabet of human thoughts. But,
as Couturat remarks,5 there are many more simple ideas
than Leibniz believed; and furthermore there is no great
philosophical interest in such. "An idea which can be
defined or a proposition which can be proved, is only of
subordinate philosophical interest."6 It is precisely the
business of philosophy to deal with the primitive, intuitive
assumptions on which any calculus must be based.
So the plain man is to some extent justified in his mis-
trust of the uncritical application of a calculus.
§3.
It is very necessary, however, to see exactly what is, and
what is not, here granted to the plain man. It is true that
in using a calculus we must be careful not to over-empha-
B L. Couturat, La logique de Leibniz, p. 431.
8 B. Russell, The Philosophy of Leibniz, p. 431.
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 4!
size the results at the risk of forgetting the premises from
which they have been obtained. But that being admitted,
thus making the final development of the universal char-
acteristic a matter not of philosophy but of a sort of gener-
alized mathematics of which formal logic7 and geometry
are special cases, it does not follow that there must be
limits to the applicability of the calculus in these spheres.
Yet that is what the modern representative of our plain
man asserts. His criticism of a logical calculus has put on
a more philosophical form, but remains essentially the
same. Henri Poincare may justly, I think, be taken as
such a representative. For he says, "I appeal only to
unprejudiced people of common sense. . . .they [the logis-
ticians] have shown that mathematics is entirely reducible
to logic, and that intuition plays no part in it whatever."8
This belief led Poincare to the view that, since he knew
from his own experience as a mathematician of great in-
sight the important part intuition plays in mathematical
discovery, therefore the nature of mathematics cannot be
logical.
This reasoning is founded on a very common fallacy
which I will call the genetic error — the error, namely,
which lies in the assumption that the origin of a thing in
some way determines its nature.9 If this assumption is
made it follows that since intuition plays a part in dis-
7 Leibniz himself foresaw this development carried out by Boole, Peano,
Frege, Whitehead and Russell and their school of symbolic logicians. In fact
he made discoveries in this field but did not publish them because they contra-
dicted certain points in the traditional doctrine of the syllogism. In some
points he even advanced beyond Boole (See Couturat, op. cit., p. 386).
8 Science et Methode, p. 155; cf. also C. J. Keyser in Bull. Amer. Math.
Soc., Jan. 1907, pp. 197, 198.
8 This error has been very common in philosophy. It underlies much
argument against rationalism, denying that knowledge reached empirically
can be anything other than empirical. (Cf. Leibniz, New Essays, IV, 1 §9,
against Locke.) It is at the basis of many criticisms leveled against any
generalization of number, since the idea of number arose from perceptual ex-
perience. It vitiates pragmatism, which inquires into the causes of our judging
things to be true in order to get at the nature of truth. (See B. Russell, Philo-
sophical Essays, p. 110.)
42 THE MONIST.
covery, the nature of mathematics cannot be purely formal,
and therefore it cannot be expressed in terms of symbolic
logic. Now all such references to the origins of mathe-
matics are irrelevant. Once the premises have been made,
and that is where intuition comes in, symbolic logic is
merely "an instrument for economizing the exertion of
intelligence."1 The mind, being relieved of unnecessary
work by a good symbolism, is set free to attack more diffi-
cult problems; for as Professor Whitehead says,11 "Opera-
tions of thought are like cavalry charges in a battle — they
are strictly limited in number." Nor is that the only ad-
vantage of this modern development of Leibniz's universal
mathematics. It "has the effect of enlarging our abstract
imagination and providing an infinite number of possible
hypotheses to be applied in the analysis of any complex
fact."1 And so it lends itself to the production of just
such novel fundamental hypotheses as are needed in sub-
jects like the dynamics of relativity.
So finally, we must say of the symbols of a universal
calculus what Hobbes said of words, "They are wise men's
counters, they do but reckon by them; but they are the
money of fools." Yet it must be recognized that when it
is confined to dealing with mathematics in its widest sense
(taken to include formal logic), — within the limits im-
posed on his own calculus by Grassmann, in fact, — it serves
as a powerful and legitimate tool.
§4.
This discussion of the neglect and mistrust of mathe-
maticians for the generalized calculus of both Leibniz and
Grassmann has, I hope, shown what the nature of such a
10 W. E. Johnson in Mind, N. S., Vol. I, pp. 3, 5. Cf. Stout, "Thought and
Language," Mind, April, 1891.
11 An Introduction to Mathematics, Home Univ. Library, p. 59. See also
P. E. B. Jourdain in The Monist, Jan. 1914, p. 141.
12 B. Russell, Our Knowledge of the External World, pp. 58, 242.
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 43
calculus is. Moreover, it accounts for the long period
which elapsed before their fruitful application of these
methods of calculation to special fields obtained the notice
they deserved.
The particular application we are here concerned with
is that to geometry. In a letter to Huygens of September
8, 1679, Leibniz complained that he was not satisfied with
the algebraic methods, and adds: "I believe that we must
have still another properly linear geometrical analysis,
which directly expresses situm as algebra expresses mag-
nitudinem. And I believe I have the means for it, and that
one could represent figures and even machines and move-
ments in symbols, as algebra represents number or magni-
tude; I am sending you an essay which seems to me
notable." This essay contained an account of his geo-
metrical calculus in which the relative position of points
is denoted by simple symbols and fixed without the help of
the magnitude of lines and angles. It differs therefore from
ordinary algebraic analytical geometry. The further de-
velopment of this calculus was the subject of Grassmann's
Geometrlsche Analyse13 which we have already noted as
being crowned by the Jablonowski Society. This was done
on the recommendation of Mobius, who found in Grass-
mann's essay a generalization and extension of his own
barycentric calculus.
We will now consider the geometrical calculus of Leib-
niz with a view to discovering if Grassmann's develop-
ment of it has fulfilled in any way Leibniz's hopes of its
ultimate importance.
§5.
The letters and papers of Leibniz in which he deals
with his project of a geometrical calculus are many, and
13 This treatise develops some of the subjects which Grassmann had in-
tended for a second part of the 1844 Ausdehnungslehre, which was never
written.
44 THE MONIST.
spread over a considerable period of time.14 The most im-
portant is the Char act eristic a geometrica, a sketch of the
notion which he made for fear it should be lost if he found
himself unable to develop it. The essay enclosed in the
letter to Huygens in 1679 was an extract from this. From
these writings it seems clear that the starting point was
his conviction of the imperfection of algebra as the logical
instrument of geometry. Thus, "Algebra itself is not the
true characteristic of geometry, but quite another must be
found, which I am certain would be more useful than
algebra for the use of geometry in the mechanical sciences.
And I wonder that this has hitherto been remarked by no
one. For almost all men hold algebra to be the true math-
ematical art of discovery, and as long as they labor under
this prejudice, they will never find the true characters of
other sciences." It must be n'oted that algebra is here used
by Leibniz in its ordinary sense, not as a general term for
any calculus.
He saw that analytical geometry only expressed geo-
metrical facts in a complicated and roundabout manner.
A figure such as the circle is not defined by its internal re-
lations, but by reference to its relations to arbitrary coor-
dinates. So a set of magnitudes foreign to the figure are
introduced and obscure the purely geometrical relation-
ships. Further, to reduce relations of position to relations
of size presupposes Thales's theorem about similar tri-
angles and the theorem of Pythagoras.15 In other words
analytical geometry is thus made dependent on synthetic.
The analysis not being pushed far enough, it has not the
logical perfection which belongs to a purely rational sci-
ence.16 He realized the want of rigor and generality of
14 An interesting bibliography of them together with an account of the
main ideas which inspired and directed his search for a geometrical charac-
teristic is given in Couturat, La logique de Leibniz, 1901.
16 Characteristica geometrica, § 5.
18 Cf. his letter to Bodenhausen.
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 45
intuitive methods, but dreamed of a method which would
be completely analytical and rational while still possessing
all the advantages of a synthetic method.
In this his aim was similar to that which he had in
mind for his universal characteristic, which was to be a
logical calculus replacing concepts by combinations of
signs, and which furthermore was not merely to furnish
demonstrations of propositions but to be the means of dis-
covering new ones. So, in like manner, his geometrical
calculus was to combine analysis with guidance of the in-
tuition.17 A fusion of analysis and synthesis being made,
the divorce between calculation and construction would
disappear. "This new characteristic. . . .will not fail to
give at the same time the solution, construction, and geo-
metrical demonstration, the whole in a natural manner and
by an analysis."1 It is clear that the final goal was a
science of form of very wide application.19 This aim we
must distinguish carefully from the manner in which he
attempted to realize it.
As Grassmann points out in the introduction to his
"Geometrical Analysis," this distinction between the dis-
tant goal and his attempt toward a new characteristic
which he connects with it to render the thought more real-
izable, is recognized fully by Leibniz. Although the char-
acteristic he provided will be seen to be only a small first
step toward the goal he had set himself, yet he had esti-
mated the essential advantages of a final geometrical anal-
ysis to an extraordinary completeness. Grassmann says:
"Just this eminent talent of Leibniz of being able to fore-
see in presentiment a whole series of developments without
being able to work it out and without dismembering and
17 Leibniz conjectured that the ancients had some natural and spontaneous
analysis of this kind resting on the abstract relations of figures, which under-
lay and helped their synthetic methods. (De analyst situs.)
18 Letter to Huygens.
19 Ibid. "I believe that one could handle mechanics by these means almost
like geometry."
46 THE MONIST.
dissecting it, yet to make it present to himself with pro-
phetic mind and to recognize the importance of its conse-
quences— it is just this talent which led him to such great
discoveries in almost all domains of knowledge."
§6.
Leibniz founds his fundamental definitions on con-
gruence, which means the possibility of coincidence. He
represents points whose positions are known by the first
letters of the alphabet, and those which are unknown or
variable by the last letters. Any two combinations of cor-
responding points are said to be congruent if both can be
brought to coincide without the mutual position of the
points being changed in either of the two combinations;
so that every point of one combination covers a correspond-
ing point in the other. Congruence (geometrical equality)
is a union of two relations — similarity and equality (quan-
titative equivalence).
All points are equal and similar, so all points are con-
gruent.20 Hence if we use = for congruence, the ex-
pression a - x, where a is fixed and x is variable, is a defi-
nition of space.
It must be noticed that in defining figures by congru-
ence the axiom of congruence or free mobility21 must be
postulated. If we do this, ax = be represents a sphere of
center a and radius be.
Also, ax = bx represents a plane which bisects ab per-
pendicularly.
The above can be taken as the definition of the sphere
and the plane respectively. Again ax = bx =cx gives the
locus of the center of all spheres which pass through a, b,
c; and so it is a straight line.
If ax^ac and bx = bc
they together give the common trace of two spheres.
20 Characteristica geometrica.
91 See B. Russell, Foundations of Geometry, Cambridge, 1897.
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 47
Combined they are written abx = abc. This therefore
represents the locus of points whose distances from the
points a, b are the same as the distances of c from a, b.
That is, it is a circle.
The economical nature of the symbolism is shown by
the fact that if we take this as a definition of the circle, it
does not imply the idea of the straight line or the plane;
nor does it require (as the circle defined by an algebraic
equation) that the center of the circle must be known.
As an example of a proof consider the proposition
that the intersection of two planes is a straight line.
Let ay = by be one plane
and ay = cy be the other.
Then ay = by = cy, and this we saw above to be
the form of the congruences representing a straight line.
In these examples is a faint foreshadowing of the side
by side development of construction, proof and analysis.
And since all kinds of spatial relationships can be devel-
oped from the line and the sphere, the method is capable
of wide extension.
§7.
There are several obvious defects in it, however. .These
appear at once if we attempt by means of it to solve the
fundamental problem in geometry of finding the expres-
sion for a straight line passing through two given points.
Leibniz had previously attacked the problem only to find
himself involved in difficulties.22
Grassmann's treatment is as follows : We saw above the
expression for a straight line was
ax = bx - ex.
If we now take three auxiliary points, a! ' , b' ', c', which
are not in a straight line, and write
** Couturat gives a clear account of this, op. cit., pp. 420-427.
THE MONIST.
'x = b'x = c'x
a'a = b'a = c'a
a'b = b'b = c'b,
then together these congruences represent the required
straight line through a, b, as the locus of x.
Combining the last two we get
$ a'x= b'x^c'x
\ aba' = abb' = abc'.
This then expresses that the auxiliary points lie on the
circle the plane of which is cut at its center by the line ab
at right angles.
If this expression is to have the necessary simplicity,
it must be possible to eliminate the arbitrary auxiliary
points which have nothing to do with the nature of the
problem, and to combine the group of formulas into one.
That being impossible, the characteristic has failed to
serve its purpose.
Indeed the failure of the method followed at once from
the choice of congruence as the fundamental relation. For,
as we have seen, this complex relation contains a quanti-
tative element, and so prevents any freeing of geometry
from considerations of magnitude. In fact, as the above
expression for the line through ab shows, we are still left
with arbitrary coordinates. Further, in this system there
is also ambiguity, as Couturat has shown.23 In other words
the analysis had not gone far enough. If what remained
of magnitude had been eliminated — not merely by taking
the relation of similarity, for Leibniz had himself shown
that to imply metrical relations24 — but by reducing figures
to their projective properties and relations, at least a real
geometry of position25 would have followed. But such a
projective geometry, while satisfying Leibniz's desire to
28 Ibid., p. 428, note 2.
24 "Elementa Nova Matheseos Universalis."
28 Developed by Staudt, Geometric der Lage, 1847.
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 49
eliminate algebraic methods from geometry, would not
have been a geometrical calculus with points as elements.
Nor could it have had the wide application which he sought
for in his calculus ; for if it was to be applicable to mechan-
ics and physics, it must at some point have been susceptible
of metrical development.
Now, throughout our discussion we have seen that
Leibniz was seeking for a characteristic particularly ap-
plicable to geometry but akin to his universal character-
istic. At the end of the letter to Huygens he says: "I
believe it is possible to extend the characteristic to things
which are not subject to imagination." In other words
he was seeking a formal calculus, an abstract mathematics
lying at the base of geometry and applicable not only to it
but also to logic. Now Grassmann had already developed
such a science of form in his Ausdehnungslehre of 1844.
So when the Jablonowski Society announced the subject
of their prize essay he took the opportunity of expounding
his science of extensive magnitudes, not as he had orig-
inally derived it, but starting from Leibniz's characteristic.
§8.
When he had proved the insufficiency of the relation
of congruence as Leibniz had left it, he tried to give it a
form in which substitution would be possible. What are
congruent to the same thing are congruent to each other,
but that does not mean that we can in a general way place
instead of a given term in a congruent expression one
congruent to it. So substitution is not possible. This can
be seen at once. All points are congruent. Therefore,
if one could substitute the congruent, one could place abc
congruent to every combination of three points — which is
absurd.
Grassmann rightly regarded the fact that substitution
was not possible as a serious defect in the calculus. So he
50 THE MONIST.
inquired what equations would hold between the points
a, b, c, d, e, f, if abc = def.
There must be some function f such that, when the
above holds,
f(a,b,c)=f(dfe,f).
So he was led to the general linear relation of collinearity.
Now in the Ausdehnungslehre Grassmann had reached
the fruitful idea of a true geometrical multiplication which
has the peculiarity that if any two factors of the product
are interchanged the sign of the product is changed, that is,
AB = — BA.
This combinatory multiplication enabled him now to give
an intrinsic definition of geometrical figures in terms of
points, and so to accomplish what Leibniz had failed to do.
Thus the product ab determines the straight line between
the points a, b ; the product of three points determines the
plane, and so on. But since the product is non-commuta-
tive these figures when so defined have a sense represented
by the signs + or — . Furthermore, he conceived the
notion of using these products to express not only relations
of position but also of magnitude. So that the same anal-
ysis which gave a geometry of position also gave, side by
side and without confusion, a metrical geometry. In making
this step he had to define (§3, Geometrlsche Analyse) what
he meant by a point magnitude. Each element (point,
line, plane) has two aspects — its position in space, and its
intensity. In the case of the point, this latter was repre-
sented by a positive or negative "mass."
By now defining a line magnitude as the combination
ab of the point magnitudes a, b — the direction of which is
through a and b, and the intensity of which can be defined ;
and also defining the point magnitude as the combination
AB of two line magnitudes, the position of which is the
intersecting point of A and B and the mass value of which
THE GEOMETRICAL ANALYSIS OF GRASSMANN. $1
can be made the subject of a definition — then by an as-
sumption which makes ab = O and AB = O represent
coincident lines and points, it is possible to write in the
form of an equation every linear dependence.
Thus (ab) (cd)e = O denotes that e is the intersecting
point of ab and cd.
So the principle of collineation can be expressed, though
cumbrously without further adaptation, by such combina-
tion equations.
In this way equality is made to include the two relations
of identity of position and equality of intensity. So pro-
jective and metrical relations can be expressed in one form,
and considered either separately or together.
§9.
It is impossible to follow Grassmann's development26
further without setting up a technical symbolism, but it
may easily be shown how brilliantly Leibniz's hopes of an
analysis specially applicable to mechanics have been ful-
filled.
In terms of this calculus the sum of n points is their
mean point. If intensities are considered, the metrical
relation follows. Thus if the intensities represent masses
at the points the sum gives the center of gravity of the
system — a point whose intensity will be the sum of the
other intensities. If the intensities represent parallel forces
acting at the point the sum gives the point of application
of the resultant. The barycentric calculus of Mobius is
thus included in this more general analysis.
Furthermore, the line magnitude of Grassmann ex-
presses a force with exactitude. Composition of forces
thus becomes the addition of line magnitudes. The general
equations of dynamics can also be represented (§ n, Geo-
26 Needless to say the above is a mere sketch of the beginning of Grass-
mann's "Analysis." In particular no mention is made of his distinction be-
tween inner and outer products.
52 THE MONIST.
metrische Analyse) by means of this calculus, as soon as
certain modes of treating infinitesimals have been evolved.
Moreover the possibility of attaching a metrical co-
efficient to each point in space opens at once many fields
of application in physics.
We must notice in addition that the "Geometrical Anal-
ysis" does not treat of the quotients of non-parallel
stretches, a subject which leads to a calculus for dealing
with powers, roots, logarithms and angles.
Grassmann can claim justly therefore, as he does, in
the concluding remarks to this work, that his mode of
treatment, if transferred to physics in general, would sim-
plify the mathematical treatment in a splendid manner.
He himself has shown the great advantages of the calculus
in many fields. In the essay we have several times referred
to, Leibniz wrote, "If it [the characteristic] were set up
in the manner I conceive, one could construct in symbols,
which would only be the letters of the alphabet, the de-
scription of any machine. . . . One could by these means
make exact descriptions of natural objects."
As an example of such descriptive power Grassmann
mentions his application of the calculus to crystallography
(cf. Ausdehnungslehre of 1844, § 171).
§ 10.
Apart from the adaptability of the geometrical cal-
culus to different provinces, there are other good reasons
for believing that it realizes the ideal toward which Leib-
niz looked forward.27 Grassmann's claim put forward in
his concluding remarks will, I think, be granted by any
one willing to master the symbolism sufficiently to under-
27 Letter to Huygens : "Algebra is nothing but the characteristic of in-
determinate numbers, or of magnitude. But it does not express exactly situa-
tion, angles and movement.... But this new characteristic. .. .cannot fail to
give at the same time the solution, the construction and the geometrical proof,
the whole in a natural manner and by analysis. That is by determined ways."
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 53
stand any of his theorems. "As in the analysis demon-
strated here every equation is only the expression, clothed
in the form of the analysis, of a geometrical relation, and
this relation expresses itself clearly in the equation without
being obscured by arbitrary magnitudes — as for example
the coordinates of the usual analysis — and therefore can be
read off from it without further trouble; and as further
every form of such equation is only the expression of a
corresponding construction, then it follows that as a matter
of fact, by means of the analysis here given, the solution
of a geometrical problem results at the same time as the
construction and the proof. As further nothing arbitrary
.... need be introduced, the kind of solution must always
be according to the nature of the problem; and as it is in
the form of analysis, therefore a necessary one in which
there can be no question of any seeking round for methods
of solution." In other words the fusion of synthetic and
analytic methods which Leibniz hoped for is fully accom-
plished.
It must be noted that in one respect Grassmann has
not only realized the prophetic vision of Leibniz but also
cleared away the inconsistency which vitiates his attempt
at making his dream come true. For Leibniz, seeing that
the fundamental analysis of geometry must rest on non-
metrical relations, yet desired its final application to me-
chanics and natural science, in which metrical relations
are all important. So he was led to a half-hearted attempt
at non-metrical analysis by means of a relation — congru-
ence— which, while showing the way to a geometry not
based on algebra, yet failed itself to travel far in that direc-
tion. The special merit of Grassmann has been to found
a geometrical analysis free of magnitude and yet so to
develop it that metrical considerations may be introduced
without disturbing the form of that analysis. Projective
geometry, therefore, only partly fulfils Leibniz's hopes;
54 THE MONIST.
their complete realization is found in Grassmann's theory
of extension.
§».
We began our discussion of the relation between Grass-
mann's calculus and the characteristic of Leibniz by an
analysis of the manner in which their work has been re-
ceived by the average mathematician. It seems to me that
we can profitably return to these historical considerations
for a moment, and look at them from another view-point.
There is some reason, as I have tried to show else-
where,28 for citing lack of historical perspective on the part
of mathematicians as the cause of the unsympathetic atti-
tude commonly taken up in regard to work of philosophical
breadth; and that if more regard were paid to historical
development in mathematical education wider and more
penetrating vision would result. The position taken up is
well expressed by Branford29 : "The path of most effective
development of knowledge and power in the individual
coincides, in broad outline, with the path historically tra-
versed by the race in developing that particular kind of
knowledge and power." At the same time, however, we
must realize that, if we alter our attitude to this slightly,
and regard it not from the point of view of the education-
alist but from that of the original worker himself, obsession
with origins seems inevitably to lead to what I have called
above the genetic error. The effective point of departure
in attaining knowledge of geometry may be from such
empirical and utilitarian experiments as form its historical
origin. But that must not be allowed to create an at-
mosphere hostile to any recognition of the a priori and
formal nature of that science.
Furthermore the historical method may lead to a cer-
tain ex cathedra manner, a reliance on authority and tra-
28 "The Neglect of the Work of H. Grassmann."
29 B. Branford, A Study of Mathematical Education, 1908.
THE GEOMETRICAL ANALYSIS OF GRASSMANN. 55
dition. It is this factor which especially concerns us in
our attempt to see the work of Leibniz and Grassmann
in true relation to each other and to mathematical thought.
For Couturat points out30 that what probably hindered
Leibniz's development of his geometrical calculus and ren-
dered abortive his attempt at its realization was the author-
ity of Euclid. He says, "Why, amidst all the relations
which Leibniz catalogued, did he give preference to the
relation of congruence and neglect the relations of simi-
larity, inclusion, situation, which serve to-day as the bases
of quite new sciences31 which he foresaw and would have
been able to found ? It is evidently because tradition, rep-
resented and embodied by the Elements of Euclid, limited
geometry to the study of the metrical properties of space.
Now the tradition is not explicable by any reason of theo-
retical order (considering that metrical relations are more
complex and less general than projective relations) but
solely by reason of historical and practical order."
I have already in the previous section shown that an-
other explanation may be held of this clinging to a metrical
relation by Leibniz. However that may be, the authority
of the Euclidean tradition may have had some influence
on his work in geometry, as the Aristotelian tradition had
in his foreshadowing of a logical characteristic.32 In fact
we shall not be laying over-emphasis on the tendencies of
an exaggerated reliance on historical method if we say that
its final result is the attitude of the young critic in Shaw's
play88 who says, in effect, "Give me the name of the author
and I'll tell you if it's a good play." If that critic held a
university chair of historical criticism he would doubtless
be able to .find valid arguments for his position — for how
80 La logique de Leibniz, pp. 438-440. Russell however attributes Leibniz's
failure to his holding the relational theory of space, Mind, 1903, p. 190.
81 Theory of aggregates, modern Analysis situs, projective geometry, etc.
82 See note 7 above.
88 "Fanny's First Play."
56 THE MONIST.
(he might ask) can one judge competently without a com-
plete set of data, and is not authorship an important datum ?
It is irritation at this standpoint which causes Mr. Bertrand
Russell, whom I have heard speak very forcibly on the sub-
ject, inveigh against this hyper-historical method. But the
objection can be stated in a much stronger form. "Erudition
often does violence to inventive power: and the proof is
that the modern discoverers of symbolic logic, Boole and
his successors, have all ignored (and rightly) the example
and precedent of Leibniz ; it has even been remarked34 that
they have almost all been ignorant of one another, and if
this ignorance has been a source of error, it has been above
all a condition of originality."3 Now it does not appear to
me that the essential defect of such an extreme anti-histor-
ical attitude has been that it caused error. Staudt realized
the ambitions of Leibniz in some degree in founding his
projective geometry, and Grassmann in still further degree
in creating his theory of extension, without knowing that
their historical origins lay in his work. No great harm
comes from this, although an original genius will, as a
general rule, be less likely to be deflected from his way
by the work of others than to find in them sources of stim-
ulation. But to the mass of us, who form the bulk of man-
kind, narrowness is a mental blinker which hides the full
splendor of the creations of genius. The real toll taken
by historical ignorance is in neglect of originality, and the
loss of power and influence consequent on it.88
A. E. HEATH.
BEDALES, PETERSFIELD, ENGLAND.
84 J. Venn, Symbolic Logic, Introd., pp.29, 30.
88 Couturat, op. cit., p. 440.
88 I have to thank Miss Vinvela Cummin for valuable help in translating
the Geometrische Analyse.
GREEK IDEAS OF AN AFTERWORLD.
A STUDY OF THE RELATION BETWEEN PRACTICE AND BELIEF.
WE have a free and easy way of generalizing the after-
world of Greek religious belief as an underworld.
This is indeed the usual form of the belief from Hesiod
onward, and it is the view generally disclosed by Homer
both in the Iliad and Odyssey. Yet the fact is that the
most deliberate and detailed Greek presentation of the
approach to that dread world, that of the eleventh book of
the Odyssey, does not at all represent it as an underworld
like the infernal regions of Vergil's fancy, but as a far
western realm. The far-wandering Odysseus sails to the
distant west, out of the sea, and across the mighty ocean
stream to its farther shore; he beaches his black-hulled
ship on a lone waste beach where stand the barren groves
of Persephone ; thence he directs his steps inland to a great
white rock at the confluence of the Styx, Pyriphlegethon
and Acheron ; and there it is that he enters the purlieus of
the many-peopled house of dark-browed Hades.
The Odyssean realm of the dead is reached neither by
descent into a cave nor by passage underneath an over-
hanging ledge. It is of the same level as the land of living
men. Its darkness is apparently due to its location beyond
the path of the western sun, which, descending into Ocean
Stream, disappears somewhere from the sight of mortal
men to be ushered in anew by rosy-fingered Eos, each
succeeding morn. To speak of Odysseus as descending into
an underworld is to have but little regard for the language
of Homer. Clearly to discern the picture that he actually
58 THE MONIST.
presents is to become aware of a striking contrast between
it and the afterworld of classic Greek and Roman belief;
and this contrast raises the problem of explaining and ac-
counting for such different views, obviously related in the
same way to the same fact, — the fact of death. An obvious
relation, I say ; if this appears to be but a bold assumption,
I trust it will be justified in the course of my argument.
A study of early man's beliefs about an afterworld in-
volves a consideration of two groups or series of facts —
mental facts and motor facts, or facts of belief and facts
of practice, — both associated with the event of death. Ap-
parently these two kinds of facts do not simply constitute
two parallel series that were mutually unrelated in life and
thought and that may therefore be studied and understood
apart from each other ; they seem to have an intimate and
genetic relationship. This, however, is not to say that they
are absolutely simultaneous in origin, or that one may not
be primary and the other secondary, both in origin and
importance. On the contrary, in their genesis, either belief
is antecedent and causal to practice, or practice is antece-
dent and causal to belief.
It is popularly supposed that belief originates and dic-
tates practice, or custom, which is thus regarded as secon-
dary to belief. Anthropologists generally confirm the sup-
position, and whole systems of social interpretation and
philosophy have been built upon the assumption. Professor
Seymour, in his Greek Life in the Homeric Age, insists
upon this relation in the case of Greek mortuary practice
and belief, and cautions the reader against assuming that
the Greeks who maintained certain customs may have "in-
herited also the beliefs on which those customs were orig-
inally based." He brings to bear upon the case the author-
ity of the German scholar Rohde, declaring that "Rohde
gives as the cause of the adoption of cremation by the
ancestors of the Homeric Greeks, a desire to rid themselves
GREEK IDEAS OF AN AFTERWORLD. 59
of the souls of the dead; and as a result of the change, the
abandonment of the old ritual and sacrifices."
According to Professor Brinton, "The funeral or mor-
tuary ceremonies, which are often so elaborate and so
punctiliously performed in savage tribes, have a twofold
purpose. They are equally for the benefit of the individual
and for that of the community. If they are neglected or
inadequately conducted, the restless spirit of the departed
cannot reach the realm of joyous peace, and therefore re-
turns to lurk about his former home and to plague the sur-
vivors for their carelessness.
"It was therefore to lay the ghost, to avoid the anger of
the disembodied spirit, that the living instituted and per-
formed the burial ceremonies; while it became to the in-
terest of the individual to provide for it that those rites
should be carried out which would conduct his own soul
to the abode of the blest."
Here again practice is regarded as secondary to belief,
and is interpreted by reference to belief. Professor Frazer,
also, the dean of living anthropologists, insists upon this
relation between our two series of facts, and cannot admit
or conceive of the opposite as being true. I intend, how-
ever, to take the other side of the question here involved,
advancing the proposition that it was mortuary practice
that constituted the motive for belief in an af terworld ; and
especially shall I endeavor to indicate the application of
this formula to the genetic interpretation of Greek ideas
of an afterworld.
The Hellenic peoples of whom we have knowledge uni-
versally believed in an afterworld, whither the souls of
mortals departed at death and where they had a continued
existence. But they entertained not merely the two con-
flicting beliefs already mentioned; they held in developed
form at least four quite different beliefs regarding the
destination and abode of the souls of the dead. According
6O THE MONIST.
to one of these beliefs the souls of dead men ascended to
Olympus, as did that of Heracles in story ; according to an-
other they descended into an underworld; in the eleventh
book of the Odyssey Homer places them in a continental
region beyond the western verge of Ocean Stream; and
Pindar places the souls of great heroes in "Islands of the
Blest" in the far Western Ocean.
It may be well at this point to note some apparently
fundamental resemblances between these last two beliefs.
Pindar places the souls of sinful mortals in an underworld,
subject to sentences reluctantly imposed upon them. Hesiod
declares that the men of the Golden, Silver and Bronze
ages were hidden away in earth ; and it is but natural, be-
cause of the different types of life imputed to them, that
he should fancy different conditions for them after death.
But the souls of his age of heroes, he says, were given a
life and an abode apart from men, and established at the
ends of the earth in "Islands of the Blest by deep-eddying
Ocean." He does not state the direction of these wondrous
islands, but undoubtedly their direction, like that of the
Pindaric Islands of the Blest and that of the Odyssean
realm, was already so fixed in the tradition of his day that
there was no need of indicating it. It would appear, then,
that in essential characteristics the continental Odyssean
realm and the Islands of the Blest are alike in being
conceived as western, and differ only in geographical form
and extent. From this it would further appear that
the notions of these two similar abodes of the dead are
variants derived from a single source. But if these two
notions did grow out of a single origin there was certainly
a reason for the divergence, which it should be part of our
task to discover. And yet, on the other hand, it may be
unnecessary or even incorrect to assign their origin to the
same people, even though we may feel compelled to assume
GREEK IDEAS OF AN AFTERWORLD. 6l
that the significant common element of direction must in-
here in a common element of antecedent cause.
Whence came these three or four differing beliefs ? That
is to say, upon what difference of psychological ground do
they severally stand ? No one man could at one time enter-
tain so many and so contradictory beliefs upon one subject;
neither could one homogeneous people, as, for example, a
single city state of the Mycenean civilization, or even the
Minoan civilization of Crete as a whole. Wherefore we
should probably look for this difference of belief either in
the several racial stocks amalgamated to form the historic
Greek people, or in part to their respective traditional be-
liefs and in part to alien streams of influence. But in either
case it will be pertinent to inquire how different races and
racial stocks should have come thus to act and believe so
differently in the face of the same fact, death. To trace a
belief or practice from one people back to another should
never be taken as an explanation; this done, the question
of real origin and motive still remains, as insistent as ever.
Neither should identity of belief or practice be taken as
necessary evidence of racial relationships, or of racial con-
tacts; nor difference of belief or practice as evidence of
difference of race. There are others besides the human
factor that enter into the origin and development of prac-
tice, as we shall presently see.
With regard to mortuary practice, the Greek world
furnishes only two types of historically attested facts. The
Homeric Achseans cremated their dead, and the practice
survived far beyond the Homeric, and even the Periclean
age. The Mycenean civilization laid its dead beneath the
surface of the earth, and this practice gradually superseded
cremation, even among the descendants of the Achaeans.
Thus the Greeks of historical times had two strongly con-
trasted modes of disposing of their dead, corresponding to
two of the contrasted beliefs we have mentioned. For there
62 THE MONIST.
is undoubtedly a genetic relation between cremation and
belief in a heavenly abode of souls, and between inhuma-
tion and belief in an underworld. But which is cause and
which effect? And how did the causal series itself orig-
inate? And how could the belief in a western abode of
souls be related to either of these, either as antecedent or
as consequent?
These two series of facts in Hellenic life give rise to
three problems of immediate significance; to say nothing
of others more remote, as for example, how man came to
believe that he had a soul at all, how nearly the belief coin-
cides with actuality, the origin of religious fears, etc. The
three special problems thus isolated for present considera-
tion are:
1. What is the genetic relation and order of precedence
between practice and belief, — between cremation and belief
in a heavenly abode of souls, and between inhumation and
belief in an underworld?
2. In case either belief or practice is found to be antece-
dent to the other, how then did this antecedent series take
its rise?
3. Whence and how came the belief in a far western
abode of souls, and why the apparently twofold differen-
tiation of this belief, which we have noted?
In the interest of brevity I may appear to be cutting
the Gordian knot rather than untying it; but I feel sure
that the drift of my argument will be caught, and that its
essential truth must make a strong appeal for assent.
In the first place, let us consider this intimate and in-
herent correspondence between mortuary practice and be-
lief about the dead, under conditions where we can see more
plainly the part played by geographical environment, and
where at the same time we can be sure of the soil on which
our two series of facts originated; for we know not yet
where the practice of inhumation originated among the
GREEK IDEAS OF AN AFTERWORLD. 63
Myceneans and Minoans, nor where cremation first de-
veloped among the Achaeans.
The ancient Egyptians and the Incas of Peru preserved
their dead by mummification, and both believed in a bodily
resurrection of the dead. We are reasonably sure that the
land where each of these peoples developed was likewise
the soil upon which their respective traditions in this mat-
ter originated; we shall be still more sure of this local
origin as we proceed. Did the belief or the practice pre-
cede?
Now no matter what we may imagine them to have
thought about soul and body and their mutual relations
before the practice began, the Egyptians and Peruvians
could not have cremated their dead ; both Egypt and Peru
lacked that abundant supply of fuel which would be neces-
sary for this practice among a numerous people. Neither
could either people long have inhumed its dead in the fertile
valley land of its abode. These restricted valleys early
became the seat of such dense populations that productive
land could not be permanently set aside for burial pur-
poses ; nor could land under cultivation be wantonly tram-
pled over for this common social purpose, even though six
feet of earth were sufficient for the individual grave. Of
necessity, therefore, the adjacent desert ridges were em-
ployed for the purpose, and the earliest mode of burial there
was inhumation. But the dry climate and the nitrous char-
acter of the upland soil, both in Egypt and Peru, tended
naturally to preserve the bodies of the dead. The action
of wind and wild animals, however, tended often to exhume
them, at the same time disclosing a high degree of preser-
vation. In order to protect their dead, especially to prevent
the work of their hands from being made of none effect,
the Egyptians, in particular, came to build rock tombs.
But this required much labor and expense. Yet it was
cheaper to build one tomb large enough for many burials,
64 THE MONIST.
for whole families, even through successive generations,
than to build many individual tombs. Hence, by mutual
suggestion and social rivalry through long stretches of
time, the mighty Pyramids of Egypt came to be developed.
But under these conditions a tomb must be entered from
time to time for new burials; and in spite of their high
degree of preservation by natural means, the bodies of the
dead within gave rise to noisome odors. Hence arose the
practice of embalming with aromatic spices, to counteract
or obscure the evil odors of decomposition. What but this
fact of unpleasant odors could first have suggested the use of
expensive spices in embalming? With the prominent Egyp-
tian nose was undoubtedly associated a keen sense of smell.
Wrappings of linen served in the first instance to retain
the spices. The embalming tended to more perfect preser-
vation of the flesh, and this result also helped to accomplish
the primary object of the practice, which was the laying
of unpleasant odors. Upon this combination of facts arose
a profession of embalmers, who developed a more and
more elaborate technique. When death and funerals had
thus become an economic burden upon the living, for which
no obvious or adequate return was received, the question of
meaning inevitably arose and persistently pressed for a
satisfactory answer. It is exceedingly difficult for man to
admit that he is spending sacred energies in vain or pur-
poseless quests, and thus making a fool of himself; and so
the practice, entailing so large an expense, insistently re-
quired a sanction, and a tremendous one at that.
Now the care lavished upon the dead body, by tending
to preserve it for an indefinite length of time, embodied
within it an inherent and obvious suggestion of the primary
sanction that actually came to be formulated. For by this
time embalming had come to take place before the process
of decomposition had set in ; and the original cause of the
practice was no longer making its appearance, even though
GREEK IDEAS OF AN AFTERWORLD. 65
from allied experience the agents may well have been
aware of what would soon happen without embalming and
burial. So now, instead of really knowing that they are
trying to forestall or allay the noisome odors of decompo-
sition, they detect but one purpose in the practice, the
preservation of the body. But why should the body of the
dead be preserved ? With this query arose the first sugges-
tion of a mystical or transcendental idea in association
with the practice, and the first attempt to formulate an
ultra-pragmatic or other-world sanction for it. This sanc-
tion was formulated as an explanation. It was from the
first employed for this purpose, and as all thinking indi-
viduals were implicated in the practice no one was in a
position to question or challenge it.
It might be urged on this latter ground that the ques-
tion of purpose or value could never have arisen; but we
must not overlook the fact of foreign contacts — especially
among the Egyptians — wherein contrasted practices would
raise the question from without, if not from within. Be-
sides this, they had always the poor with them, who,
from contrast with their own meager efforts in the same
field, would be forced to think about values. And above
all, there was always growing up among them the supreme
pragmatist, — the eager, curious child.
Thus this question of values, like the ghost of Banquo,
was ever likely to confront the living, and only a powerful
sanction would serve to lay it. The priesthood and the
professional embalmers, in particular, had constant need
of the sanction, as a means of justifying their existence.
Thus it is that this sanction arose, and that it has been
passed on and received as an explanation even by the
wisest, even unto the present day. And that is in brief the
story of the Egyptian and Peruvian practice of mummi-
fication, and of their belief in a bodily resurrection. It all
comes back in the last analysis to the fact of decomposition
66 THE MONIST.
and the despised sense of smell, which would move men to
acts of aversion and riddance.
But, one may ask, is it not after all just possible that
this practice arose out of an antecedent idea of souls and
the notion that the body must be preserved against a future
resurrection and a reincarnation of the soul? Rather is
it not far more reasonable to see that the belief arose out
of the practice, as a sanction for the care and expense in-
volved in it? On the first alternative we must certainly
congratulate the Egyptians, and the Peruvians too, on
having found a geographical location so congenial to their
belief. What would they have practiced, or how could this
belief have survived, had they lived in the valley of the
Congo or Amazon, or even in Greece ? Or how could they
have come to believe in a heavenly abode of souls, when
they did not cremate ? And if the belief in a bodily resur-
rection came before the practice of mummification, then
how did this notion and belief arise?
Now let us take a look at barren, hungry, frost-bitten
Tibet. What burial practices and what cognate beliefs
about the dead have from the first been inherent in the
natural environment of man presented by the Himalayan
highland? Let us picture to ourselves a people making
here its arduous ascent from lowest savagery to barbar-
ism. As they come to have a settled place of abode, how
shall they secure for themselves riddance from the discom-
forting odors of decomposition that follow in the train of
death? Suppose that they have attained to such a degree
of economic efficiency as to have left behind the practice
of cannibalism, and that they are as yet without any meta-
physical or transcendental ideas and beliefs; how then
shall they dispose of their dead ? Or what shall they believe
about their dead, if they have as yet paid no attention to
them save by the simplest modes of seposition and aban-
donment ?
GREEK IDEAS OF AN AFTERWORLD. 67
Here in Tibet is a people that could not cremate its
dead ; for here, too, fuel is scarce. Neither could it inhume
its dead ; for during a considerable portion of the year the
deeply frozen ground is proof against even the tools of civil-
ized man. Here preservation of the dead by natural means,
that of freezing, may be assured for a season ; but should
this be relied upon temporarily, final burial by one means
or another would become imperative with the advent of
spring. Shall the Tibetans preserve the bodies of their
dead through the long winter, to the end that they may
give them some sort of approved burial in the spring?
What could originally have suggested to them the notion of
an approved form of burial, and of the preservation of their
dead against the time when this should become possible?
The primary function of burial by whatever means is avoid-
ance or riddance of certain after effects of death ; and with
an abundance of carnivorous animal life scouring the coun-
try for the means of subsistence, how could the immediate,
practical function of burial be more readily or more easily
secured than by calling in the aid of dogs and vultures that
infest the land ? Now this is exactly what the Tibetans do,
even to-day. And from their own hard struggle for ex-
istence they furthermore feel it an act of charity thus to
minister to these scavengers of their land. There is no
other people on earth with whom charity is so highly es-
teemed as a virtue, and so universally encouraged. Under
the hard conditions of life, charity, generosity, is a neces-
sary practice among their own kind. And furthermore,
the leisure-class priesthood, which is very numerous, in
its own self-interest has need of encouraging this funda-
mentally necessary virtue ; and finally, this virtue is invoked
as a sanction for the feeding of their dead to the beasts of
the field and the birds of the air. Without some notion of
other ways of securing this same object, they could feel
no need of this or of any other sanction.
68 THE MONIST.
In the case of a very few individuals of the highest rank
cremation is allowed as a special honor, and naturally this
privilege is mostly restricted to the religious hierarchy. It
is evidently not the native Tibetan practice, but was plainly
introduced into Tibet by the Buddhists of India, with whom
it was native. But the great majority of the Tibetan dead
go to feed the hungry dogs and vultures, which are highly
esteemed for this purpose; and this, despite the fact that
Tibet has for a dozen centuries been subjected to Buddhist
influence, which would naturally favor cremation, its own
native mode of burial, if this were economically possible.
Here in Tibet the native mode of burial is directly apposite
to geographical conditions, even as it was in Egypt and
Peru; and the beliefs by which it is explained are merely
so many sanctions, or justifications, which have been de-
veloped out of the practice itself.
But what are the Tibetan beliefs about the dead ? When
once they have acquired the notion of a soul that survives
the event of death, whether originally or by adoption from
other peoples, we should expect them to hold a belief in
some sort of transmigration. From seeing the bodies of
the dead devoured by animals, they would seem naturally
to think that souls also passed into the bodies of these liv-
ing sepulchers. This is exactly what they believe. We
should furthermore expect them to have a preference for
transmigration into the winged vulture that sails so easily
through the air, to taking up their abode within the body
of a lazy, grunting pig or snarling dog. Here too our sur-
mises are correct. In the course of centuries, as the rela-
tion between practice and belief has become obscured, their
beliefs have been elaborated and graduated, so that even
non-carnivorous animals are included in certain cycles of
transmigration. But in this fact of feeding their dead to
animals is certainly to be found the original germ and sug-
gestion of their belief in transmigration. Tibetan religious
GREEK IDEAS OF AN AFTERWORLD. 69
ideas and beliefs are not so definitely conceived nor so sys-
tematically organized as are those of some other peoples,
because their authors have never devoted so much personal
care and energy to the disposal of their dead. They have
not felt so strong a necessity for justifying their practice
as have the Egyptians and some other peoples.
Again, let us consider the case of India, where Brah-
manism and Buddhism have their origin and home. The
Indians, like the Tibetans, hold a belief in transmigration,
and of course for that same fundamental reason. That a
mighty, far-scattered people like the Indians exhibits a
characteristic belief or practice does not mean that all in-
dividuals of the group hold it in common. It would be too
much to expect such a people, or any people at all, to be
really homogeneous in belief and practice from the early
stage when human burial began among their forebears
until the present time. Thousands of families in India to-
day are too poor to afford the most characteristic tradi-
tional form of burial for their dead, and throw them into
rivers, or otherwise dispose of them. In Indo-China those
to poor tu afford cremation commonly carry out their dead
to be eaten by the beasts of the jungle. On the Ganges,
"When the pyre is built the nearest relative of the deceased
goes to the temple and haggles with the keeper of the
sacred fire over the price of a spark ; and having paid what
is required he brings the fire down in smouldering straw
and lights the pile. If the family can afford to buy enough
wood, the body is completely consumed; in any case the
ashes or whatever is left on the exhaustion of the fire is
thrown into the sacred river ; . . . . and any failure on the
part of the fire to do its full duty is made good by the fish
and the crocodiles."1 Thus it is easy to see how in bygone
days the Indian, at least in the lower social strata, became
possessed of a belief in transmigration, and how, through
1 Pratt, India and its Faiths. New York, 1915, p. 44.
7O THE MONIST.
ignorance of its primary source and relationships, carried
it over into relationships bearing little or no connection
with its parent practice, as in his abstinence from eating
flesh.
And yet India, with its wide extent and countless popu-
lation, has more constant elements of religious and philo-
sophical belief than would at first seem possible, — a result
of mutual contacts and social cooperation through long
stretches of time. "The central point of Hindu thought is
the soul. It is from the soul or self that all the reasoning
of the Hindu starts and to it that all his arguments finally
return."2 Probably the most widely known characteristic
of Indian religious philosophy is the doctrine of the im-
manence and absoluteness of the supreme soul Brahman,
with its correlate doctrine of the oneness of the individual
self with the All, — the merging of the objective, phenom-
enal world into the universal absolute, which is Brahman.
Yet it is plain that this interest in the objective world
begins with the individual human self. "This unity of the
soul with God is at the foundation not only of Hindu meta-
physics, but of Hindu ethics as well. The great aim of life
is the full realization of that God-consciousness, the sig-
nificance of which forms the central point of Hindu
thought. Before this can be fully attained, the soul must
be liberated from the mass of particular interest and petty
wishes and self-born illusions which weigh it down and
hide from it the beatific vision. Hence liberation and reali-
sation may be called the twin ideals of Hinduism, and it is
these that determine all its ethical theory."3
The doctrine of "liberation" and "realization," the doc-
trine of Nirvana, the yoga-systems, and other character-
istic Indian notions would be meaningless and impossible
without the basic body of "religious intuitions" that make
2 Pratt, p. 91.
8 Pratt, p. 92.
GREEK IDEAS OF AN AFTERWORLD. 7!
up the Brahmanistic doctrine of the Upanishads. But an
"intuition" has in primordial genesis some sensuous basis,
direct or indirect; and so, instead of seeking for the idea
or philosophy back of the practices associated with these
and other beliefs, we should undoubtedly seek in practice
for the sensuous elements of suggestion that formed the
basis of the beliefs, and then seek in turn for the sensuous
motive of the practice itself.
We may admit that the Indians are a peculiar people;
yet, when we pin ourselves down to minute details, we note
that the testimony of their senses, the ultimate constituent
of all intellectual forms, is the same as our own. Their
intellectual peculiarity consists not in their physical or psy-
chological selves, but in the differences of their objective
environment, part of which they themselves make, and in
the various ways in which the sensuous details of expe-
rience with it have been combined through generations of
spontaneous social collaboration.
If, then, we consider these doctrines of the "infinite
ocean of the absolute Brahman"; of the essential oneness
of the one with the All; of the soul's struggle for liberation,
to realize and complete this oneness in "Nirvana, or re-
absorption into the eternal light" : as we contemplate these
doctrines, seeking to discover their source in sensuous ex-
perience at a time antedating the rise of science with its
theories of atoms and corpuscles, can we not almost see
before our eyes the primitive populace of India cremating
its dead and beholding the body ascending in the form of
flame and smoke, thus becoming absorbed in the ocean of
air, which to them, at that time, seems infinite ?
We examine Indian burial practices, both present and
past, and we find that from time immemorial cremation
has been a characteristic Indian mode of burial. When
men actually beheld the body of a deceased friend dissolve
and mingle with the elements, they were bound to have
72 THE MONIST.
different thoughts about the destiny of the individual than
if it were laid away in earth to decompose by degrees for
an unknown length of time, or if it were altogether pre-
served by embalming against decomposition. And from
seeing the individual thus pass so visibly from a corporate
existence into thin air, they would also be moved more
strongly to contemplate the other end of individual exist-
ence, the whence as well as the whither. There could be no
doubt that the deceased had attained to freedom from the
bonds and ills of terrestrial existence ; and the living, from
their own desires to live beyond the usual limits of life,
would be brought face to face with the question whether
they should ever live again and how their scattered selves
could realize another conscious existence. To hold before
them the notion of another life as something to be desired
was to believe in it; and from this point it was an easy
matter to identify the conditions of existence before birth
and after death, whence Brahman becomes the source, the
end, and the essential constituent of individual existence.
Add to all this the practice of feeding to animals either
the entire body or the remains of partial cremation, already
noted — the differences of practice being characteristically
in agreement with differences of social rank — and we have
the proper sensuous background in practice for the doctrine
of transmigration, which we find embodied in the doctrine
of Karma and fused with the doctrine of Nirvana.
Geographical conditions undoubtedly favored crema-
tion in India in the days when fuel was abundant and
easily secured. But with a numerous population making
large demands upon the wood-supply through scores of
generations, the practice has become more and more ex-
pensive, and the demand for sufficient sanction has become
more imperative. Thus in the course of many centuries
the beliefs genetically inhering in these practices have be-
come much elaborated ; and, by the development of an elab-
GREEK IDEAS OF AN AFTERWORLD. 73
orate logic and metaphysic, they have in turn modified the
practice itself. It is in this way that the religious institu-
tion has justified its ways and made itself indispensable
to men.
Here again we may claim without fear of successful
contradiction that burial practice arose as a purely prac-
tical matter and by its form dictated the form that belief
about souls must take, when once the notion of soul itself
arose out of the practice. The sense of smell together with
the simple, practical knowledge of the purifying agency
of fire suggested and motivated the practice; here it is
that we find the sensuous motive behind the practice, which
in turn motivated the belief. Primarily, the belief is a
supposed explanation of the practice, invented when the
practice had become so highly elaborated as to conceal its
real cause and thus to demand justification. Men do not
feel the need of explaining or justifying the obviously prac-
tical.
But the explanation given of this and other kinds of
practice is not an explanatiaon of the covert act; rather is
it intended to explain or justify the care and energy de-
voted to it or required by it in the name of social form.
The overt act merely affords suggestions toward the ex-
planation that is evolved. It is only after a long lapse of
time during which a practice has by social concurrence
become highly elaborated that a justification is required.
Men acting in unison, with a common sense or emotional
interest, will do extravagant things not dreamed of in indi-
vidual life. But, having participated in such an act, un-
sophisticated man can easily find a justification for his act,
suggested by the act itself. It seems to be a characteristic
of universal human nature, in the absence of a true, an-
tecedent cause for specified conduct, to seek about for some
consequent justification ; and the race seems equally prone
74 THE MONIST.
to accept such a justification as a statement of antecedent
cause.
And now we may return to the case of Greece. We
do not find there that close, almost necessary relation be-
tween practice and environment which we have seen in
Tibet, Egypt and Peru; in fact, we cannot say with cer-
tainty where the two historic Greek forms of burial orig-
inated. Already some 3000 years before the Christian era
we find the Minoan civilization in the ^Egean world, prac-
ticing inhumation. And the northern Achaeans, from
whatever source they came, were already at their arrival
in Greece practicing cremation. As to the relation between
the beliefs and practices that prevailed on Hellenic soil, we
can argue only by analogy, or homology, with what we
have seen to be true in Egypt, Peru, Tibet and India ; but
it is far more reasonable to believe that the same relation
holds true here than to defend the other horn of the di-
lemma.
With regard to the Achaean belief in a heavenly abode
of souls, we may cut the matter short by asserting its
rise out of the practice of cremation. In the course of time,
after this practice had become the rule among the ancestors
of the Homeric Achaeans, they probably came to feel much
the same regarding it as did the Indian of California. "It
is the one passion of his superstition to think of the soul of
his departed friend as set free, and purified by the flames ;
not bound to the mouldering body, but borne up on the soft
clouds of smoke toward the beautiful sun."4 I say the
Achaean may have come to feel in this way, much as did
the Hindu; but this was not the original motive of his
practice. His thoughts about the mouldering body of his
departed friend and his fancies about purification were not
in the first instance inspired by a desire for the friend's
welfare after death; he was first of all concerned for the
* Powers, The Indians of California, pp. 181, 207.
GREEK IDEAS OF AN AFTERWORLD. 75
living, especially with regard to the sense of smell. And
however transcendental the notion of purification came to
be by reinterpretation of the practice, after its original
motive had ceased to prevail — because burial came to be
practiced before decomposition had set in — the very associa-
tion of purity with cremation betrays the original motive
of the practice, just as did the use of spices by the Egyp-
tians.
As with cremation among the Achseans, so in the case
of inhumation among the Minoans and Mycenaeans we
may assert that the practice was suggested, and passed
through its primary stage of development, as a means of
escape from the discomforting odors of decomposition.
And as the belief in an upper-world abode of souls devel-
oped as an explanation and sanction for cremation, so
belief in an underworld developed by suggestion from the
practice of inhumation. To make good the claim that be-
lief came first and suggested practice, one must show satis-
factorily how any people ever could have associated souls
with a heavenly or with an underworld abode without the
practice of cremation or inhumation, respectively, or at
least contact with some people who did practice this mode
of burial.
The belief associated with cremation never became so
highly elaborated in Greece as it did in India, and for very
good reasons. For in the first place, Greece never came so
completely into the power of a priestly class as did India;
and in the second place, the practice on which it depended
here came into rivalry with the already established prac-
tice of inhumation, which on the whole was cheaper. To
this we should add the fact that the social institutions of
the older race proved to be the more persistent, as with
the Normans and Saxons in England, whence this must
have been especially true of such ideas as we are discussing.
And however spectacular and interesting the act of crema-
76 THE MONIST.
tion became among the Hellenes, as reflected in the Ho-
meric picture of the funerals of Patroclus and Hector, the
accompanying conception of the soul after death could be
but very vaguely imaged, as in the case of India ; while the
same idea accompanying burial in the ground, in cave-
tombs, cist-tombs, and rock-tombs, as the so-called "treas-
ury of Atreus" was capable of very definite imagery. Thus,
although cremation continued to be practiced side by side
with inhumation, it was the belief associated with the latter
practice that possessed the more definite imaginative ap-
peal, and that finally prevailed.
Yet the upper-world conception of the soul persisted
and influenced the belief of later generations. As in the
first instance it was only the Achaean masters of Hellas
who practiced cremation, while the subject populace in-
humed its dead; and since in the classical age it was only
the wealthy who could afford cremation; so it came to be
believed that the "good" — the worthy and the proud — at
death went to heaven above, while the poor in purse and
spirit descended into hell. Various modifications of this
composite belief have grown up by internal suggestion and
by accretions from foreign practices and beliefs; but in
the last analysis each belief grew out of a practice, and the
practice originated as an obvious and immediately practical
necessity.
While we cannot say just where or why the Minoans
developed inhumation and the Achaeans cremation, or why
some other practice did not arise and prevail among each
people, yet it is perhaps significant that cremation was the
practice of the northern race, like the aboriginal Hindus,
— a people who had more need of fire on a large scale, such
as would be necessary for the cremation of human bodies,
a people with whom fire was necessarily a more continuous
object of experience and therefore a more constant agent
GREEK IDEAS OF AN AFTERWORLD. 77
of purification in other ways also, than it was in the sunny
southland of Crete and Hellas.
Homer was the poet of the Achaean overlords of Hellas.
Yet he was apparently not of the Achaean race. Although
he quite consistently presents to us the Achaean mode of
burial, his idea of the soul and its abode is not consistent
with the practice of cremation. He thinks of the cremated
Heracles as having a corporate existence in Olympus, with
lovely-ankled Hebe at his side; yet Heracles must also be
seen of Odysseus in the house of Hades. Homer is him-
self aware of the contradiction, and declares it to be but
a phantom that Odysseus sees there. On the other hand,
Achaean heroes — Patroclus and others such as would nat-
urally have been cremated — he unequivocally represents as
being in the populous realm of Hades in the distant west.
In Homer's references to the realm of the dead we discern
the unconscious and inextricable mingling of at least three
traditional views on the subject. Nor should we be sur-
prised at this when we note that the entire period from the
Trojan War to the final completion of the Homeric tales
was one of ethnic amalgamation between at least the two
races we have already mentioned. Our view of this process
is still further complicated, and yet perhaps much illumi-
nated, by the knowledge of a continuous intercourse with
the west coast of Asia Minor during this time, such that
most of the cities that laid claim to Homer were of this
region.
And this prompts us to consider how the notion could
have arisen that the dread abode of souls was in the west.
It would perhaps be interesting to point to the west as the
region of the setting sun, to associate it with the death of
the day, and to conjure up some fancied analogy as having
been indulged in by the aboriginal authors of this tradition.
Yet in the face of such a procedure stands the fact that the
west has always been the land of allurement and promise
78 THE MONIST.
to which Greek no less than Teuton has ever turned his
eyes. The fact is that if the association of the west is an
essential element of the belief, as it appears to be, then
thoughts of the west were inherently involved in the form
of burial with which the belief was genetically associated.
We might look to cremation for the source of the asso-
ciation, if anywhere in the yEgean world the prevailing
winds blew to the westward, thus bearing the smoke of the
funeral pyre in that direction. But such is not the case;
and besides, neither the earthly location of the Odyssean
afterworld and the Islands of the Blest, nor the substantial,
corporeal nature of the spirits dwelling there would permit
of this conclusion.
I know not what may be the value of the suggestion I
am about to make upon this subject ; I simply present it as
the most plausible explanation I can imagine for the con-
ception of a western realm of the dead. I have by no means
enumerated all the methods that man has employed for
the disposal of his dead. Fundamentally there is but one
reason for disposing of the dead by any means, and that is
to secure a separation between the dead and the living.
Inhumation and cremation are merely the most obvious
and most universally practicable means of securing this
one end.
Now one of the simplest modes of accomplishing this
object, where natural facilities permit, is what is called
canoe-burial, — a mode in which the body of the dead is
placed upon a log, or raft, or boat, and set adrift upon the
sea, or down a stream. In the course of time this practice,
just as any other, is subject to elaboration and refinement,
and finally to mythical, transcendental interpretation. I
suggest that this Hellenic notion of a western realm of the
dead originated on the western coast of Asia Minor. Here
all rivers flow to the west; out to the westward over the
sea are beautiful islands which could once have been imag-
GREEK IDEAS OF AN AFTERWORLD. 79
ined as the destination of bodies set adrift on the rivers of
this coast ; and finally, when these islands had been visited
and explored and the fancy exploded, it was but natural
to set the place of destination of the dead still farther to
the west beyond the ^Egean archipelago. And since even
by Homer's time the Hellenes had dim fancies, more or
less substantiated, of extensive coasts in the distant west,
it was but natural that the earlier notion of an island abode
for the dead had to give way to fancies of a more con-
tinental region. But as the primitive occupants of this
Asiatic coast had grown bolder and put out to sea, they had
perhaps found on the coasts of the y£gean islands the un-
sightly wrecks of their death-craft, and so had come to dis-
continue the practice. It is not necessary to suppose that
this practice was current in the time of Homer, or even
of the Trojan War; mythical fancies may survive long
after the conditions that fathered them have ceased to
exist.
Such is my suggestion for explaining the notion of a
western abode of souls, presented on the assumption that
both these traditions go back to a single local source. Yet
I am not unmindful that the coast of Epirus and Illyria
furnish the natural conditions in which either one or both
may have arisen; whence we should have to suppose that
they were brought into Greece by the Achaeans. On this
assumption we should have to suppose further that these
Achaean adventurers, after leaving their native abode and
the conditions supporting their native mortuary practice,
took to cremation as a new means of disposing of their
dead, and yet retained the tradition associated with the
native practice of canoe burial. This would help to account
for the incongruities in the Homeric conception of the con-
dition of souls whose bodies had been burned; it would
mean that they had not yet maintained the practice long
enough to have invested it with a systematic sanction and
8O THE MONIST.
philosophy. As between these two suggestions, I should
probably prefer the former. As yet I see no way in which
archeology may help us here.
In any case the tradition of a western abode of the dead,
which had already been started and which had by this time
lost all direct association with the practice, continued and
gathered to itself the Homeric, and Hesiodic, and Pindaric
refinements and differentiae which we have already noted.
Such is the regular course of tradition. It is undoubtedly
in this way, and by reference to the same kind of burial
practice in Britain that the traditional pictufe came to be
built up of the black-hulled ship that bore "Elaine the fair,
Elaine the beautiful" down the Thames to Westminster;
and of that other dusky barge that bore out into the mystic
lake beyond the ken of mortal man all that was mortal of
good King Arthur. Such a social background is probably
necessary for the historical interpretation of the death voy-
age of Sinfiotli, son of Sigmund, away "to the west" ; and
of Balder and his faithful wife Nanna, laid on their funeral
pyre on the deck of the stately ship Ringhorn. We can
understand and explain how a traditional practice arises
and grows by social concurrence, and how a belief arises
in association with it, all conscious association with the
practice being gradually lost. But to explain how prac-
tice should arise out of an antecedent belief, and how that
belief should first have arisen as a purely intellectual con-
ception without sensuous motivation — as the grin without
the cat, as one might say — in spite of some three thousand
years of effort upon this problem, we are quite as far from
a satisfactory solution as ever.
To conclude, then, the act of burial by early peoples is
an act of aversion and riddance, even as the traditional
interpreters of the act have claimed ; but the primary object
of the riddance, instead of being a metaphysical, or spirit-
ual object, is a real, concrete, sensuous reality, which is
GREEK IDEAS OF AN AFTER WORLD. 8 1
exactly the necessary and apposite kind of motive that we
should expect. If only Hobbes had hit upon this formula!
But he had not at hand the rich accumulation of anthro-
pological data that we now possess. And even Spencer
and Tylor, with all the data at their command and with all
their ability to analyze and organize their essential ele-
ments, made the same mistake as Hobbes. For in the first
place they made belief about the dead a result of secondary
sensuous experience, instead of primary; and secondly,
they made it to depend upon visual instead of olfactory ex-
perience. The sense primarily concerned in the evolution
of religious aversions associated with ideas of the dead is
undoubtedly that of smell. This primary aversion, by a
traditional transfiguration, becomes a dread or fear of the
dead and places of burial ; and only when man requests of
his most-used sense to show him the cause of the aversion
does it become visualized. And then only is it that dreams,
visions, apparitions, reflections and other illusory visual
phenomena gain a superstitious meaning.
Thus it is only by misinterpretation of the act of avoid-
ing or allaying the noisome odors of decomposition, when
the real motive to the act has disappeared from view, that
a people can ever explain its burial practice as a spiritual
"riddance" or "aversion," or as a "laying of the ghost."
For the anthropologist to accept this secondary aspect of
the relation between belief and practice as being primary,
and to proceed upon this assumption to the explanation
of burial practices is to put the cart before the horse. Such
reasoning is all of a piece with myth; it is reasoning in a
circle, and will never get us anywhere in the realm of scien-
tific knowledge.
For such reasons as I have given above, which I believe
to be sound, I feel reasonably certain that my primary
assumption of an obvious and constant relation between
the fact of death and beliefs about the dead is justified;
82 THE MONIST.
that geographical conditions have played a hitherto un-
recognized part in the development of burial practice and
belief about the dead; that the sense of smell has had an
unrecognized share in the development of religious notions
and especially religious fears ; that the Greek notion of an
underworld abode of the dead grew out of the practice of
inhumation, and that the notion of a heavenly abode of
souls in like manner grew out of the practice of cremation.
And it is by reason of the satisfactory corroboration of
my reasoning with regard to inhumation and cremation
that I suggest a primitive practice of canoe-burial on the
west coast of Asia Minor — or possibly the Balkan penin-
sula— as the primary motive to the conception of a western
abode of souls, whether as Islands of the Blest or as a con-
tinental realm of dark-browed Hades.
ORLAND O. NORRIS.
YPSILANTI, MICHIGAN.
BERNARD BOLZANO.*
(1781-1848.)
IN BOLZANO we find the virtues of human sympathy
and insight coupled with the austerer virtues of the
metaphysician and logician. He was a man of action as
well as a man of ideas. He was well known for his kindly
disposition and his broadmindedness. He possessed not
only the sympathy with the poor necessary for a social
reformer, but the ability to develop his ideas of social re-
construction on practical lines. Not only did he elaborate
a theory of an ideal state, but he also introduced numerous
reforms in the actual state of which he was a member. He
studied theology very earnestly as a young man and later
wrote a great deal on the subject. Even though his liberal
views brought him into collision with those on whom his
livelihood depended, yet he courageously continued his
teaching and writing, always making it his aim to seek
for truth. He was a metaphysician of some importance
and his treatises on metaphysics are valuable, not only for
the original thought which they contain, but also for his
important criticisms of Kant. In esthetics his work is by
no means without interest, and to the psychology and
ethics of his day he made very valuable contributions. But
preeminently he was a mathematician and logician. In his
* We regret that owing to limited time and the uncertainties of trans-
atlantic mail service The Monist is compelled to go to press without receiving
the author's imprimatur.
84 THE MONIST.
work on mathematical analysis and mathematical logic,
he stood out from all the other thinkers of his day. He
was a man of many ideas and his intellectual equipment
made him able to indicate to his followers the most fruitful
lines of inquiry. All through his life he worked for the
good of mankind, helping it on in its search for truth.
Bernard Bolzano was born on October 5, 1781, at
Prague.1 He was the fourth son of Bernard Bolzano, an
upright and philanthropic member of the Italian commun-
ity at Prague. His mother was a very pious women. He
had a large number of brothers and sisters, the majority
of whom perished in childhood; he himself was a sickly
child. In his early youth he was very much interested in
mathematics and philosophy. His education was of the
type usual at the end of the eighteenth century. He tells
us that as a child he used to let passion completely over-
master him because he believed that he was raging not at
people but at Evil itself. Bolzano was sent to one of the
gymnasia of his native city, where he did not distinguish
himself very much, and later proceeded to the university
there. At the university he studied philosophy and sub-
sequently theology. It was his father's wish that he should
be a business man, and though his father finally gave way
he showed his disapproval of his son's desire to continue
his studies in various ways.
Bolzano had been brought up a Roman Catholic and
he was much troubled with doubts as to whether he should
take orders. Finally, however, he became convinced that
difficult problems, such as the authenticity of the miracles,
were not essential parts of the Catholic faith, and as in
his opinion the office of priest offered the best opportunity
of doing good, he took orders in 1805. At the same time
he became doctor of philosophy at Prague University, and
1 Lebensbeschreibung des Dr. B. Bolzano mit einigen seiner ungedruckten
Aufs'dtze und dem Bildnisse des Verfassers; eingeleitet und erldutert von dem
Herausgeber (J. M. Fesl), Sulzbach, 1836.
BERNARD BOLZANO. 85
was appointed professor of the philosophic theory of re-
ligion.
As professor, Bolzano suffered many cramping indig-
nities which surrounded all teachers in Roman Catholic
countries at that time. To a man with Bolzano's sympathies,
the position must have been a peculiarly trying one. He
had a great love for young people2 and mixed freely with
the students. He was particularly sought after by the
students because of his liberal views. His broad-minded
interpretation of the dogmas of the Catholic faith, while
provoking the distrust of the authorities, recommended him
to the younger generation, and he wielded a great influence
in their revolutionary schemes and was thought by many
to have supported them with an enthusiasm unbecoming
in a professor. At any rate, relations between Bolzano
and the authorities grew more and more strained, and
finally, as he would not recall what they were pleased to
call his "heresies," he was dismissed on the grounds that
he had "failed grievously in his duties as priest, as precep-
tor of religion and of youth, and as a good citizen."
After his dismissal from Prague, two t ecclesiastical
commissions were successively appointed by the Archbishop
of Prague to inquire into the orthodoxy of his teaching.
In the first commission, the majority declared that Bol-
zano's teaching was entirely Catholic, but the word "en-
tirely" was deleted at the wish of the minority — which
consisted of one person. This decision so enraged the ob-
scurantist party that a large amount of evidence (not a
small amount of which was "faked" for the purpose) was
collected and put before the second commission. In 1822
Bolzano made two declarations in writing in which he
stated that he held it "dangerous, even with the best in-
tentions, for a man to seek and teach new points of view
2 See A. Wishaupt, Skizzcn aus dem Leben Bolsanos: Beitrdge zu seiner
Biographic von dessen Arzte, Leipsic, 1850, pp. 19ff.
86 THE MONIST.
as proofs of the truth and divine nature of the Christian
Religion."3 The commission then finally collapsed. Two
years later Bolzano was pressed for a public recantation.
The Archbishop of Prague brought illicit pressure to bear
on him by pleading his affection for him and by declaring
that a refusal would bring him to the grave. Bolzano,
however, refused to recant publicly, but solemnly declared
his orthodoxy in writing.
The main points of his teaching on religion are set out
at some length in his Lehrbuch der Religionswissenschaft.*
He defines religion as the aggregate of doctrines which
influence man's virtue and happiness. He then proceeds
to discuss what seemed to him the most perfect religion,
viz., the Catholic faith. His reason for so regarding the
Catholic faith is that it is, in his opinion, revealed by God.
A religion is divinely revealed, according to Bolzano, if it
is morally beneficial and if connected with it there are
supernatural events which have no other use than that
they serve to demonstrate this religion. In the first chapter
the concepts of religion in general, and organized religion
in particular, are discussed. In the third chapter he main-
tains that for a religion to be true it must be revealed, and
then he proceeds to enunciate the characteristics of a reve-
lation. In the second volume, he sets out to prove that the
Catholic religion possesses the highest moral usefulness
and that its origin has the attestation of supernatural oc-
currences. He discusses the evidence for Christ's miracles
and the genuineness of the sources and points out the pres-
ence in Christianity of the external characteristic of reve-
lation. He then passes on, in the third volume, to demon-
strate in some detail the moral usefulness of the faith.
After a discussion of the Catholic doctrine of the sources
of knowledge he examines the various doctrines of the
3 Published 1836 (Sulzbach) with autobiography.
* Sulzbach, 1839 (4 volumes).
BERNARD BOLZANO. 87
Catholic church. It is interesting to notice that he regards
the doctrine of the Trinity as entirely reasonable, and com-
pares the Father to the All, the Son to humanity, and the
Holy Ghost to the individual soul. In the last chapter of
this volume Bolzano is concerned with the Catholic system
of morals. In his investigation he discusses first Catholic
ethics and then the various means of salvation recom-
mended by the church. He examines each of the sacra-
ments in turn.5
After his dismissal from Prague, Bolzano wrote a very
great deal, but the internal censorship prohibited all publi-
cations in his name and even in some cases retained the
manuscript. Bolzano once expressed the pious hope that
some day he might be allowed to publish some work of a
purely mathematical nature ! After he left Prague he lived
chiefly with friends at Techobuz. He came back, finally,
to his native city in 1841 and continued his work with
vigor until his death in 1848.
Though it was in mathematics that Bolzano did his
most important work, yet in other subjects, notably in
political science, his work is of considerable value. He had
very great sympathy with the poor and was anxious to
abolish class differences. He was convinced that the in-
adequacy of social organizations was the cause of poverty.
He never wrote very much on the matter, but made it the
subject of many of his professorial addresses. There is,
however, one short manuscript8 in which he sets out the
main points of his political theory. Bolzano himself thought
a great deal of this manuscript for he says in the intro-
duction: "And small as is the number of these pages, yet
the author thinks he may be allowed to attribute some
value to them. Nay, he considers that this little book is
8 For a complete list of his theological works see Bergmann, Das philo-
sophische Werk Bernard Bolzanos, Halle, 1909, p. 214.
6 "Vom besten Staate, MS. in the Royal Bohemian Museum. For a con-
venient summary of the MS. see Bergmann, op. cit., pp. 130ff.
88 THE MONIST.
the best and most important legacy that he can bequeath
to his fellow men if they are willing to accept it."
In Bolzano's ideal state, men and women alike are to
have the privilege of voting, but a person is only allowed
to vote on a matter of which he has some knowledge and
in which he has some interest. Further, the right of voting
is liable to forfeiture in the case of misconduct. Any citi-
zen may put forward a suggestion. The suggestion is
examined by six independent citizens, each one examining
it privately, and it is only rejected if all six of the citizens
reject it — and even then it is retained by the state for
further reference. If it is not rejected, a general vote is
taken, and if there is a majority in favor of it, it goes to a
council7 which is composed of men and women over sixty
years of age, who are chosen by the people every three
years. The council can only veto the decisions of the people
if ninety percent of the council are against it. The govern-
ment is the administrative body, its members are paid and
elected by the people, and there is a strict limit to the
length of time that they may remain in office. The govern-
ment takes special care to prevent private individuals com-
bining in their own interest. Bolzano looked upon war as
a dreadful misfortune and in his Utopia war is only to be
used as a defensive measure. Bolzano points out that
internal revolutions are unlikely, for they arise in general
from one of two causes — a bad constitution or poverty.
Of these, poverty is to be non-existent and a revolution
due to the first cause is improbable because it could only
be brought about if the council opposed a change in the
constitution which the people considered advisable. But
the council in its wisdom would not taunt the people but
would give reasons for its decision. It therefore seems
unlikely that the people would rise in revolt, all the more
because it is early impressed upon the young that a good
7 The council is called the "Rat der Gepriiften."
BERNARD BOLZANO. 89
citizen does not work against the government, for the
government's object is to work for the good of the whole
state.
One of the most interesting parts of the manuscript
deals with the idea of property. In the ideal state property
is only desired in so far as the possession of it contributes
to the common good. The only valid claim of a man to
property is, therefore, that he can make it more useful to
the state than any one else could. The fact that a man
may possess a certain thing at a certain time is not a
necessary or sufficient reason that he shall possess it alto-
gether. The right of inheritance is not recognized. Things
such as books, paintings, furniture or jewels, are given to
a citizen to use but not to possess. Further, even though he
may have established his claim to a certain object, yet, if
at any subsequent time another citizen can make more use
of it, the title of the first citizen to it is gone. Moreover,
the state does not offer any compensation to a man for
depriving him of anything. Thus a man whose eyesight
has been cured has his glasses taken away and no compen-
sation is made. In all the distribution of goods the govern-
ment is guided entirely by the principle that the use of a
certain thing should be granted to the citizen who can
render it most useful to the state as a whole.
The ideals of the state are freedom and equality. There
is no unequal distribution of wealth. However there is not
an absolute equality of owners, for, as Bolzano points out,
the possibility of increasing one's property is a powerful
incitement to work. But there are limits beyond which
a man cannot increase the extent of his property, and these
limits are determined by the consideration of the good of
the state as a while. There are "equal" right for all citi-
zens, but the word "equal" is not to be interpreted in any
narrow sense. Rather there is an adjustment between the
rights of a citizen and his obligations, between his strength
9O THE MONIST.
and his need. The government aims at promoting religious
freedom. No religion is given preferential treatment by
the state. People choose their own ministers of religion and
support them. But a new religion may not be preached
without permission, for some might not be able to grasp
all the consequences of accepting certain doctrines and be-
liefs. Further, a citizen may change his religion, but he
must first bring proof that he has studied with earnestness
the principles of the religion he is about to leave, as well
as of the one which he desires to embrace.
In the education of children the special aim is the de-
velopment of the mind. The teachers do not have complete
freedom in the choice of what the children are taught.
The Council, if it is unanimous, has the power to prevent
the teaching of any particular doctrine. The children's
books are censored. The censor is responsible directly to
the government. And not only the children's books, but
all the books in the state are censored strictly.
The question of rewards and punishments in the state
is treated in a practical way. Rewards are to consist in
public recognition of merit, and punishments are not ar-
ranged on a definite plan but are modified so as to suit
individual cases. There is however a special proviso that
no citizen is under any circumstances to be imprisoned for
life.
Bolzano has some very interesting ideas on the occu-
pations of the people in his Utopia. To begin with, the
state is to support those who are not fit to work. From
those who are fit, the state demands a certain fixed amount
of work — the fixed amount, of course, varying from one
individual to another. In return for the work the state
distributes goods. Citizens are not allowed to waste their
time in useless or pernicious occupations — Bolzano con-
sidered newspapers pernicious. Neither are they allowed
to do things in any but the quickest and most satisfac-
BERNARD BOLZANO. 91
tory way. Thus they are not allowed to thresh with a
flail when a threshing machine has been invented, nor,
presumably, to walk when there is a tram. One interesting
point is that the state is to pay compensation for damage
done by nature. Bad weather would quickly lose its terror
for farmers in Bolzano's ideal state. Finally, those who
wish to devote their lives to art or some branch of learning
are supported by the state if they can produce evidence to
show that it will be in the state's interest that they shall
be employed in this way. The whole theory of the state
is peculiarly fresh and in many respects suggestive.
But Bolzano's Utopia is only a practical illustration of
his general ethical principles. The guiding principle of
his inquiry may be enunciated as follows: Of all possible
actions, one should always choose that one which, when
all consequences have been considered, produces the great-
est amount of good or the least amount of evil, for the
human race as a whole, and in this estimate the good of
individuals, as such, is to be left out of consideration. But
Bolzano points out that if this principle is to be the highest
moral law, it would be necessary to frame a definition of
good and bad before any practical applications could be
made. Further since he holds that an action is good if it
is an action which we ought to perform, he gets back im-
mediately to the question: What ought I to do?8
There then remains only the effects of action on the
faculty of sensation. Bolzano argues that, since one can
excite only either pleasant or unpleasant sensations and
since no one would hold that it is one's duty to excite un-
pleasant sensations, it is obviously one's duty to excite
pleasant sensations. By this process of eliminating every-
thing except the faculty of sensation, Bolzano comes to the
conclusion that the highest moral duty is the excitement
8 For an interesting and valuable criticism of Bolzano's assertions and
deductions mentioned here, see Bergmann, op. cit., Part V, § 958.
92 THE MONIST.
of pleasant sensations. Not the least interesting part of
his work in ethics is his criticism of Kant's categorical
imperative. He urges the necessity for a modification in
Kant's principle and points out the invalidity of Kant's
theory that the opposite of a duty involves a contradiction.
Bolzano's work in esthetics is not without interest.9
His theory of esthetics is the result, not of his own esthetic
sensations, but of a painstaking analysis of the abstract
idea. His definition of the scope of the subject does not
make it coincide with the theory of beauty unless we include
in that theory not only the sum total of truths directly con-
cerned with beauty but also all those which stand in such
a relation to them that either the former cannot be thor-
oughly understood without the latter or the latter without
the former. To get at his concept of beauty, he eliminates
goodness and attractiveness, and by this process obtains
a first criterion of beauty, viz., all beauty is pleasant, i. e.,
it produces pleasure and this pleasure arises solely from the
contemplation of the object. Further, since animals are
to be excluded from esthetic enjoyment, qualities must be
introduced which they do not possess, e. g., intelligence,
judgment and reason. Bolzano then comes to the conclu-
sion that it is the growth of these qualities in us that is
responsible for the pleasure we find in beauty. Together
with the "Ueber den Begriff des Schonen" in the Royal
Bohemian Museum, there is another short treatise of Bol-
zano's in which a theory of laughter is elaborated.10 Bol-
zano thought that laughter was caused by the rapid alter-
nation of pleasant and unpleasant sensations and from the
fact that animals and infants do not laugh he deduces that
laughter is not entirely physical.11
In his metaphysics, Bolzano reveals himself as "one of
9 See Ueber den Begriff des Schonen, Prague, 1843.
10 Ueber den Begriff des L'dcherlichen, 1818.
11 See Bergmann, op. cit., Part IV, § 56.
BERNARD BOLZANO. 93
the acutest critics of the Kantian philosophy and the 'ideal-
ist' development from Fichte to Hegel."1 He also did
some important original work. His chief book on the
subject,13 entitled Wissenschaftslehre: Versuch einer aus-
fiihrlichen und grosstenteils neuen Darstellung der Logik,1*
is divided into five sections. In the first of these he sets
out to prove that objective truth exists and that it is pos-
sible for us to have knowledge of it ; but he allows that in
the development of the science of knowledge, which is the
most fundament0! of the sciences, it is necessary to use
some psychological methods of treatment. In the second
part, the "Theory of Elements," he treats successively
ideas-in-themselves, their combination into propositions-
in-themselves, the theory of true propositions-in-themselves,
and finally their combination into syllogisms. He is ex-
tremely careful to distinguish between the idea-in-itself
and the conceived idea. The concept of a proposition-in-
itself is produced by a double abstraction. First the mean-
ing of the proposition and the words conveying the mean-
ing have to be separated from each other, and then one has
to forget that the proposition has ever been in anybody's
mind. By this means we get to the concept of a proposition-
in-itself.
In the distinction that he draws between perception
and conception, Bolzano himself says that he owes very
much to Kant, but Bolzano disagrees with him in the use
he makes of this distinction in his theory of time and
space. Bolzano examines in some detail Kant's theory of
time and space and his theory of the categories, making
some very acute criticisms. After an investigation into the
theory of the syllogism and a discussion of the function
12 A. E. Taylor, Mind, October, 1915.
For a criticism
1905.
Sulzbach, 1837.
13 For a criticism of Bolzano's theories see M. Palagyi, Kant und Bolzano.
Halle, 1905.
94 THE MONIST.
of the linguistic expression of a proposition, the "Theory
of Elements" closes with a criticism of previous works on
the subject. Next Bolzano considers the appearance in the
mind of propositions-in-themselves. And it is in this part
of his work in particular that we see the extent and depth
of his learning. He treats first our subjective ideas, then
our judgments, then the relation of our judgments to
truth, and finally their certainty and probability. In this
investigation Bolzano uses psychological methods to some
extent. Then after the fourth part, the "Art of Inventing,"
he comes at last in the fifth part to the "Science of Knowl-
edge Proper." The book is remarkable as much for its
wealth of original thought and the clearness of expression
as for the important criticisms of earlier works on the
subject.
But important as is Bolzano's work in metaphysics,
ethics, esthetics, and theology, it is preeminently as a math-
ematician that he should be remembered. Now there are
two ways of looking at mathematics. One can look upon
it as Huxley did: "Mathematics may be compared to a
mill of exquisite workmanship, which grinds you stuff to
any degree of fineness." On the other hand, one can look
upon mathematics as a real and genuine science and then
the applications are only interesting in so far as they con-
tain and suggest problems in pure mathematics. From the
second point of view the most important business of the
mathematician is to examine and strengthen the founda-
tions of mathematics and to purify its methods. In addi-
tion to these points of view which may be called the prac-
tical and the philosophical, a third point of view has sprung
up in the last century which may be called the purely logical
point of view. Whitehead describes this new point of view
in the words, "Mathematics in its widest significance is the
development of all types of formal, necessary, deductive,
BERNARD BOLZANO. 95
reasoning/'1 In this purely logical system, it is proposed
to treat any special development of mathematics with the
help of a definite, logically connected complex of ideas,
and the mathematician is not to be satisfied to solve par-
ticular problems with the help of any methods which may
casually present themselves, however ingenious these meth-
ods may be. Clear definitions and unambiguous axioms
must be framed and then by rigorous reasoning the the-
orems of the subject are to be deduced.
We find examples of the first and second points of view
among the Greeks. It is said of Pythagoras that "he
changed the occupation with this branch of knowledge into
a real science, inasmuch as he contemplated its foundation
from a higher point of view and investigated the theorems
less materially and more intellectually,"15 and of Plato
that "he filled his writings with mathematical discussions,
showing everywhere how much geometry there is in phi-
losophy." Just as mathematics among the Greeks had its
origin in the geometry invented by the Egyptians for
practical surveying purposes, so the mathematics of the
seventeenth and eighteenth century received its stimulus
from the practical researches of Kepler, Newton and La-
place. But in this same fragment of Eudemus we find
it recorded that Euclid tried to revise the methods used
and "put together the elements, arranging much for Eude-
mus, finishing much for Thaetetus ; he moreover subjected
to rigorous proofs what had been negligently demonstrated
by his predecessors."
This same work that Euclid did for Greek mathematics
three hundred years B. C, the new school of nineteenth
century mathematicians performed for European mathe-
10 A. N. Whitehead, A Treatise on Universal Algebra, Cambridge, 1898,
preface, p. vi.
16 Extract from a fragment preserved by Proclus ; generally attributed
to Eudemus of Rhodes who belongs to the peripatetic school and wrote treat-
ises on geometry and astronomy. See extracts in J. T. Merz, History of
European Thought in the Nineteenth Century, Vol. II, p. 634.
g THE MONIST.
matics. The researches of Newton had suggested a wealth
of material for mathematical treatment. Newton a'nd
Leibniz had stumbled across the powerful methods of the
calculus, which were of tremendous practical importance;
but as Klein says, "the naive intuition was especially active
during the period of the genesis of the calculus,"17 and in
the great call for powerful methods the theoretical side was
almost entirely overshadowed. For example Newton as-
sumed the existence of the velocity of a moving point at
every point of its path, not troubling whether, as subse-
quent investigation has shown to be the case, there might
not be continuous functions having no derivative. The
great work then of this new school was to investigate the
validity of the methods used in the two previous centuries.
This was no easy task, and it is only now after one hundred
years that the theory of the subject is being put on a logic-
ally satisfactory basis. The most important ideas round
which the greater part of the work in mathematics cen-
tered, are those of continuity and infinity. The importance
of these concepts became apparent from the work done on
infinite series. A particularly simple example of series,
viz., decimal fractions, was in use as early as the sixteenth
century, but Leibniz was the first mathematician to have
any idea of the importance of series in mathematics. Be-
fore his time it had not been realized that an infinite series
can only have a meaning under certain circumstances. Un-
fortunately Leibniz came to the conclusion that the sum
of the series
i — i -|- i — i. . . .ad inf.
is 5^2 >18 and so exercised a somewhat baneful influence on
17 Evanston Colloquium; Lectures on Mathematics delivered September,
1893, Lecture VI.
18 Euler in 1755 (Instit. Calc. Diff.) defined the sum of this series to be
J4- In the recent theory of divergent series (due in great measure to E. Borel
see his Legons sur les series divergentes, Paris, 1901) one way of denning the
formal sum of a divergent series So/i is as the limit, when it exists, of *2>anXn
as x tends to unity through values less than unity. This definition has the
BERNARD BOLZANO. 97
subsequent mathematical developments of the theory of
infinite series. However it was left to the genius of Bol-
zano19 to enunciate for the first time the necessary and
sufficient conditions for the convergence of an infinite se-
ries. In 1804 Bolzano published his Betrachtungen iiber
einige Gegenstdnde der Element ar geometric (Prague), and
in 1810 his Beytrdge zu einer begrundeteren Darstellung
derMathematik (Prague). In 1816 he published an impor-
tant tract on the binomial theorem. In this tract his work
on convergency is of great value and his investigation for
a real argument (which he everywhere presupposes) is very
satisfactory. Bolzano comments on the unrestricted use
of infinite series which was common at the time. In 1812
Gauss had published an investigation into the circum-
stances under which the hypergeometric series converges,
and in 1820 Cauchy delivered some extremely important
lectures on analysis at the College de France, where he
was the leader of a group of young mathematicians. Thus
Bolzano, Gauss and Cauchy were the pioneers. In his
book, Der binomische Lehrsatz und als Folgerung aus
ihm der polynomische und die Reihen, die zur Berechnung
der Logarithmen und Exponential gross en dienen, genauer
als bisher erweisen (Prague), Bolzano has made a valu-
able criticism of earlier investigations. It is remarkable
that his writings, though of great importance, received
comparatively little attention at the time. According to
Merz, he had not, like Cauchy, "the art peculiar to the
French of refining their ideas and communicating them in
merit of simplicity and also of "consistency," i. e., When the series So» con-
verges, its sum is still the limit as x tends to unity through smaller values, of
^anxn if this limit exists.
Defining the formal sum in this way the sum of the series 1 — 1 + 1...
ad inf. is Vz.
19 Accounts of Bolzano's mathematical work were given by Otto Stolz
(Math. Ann., Vol. XVIII, 1881, pp. 255-279; Vol. XXII, 1883, pp. 518-519)
and on pp. 37-39 of the notes at the end of the reprint of Bolzano's "Rein
analytischer Beweis" of 1817 in No. 153 of Ostivald's Klassiker.
98 THE MONIST.
the most appropriate and taking manner."2 In his Rein
analytischer Beweis (1817) Bolzano tells us that it is very
much better to publish one's mathematical work in separate
treatises ; in this way there is more chance of getting acute
criticism. Consequently we find his mathematical work
scattered about in various small treatises.21 Also he tells
us that one of his treatises had the misfortune not to be
noticed by some of the learned periodicals and in others to
be criticized only superficially.
In 1842, in the course of some work on the undulatory
theory of light, he made a prophecy which is extremely
interesting in the light of the invention of spectrum anal-
ysis and the researches of Sir W. Huggins, Kirchhoff, and
others. He said: "I foresee with confidence that use will
hereafter be made of it in order to solve, by observing
the changes which the color of stars undergoes in time,
the questions as to whether they move, with what velocity
they move, how distant they are from us and much else
besides." But let us return to the most important part of
Bolzano's mathematical investigations.
In 1817 Bolzano published a paper we have already
mentioned entitled "Rein analytischer Beweis des Lehr-
satzes: dass zwischen je zwei Werthen, die ein entgegen-
gesetztes Resultat gewahren, wenigstens eine reele Wurzel
der Gleichung liege." This paper is, in a way, his most
important work and is a triumph of careful and subtle
mathematical analysis. His central theorem, as the title
indicates, is as follows : If in an equation f(x) — o, x = a
makes f(x) positive and x = p makes f(x) negative, then
there is at least one real root of the equation f(x) = o
between a and p. Before he begins his constructive work
he criticizes very acutely the previous attempts of La-
grange and others. He points out the errors that had
»o Op. cit., Vol. II, p. 709.
81 For complete list see Bergmann, op. cit., pp. 213-214.
BERNARD BOLZANO. 99
been made by previous investigators and he emphasizes
once more the great importance of freeing mathematical
analysis from the intuitional treatment to which it had
formerly been subjected. In order to prove his main theo-
rem, Bolzano found it necessary to introduce the concept
of the continuity of a function, the notion of the upper
limit of a variate and some important work on infinite
series. His method is briefly as follows:
1. He introduces the concept of "continuity." A func-
tion is said to be "continuous" for the value x if the differ-
ence between /(jr-f-co) and f(x) can be made less than
any assigned number, however small, if only CD is taken
sufficiently small.
2. He discusses the convergence of infinite series and
makes the following important statement. "If the differ-
ence between the value of the sum of the first n terms and
the first n-\-r terms of a series can be made as small as we
please, for all values of r, if only we take n large enough,
then there is one number X and only one such that the
sum of the first p terms approaches ever more and more
nearly to X as p increases." Unfortunately his proof of
this theorem is not rigorous and his discussion only renders
the existence of X probable.
3. From his work on infinite series Bolzano passes on
to an extremely important theorem in which he introduces
the new idea of an upper limit. And the theorem, as it
occurs in this paper, gains in importance from the fact that
the method used is one of fundamental importance in anal-
ysis. The theorem runs as follows : "If un be such a num-
ber that the property M holds for all values of x which
are less than un, and if the property does not hold for
all values of x without exception, then of all the num-
bers un satisfying this condition there is one (say U) which
is greater than all the others." This theorem, which might
appear obvious to those who allow their geometrical in-
IOO THE MONIST.
tuitions to cloud their mathematical ideas, is proved by
Bolzano with great care and completeness. The method
used in the proof was used a great deal by Weierstrass
and is now known as the "Bolzano-Weierstrass" process.
As the method is of such great importance, we will indicate
roughly the way it is used in the proof of this theorem.
It will be convenient to call ;tr's which have the property M
"suitable" x's and ^r's which do not have the property M
"unsuitable" .ar's; and further to call a number N a "suit-
able" number if all x's which are less than N have the
property M, and to call a number N an "unsuitable" num-
ber if there are some values of x, less than N, which do
not have the property M. Now it is obvious that there is
a positive number D, such that un -)-D is an unsuitable
number. Then, bisecting the interval between un and
un -f- D, we get the number un -\- D/2 ; bisecting the inter-
val between un and un -f- D/2 the number un -\- D/22 ; and
so on. When either all the numbers un -\- D/2r for r =
i, 2, 3. .. are unsuitable or there is a number R such
that un -f D/2R is an unsuitable and un -f D/2R— l a suit-
able number. In the first case the existence of U is estab-
lished, U being equal to un. In the second case we repeat
the process, dividing the interval between un -\- D/2R— l
and un -f D/2R. Again, either all the numbers un + D/2R
+ D/2R+J, s — i, 2. . . . are unsuitable or there is a num-
ber S such that un -f- D/2R -f- D/2R— s is an unsuitable
and un + D/2R -f D/2R+S — i a suitable number. We
continue the same process : if it does not terminate we get
finally to an infinite series
UH + D/2R + D/2S -f D/2? -f . . .
and since R, S, T. . . are positive integers the series ob-
viously satisfies the conditions of the theorem in paragraph
(2) above, and so there is a definite limit to which it tends.
BERNARD BOLZANO. IOI
this limit being the "upper limit" U in question. The
existence-theorem for an upper limit is thus established.
4. Bolzano next attacks the following theorem: "f(x)
and <p(.tr) are continuous functions of x and for x = a,
f(x) <cp(.r) and for x — |3, f(x) >q>(^r) : then there is
a value of x between a and (3 for which f(x) = <$(x)"
We will indicate the method Bolzano uses to prove it and
we shall see exactly why he found it necessary to establish
the existence of an "upper limit." Bolzano shows that,
since f(x) and q>(x) are continuous, there is a number co
such that all numbers less than it satisfy the relation
cp(<x + co) > /(a + co). Such a number we may call as
in paragraph (3) a "suitable" number. Then from a direct
application of the theorem about an upper limit he estab-
lishes the existence of an upper limit, say U, for all suit-
able numbers. It is then easy to show that /(a -}- U)
cannot be less than cp(a -f- U) and cannot be greater than
qp(a + U) and is therefore equal to (p(a -f- U). In this
kind of way Bolzano proves the existence of the value of
x between a and P giving f(x) = (p(^).
5. Finally Bolzano proves that an expression of the
form
a + bxm + cxn+ . . . + pxr,
in which m, n,. . .r are positive integers, is continuous.
Then by means of an easy application of a slightly modi-
fied form of the theorem in (4) he proves that there is at
least one real root between a and [3. The whole paper is
extremely valuable and it is interesting to see how Bolzano
was led from his central theorem to the theorem in (4), to
the concept of "continuity" and the idea of an "upper
limit," and in the existence-theorem for the upper limit to
the question of the convergence of series.
In mathematical logic and in the theory of infinite num-
bers, Bolzano's work was also of great importance. Bol-
IO2 THE MONIST.
zano's definition of the continuum is of some interest in
itself. He defines a continuum as a set of points such that
every point has another point also belonging to the set as
near to it as we please." This is expressed in modern
phraseology by saying that the continuum is a set of points
which is "everywhere dense." The name continuum is
now used (after Cantor) only for a set of points which is
not only "everywhere dense" but also "perfect." A set of
points is "perfect" when every convergent sequence has a
limit which is itself a number belonging to the set, and
conversely when every number is the limit of properly
chosen convergent sequences of numbers themselves be-
longing to the set.23 Thus Bolzano would call the set of
rational numbers a "continuum," but this set is not perfect
and is therefore not a "continuum" in the modern sense
of the word. In his work on infinite numbers Bolzano
anticipated to some extent the work of Georg Cantor. An
"infinite" collection is defined to be a collection which has
no last term.24 He proves that the number of natural
numbers and the number of real numbers is infinite, and
he sees (§49) that the number of these two collections is
different. Bolzano also recognizes the fact that it is pos-
sible to arrange the points in two lines of different lengths
so that each point of one collection corresponds to one
single point of the other collection and vice versa, no point
being left without a corresponding point. This brilliant
idea of a one-one correspondence went a long way toward
dispersing the cloud of mystery which hung over the con-
temporary infinite number. Leibniz had stated the diffi-
culty quite plainly. Every number can be doubled, he said,
therefore the number of natural numbers and the number
of even natural numbers is the same. Therefore the whole
22 Paradoxien des Unendlichen, Leipsic, 1851, 2d ed., Berlin, 1889, § 38.
23 See E. W. Hpbson, The Theory of Functions of a Real Variable and
the Theory of Fourier's Series, Cambridge, 1907, p. 49.
24 Paradoxien des Unendlichen, § 9.
BERNARD BOLZANO. 1 03
is equal to the part — which is absurd. Bolzano realized
that there is no real contradiction in this. This same idea
of the one-one correspondence between points belonging
to certain sets of points has led to the modern idea of "re-
flexiveness" of infinite numbers. The property of "re-
flexiveness"20 together with that of "non-inductiveness,"a
which disposes of all attempts to count up infinite collec-
tions or identify the number of terms in an infinite collec-
tion with the ordinal number of the last, has removed all
serious difficulties and has helped to make it possible to
put the concept of an infinite number on a logical founda-
tion.27 Defining "similar" classes as classes whose terms
have a one-one relation to each other and the "cardinal
number" or "power" of a class as the class of all similar
classes, we see immediately that the class of natural num-
bers and the class of even natural numbers have the same
cardinal numbers. Thus Bolzano was quite right in seeing
no contradiction in Leibniz's statements.
From these few references to isolated theorems and
statements in Bolzano's work, it is seen that he had most
of the ideas essential in the modern view of mathematics,
and that in mathematics at least Bolzano's work has been
a source of inspiration to those who came after him.
Whether in his theology, his ethics, his political science,
his metaphysics, or his mathematics, the desire for clear-
ness of concepts was always his aim. Even the parts of
his work which are no longer of intrinsic interest, e. g.,
his esthetics or his theory of laughter, have an interest
for us in that they show us the methods he used in seeking
25 A number is said to be "reflexive" if it is not increased by adding one
to it. See B. Russell, Our Knowledge of the External World as a Field for
Scientific Method in Philosophy, Chicago and London, 1914, p. 190.
28 A number is said to be "non-inductive" if it does not possess deductive
properties. See B. Russell, op. cit., p. 195.
27 Cf. the definitions "that which cannot be reached by mathematical in-
duction starting from 1" and "that which has parts which have the same num-
ber of terms as itself," B. Russell, The Principles of Mathematics. Cambridge.
1903, Vol. I, p. 368.
IO4 THE MONIST.
for truth. That there is objective truth and that we can
have knowledge of it — this was the thesis which he set
before him in his work. In mathematics especially his
work was needed, for whereas idealists maintained that
mathematics deals only with appearances, empiricists in-
sisted that mathematics could only approximate to the
truth. Bolzano's life work was to start mathematicians
on the right way to refute both the idealists and the em-
piricists. His method of strictly logical analysis of the
ideas of continuity and the infinite was the clue which was
followed up by all the great mathematical logicians and
mathematical analysts of the nineteenth century, until
finally the fundamental thesis has been proved that all
concepts of pure mathematics are wholly logical. Thus
Bolzano was one of the first to suspect and in this he was
a worthy successor of the great Leibniz. Unlike most
mathematicians of his day, Bolzano did not in his thirst
for results succumb to D'Alembert's maxim, Allez en
avant, la foi vous viendra.
We live in days when some of the contradictions and
paradoxes which have perplexed the human race since the
days of Zeno are being finally cleared up. Do not let us
forget the work of Bolzano who with painstaking endeavor
sowed the seeds of this great revolution in mathematical
ideas.
DOROTHY MAUD WRINCH.
CAMBRIDGE, ENGLAND.
A MEDIEVAL INTERNATIONALIST.
A RBITRATION, a league of peace and a council of
1~Y conciliation seem to be very modern suggestions as
methods of avoiding war between civilized nations. Some
hints of these, however, can be found in Kant's Perpetual
Peace and in the grand dessein as expounded by the Abbe
de S. Pierre. These schemes belong to the Revolutionary
and Renaissance periods. But even before, in the Middle
Ages, similar schemes are to be found in the work of
Petrus de Bosco (Pierre Dubois).
The political acuteness of this brilliant thinker can only
be understood by allowing for the fact that he had listened
at Paris to "that most prudent friar Thomas Aquinas"1
and by remembering that he wrote while the official poli-
ticians were engineering war after war for no purpose.
His work on international politics is contained in the un-
printed Summaria brevis abbreviations guerrarum
and in the "De recuperatione Terre Sancte," published
(1891) in the Collection des Textes. I propose to sum-
marize and comment upon the latter, not as of merely
archeological interest, but as an early attempt to grapple
with the same political problem which we now face.
The treatise is supposed to deal with a plan for recov-
ering the Holy Land and is addressed in 1306 to Edward I,
"King of England and Scotland, Lord of Ireland and Duke
of Aquitaine," as a great legislator and one who was
1 Par. 63, De recup. Terre Sancte. (In medieval Latin final ae becomes <?.)
IO6 THE MONIST.
specially interested in a new crusade. But this initial pur-
pose of the treatise, even if it was intended by the author
as more than a mere captatio benevolentiae , is certainly
subordinated to the general problem of international policy
among the European states.2 The order of the argument
is confused, the author continually going back to a subject
after he has left it for some other. He writes well, but too
eagerly to be as exact as the philosophers of his day. He
is genuinely excited by the pressing importance of estab-
lishing peace. I shall, therefore, not follow the order of
the treatise, but state first the nature of the problem as it
appears to Dubois and then his suggestions for solution.
War between European countries and kings, says Du-
bois, is the chief hindrance to "having time for progress
in morality and knowledge." War breeds war until war
becomes a habit.8 The deaths of one war cause speedy
preparations for revenge.4 "We should seek a general peace
and pray God for it, that by peace and in time of peace we
may progress in morality and the sciences, since we cannot
otherwise ; as the Apostle feels when he says : 'The peace of
God which passeth all understanding keep your hearts and
your minds:' your minds, which are reasonable souls, are
not kept but are often destroyed by wars, discords and civil
brawls which are like wars, and by the continuance of all
such. Therefore, as far as he can, every good man should
avoid and flee them ; and when he takes to war, being un-
able otherwise to obtain his rights, he ought as much as
possible to shorten it .... Thus universal peace is the end
we seek."8
2 Guillaume de Nogaret uses the same pious cover for his scheme of
social reform. One had to bow, so to speak, to the crusading ideal and then
one was free to suggest anything !
8 Quanto frequentius bella committunt, tanto magis appetunt committere,
hoc consuetudine magis quam emendatione deputantes." Par. 2.
4 "Ad bellum et vindictam voluntariam se preparant."
6 Par. 27, in line.
A MEDIEVAL INTERNATIONALIST. IO7
It is agreed that peace is desirable; but, says Dubois,
"since it is proved that neither the Scriptures, nor sermons
drawn from the Scriptures, nor the elegant lamentations
and exhortations of preachers have been sufficient to stop
frequent wars and the temporal and eternal death of so
many human beings which have resulted, why should there
not be found at last a new remedy for militarism (reme-
dium manus militaris), as for example a judiciary backed
by force (justicia necessario compulsiva) ?" (par. 109).
"This is an argument," he declares, "to which a reply is
impossible morally and politically speaking." Peace has
come within states by vis coactiva: so also it will come
between states. One could not have a clearer statement
of political judgment upon the evidence. The author him-
self says that he depends upon experience for his opinions :
and he declares that exhortations to peace and praise of its
excellencies and even rhetorical attacks on war are polit-
ically valueless. They have been tried and they have failed.
Before speaking, however, of the means by which peace
is to be established between states, we must notice the
plan which is not suggested by Pierre Dubois. The gov-
erning ideal of medieval politics, unity, led many to look
for peace through subordination to one overlord. "Now
there is no sane man, I think," Dubois writes, (par. 63),
"who could think it likely that in this latest age (in hoc
•fine saeculorum) there could be one monarch of the whole
world in temporal affairs who would rule all and whom
as superior all would obey. For, if there were any attempt
at this there would be wars, seditions and discords without
end; nor would there be any one who could allay them by
reason of the number of different nations, the distance and
distinction between countries and the natural inclination
of men to diverge. Although some have been popularly
called "lords of the world" nevertheless I think that since
the countries were settled there never has been any one
IO8 THE MONIST.
whom all obeyed." That passage, if it seems to condemn
Dante as a homo non sane mentis, certainly shows an his-
torical acumen and a political judgment far superior to the
opinions of the De Monarchia. Dubois recognizes the im-
possibility of arriving at peace by means of the conquest by
one state of all other states. He sees that world-power is
nonsense.
It must be admitted, however, that from the passages
of the Summaria brevis which have been commented upon
by M. de Wailly and Ernest Renan, one might judge that
Dubois hoped for a domination in Europe of the French
king. He held, indeed, that it should be arrived at by dip-
lomacy and not by war, but in the above passage of the
De recuperatione he seems to condemn not merely any
special means, but dreams of domination by a single lord.
Inconsistency may be urged against him, and yet it must
be remembered that here he is writing to the English king
and also that he may very well have felt uncertain as to
how the vis coactiva above the warring states might be
established, even if he held quite clearly to the notion that
the ultimate supremacy of one monarch was impossible.
But let us turn to the definite political means he suggests
for establishing peace between European states.
The means by which such peace is to be arrived at are :
First: International arbitration and the establishment of
an international judiciary. This is to begin by a general
council (par. 3), a preliminary to all medieval and early
Renaissance plans for reform. But what is unusual in
Pierre Dubois is the statement that the difficulty of arran-
ging matters is due to the fact that the cities of Italy, for
example, and the various princes acknowledge no superior.
"Before whom then," he asks, "can they bring their dis-
putes? It can be answered that the council should estab-
lish elected arbiters (arbitros) religious or others, prudent,
experienced and trustworthy men." These are to select
A MEDIEVAL INTERNATIONALIST.
three prelates and three others for either party to the dis-
pute. They are to be well paid and such as are not likely
to be corrupted by affection, hate, fear, greed or otherwise.
They are to meet at a suitable place, to have presented to
them in a summary and clear form, without minor and
unimportant details, the pleas of either side. They are
to take evidence from witnesses and documents, each wit-
ness being examined by at least two trustworthy and care-
ful members of the "jury." The depositions are to be writ-
ten and preserved. "For the decision, if it is expedient,
they are to have assessors (assessores) who are thought
by them most trustworthy and best trained in the divine,
the canon and the civil law."6
Secondly, these decisions must be made effective. The
Holy See is recognized as an influnce, but excommunica-
tion had better not be used. "Temporal punishment, al-
though incomparably less than eternal, will be more
feared."7 The suggestions in detail of Pierre Dubois are
perhaps a little comic, but we must allow for the situation.
In the first place any group making war shall, after the war
is over, be removed bodily and sent to colonize the Holy
Land ! If they do not oppose the movement, they may take
some of their property with them. The author feels that
it may be difficult. He then goes on as to other measures.
Suppose, he says, that the Duke of Burgundy declares war
against the King of France, — the king should then institute
an economic boycott8 and by a general council the same
boycott should be declared by all Europe. Active military
measures should be taken to devastate the country so that
the whole people should feel it: Dubois, it seems, would
adopt extreme measures to prevent war spreading, his main
8 Par. 12, De recup. Terre Sancte.
7 Excommunication is to be used (§ 101) but not depended upon by itself.
Any one refusing to enter the league of peace (pacts universalis federa) is to
be immediately attacked.
8 Prohibebit quod nullus ad terras eorum deferat victualia, arma, merces
et alia quaecumque bona, etiam quacumque causa sibi debita," (par. 5).
I IO THE MONIST.
point being that in whatever corner it broke out the whole
of Europe should act together and at once to stop it.
The reader may feel that this is hopelessly unpractical,
since we could not act thus against any great country or
against any combination of countries. But we must re-
member ( i ) that Dubois supposes Europe to be one polit-
ical system (respublica Christicolarum) able to act in con-
cert at least in some issues, and (2) that every war begins,
according to him, in some comparatively small group. Thus
practically, if Europe had adopted strong economic, even
without military, action during the Balkan wars of 1912
and 1913, the war of 1914 might never have occurred.
And surely it is not unpractical to suggest that all civilized
countries should act together in the case of any conflict
breaking out such as that of 1912. Deal effectively with
the small conflicts and the first difficulty is met with regard
to the larger. But one can imagine the horror of medieval
diplomatists if all the states were asked to prevent any small
wars by direct intervention of enforcing arbitration. Even
to-day all the schemes for rearranging international politics
start from the present almost universal war. I cannot help
feeling, however, that Dubois was right. Our schemes for
doing without war must inculcate combined action in small
wars. Deal effectively with them and we may never have
to deal at all with war between great states. It is the
spark, not the conflagration, that we must consider first:
and perhaps European diplomacy was more futile in 1912
than in July 1914, although the results of inaction did not
show themselves till August, 1914. But let us return to
the general thesis and omit further applications of it.
After details as to raising funds for a common force
and plans for a common advance on the Holy Land, Dubois
recalls himself to his main interest, "a general peace." In
the third place therefore, he says that no external measures
will be effective until the religious attitude is changed.
A MEDIEVAL INTERNATIONALIST. Ill
This opens an elaborate project for the reform of the
Roman church. Dubois says (par. 29) if the pope really
wants to stop war "he must begin with his brothers the
cardinals and his fellow bishops." They are always going
to war (ipsi guerras movent). Their attitude is quarrel-
some even in England and France where they do not ac-
tually fight. The monks are as bad. But the whole attack
is common to many writers of the date of Pierre Dubois.
His remedies are extreme. First he suggests that if the
pope had no "temporal power/' no one need to go to war
for him and that would be a beginning; and next, he
actually proposes the confiscation of ecclesiastical property
by states and the use of the wealth for common European
civilization !8 But how ?
The fourth suggestion of Pierre Dubois is that the
money should be spent in education.10 The purpose of the
education, according to the general thesis as to the taking
of the Holy Land, is directed by the general need of non-
military contact with the East. It is urged that you can only
hold the East effectively by intellectual superiority to it.
Then begins a long and elaborate scheme of education,
primary and secondary. University education is implied
but not dealt with in detail. All this is to occur in the Holy
Land. It is a well-known medieval trick for writing a
Utopia. In 1223 'The Complaint of Jerusalem" gave a
plan for reconstructing European society under the guise
of a scheme for an Eastern kingdom. So here Dubois,
appearing to speak of what ought to be done when the Holy
Land is established as a state, is really speaking of the
remedies which ought to be applied in Europe. In the
matter of education he is as original as in politics, but what
is most interesting to us now are the hints for bringing
• Par. 57. "Que tendit ad reformationem et unitatem veram totius rei-
publice catholicorum."
10 Par. 60, "Studentcs et corum doctorcs vivent de bonis dictorum priora-
tum. etc,"
112 THE MONIST.
the European nations together. Colleges for boys and for
girls are to be established where "modern languages" are
to be taught — "the literary idioms, especially of Europe,
that by these scholars trained to speak and write the lan-
guages of all, the Roman church and the princes of Europe
should be able to communicate with all men." Some are
also to be taught medicine, some surgery — the girls also
(par. 61) ; and these girls, in the medieval fashion perhaps,
are to be married to foreigners, even Orientals (ditioribus
Orientalibus in uxores dari). I need not detail the plans
for intermarriage and colonization, among which is in-
serted a suggestion for a married clergy (par. 102). A
long section follows upon the utility of scientific knowledge
"according to brother Roger Bacon" (par. 79) and upon
the variety of human knowledge in general. There are
interesting hints as to the transformation of convents into
girls' schools, and as to military reform, for example the
institution of definite uniforms (par. 16). But all these
do not bear directly upon his plans for peace and we may
therefore omit them here. His boldness of conception is clear.
The other element in his Utopia, which is to establish
peace, is a modification of the processes of law (par. 90 f.).
The processes must be shortened according to a definite
plan; but the detail need not concern us here. The fact
remains that he saw that social, educational and religious
reform within the state are all means for the attainment
of international peace.
The closing section of the work (110-142) are ad-
dressed to Philip, king of France, who is asked to send the
preceding to Edward I. Dubois urges the economic gain
from the abolition of wars, and in the meantime the insti-
tution of various military reforms — as for example the
regular payment of troops. It is amusing to note that the
author feels the danger to himself from the powers that
be, if his projects are made too public. He therefore asks
A MEDIEVAL INTERNATIONALIST. 113
both Edward I and Philip to consider his ideas more or less
privately; and he hints that one who does not happen to
hold popular opinions may suffer even physical assault.
So far as we know nothing evil happened to Pierre
Dubois. He was a lawyer who worked first for the king of
France and afterward, when he wrote the De recupera-
tione, in the service of Edward I in Guyenne. He seems
to have represented the central government in either case,
and to have found his chief opponents among the church-
men. He is known as the author of a popular pamphlet
in French against papal claims, as the writer of a few
short Latin treatises, and as the elected representative of
Coutances at the Etats Generaux which met in Tours in
1308. After that nothing is known of him.
More than six hundred years have gone since the trea-
tise of Pierre Dubois was forgotten : and one may well rub
one's eyes in wonder at what is now occurring in Europe.
Perhaps we are dreaming. The practical man will say
that the old plans for political reform are by current events
proved to be valueless ; that the internationalists are shown
by the facts to be unable to understand real politics. And
yet one would have thought that any plan might have been
better worth trying than one which has brought us to our
present pass. However that may be we should not despair
too soon. Ecclesiastical reformation was suggested for
hundreds of years before Europe arrived at the compara-
tively tolerant situation in religion now established. Polit-
ical reformation may be more difficult, but the work of its
forerunners is important. Si Lyra non lyrasset, Lutherus
non saltasset: so also in politics, the effective reformer is
taught by his predecessors who found the circumstances of
their time too strong for them.
C. DELISLE BURNS,
LONDON, ENGLAND.
CLASS, FUNCTION, CONCEPT, RELATION.1
IN my Grundlagen der Arithmetik of 1884 I have tried
to make it seem probable that arithmetic is a branch
of logic and need not borrow any ground of proof what-
ever from experience or intuition. The actual demonstra-
tion of my thesis is carried out in my Grundge seize of 1893
and 1903 by the deduction of the simplest laws of numbers
by logical means alone. But to make this proof convincing,
considerably higher claims must be made for deduction
than is habitually done in arithmetic.2 A set of a few
methods of deduction has to be fixed beforehand, and no
step may be taken which is not in accordance with them.
Consequently, when passing over to a new judgment we
must not be satisfied, as mathematicians seem nearly al-
ways to have been hitherto, with saying that the new judg-
ment is evidently correct, but we must analyze each step of
ours into the simple logical steps of which it is composed,
— and often there are not a few of these new steps. No
hypothesis can thus remain unnoticed. Every axiom which
is needed must be discovered, and it is just the hypotheses
which are made tacitly and without clear consciousness
that hinder our insight into the epistemological nature of
a law.
In order that such an undertaking be crowned with
success, the concepts which we need must naturally be con-
1 [Translated from the Grundgesetze der Arithmetik by Johann Stachel-
roth and Philip E. B. Jourdain.]
2 Grundlagen, pp. 102-104.
CLASS, FUNCTION, CONCEPT, RELATION. 1 15
ceived distinctly. This is true especially in what concerns
the thing that mathematicians denote by the word "ag-
gregate" (Menge). It seems that Dedekind, in his book
Was sind und was sollen die Zahlenf3 of 1888, uses the
word "system" to denote the same thing. But in spite of
the exposition which appeared four years earlier in my
Grundlagen, a clear insight into the essence of the matter
is not to be found in Dedekind's work, though he often gets
somewhat near it. This is the case in the sentence:4 "Such
a system S is completely determined if of everything it is
determined whether it is an element of S or not. Hence
the system S is the same as the system T (in symbols
S = T) if every element of S is also element of T and every
element of T is also element of S." In other passages, on
the other hand, Dedekind strays from the point. For in-
stance:5 "It very frequently happens that for some reason
different things a, b, c, . . . can be considered from a com-
mon point of view, can be put together in the mind, and we
then say that they form a system S." Here a presentiment
of the correct idea is contained in the words "common
point of view"; but the "putting together in the mind" is
not an objective characteristic. In whose mind, may I
ask? If they are put together in one mind and not in
another do they then form a system? What is to be put
together in my mind must doubtless be in my mind. Then
do not things outside myself form systems? Is a system
a subjective formation in each single soul? Is then the
constellation Orion not a system? And what are its ele-
ments? The stars, the molecules, or the atoms? The fol-
lowing sentence8 is remarkable: "For uniformity of ex-
pression it is advantageous to admit the special case that a
system S is composed of a single (one and one only) ele-
ment a: the thing a is an element of S, but every thing
8 [English translation under the title Essays on the Theory of Numbers,
Chicago and London, 1901. See especially p. 45.]
* [Ibid., p. 45.] 6 [Ibid.] e
Il6 THE MONIST.
different from a is not an element of S." This is after-
ward7 understood in such a way that every element ^ of a
system S can be itself regarded as a system. Since in this
case element and system coincide, it is here quite clear that,
according to Dedekind, the elements are the proper con-
stituents of a system. Ernst Schroder in his lectures on
the algebra of logic8 goes a step in advance of Dedekind in
drawing attention to the connection of his systems with
concepts, which Dedekind seems to have overlooked. In-
deed, what Dedekind really means when he calls a system
a "part" of a system9 is that a concept is subordinated to a
concept or an object falls under a concept. Neither Dede-
kind nor Schroder distinguish between these cases because
of a mistake in point of view which is common to them
both. In fact, Schroder also, at bottom, considers the ele-
ments to be what really make up his class. An empty class
should not occur with Schroder any more than an empty
system with Dedekind. But the need arising from the
nature of the matter makes itself felt in a different way
with each writer. Dedekind says:10 "On the other hand,
we intend here for certain reasons wholly to exclude the
empty system, which contains no element at all, although
for other investigations it may be convenient to invent
(erdichten) such a system." Thus such an invention is
permitted; it is only desisted from for certain reasons.
Schroder dares to invent an empty class. Apparently then
both agree with many mathematicians in holding that we
may invent anything we please that does not exist, — even
what is unthinkable; for if the elements form a system,
then the system is annulled at the same time as the ele-
7 [Ibid., p. 46.]
8 Vorlesungen uber die Algebra der Logik (exakte Logik), Vol. I, Leipsic,
1890, p. 253. [This reference of Frege seems wrong and it should perhaps
rather be to such a page as p. 100. Cf. also Frege's later critical study : "Kriti-
sche Beleuchttmg einiger Punkte in E. Schroders Vorlesungen uber die Al-
gebra der Logik," Archiv fur systematische Philosophic, Vol. I, 1895, pp. 433-
456.]
9 [Op. cit., p. 46:] 10 [Ibid., pp. 45-46.]
CLASS, FUNCTION, CONCEPT, RELATION.
ments. As to where the limits of this license lie and
whether indeed there are any such limits, without any
doubt we will not find much clearness and agreement; —
and yet the correctness of a proof may depend on such
questions. I believe I have settled them in a way that is
final for all intelligent persons, in my Grundlagen11 and in
my lecture "Ueber formale Theorien der Arithmetik."1
Schroder invents his zero-class and thus gets into diffi-
culties.13 We do not find, then, a clear insight into the
matter with either Schroder or Dedekind ; but still the true
position of affairs is seen whenever a system is to be de-
termined. Dedekind then brings forward properties which
a thing must have in order to belong to a system, i. e., he
defines a concept by its characteristics.14 If now a concept
is made up of characteristics and. not of the objects falling
under the concept, there are no difficulties to be urged
against an empty concept. Of course in this case an ob-
ject (Gegenstand) can never also be a concept, and a con-
cept under which only one object falls must not be confused
with this object. Thus we are finally left with the result
that the number datum contains an assertion about a con-
cept.10 I have traced back number to the relation of simi-
larity18 (Gleichzahligkeit} and similarity to univocal cor-
respondence (eindeutige Zuordnung). Of "correspond-
ence" much the same holds as of "aggregate" (Menge).
Nowadays both words are often used in mathematics, and
" Pp. 104-108.
l2Sit2ungsberichte der Jenaischen Gesellschaft fur Medicin und Natur-
wissenschaft, July 17, 1885.
13Cf. E. G. Husserl, Gottinger gelehrte Anzeigen, 1891, No. 7, p. 272,—
where, however, the difficulties are not solved.
14 On concept, object, property, and characteristics, cf. my Grundlagen,
pp. 48-50, 60-61, 64-65, and my essay "Ueber Begriff und Gegenstand," Viertel-
jahrsschrift fur wissenschaftliche Philosophic, Vol. XVI, 1892, pp. 192-205.
15 See Grundlagen, pp. 59-60.
16 [The same idea and word were used by Dedekind (op. cit., p. 53) ; and
the same idea but with the name "equivalence" was used by Georg Cantor (cf.
Contributions to the Founding of the Theory of Transfinite Numbers, Chicago
and London, 1915, pp. 40, 86).]
Il8 THE MONIST.
very oftep there is lacking an insight into what is intended
to be denoted by them. If my opinion is correct that arith-
metic is a branch of pure logic, then a purely logical ex-
pression has to be chosen for "correspondence." I choose
the word "relation." Concept and relation are the founda-
tion stones upon which I erect my structure.
But even when concepts have been grasped quite pre-
cisely, it would be difficult — nearly impossible in fact — to
satisfy the demands we have had to make of a process of
proof without some special means of help. Now such a
means is my ideography (Begriffsschrift), the explanation
of which will be my first problem. The following remarks
may be noticed before we proceed farther. It is not pos-
sible to define everything, hence it must be our endeavor
to go back to the logically simple which as such cannot
properly be defined. I must then be satisfied with referring
by hints to what I mean. Before all I have to strive to be
understood, and therefore I will try to develop the subject
gradually and will not attempt at first a full generality
and a final expression. The frequent use made of quota-
tion marks may cause surprise. I use them to distinguish
the cases where I speak about the sign itself from those
where I speak about its denotation. Pedantic as this may
appear, I think it necessary. It is remarkable how an
inexact mode of speaking or writing which perhaps was
originally employed only for greater convenience or brev-
ity and with full consciousness of its inaccuracy, may,
when that consciousness has disappeared, end by confusing
thought. Has it not happened that number signs have been
mistaken for numbers, names for the things named, the
mere auxiliary means for the real end of arithmetic ? Such
experiences teach us how necessary it is to make the high-
est demands of exactitude in manner of speech and writing.
And I have taken pains at least to do justice to such de-
mands wherever it seemed to be of importance.
CLASS, FUNCTION, CONCEPT, RELATION.
If17 we are asked to give the original meaning of the
word "function" as used in mathematics, we easily fall
into saying that a function of x is an expression formed by
means of the notations for sum, product, power, difference,
and so on, of "x" and definite numbers. This attempt at a
definition is not successful because a function is here said
to be an expression, a combination of signs, and not what
the combination stands for. Then probably another at-
tempt would be made with "denotation (Bedeutung) of an
expression" instead of "expression." But there appears
the letter "x" which indicates a number, not as the sign
"2" does, but indefinitely. For different number-signs
which we put in the place of "#•", we get, in general, differ-
ent denotations. Suppose for example, that in the ex-
pression "(2 + 3.^)^", instead of 'V we put the num-
ber-signs "o", "i", "2", "3", one after the other; we then
get as corresponding denotations the numbers o, 5, 28, 87.
Not one of these denotations can claim to be our function.
The essence of the function is in the correspondence that
it establishes between the numbers whose signs we put for
"#•" and the numbers which then appear as denotations
of our expression, — a correspondence which is represented
to intuition by the course of the curve whose equation is,
in rectangular coordinates, "y= (2 -f- 3--*^)^'". In gen-
eral, then, the essence of the function lies in the part of the
expression which is outside the "x". The expression of
a function needs completion (ist ergdnzungsbedurftig) and
is not satisfied (ungesdttigt) . The letter "x" only serves
to keep places open for a numerical sign which is to com-
plete the expression, and thus makes known the special
kind of need for completion that constitutes the peculiar
nature of the function indicated above. In what follows,
17 Cf. my lecture Funktion und Begriff, Jena, 1891, and my essay "Ueber
Begriff und Gegenstand" cited above. My Begriff 'sschrift of 1879 now no
longer represents my standpoint, and thus should only be used with caution
to illustrate what I said here.
I2O THE MONIST.
the Greek letter "£" will be used18 instead of the letter
'V. This keeping open is to be understood in this way:
All places in which "!•" stand must always be filled by the
same sign and never by different ones. I call these places
argument-places and that whose sign or name takes these
places in a given case I call argument of the function for
this case. The function is completed by the argument;
I call what it becomes on completion the value of the func-
tion for the argument. We thus get a name of the value
of a function for an argument when we fill the argument-
places in the name of the function with the name of the
argument. Thus, for example, "(2H~3-12)1" is a name
of the number 5, composed of the function-name "(2 +
3-?2)l" and "i". The argument is not to be reckoned in
with the function, but serves to complete the function which
is unsatisfied by itself. If in the following an expression
like "the $(?)" is used, it is always to be observed that
the only service rendered by "\" in the notation of the
function is that it makes the argument-places recognizable ;
it does not imply that the essence of the function becomes
changed when any other sign is substituted for "§".
To the fundamental operations of calculation mathe-
maticians added, as function-forming, the process of pro-
ceeding to the limit as exemplified by infinite series, differen-
tial quotients and integrals ; and finally the word "function"
was understood in such a general way that the connection
between value of function and argument was in certain
circumstances no longer expressed by signs of mathemat-
ical analysis, but could only be denoted by words. Another
extension consisted in admitting complex numbers as argu-
ments and consequently also as function-values. In both
directions I have gone still farther. While, indeed, the
18 Nothing, however, is fixed by this for our ideography. The "£" never
appears in the developments of the ideography itself, and I only use it in my
exposition of it and in illustrations.
CLASS, FUNCTION, CONCEPT, RELATION. 121
signs of analysis were hitherto on the one hand not always
sufficient, they were on the other hand not all employed
in the formation of function-names. For instance, "^ = 4"
and "| > 2" were not allowed to count as names of func-
tions; but I do so allow them. But that indicates at the
same time that the domain of function-values cannot re-
main limited to numbers ; for if I take as arguments of the
function |2 — 4 the numbers o, I, 2, 3, in succession, I do
not get numbers. I get: "o2 = 4", "i2 = 4", "22 = 4",
"3* = 4", which are expressions of one true and some false
thoughts. I express this by saying that the value of the
function |2 — 4 is either the "truth-value (Wahrheits-
werth) of the true or of the false."1 From this it can be
seen that I do not intend to assert anything by merely
writing down an equation, but that I only designate (be-
zeichne) a truth-value, just as I do not intend to assert
anything by simply writing down "22" but only designate
a number. I say: "The names "2* = 4" and "3 > 2" denote
the same truth-value" which I call for short the true. In
the same manner "3* = 4" and "i>2" denote the same
truth-value, which I call for short the false just as the
name "22" denotes the number 4. Accordingly I say that
the number 4 is the "denotation" of "4" and of "22", and
that the true is the "denotation" of "3 > 2". But I dis-
tinguish the "meaning" (Sinn) of a name from its "de-
notation" (Bedeutung). The names "22" and "2 + 2"
have not the same meaning, nor have "22 = 4" and "2 -f- 2
= 4". The meaning of the name of a truth-value I call
a "thought" (Gedanken). I say further that a name "ex-
presses" (ausdriickt) its meaning and "denotes" its de-
notation. I "designate" (beseichne) by a name what it
means.
The function ?2 = 4 can thus have only two values, the
19 I have shown this more exhaustively in my essay "Ueber Sinn und Be-
deutung" in the Zeitschrift fur Phihs.und phil. Kritik, Vol. C, 1892, pp. 25-50).
122 THE MONIST.
true for the arguments -f- 2 and — 2 and the false for all
other arguments.
Also the domain of what is admitted as argument must
be extended, — indeed, to objects quite generally. Objects
(Gegenstdnde) stand opposed to functions. I therefore
count as an object everything that is not a function ; thus,
examples of objects are numbers, truth-values, and the
ranges (Werthverldufe) to be introduced further on. The
names of objects — or proper names — are not therefore
accompanied by argument-places, but are satisfied like the
objects themselves.
I use the words, "the function $(|) has the same range
as the function ^P(^)", as denoting the same as the words,
"the functions $(|) and ^(\) have the same value for the
same argument." This is the case with the functions
|2 = 4 and ^.^2=i2, at least if numbers are taken as
arguments. But we can also imagine the signs of evolution
and multiplication denned in such a manner that the func-
tion (|2 =4) = (3.1= 12) has the value of the true for
any argument whatever. Here an expression of logic may
be used : "The concept square-root of 4 has the same ex-
tension as the concept something of which three times its
square is 12." With those functions whose value is always
a truth-value we can therefore say "extension of the con-
cept" instead of "range of the function," and it seems suit-
able to say that a concept (Be griff) is a function of which
the value is always a truth-value.
Hitherto I have only dealt with functions of a single
argument, but we can easily pass over to functions with
two arguments. Such functions are doubly in need of
completion. A function with one argument is obtained
when a completion by means of one argument has been
effected. Only by means of a repeated completion do we
arrive at an object, and this object is then called the "value"
of the function for the pair of arguments. Just as the
CLASS, FUNCTION, CONCEPT, RELATION. 123
letter "|" served with functions of one argument, I use
here the letters "|" and "£" in order to indicate the two-
fold non-satisfaction of a function of two arguments, as,
for example, in "(| + £)* + £"• By replacing "£" by "i",
for example, we satisfy the function in such a way that we
have in (|-|-i)2-fi a function with only one argument.
This manner in which we use the letters "§" and "£" must
always be kept in mind when an expression like "the func-
tion ¥(1, £)" occurs.20 I call the places in which "!="
stands "^-argument-places", and those in which "£" stands
"^-argument-places". I say that the ^-argument-places
are "related" (verwandt) to one another, and also the
^-argument-places to one another, and I say that a ^-
ment-place is not related to a ^-argument-place.
The functions with two arguments £ = £ and
have as value always a truth-value — at least if the signs
"=" and ">" are defined in a suitable manner. I shall
call such functions "relations". In the first relation, for
example, I stands to I, and in general every object to itself;
in the second, for example, 2 stands to I. I say that the
object F "stands in the relation *P(|, £) to" the object A,
if $(T, A) is the true. I say that the object A "falls
under" the concept $(?)» if 3>(A) is the true. It is pre-
sumed, of course, that both the functions $(£) and
have always truth-values as values.21
I have already said above that no assertion is to lie as
2<> Cf . note 18.
21 Here there is a difficulty which may easily obscure the true position of
things and thus rouse distrust of the correctness of my view. If we compare
the expression "the truth-value of the circumstance that A falls under the con-
cept *(£)" with "4>(A)", we see that to the "*(A)" properly corresponds "the
truth-value of the circumstance that (A) falls under the concept $( I)" and not
"the concept *(£)"• The last words do not therefore really designate a concept
(in my sense of the word), though they have the appearance of doing so in
our linguistic form. With regard to the constrained position in which language
here finds itself, cf. my essay "Ueber Begriff und Gegenstand" mentioned in
note 14.
124 THE MONIST.
yet in a mere equation; by "2 -{-3 = 5" only a truth-value
is designated and it is not stated which of the two it is.
Again, if I write "(2-f3 = 5) = (2 = 2)" and presup-
pose that we know that 2 = 2 is the true, yet I would not
have asserted by that the sum of 2 and 3 is 5, but I would
only have designated the truth-value of the circumstance
that "2 + 3 = 5" denotes the same as "2 = 2". Thus we
need a special sign to assert that something or other is
true. For this purpose I write what I call a "sign of
assertion" just before the name of the truth-value, so that
if this sign is written just before "2* = 4"," it is asserted
that the square of 2 is 4. I make a distinction between
"judgment" (Urtheil) and ''thought" (Gedanken), and
understand by "judgment" the recognition of the truth
of a "thought." I shall call the ideographic representation
of a judgment by means of the sign of assertion an "ideo-
graphic theorem" or more shortly a "theorem." I regard
this sign of assertion as composed of a vertical line, which
I call "line of judgment" (Urtheilsstrich) , and a short
straight horizontal line proceeding from the middle of
the vertical line and going toward the right, which I will
simply call the "horizontal line" (Wagerechte). In my
Begriffsschrift I called this last line the "line of content"
(Inhaltsstrich) and at that time I expressed by the words
"judicable content" (beurtheilbarer Inhalt) what I have
now arrived at distinguishing into truth-value and
thought.23 The horizontal line most often occurs in com-
bination with other signs, as it does here with the line of
judgment, and is thus guarded against confusion with the
minus sign. Wherever it occurs by itself it must be made
somewhat longer than the minus sign for purposes of dis-
22 1 often use here the notations of sum, product, and power in order
conveniently to form examples and to facilitate understanding by means of
hints, although these signs are not yet defined in this place. But we must keep
in view the fact that nothing is founded on the denotations of these signs.
23 Cf. my essay "Ueber Sinn und Bedeutung" cited above.
CLASS, FUNCTION, CONCEPT, RELATION.
tinction. I regard it as a name of a function in the way
that "A" preceded by this sign denotes the true if A is the
true, and the false if A is not the true. Of course the sign
"A" must denote an object] names without denotation
may not occur in our ideography. The above arrangement
is made so that "A" preceded by a horizontal line denotes
something under all circumstances if only "A" denotes
something. If not, "\" preceded by a horizontal line would
not denote a concept with sharp boundaries, — and thus
would not denote a concept in my sense. I here use capital
Greek letters as names denoting something without my
saying what their denotations are. In the actual develop-
ments of my ideography they will not occur any more than
"|" and "f '. The above "\" preceded by a horizontal
line denotes a function whose value is always a truth-
value or, by what I have said, a concept. Under this con-
cept falls the true and this only. Thus "2* = 4" preceded
by a horizontal line denotes the same thing as "2* = 4",
namely the true. In order to do away with brackets,
I lay down that all which stands to the right of the
horizontal line is to be regarded as a whole which stands
at the argument-place of the function denoted by "5" Pre~
ceded by a horizontal line, unless brackets forbid this.
The sign "2* — 5" preceded by a horizontal line denotes
the false and thus the same as "22 = 5", whereas "2" pre-
ceded by a horizontal line denotes the false, and thus some-
thing different from the number 2. If "A" is a truth-
value, A preceded by a horizontal line is the same truth-
value, and thus the equation of "A" to "A" preceded by a
horizontal line denotes the true. But this equation denotes
the false is A is not a truth-value ; so that we can say that
it denotes the truth-value of the circumstances that A is a
truth-value.
Thus the function "$(!)" preceded by a horizontal
line, denotes a concept and the function "*?(!,£)" pre-
126 THE MONIST.
ceded by a horizontal line, denotes a relation, whether or
not $(!) is a concept and \P(|,£) is a relation.
Of the two signs out of which the sign of assertion is
composed the line of judgment alone contains the assertion.
We need no sign to declare that a truth-value is the
false, if only we have a sign by which either truth-value
is changed into the other. This sign is also indispensable
on other grounds. I now lay down that the value of the
function denoted by "|" preceded by a horizontal line from
the middle of which hangs a small vertical line directed
downward and called the "line of denial" (V erneinungs-
strich), so that the whole is like a sign of assertion turned
round on its face, is to denote the false for every argu-
ment for which the value of the function denoted by "|"
preceded by a horizontal line is the true. For all other
arguments the function under definition is to be the true.
The function thus defined may be called "the negation of
§", and thus its value is always a truth-value; it is a con-
cept under which all objects fall with the single exception
of the true. From this it follows that horizontal lines,
whether or not they form part of a sign of negation, can
be combined with immediately preceding or following
simple horizontal lines in such a way that the latter, so to
speak, lose their separate existence and melt into the former
(Verschmelzung der Wagerechten).
Thus "the negation of 22 = $" denotes the true; and
thus we may put the sign of assertion so as to join on to the
left of the sign of negation. We may assert, too, the
negation of 2.
I have already used the sign of equality to form ex-
amples, but it is necessary to lay down something more
accurate about it. The sign "F = A" is to denote the
true if F is the same as A, and the false in all other cases.
In order to dispense with brackets as far as possible,
I lay down that all which stands on the left of the sign
CLASS, FUNCTION, CONCEPT, RELATION. 127
of equality as far as the nearest horizontal line is to denote
the ^-argument of the function § = £, in so far as brackets
do not forbid this; and that all which stands on the right
of the sign of equality as far as the next sign of equality is
to denote the ^-argument of that function in so far as
brackets do not forbid this.
GOTTLOB FREGE.
JENA, GERMANY.
A CHINESE POET'S CONTEMPLATION OF LIFE.
INTRODUCTION.
MY attention has repeatedly been called to the poetry of Su
Tung P'o (also briefly named "Su Hsi"), especially to his
thoughtful meditation on an excursion by boat to the Scarlet Cliff.
In this poem he comments on the transiency of life, and referring to
the law of change as represented by the phases of the moon he finds
the underlying permanence symbolized by the river which remains
the same although its waters pass on without a halt.
The original was kindly furnished me by Mr. Sawland J. Shu,
president of the Technological College at Nanking, while a literal
translation was procured through Prof. Frederick G. Henke from
Mr. W. T. Tao and another one from Prof. King Shu Liu, of the
University of Nanking. Professor Henke further informed me
on the authority of Prof. William F. Hummel that a prose trans-
lation by Prof. Herbert A. Giles was published in the University
of Nanking Magazine and republished together with other Chinese
poems collected in the volume entitled Gems of Chinese Literature.
Professor Giles says that Su Tung P'o was "even a greater
favorite with the Chinese literary public" than the famous Ou-Yang
Hsiu.1 So we may regard Su Tung P'o as easily a genius of first
rank. Professor Giles says of him:
"Under his hands, the language of which China is so proud
may be said to have reached perfection of finish, of art concealed.
In subtlety of reasonings, in the lucid expression of abstractions,
such as in English too often elude the faculty of the tongue, Su
Tung P'o is an unrivalled master."
Even a rough translation of his poems will impress the reader
1 Ou-Yang Hsiu lived 1017-1072 A. D. Professor Giles says of him: "A
leading statesman, historian, poet, and essayist of the Sung dynasty. His
tablet is to be found in the Confucian temple, an honor reserved for those
alone who have contributed to the elucidation or dissemination of Confucian
truth."
A CHINESE POET'S CONTEMPLATION OF LIFE. I2Q
with the versatility as well as the profundity of his poetic flights,
and here I venture to present his famous poem on "The Scarlet
Cliff" in English blank verse which seems to be the appropriate
form for this kind of thought. I hope that it will be a fair example
of Chinese literature in its noblest accomplishment.
There are some people who have little appreciation of the
beauties of Chinese literature and have nothing but ridicule or even
contempt for it. With reference to one of these haughty scoffers
Professor Giles adds with grim humor:
"On behalf of his (Su Tung P'o's) honored manes I desire to
note my protest against the words of Mr. Baber, recently spoken
at a meeting of the Royal Geographical Society, and stating that
'the Chinese language is incompetent to express the subtleties of
theological reasoning, just as it is inadequate to represent the
nomenclature of European science.' I am not aware that the nomen-
clature of European science can be adequately represented even in
the English language ; at any rate, there can be no comparison
between the expression of terms and of ideas, and I take it the
doctrine of the Trinity itself is not more difficult of comprehension
than the theory of 'self -abstraction beyond the limits of an external
world,' so closely reasoned out by Chuang Szu. If Mr. Baber merely
means that the gentlemen entrusted with the task have proved
themselves so far quite incompetent to express in Chinese the subtle-
ties of theological reasoning, then I am with him to the death."
Mr. K. S. Liu sends with his translation these further remarks
concerning Su Hsi, the classical philosopher of Chinese belles lettres :
"This poem was composed by Su Hsi, a famous Chinese poet
who flourished 1036-1101. Owing to the intrigues of his political
enemies he was exiled to Hwang-Cheo, a place in the province of
Hu Peh. While there he made a visit to a place called Chi Pi
(literally Red Wall), made famous by the battle which took place
there between Tsao-tsao and Cheo-yu (two historical characters in
the period of the Three Kingdoms). The poem is an account of
this visit and a description of the feelings it aroused in him. Like
many other poets who consider poetry an embodiment in symbols
of one's inner spiritual experiences, he shows in the poem, first,
the ephemeral nature of human existence with all its paraphernalia,
and then how in the contemplation of nature one can transcend
the mutations of time and be one with the eternal order. In this
state one can rise above the vicissitudes of life."
I3O THE MONIST.
The poem begins by giving the date of Su Hsi's excursion to
the Scarlet Cliff. The year reads in Chinese characters fan siih,
and we here encounter the difficulty of reproducing the Chinese
3
V ^t a * **
w
C/5
w
£ <•$ TJQ *g *$£
o 'A
a \
method of determining chronology. For this they make use of the
sexagenary cycle by repeating five times the twelve branches and
six times the ten stems (see the author's Chinese Thought, p. 4).
A CHINESE POET'S CONTEMPLATION OF LIFE. 13!
The meaning of jan (pronounced zhan) is the "germ in the
womb," and it "denotes the ninth of the ten stems ; it is connected
with the north and running water." It means "great, full" and also
•=:
"to flatter and adulate." As the ninth of the ten stems it denotes
swollen water, hence we translate it "billow." The other character
siih which is the eleventh of the twelve branches denotes in its
132 THE MONIST.
horary significance the hour 7-9 P. M., called the "dog hour." We
here translate it by "hound." To Chinamen this denotation of the
year is very familiar, but it is difficult to reproduce its exact sig-
nificance in a poetic translation in English. The "billow hound"
year corresponds in our chronology to 1082 A. D., which is the
fifty-eighth year in the sexagenary cycle under the Sung dynasty.
The latter being a matter of course in the poet's day is not men-
tioned in the Chinese text.
39 38 37 36 35 34 33
*•
» t
>
*' J *
' i
CHINESE TEXT.
The songs "To the Bright Moon" and "To the Modest Maid"
mentioned in the poem are probably the odes known as I, XII, 8
and I, III, 17 of the Shih King, the canonical collection of ancient
Chinese songs. In the translation of William Jennings (The Shi
King, pages 151 and 69) they read as follows:
To the Bright Moon.
O Moon that climb'st effulgent!
O ladylove most sweet!
A CHINESE POET'S CONTEMPLATION OF LIFE. 133
Would that my ardor found thee more indulgent!
Poor heart, how dost thou vainly beat!
O Moon that climb'st in splendor!
O ladylove most fair!
Couldst thou relief to my fond yearning render!
Poor heart, what charing must thou bear!
0 Moon that climb'st serenely!
O ladylove most bright!
Couldst thou relax the chain I feel so keenly!
Poor heart, how sorry is thy plight!
To the Modest Maid.
A modest maiden, passing fair to see,
Waits at the corner of the wall for me.
1 love her, yet I have no interview: —
I scratch my head — I know not what to do.
The modest maid — how winsome was she then,
The day she gave me her vermilion pen !
Vermilion pen was never yet so bright —
The maid's own loveliness is my delight.
Now from the pasture lands she sends a shoot
Of couchgrass fair; and rare it is, to boot.
Yet thou, my plant (when beauties I compare),
Art but the fair one's gift, and not the Fair!
There is some doubt, according to Professor Giles, whether
the Scarlet Cliff visited by Su Hsi was really the place of battle as
the latter assumes, but the poem remains of the same significance
even if Su Hsi was mistaken, and we need feel no concern about it.
p. c.
THE SCARLET CLIFF.
It was the Billow-Hound year of House Sung:
The seventh moon was on the wane, when I
Was down stream drifting in a boat with friends
On an excursion to the Scarlet Cliff.
134 THE MONIST.
The evening breeze so gently blew that scarce
The water rippled on its smooth expanse.
I rilled the cups and bade my friends to sing
The ode 'To the Bright Moon," and then they chanted
The lay melodious "To the Modest Maid."
Slowly the moon rose o'er the eastern hills,
And passed between the Wain and Capricorn,
Shedding her silver beams upon the water,
To link our world below with heaven above.
In such surroundings, infinite in charm,
Our skiff was freely gliding, — traveling
Unchecked through space, unmindful whither bound;
Like gods we moved in a transcendent realm:
I poured out a libation for our joy,
And beating time on our boat's wooden rim,
I sang these verses in sad exaltation :
"Our olive boat with orchid oars propelled,
Breaks splashing through the moonlit glittering
wave;
In lovelorn loneliness I here am held,
From friends who now lie buried in the grave."
One of my guests accompanied the song
Upon his flageolet, with proper notes
To suit the music to the sentiment
Of plaintive moods, in sounds that wove unbroken
Their silken threads around our company.
The music stirred the dragon in the deep
And moved the the boatswain's widow unto tears.
"And why is that?" I asked in pensive query
My cherished guest. "Why does thy magic art
So powerfully affect us all?" Said he:
A CHINESE POET'S CONTEMPLATION OP LIFE. 135
"Few stars are seen and yet the moon shines bright,
To southern lands the raven wings his flight.
"Was this not uttered here by Tsao Meng Te,
Here, eastward of Hsia-K'ou, west of Wu-Chang,
Where hill and stream in wild luxuriance blend?
'T is here Meng Te was routed by Chou Yii.
Before him lay Ching-chou. Kiang-ling he conquered,
And eastward did he push upon the river;
His warships, prow to stern, stretched thousand miles,
The banners of his troops darkened the sun.
Then a libation he poured out, and nearing
The Scarlet Cliff, the hero of his age,
On horseback, clad in armor, spake those words!
Yet where is he to-day? And what are we?
To-day we fish and gather fuel here
On river isles where shrimps are our companions
And deer our. friends. We paddle here about
In frail canoe and drink companionship
From flasks of gourd. How transient is the life
Of creatures as ephemeral as we.
Tossed o'er the ocean like a husk of straw,
We are mere twinklings on the river Time;
Oh, could I be the stream itself which rolls
Incessantly and without end! Alas!
Could I but clasp the bright and beauteous moon
Close to my heart and dwell with her in heaven !
Yet unfulfilled remain my deep-felt yearnings
Which find expression in melodious strain."
"But you my friend," replied I questioning,
"Do you well comprehend the mystery
Of this great river and the changing moon?
Past flows the water but 'tis never gone;
The moon is waning, but again 'twill wax.
136 THE MONIST.
So I with this great world, all in a change —
E'en Heaven and Earth are transient constantly —
Myself, and also thou, in this same sense
Viewed as a whole, live on eternally.
Why then lament? Thou long'st for what thou hast!"
And further musing on life's complex problems
Continued I: "Whate'er our senses hold
Is owned by him who feels it, who enjoys it.
For nothing can I take unless I own it,
The bracing breeze, the landscape of the river,
The moon above the valleys, gorgeous sights
Enrapturing the eye, and all the sounds
Which greet the ear, all are enjoyed by me.
All these are mine, and without let or hindrance
Are they the gifts of God, unstintedly
Given to man — indeed to all mankind.
And we enjoy them now."
He smiled approval —
My friend; he threw away the dregs of wine
And had his cup refilled up to the brim.
Thus finishing our feast we laid us down
To rest among the scattered cups and plates,
While in the distant east dim streaks of light
Appeared as heralds of another day.
CRITICISMS AND DISCUSSIONS.
LEIBNIZ AND LOCKE.
John Locke, the founder of the sensationalist school, who form-
ulated the principle of his philosophy in the statement Nihil est in
intellectu quod non antea fuerit in sensu, and who therefore on the
one hand denied innate ideas and on the other claimed that all
knowledge rises from experience, devotes to an investigation of
truth Chapter V, and also part of Chapter VI of his famous work
On the Human Understanding from which we make the following
extracts :
" 'What is truth?' was an inquiry many ages since; and it being
that which all mankind either do or pretend to search after, it cannot
but be worth our while carefully to examine wherein it consists ; and
so acquaint ourselves with the nature of it, as to observe how the
mind distinguishes it from falsehood.
"Truth then seems to me, in the proper import of the word, to
signify nothing but the joining or separating of signs, as the things
signified by them do agree or disagree one with another. The join-
ing or separating of signs here meant, is what by another name we
call 'proposition.' So that truth properly belongs only to propo-
sitions : whereof there are two sorts, viz., mental and verbal ; as there
are two sorts of signs commonly made use of, viz., ideas and words...
"We must, I say, observe two sorts of propositions that we are
capable of making:
"First, Mental, wherein the ideas in our understandings are,
without the use of words, put together or separated by the mind
perceiving or judging of their agreement or disagreement.
"Secondly, Verbal propositions, which are words, the signs of
our ideas, put together or separated in affirmative or negative sen-
tences. By which way of affirming or denying, these signs, made
by sounds, are, as it were, put together or separated one from an-
138 THE MONIST.
other. So that proposition consists in joining or separating these
signs, according as the things which they stand for agree or dis-
agree ....
"When ideas are so put together or separated in the mind, as
they or the things they stand for do agree or not, that is, as I
may call it 'mental truth.' But truth of words is something more,
and that is the affirming or denying of words of another, as the
ideas they stand for agree or disagree: and this again is twofold;
either purely verbal and trifling or real and instructive, which is
the object of real knowledge. . . .
"Though our words signify nothing but our ideas, yet being
designed by them to signify things, the truth they contain, when
put into propositions, will be only verbal when they stand for ideas
in the mind have not an agreement with the reality of things.
And therefore truth, as well as knowledge, may well come under
the distinction 'verbal' and 'real'; that being only verbal truth
wherein terms are joined according to the agreement or disagree-
ment of the ideas they stand for, without regarding whether our
ideas are such as really have or are capable of having an existence
in nature. But then it is they contain real truth when these signs
are joined as our ideas agree; and when our ideas are such as we
know are capable of having an existence in nature: which in sub-
stances we cannot know but by knowing that such have existed.
"Truth is the marking down in words the agreement or dis-
agreement of ideas as it is. Falsehood is the marking down in words
the agreement or disagreement of ideas otherwise than it is. And
so far as these ideas thus marked by sounds agree to their arche-
types, so far only is the truth real. The knowledge of this truth
consists in knowing what ideas the words stand for, and the per-
ception of the agreement or disagreement of those ideas, according
as it is marked by those words ....
"Certainty is twofold; certainty of truth, and certainty of
knowledge. Certainty of truth is, when words are so put to-
gether in propositions as exactly to express the agreement or dis-
agreement of the ideas they stand for, as really it is. Certainty of
knowledge is, to perceive the agreement or disagreement of ideas,
as expressed in any proposition. This we usually call 'knowing,' or
'being certain of the truth of any proposition.' "
His great critic Leibniz wrote a voluminous book1 to refute
1New Essays Concerning Human Understanding. Translated by A. G.
Langley. 2d ed., Chicago and London, 1916.
CRITICISMS AND DISCUSSIONS. 139
Locke's sensationalism, pointing out that what Locke called re-
flection was not a product of sensation. He amended Locke's prin-
ciple to read : Nihil est in intellectu quod non antea fuerit in sensu,
nisi intellectus ipse, and this amendment upset Locke's very lucid
but superficial arguments. According to Leibniz the senses furnish
us the material for positive knowledge but they offer nothing but
particular instances, not methods, nor principles, nor general truths.
Brutes have the same sensations as man, but brutes can never attain
to necessary propositions. These conceptions of necessary propo-
sitions are innate in the human mind. The human mind is not a
tabula rasa, but contains certain principles which, in the measure
that experience furnishes the occasion, develop into ideas of eternal
and necessary verities.
From this standpoint Leibniz distinguishes two kinds of truths,
necessary truths and contingent truths ; the former are the eternal
verities as instanced by mathematics, the latter the knowledge of par-
ticular facts furnished by experience. God is the ultimate source
of both kinds of truth; the eternal verities correspond to his in-
tellect, the contingent truths to his will. The former are such and
can not be different because God is such ; the latter could be different
but are not because God willed them to be as they are and not other-
wise. Necessary truths reveal to us what is possible and what impos-
sible. Thus, e.g., a regular decahedron (i.e., a figure bounded by ten
equal plane surfaces) is impossible, and "all intelligible ideas have
their archetype in the eternal possibilities of things."
In reply to Locke's view of certainty, Leibniz says:
"Our certitude would be small, or rather nothing, if it had no
other basis of simple ideas than that which comes from the senses.
Have you forgotten, sir, how I have shown that ideas are originally
in our mind, and that indeed our thoughts come to us from the
depths of our own nature, other creatures being unable to have an
immediate influence upon the soul? Besides, the ground of our
certitude in regard to universal and eternal truths is in the ideas
themselves, independently of the senses, just as ideas pure and in-
telligible do not depend on the senses, for example, those of being,
unity, identity, etc. But the ideas of sensible qualities, as color,
savor, etc., (which in reality are only phantasms) come to us from
the senses, i. e., from our confused perceptions. And the basis of
the truth of contingent and particular things is in the succession
I4O THE MONIST.
which causes these phenomena of the senses to be rightly united
as the intelligible truths demand."
It is not our intention to criticize any one of the philosophers
but we wish to point out how far and in what respect we agree with
Leibniz's views as here outlined. We select Leibniz because his
philosophy is less onesided than any other, and has incorporated all
considerations, religious, scientific, mathematical and historical.
What he calls innate ideas reflecting the eternal and necessary truths
whose source lies in God, we denote as the purely formal and we
have shown that purely formal conceptions have been gained by
abstraction. Man alone has the faculty of abstraction and so he
alone is capable of producing and operating with purely formal con-
ceptions such as numbers, geometrical figures, the notion of mathe-
matical or pure space, logical syllogisms, the formulas of causation
and of the conservation of substance and energy. The principle
pervading the function of these concepts is called reason, and reason
truly reflects the cosmic order, which is due to the efficiency of
purely formal interrelations — the so-called purely formal laws. Our
senses furnish us particulars only, and these particulars, which are
innumerable isolated sense-impressions, would remain a chaos of
disconnected items if they were not classified and systematized ac-
cording to purely formal laws. The point overlooked by Leibniz
and also later on by Kant is the question as to the origin of mind.
The framework of reason, man's logical faculty, his notion of
numbers and of space relations have indeed originated through ex-
perience as Locke claimed, but it was experience in a wider sense
than either Locke or Leibniz conceived it to be. Experience in those
days meant sense-experience, or the purely sensory element of sen-
tient creatures. In this sense Leibniz is right that no amount of
sense-impressions can bring forth an eternal or universal or neces-
sary idea. Locke on the other hand, conscious of the fact that man
was in possession of universal and necessary concepts and admitting
no other source of knowledge than experience, insisted on the prop-
osition that all ideas, even the most complicated ones, were derived
from sensations, as which he understands experience to be.
Now it is obvious that there is nothing purely sensory, Sen-
sations are possessed of forms and the formal impresses itself to-
gether with sense impressions upon sentient creatures. We have on
CRITICISMS AND DISCUSSIONS. 14!
other occasions set forth how sensory impressions are by a mechan-
ical necessity so grouped that they are registered together, the par-
ticular ones being subsumed under the more general so that all of
them build up a well-arranged system constituting a logical frame-
work of types. This framework is the mind which is built up not
of mere sensations, but of the interrelations of sense-impressions
according to their various forms. Experience in the current sense
includes the form of the sensory, and in this sense the faculty of
conceiving purely formal relations has indeed arisen from expe-
rience.
The sensationalist school identifies the sense element of our
knowledge with the formal and overlooks their radical difference.
We must insist against the sensationalist school that everything
formal is radically different from the sensory. The sensory is al-
ways particular while the formal can be generalized. By leaving
out of sight everything particular our thought can operate in a field
of pure relations, and we can exhaust all their possibilities. We can
say what is possible as well as what is impossible and (all inter-
ference of unexpected particulars being excluded) we can also say
what result will always be obtained under definite given conditions.
We can exhaust all possibilities of the purely formal and can sys-
tematize the whole field. What will always be, is called "necessary,"
and so these propositions which are inevitable are called by Leibniz
"eternal truths."
We agree with Leibniz that the source of these eternal truths
is God; nay we go one step further in definiteness and claim that
the eternal verities, of which our human notions of eternal truths
are mental reflections, are God himself. All depends on our defi-
nition of God. Together with the whole cosmic order the necessary
truths constitute an eternal omnipresence, an efficient system of
norms which mould the world and determine all things. They form
a kind of spiritual, or purely formal organism, a superpersonal
presence which is the ultimate raison d'etre and determinant of all
things, the cosmos in its entirety as well as all particular events that
happen in the course of its being.
Any one who has once grasped the deep significance of the
purely formal will have liberated his mind forever of the super-
stitious, mystical or allegorical conceptions of the deity, but he will
at the same time understand the truth that underlies the God-idea
and thus he will know the real nature of the true God, whose exist-
142 THE MONIST.
ence is not a matter of belief, but a scientific certainty. All former
proofs of the existence of God were necessarily failures, because in
all cases the attempt was made to prove the existence of an anthro-
pomorphic God with arguments that prove the true God, the eternal
norm of being, and here the argument breaks down, because it no
longer applies to the idea of an anthropomorphic God.
Leibniz has not overcome the mystical conception both of God
and truth. He has unfortunately adopted the very primitive con-
ception of an atomic nature of reality which is described in his
monadology. It is strange that a man of his caliber did not see
how contradictory is the idea of God as the central monad. On the
other hand his theory is vindicated if we interpret his God to be the
universal and omnipresent norm that regulates every event and
constitutes the cosmic order of the world.
Insisting on the unity of the soul, Leibniz conceived all unities
as local units, and these innumerable local units, the monads, were
conceived as centers of force endowed with feeling and an entelechy,
which means that they were capable of pursuing purposes. At the
same time Leibniz held them to be separate entities, so as to render
their cohesion and interaction a profound problem which could be
solved only by the bold hypothesis of the preestablished harmony.
The problem of unity together with all problems of combina-
tion and configuration belongs in the domain of pure form. Com-
bination of several parts working in cooperation constitute a unity
and introduce something new. It did not exist before and will break
to pieces again, but the law of its combination remains forever and
constitutes the eternal background of its existence. The sensation-
alist school misses the main point of all philosophical considerations
and thus loses the essence of the significance of religion ; but Leib-
niz who discovers the weak spot in their arguments has not suc-
ceeded in persenting a satisfactory solution of the problem but ends
in proclaiming a mystical God-conception and a dogmatic proclama-
tion of a preestablished harmony. p. c.
EXISTENTS AND ENTITIES.1
That we must distinguish between what we may call "having
existence" and "having entity or being" becomes evident when we
look somewhat closely at ordinary mathematical propositions. A
class (or system, or aggregate) M is said to "exist" when it has
'Cf. Monist, Jan. 1910, Vol. XX, p. 114, note 85.
CRITICISMS AND DISCUSSIONS. 143
at least one member;2 whereas, when mathematicians speak of,
for example, "the existence of roots of an equation" or "the exist-
ence of the definite integral of a continuous function," they use
the word "existence" in another sense : the roots or the integral are
not classes, but individuals constructed out of mathematical con-
cepts to supply an answer to certain questions. We can, of course,
consider such an individual as the member of the class (N)^
whose sole member is this individual, and can then consider the
second kind of mathematicians' existence-proofs as proofs of the
existence of the class N ; but we should, for the sake of clearness,
avoid speaking of the "existence" of the member3 of N, and use
some such word as "entity" or "being" instead.
Mr. B. Russell4 has thus distinguished being and existence in
1901 : "Being is that which belongs to every conceivable term, to
every possible object of thought — in short to everything that can
possibly occur in any proposition, true or false, and to all such
propositions themselves. Being belongs to whatever can be counted.
If A be any term that can be counted as one, it is plain that A is
something, and therefore that A is. 'A is not' must always be
either false or meaningless. For if A were nothing, it could not
be said not to be; 'A is not' implies that there is a term A whose
being is denied, and hence that A is. Thus unless 'A is not' be an
empty sound, it must be false — whatever A may be, it certainly is.
Numbers, the Homeric gods, relations, chimeras, and four-dimen-
sional spaces all have being, for if they were not entities of a kind,
we could make no propositions about them. Thus being is a gen-
eral attribute of everything, and to mention anything is to show
that it is.
"Existence, on the contrary, is the prerogative of some only
amongst beings. To exist is to have a specific relation to existence
— a relation, by the way, which existence itself does not have. This
shows, incidentally, the weakness of the existential theory of judg-
ment— the theory, that is, that every proposition is concerned with
something that exists. For if this theory were true, it would still
be true that existence itself is an entity, and it must be admitted that
existence does not exist. Thus the consideration of existence itself
fCf., e.g., Dedekind, Was sind und was sollcn die Zahlen? 2d ed., Braun-
schweig, 1893, pp. 5, 12; or Essays on the Theory of Numbers, Chicago, 1901,
pp. 49, 58; Russell, The Principles of Mathematics, Cambridge, 1903, pp. 21,32.
* Of course, the member of N may be itself a class and may thus "exist,"
but we obviously need not consider this further.
'Mind. N. S., Vol. X, No. 39, 1901, pp. 310-311.
144 THE MONIST.
leads to non-existential propositions, and so contradicts the the-
ory...."
This doctrine was repeated in Mr. Russell's Principles of
Mathematics;5 the existence-theorems of mathematics were said8
to be "proofs that the various classes defined are not null," and the
earlier statement7 that these theorems are proofs "that there are
entities of the kind in question" must not be taken to mean what it
apparently expresses.
While Mr. Russell emphasized the distinction between entity
and existence, it does not seem that at that time he quite realized
the full bearings of the question, at least in mathematics. He at-
tributed a denotation to every term that can possibly occur in a
proposition. Thus "the round square" had a denotation, and the
only further existence-question in logic and mathematics was
whether the numbers — at least such as were defined as classes — ,
classes of spaces, and so on, could be proved to "exist," — whether
members of the classes in question could be constructed by logical
methods provided that the initial postulates are granted.
Before going on to discuss the clear separation of the impor-
tant question of entity from the less important question of existence,
which came in Mr. Russell's later works, we will refer to the very
strong tendency, even among logicians and mathematicians, to at-
tribute a denotation to every denoting phrase.
Thus, H. MacColl8 remarked that a symbol which corresponds
to nothing in our universe of admitted realities, has, nevertheless,
"like everything else named," a symbolical entity. In his sixth
paper on "Symbolic Reasoning,"9 MacColl attempted to give a
simple theory of the existential import of propositions.
By elt e2, e3,. . . ., he denoted "our universe of real existences,"
and by oi} oz, o3,. . . ., "our universe of non-existences, that is to
say, of unrealities, such as centaurs, nectar, ambrosia, fairies, with
self-contradictions, such as round squares, square circles, fiat spheres,
"Pp. 449-450; cf. pp. 43, 71.
'Ibid., p. 497.
''Ibid., p. vii.
'Symbolic Logic and its Applications, London, 1906, p. 42; MacColl here
and elsewhere used the word "existence" where we use "entity." Cf. Mind,
N. S., Vol. XI, 1902, pp. 356-357.
•Mind, N. S., Vol. XIV, 1905, pp. 74-81 ; cf. Symbolic Logic and its Appli-
cations, pp. 5, 76-78.
CRITICISMS AND DISCUSSIONS. 145
etc." ; the "symbolic universe, or universe of discourse," S, may
consist either wholly of realities, wholly of unrealities, or partly of
realities and partly of unrealities. ... If A denotes an individual or
a class, any intelligible statement 0(A) containing the symbol A,
implies that the individual or class represented by A has a symbolic
existence; but whether the statement <£(A) implies that that which
A denotes has a real or unreal or (if a class) partly real and partly
unreal existence, depends upon the context."
We will pass over the discussion between Messrs. MacColl
and A. T. Shearman10 on the interpretation of the Boolian equation
"O = OA," and come to Mr. Russell's articles of 1905,11 in which
the theory of non-entity was, it seems, for the first time treated
satisfactorily.
The sense in which the word "existence" is used in symbolic
logic is a definable and purely technical sense. To say that A
exists means that A is a class which has at least one member. Thus
whatever is not a class does not exist in this sense; and among
classes there is just one that does not exist, namely, the null-class.
MacColl's two universes of existences and non-existences are not
to be distinguished in symbolic logic, and each of them is identical
with the null-class. There are no centaurs ; "x is a centaur" is
false whatever value we give to x, even when we include values
which do not "exist" in the meaning which occurs in philosophy
and daily life, such as numbers or propositions.
"The case of nectar and ambrosia is more difficult, since these
seem to be individuals, not classes. But here we must presuppose
definitions of nectar and ambrosia : they are substances having such
and such properties, which, as a matter of fact, no substances do
have. We have thus merely a defining concept for each, without
any entity to which the concept applies. In this case, the concept
is an entity, but it does not denote anything. . . .These words [such
as nectar and ambrosia] have a meaning, which can be found by
looking them up in a classical dictionary, but they have not a deno-
tation: there is no entity, real or imaginary, which they point out."
"Mind, N. S., Vol. XIV, 1905, pp. 78-79, 295-296, 440, 578-580; Vol. XV,
1906, pp. 143-144; and Shearman's book The Development of Symbolic Logic;
a Critical-Historical Study of the Logical Calculus, London, 1906, pp. 161-171.
u "The Existential Import of Propositions," Mind, N. S., Vol. XIV, 1905,
pp. 398-401 ; "On Denoting," ibid., pp. 479-493.
146 THE MONIST.
The last sentence refers to Frege's12 distinction of Sinn (meaning)
and Bedeutung (denotation).
A point of passing interest in connection with an attempt at the
solution of a mathematical paradox, referred to later, is this sen-
tence in MacColl's reply:13 "I may mention, as a fact not wholly
irrelevant, that it was in the actual application of my symbolic sys-
tem to concrete problems that I found it absolutely necessary to
label realities and unrealities by special symbols e and o, and to
break up the latter class into separate individuals, o1} oz, o3, etc.,
just as I break up the former into separate individuals el} e2, e3, etc."
When a phrase which in form is denoting, and yet does not
denote anything, — e. g., "the present king of France," — occurs in the
statement of a proposition, the question as to the interpretation
of propositions in whose verbal expression this phrase occurs arises,
and Mr. Russell, in the article "On Denoting" referred to, suc-
ceeded in assigning a meaning to every proposition in whose verbal
expression any denoting phrases — whether they appear to denote
something or nothing at all, e. g., everything, nothing, something,
a man, every man, no man, the father of Charles II, the present
king of France — occur. It is not necessary to assume that denoting
phrases ever have any meaning in themselves.
The theory of MacColl and the allied theory of Meinong were
rejected by Mr. Russell1* because they conflict with the law of con-
tradiction. If any grammatically correct denoting phrase stands for
an object although such objects may not subsist, such objects are
apt to infringe the law of contradiction. Thus it is contended that
the round square is round, and also not round.
To solve the paradoxes that appear in the mathematical theory
of aggregates, Mr. Russell treated classes and relations in the same
way as he treated denoting phrases.15
Poincare, among others, recognized that all the paradoxes of
the modern theory of aggregates, such as those of Burali-Forti,
Russell and Richard, arise from a kind of vicious circle which may
be expressed, in the language of Peano, thus: Everything which
u "Ueber Sinn und Bedeutung," Zeitschr. fur Phil und phil. Kritik, Vol.
C, 1892, pp. 25-50.
"Mind, N. S, Vol. XIV, 1905, p. 401.
14 Ibid., pp. 491, 482-483.
u "On Some Difficulties in the Theory of Transfinite Numbers and Order
Types," Proc. Land. Math. Soc. (2), Vol. IV, 1906, pp. 29-53 (cf. especially
the part on the "No-Classes Theory") ; "Les Paradoxes de la Logique," Rev.
de Metaphys. et de Morale, Vol. XIV, 1906, pp. 627-650.
CRITICISMS AND DISCUSSIONS. 147
contains an apparent variable must not be one of the possible
values of this variable.16 But Poincare did not»perceive that if we
wish to avoid such vicious circles we must have recourse to a
fundamental re-moulding of logical principles, more or less anal-
ogous to the "no classes" theory. To have shown this seems to
be one of Mr. Russell's greatest merits ; simply because practically
all the other mathematicians who have interested themselves in the
paradoxes did not realize this important fact. Thus, said Mr.
Russell,17 the method by which Poincare tried to avoid the vicious
circle consists in saying that when we assert that "all propositions
are true or false," which is the law of the excluded middle, we
exclude tacitly the law of the excluded middle itself. The difficulty
is to make this tacit exclusion legitimate without falling into the
vicious circle. If we say, "All propositions are true or false, ex-
cepting the proposition that every proposition is true or false," we
do not avoid the vicious circle. For this is a judgment bearing on
all propositions, viz.: "All propositions are either true or false, or
identical with the proposition that all propositions are true or false."
And that supposes that we know the meaning of "all propositions
are true or false," where all has no exception. That comes to de-
fining the law of the excluded middle by : "All propositions with the
exception of the law of the excluded middle are true or false,"
where the vicious circle is flagrant. We must, then, find a means
to formulate the law of the excluded middle in such a way that it
does not apply to itself.
On the details of the new construction of logic in such a way
that the paradoxes are avoided while nearly all of the work of
Cantor on the transfinite is preserved, we must refer to Mr. Russell's
works of 1908 and 1910.18 Mr. Russell's method of avoiding the
paradoxes in question is by what he called the "theory of types,"
and the object of this theory was shortly described by Dr. White-
head and. Mr. Russell19 as follows : "The vicious circles in question
arise from supposing that a collection of objects may contain mem-
bers which can only be defined by means of the collection as a
MWe may also express this principle as follows: A collection of objects
may not contain members which can only be defined by means of the collection
as a whole.
v Rev. de Metaphys. et de Morale, Vol. XIV, pp. 644-645.
""Mathematical Logic as Based on the Theory of Types," Amer. Journ.
of Math., Vol. XXX, 1908, pp. 222-262; A. N. Whitehead and B. Russell,
Principia Mathematica, Vol. I, Cambridge, 1910, pp. 39-88.
" Op. cit., p. 39.
148 THE MONIST.
whole. Thus, for example, the collection of propositions will be
supposed to contain a proposition stating that 'all propositions are
either true or false.' It would seem, however, that such a statement
could not be legitimate unless 'all propositions' referred to some
already definite collection, which it cannot do if new propositions
are created by statements about 'all propositions.' We shall, there-
fore, have to say that statements about 'all propositions' are mean-
ingless. More generally, given any set of objects such that, if we
suppose the set to have a total, then such a set cannot have a total.
By saying that a set has 'no total/ we mean, primarily, that no
significant statement can be made about 'all its members.' Propo-
sitions, as the above illustration shows, must be a set having no
total. The same is true, as we shall shortly see, of propositional
functions, even when these are restricted to such as can significantly
have as argument a given object a. In such cases, it is necessary
to break up our set into smaller sets, each of which is capable of a
total. This is what the theory of types aims at effecting."20
* * *
In the next place, we shall go back four or five years in time,
and see how the distinction between entity and existence became
necessary in a mathematical investigation which is somewhat famil-
iar to me. If I consider, at rather greater length than it deserves,
my own work of 1903 and 190421 on the contradiction of Burali-
Forti and its bearings on the theory of well-ordered aggregates,
it is merely because familiarity with this investigation enables me
to point out a small, unobserved merit which it has, in distinguishing
entity from existence, and also to give yet another illustration of
the tendency — which seems particularly common with mathemati-
cians— of holding to the belief in the being or existence or sub-
sistence in some sense, of a non-entity.
Burali-Forti had found, in 1897, the now well-known contra-
diction arising from the fact that 'the ordinal type of the whole
series of (finite and transfinite) ordinal numbers' appears both to
be and not to be the greatest ordinal number. From this I con-
cluded, in 1903, that there are no such things as "the type" and
"The theory of logical types was described, in ordinary language, in op,
cit., pp. 39-68; and the theory of denoting was explained in the chapter on
"Incomplete Symbols" (ibid., pp 69-88).
"A general account of these investigations is contained in my paper,
written in Peano's international (uninflected) Latin: "De Infinite in Mathe-
matica," in Revista de Mathematica, VoL VIII.
CRITICISMS AND DISCUSSIONS. 149
"the cardinal number" of the series just referred to. Hence, by a
tacit use of an axiom afterwards stated explicitly by Zermelo, I con-
cluded that every aggregate which has a cardinal number and every
series which has a type can be well-ordered. The use of Zermelo's
axiom was, with me as with most mathematicians, unrecognized;
it occurred in some work of Mr. G. H. Hardy's on which I based
my argument; and I was really concerned, not so much with the
proof that every aggregate can be well-ordered, as with the proof
that the series (W) of ordinal numbers has no type.
The matter becomes simpler to express when we consider
classes instead of series. My contention, then, was that there is
no such thing as "the cardinal number of the class of ordinal num-
bers" seems to represent. But if we adopt, as I adopted, the Frege-
Russell definition of the cardinal number of a class u as the class
of those classes which are similar to (can be put in a one-one cor-
respondence with) u, there arises a difficulty. The cardinal number
of the class w of ordinal numbers is the class of those classes which
are similar to w, and this class certainly exists, for we can point
out at least one member of it, namely, w itself, for w is similar to
w. On the other hand, we have reason to deny that there is such
a class as the cardinal number of w, and most mathematicians ex-
press this by saying that the cardinal number in question does not
"exist." Of course, the solution of this apparent contradiction is
that "the cardinal number of w" is a phrase denoting nothing —
there is no such entity as the cardinal number of w. If it did
denote a class, that class would be existent.
So, in my above-quoted paper, I distinguished between the
existence of a class u from the entity of a thing v. The symbol
"3.u" was used, following Peano, to denote that u exists, and the
symbol "Ez>" was used to denote the proposition that v is an entity.
The symbol "Ez/" was defined by the definition of "not-Ez/" as
"v is a member of the null-class." Since the null-class has no
members, and is defined as the x's satisfying a prepositional func-
tion, such as x is not identical with x, which is always false, this
is a most paradoxical way of stating the case about non-entity,22
and the paradox results from the assumption that, in some sense,
there is a v, — that, as MacColl would have said, v has a "symbolical
"On printing the above article, Professor Peano wrote to me, on Jan. 1,
1906, as follows: " I see the new symbol E, which you do not define sym-
bolically, but the importance of which I believe I have understood It would
be necessary to 'introduce many kinds of null-class (A): AO = that of the
Formulaire ; Aj = the class of classes, which has no classes ; A, for the classes
I5O THE MONIST.
existence." But, as Dr. Whitehead and Mr. Russell23 pertinently
remark: "We cannot first assume that there is a certain object, and
then proceed to deny that there is such an object." Russell's solu-
tion of the difficulty about propositions asserting that "the so-and-so
is not an entity" is to reduce all such propositions to a form not
involving the assumption that "the so-and-so" is a grammatical
subject. "The so-and-so," whether it appears to denote something or
not, is an incomplete symbol, like the d/dx of mathematics.
* * *
It has, I trust, been not quite without interest to see how the
important distinction of existence and entity in mathematics strug-
gled into clearness. We have seen before24 that the discussions on
"existence" of MM. Poincare and Couturat were conducted in ob-
scurity. This obscurity was produced by the confusion of the two
notions of existence and entity, and the consequent use of one word
to denote both.
When, in a paper published in 1904, I used the badly chosen
term "inconsistent" for an aggregate whose cardinal number is a
non-entity — "does not exist," I said then — Mr. Russell rightly ob-
jected that, given a class u, its cardinal number must exist, since u
is a member of the class called the cardinal number of u. And yet
there was an undoubted difficulty about what I called "inconsistent"
classes. We know now that — at any rate when the number of a
class is defined logically — it is a delusion that there are such "in-
consistent" classes, — they are non-entities. If they were entities,
their cardinal numbers would "exist."
There is one more thing to be noticed: it is the entity of
a number that is most important, the proof of its existence is
less so. In his Principles of 1903, Mr. Russell laid great stress on
the existence-proofs of numbers and classes of spaces. Let us con-
sider the case of real numbers. A real number is, according to Mr.
of classes; A, for the classes of classes of classes;.... An,. . . .A-^
There is the generation of the transfinite numbers, in the principles of logic.
There results this rather laughable consequence, that the new philosophers
have decomposed nothing into a transfinite number of classes!"
* Op. cit., p. 69. We may remark here, as I have done in a review of
Whitehead and Russell's Principia in the Cambridge Review for 1911, that the
authors (cf. pp. 32, 69, 182, 229) use the word "existence" ambiguously;
though, of course, there is no ambiguity when the proper technical symbols
(a and E; E only occurring in a phrase involving incomplete symbols) are
used.
"Monist, Jan. 1910, Vol. XX, pp. 113-116.
CRITICISMS AND DISCUSSIONS.
Russell, a certain class of rational numbers ; its existence can be
proved, and one feels satisfied. But a rational number or a negative
number, being a relation, does not "exist," and yet one would have
thought existence quite as important in these cases as in the case
of real numbers.25 I hope to go more fully into this question on
another occasion.
PHILIP E. B. JOURDAIN.
CAMBRIDGE, ENGLAND.
IDEALISM AS A FORCE.
A MECHANICAL ANALOGY.
In the present state of knowledge the man of intelligence has
much difficulty in deciding what course of conduct he should adopt
in regard to beliefs and social and religious practice without at the
same time violating these principles which he has obtained from
science and critical philosophy. Before venturing to suggest exactly
what position he should (and eventually must) take up, a little con-
sideration of the importance of the older ideas and their relation to
new ones would be advisable. I propose to introduce various me-
chanical analogies in this sketch, for two reasons. First, because I
think they show forth more clearly the nature of the phenomena
described, and second, a training in scientific thought soon shows one
that mechanical laws pervade the whole universe, mental, moral and
physical. I do not use the word "mechanical" in at all a derogatory
sense. As a matter of fact, although it seems at first contrary to
our ideas of perfection realized by a continuous process of adjust-
ment, the really perfect state is the mechanical one, where each part
has a definite and unchanging relation to all the other parts, so that
a change in its condition is accompanied by a change in all other
parts in accordance with the nature of that mutual relation. Surely
this is what is meant by "correspondence with environment," if
there is the proviso of stability. All moral philosophers have more
or less directly stated that the key to morality is the Golden Rule,
"Do as you would be done by," or as K'ung-f u-tze puts it, in one word,
"Reciprocity," i. e., mutual bearing upon one another. This condi-
tion of mutual bearing is essentially, when complete, a mechanical
* Frege (Grundlagen dcr Arithmetik, Breslau, 1884, pp. 114-115) indicated
such definitions of all the numbers of analysis as would enable him to prove
the existence in every case.
152 THE MONIST.
state. Similarly in matters of thought consistency is the great prin-
ciple, and what is consistency but a mechanically perfect state of
balance? As to the mechanical character of physical conditions
there can be no question, provided we do not necessarily limit the
concept to the Newtonian exposition.
I wish to use frequently the idea of force. In natural philos-
ophy a force is that which tends to produce or hinder motion, and
it is the characteristic of all natural phenomena that the forces
acting on them shall be in a state of balance. Whether they are still
or moving, this balance exists either in the form of opposed pulls,
pushes, stresses or accelerations of mass. It is the criterion in the
light of which all mechanical problems may be attacked. I wish to
extend this idea of force to matters of thought and ideal, by a defini-
tion such as the following: A mental force is that which produces
or tends to produce change of thought.
The ever-famous Newton, in studying natural forces, announced
three laws of motion. There is no definite proof of these, but we
have no experience which contradicts them.
With the suggested psychical analogues these laws are as fol-
lows:
1. Any body tends to remain in its condition of rest or motion
until acted on by some force.
To extend this to matters of thought we can say:
Any idea (group of concepts) tends to remain in its state of
rest or change along certain lines until acted upon by some mental
force.
2. Change of motion is proportional to the magnitude of the
applied force.
This becomes:
Change of thought is greater or less according to the effective
importance of the mental force.
3. To every action there is a reaction, i. e., whenever a force
acts upon a body there is called out in that body a force opposed
to (and equal to) the first force which manifests itself as internal
stress or acceleration of mass.
In mental matters this notion is expressed by the change in
thought which takes place as the result of applying mental force,
appearing either as a new formation of ideas or a reaction of old
ideas on the new mental force.
It must be understood at this point that I do not mean anything
CRITICISMS AND DISCUSSIONS. 1 53
extremely mystical or undiscovered by this term "mental force."
I simply give this name to a set of ideas, in the first place external
to the mind in question, then received through the ordinary chan-
nels of sense, and acting upon the ideas already existing there,
either producing resistance or modifying those ideas. The tech-
nical word "suggestion" is almost identical in meaning.
The engineer, in the spirit of Newton, takes our above-described
three laws into one equivalent, as follows :
Force is the rate of change of motion attached to matter (tech-
nically "momentum").
This simply means that wherever and whenever a force acts
upon a body it produces a change in its motion, or, vice versa, a
change in motion is caused by a force.
This can be made the basis of a more sweeping statement which
describes mental force thus:
Mental force is the rate of change of thought attached to mind.
(Brain-matter is perhaps not to be regarded as the absolute medium
of thought, since psychologists regard the latter as contemporaneous
with, but not necessarily the same as, change in cerebral substance) .
Idealism I wish to describe as a particular type of mental
force proceeding in the first place from some external source, and
then by its action on different minds in accordance with the above
laws and by the reactions of such minds on physical and moral
actions, producing an effect tending to the realization of certain
progressive states which are for the time being regarded as perfect.
In the light of this conception all religions are forms of ideal-
ism.
If we examine any religion from its commencement we usually
find some such development as this:
1. Absorption by a master mind (the founder) of certain older
ideals, the mutual reactions of which together with the mental con-
tition induced in him by his surroundings (physical and social) pro-
duce a new system with one central ideal.
2. This result in many cases is accompanied by very severe
mental strain, and in some cases by nervous disease (cf. Mohammed
who is believed to have suffered from epilepsy) after which this
ideal takes the leading part in his thought and life (monoideism).
3. The ideal now works through him to the minds of certain
followers or disciples who receive it according to their previous
154 THE MONIST.
training and heredity, and so is formed a circle of minds in which
the ideal circulates for a time, gaining an ever increasing potential.
4. The widening of the circle and frequently the loss by decease
of the founder, causes the ideal to cease its original evolution and
take on certain new features according to the reactions in the minds
of its various adherents. Hence we have lesser circles forming, to
which certain new phases have more and more relation, until there
is a schism of the original community and the most energetic minds
found sects.
5. These sections expand or not according as the ideal is re-
sisted or absorbed by the further minds upon which it acts, and we
may finally have a large community with the ideal (usually much
modified by reaction) controlling and connecting the units. This
arrangement persists until external ideas of a different kind or in-
ternal resistances destroy its energy and it is replaced by other
ideals or a great modification of the old one.
The mechanical analogy to the action of external forces on
matter already possessing kinetic energy is so obvious if the lines
previously indicated are followed, that I will not trace out each
link of the chain, but merely point out the steps in which we draw
a comparison.
1. Composition (i. e., combining together) of various forces
(ideals) in one point (mind) which possesses considerable freedom
(enthusiasm).
2. Acceleration in this point (mind) under the resultant force
(new ideal) finally acting on other bodies (minds) in a greater or
less degree according to their condition of stability (environment).
3. Composition of the forces in these individual bodies (minds)
resulting in a balanced but unstable system (idealist community).
4. Splitting up of systems into smaller systems (sects) balanced
in themselves with moderately high stability (sects) and balanced
as a whole (unstably) as a general system (national religion).
5. Modification of system by new forces (ideals) finally re-
sulting in a new system (religion).
At this point it is necessary to discuss the importance of ideal-
ism in its effect on the social life. Once a definite ideal or system
of ideals has become established among a set of minds it acts as a
"superhuman" power (not in the accepted sense of "supernatural"
but as the simple result of evolution) whose magnitude is the re-
sultant of the various forces which it has impressed on individual
CRITICISMS AND DISCUSSIONS. 1 55
minds and whose direction (i. e., tendency to progress or degen-
erate) is determined by the manner in which it has combined with
the mental forces previously impressed on these minds.
We see then that it has a definite (but fluctuating) value, a
more or less constant direction (for the time) and it is attached
to a certain number of unit minds.
It may be compared with the constitution of the atom in which
there are a number of electrons each possessing a peculiar resultant
motion of its own but at the same time coordinating with other
electrons to confer on the atom as a whole certain dynamic properties
which manifest themselves as polarity or chemical attraction, which,
although the equivalent of the electronic energy, are different in
kind.
Similarly our ideal may be attached to a large number of minds
of varying caliber, force and direction, but as a whole organism the
system will be possessed of properties differing from those of its
units.
Such a force as this centered in a community constitutes a
divine being controlling and working through its members, just as
according to modern psychology, the soul is a centering of nervous
energy. The Christian church in which the members are said to
belong to the mystical body of Christ exemplifies this. The whole
of the church is, so long as homogeneity prevails, a force whose mag-
nitude is the resultant of the mental and moral efforts of the units.
These efforts may be distinct in kind, amount and object, but never-
theless on the whole they are cumulative and there is a resultant
which may be well called the living Christ, for it is an intelligent
force realizing within itself to some extent the ideal which the
master-mind of Jesus impressed on his disciples to such a degree
as their capacities permitted.
In this way the doctrines of salvation (i. e., separation from
anti-Christian community and ideals) and grace (impression of
idealism according to capacity for receiving it) become explicable
and even reasonable. Of this more later.
I am of course aware that I at once lay myself open to severe
criticism from the adherents of all faiths who conceive their deity
to be omnipotent and omniscient. To this notion I would say that
such a force as described above has within itself the means of doing
and knowing all those things which come within the ken of the
units, and that further it combines with the resultant forces of the
156 THE MONIST.
universe, being either decreased or increased in effect according as it
is opposed to or in line with such world forces. So long as a religion
progresses (apart from the consideration of certain artificial condi-
tions such as politics) it must be to some extent in conformity with
the laws of the universe, known and unknown. So soon as it
directly opposes those laws (still subject however to certain socio-
logical factors) it must degenerate. The gods of a religion live and
die with it, their energy appearing in other faiths after reaction
has taken place in the minds of the interregnum. The only case in
which they (or he) are immortal is when they are definitely identi-
fied with some permanent force in the universe so that the mental
force runs contemporaneously with a natural one, each producing
proportionate effects on mind and matter. It is from this cause that
Judaism has ensured its immortality. About the time of the Cap-
tivity it definitely connected its tribal deity Yahweh not only with
the ideal of tsedek (righteousness) but with that unitary world-
power which under various names (such as "the eternal energy")
all philosophers and scientists recognize, with or without moral
attributes. This element of permanence has been transmitted to
Christianity and Islam so that these three are probably the most
stable of all faiths. It does not however necessarily follow that
because the force survives, the attachment of the community to the
ideal force will also survive. Its energy may be transferred to other
minds, possibly in other forms, but practically never losing all con-
nection with the primal natural force with which it has been asso-
ciated.
In order that the idealism of a community shall have a perma-
nent effect it is necessary:
1. That there should be a continual supply of mental energy
on the part of unit minds ;
2. That the individual energies shall be so directed generally
and of such amount that there always is an external resultant pro-
ducing progress by its reaction on the minds of both the units of
the community and those outside of the community.
In order to assure the first condition some definite "cult" is
required, which by the repetition of various practices concentrates
the mind on the ideal tending to develop its realization in that mind
and directing the energy of the mind to that end, both within and
without.
In the second condition it is essential that certain agreements
CRITICISMS AND DISCUSSIONS. 157
concerning the ideal shall be established, so that the energies put
forth are not contrary in tendency. This is the foundation of dogma,
which states as far as possible the ideal in words and symbols, which
produce in the various minds a more or less homogeneous concep-
tion of the ideal.
Further, it is necessary in order that the mental forces shall not
equilibriate, that all the members of the community shall, as far as
practicable within the limits of the competition necessitated by the
law of selection and survival, support one another, so that the
mutual stress between them is minimized and the external resultant
increased.
To return to our electron analogy, if electrons move at right
angles to the general path, collisions will occur which reduce the
external force exerted by the atom, and if sufficiently numerous
may be conceived quite to destroy that force and even disintegrate
the atom. (Cf. "The house divided against itself.")
This necessity for internal balance gives rise to ethics, which
is summarized by the Golden Rule.
HERBERT CHATLEY.
CHINESE GOVERNMENT ENGINEERING COLLEGE.
TANG SHAN, CHIH-LI.
CLASSICAL CONFUCIANISM.
Sinology has so far not yet passed the stage of crude and
amateurish translation. No interpretative work worthy of serious
consideration has yet appeared. Mr. Miles Menander Dawson's
recently published book, The Ethics of Confucius: The Sayings of
the Master and his Disciples upon the Conduct of the "Superior
Man/'1 is an attempt in the direction of interpreting Confucianism
to the West. We congratulate him on his highly successful exposi-
tion of one of the greatest ethical systems of the world. His work
has at least met a need which has long been felt by all who desire
to bring about a better understanding of Chinese civilization in the
occidental world. For ever since the days of Marshman and Legge
the true meaning of Confucianism has been lying hidden in those
painstaking but unfortunately too expensive and out-of-print trans-
lations ; and the general public have long had to swallow what super-
ficial and biased writers are pleased to call "Confucianism." Mr.
Dawson's book is based entirely on Legge's translation of The
i New York, G. P. Putnam's Sons. Pp. xviii, 305. Price, $1.50 net.
158 THE MONIST.
Chinese Classics, and he has so classified and arranged his material
that the reader can easily comprehend what Confucius and the
early Confucians actually said on the various fundamental prob-
lems of life.
This book has many notable merits. First, the handling of the
immense quantity of material is excellent. The work is divided
into seven chapters : I. What Constitutes the Superior Man ; II.
Self -Development ; III. General Human Relations ; IV. The family ;
V. The State; VI. Cultivation of the Fine Arts; VII. Universal
Relations. Mr. Dawson has seized upon a very important point in Con-
fucianism when he arranges his book in accordance with the scheme
of The Great Learning. For the Confucian ethics is essentially a
system of human relations: all extension of knowledge contributes
to the cultivation of individual conduct, and from the individual
there radiate the relationships of the family, the state and the world.
Secondly, the illustrative quotations from the Confucian classics
are, with a few exceptions, very well chosen. The quotations are
all accompanied by the name of the book, the number of chapter,
paragraph and verse. The carefulness and patience with which the
numerous passages are selected and classified, certainly commands
our admiration. The index appended to the book also enhances its
usefulness.
Thirdly, the first two chapters in particular constitute the best
portion of the book. In these chapters Mr. Dawson sets forth the
Confucian ideal man, "the Superior Man," which forms the sub-
title of the book. The Superior Man, which can be more literally
translated as "the lordly man" or better still as "the gentleman,"
is quite different from the dianoetic man of the Greeks; neither
does he aspire to the Nirvanic life of Buddhism, nor aim at the
attainment of a union with God, which forms the ideal of Chris-
tianity. The Confucian ideal is simply a life made ever nobler and
richer by individual reticence and by a conscious adoption as one's
own of the social moral institutions which constitute the li (trans-
lated "rules of propriety") or what the Hegelians call Sittlichkeit.
In expounding these basic elements of Confucianism Mr. Dawson
has exhibited a high degree of clarity of exposition and richness
of illustration.
Lastly, we believe that the greatest merit of the book lies in
its objectivity, by which is meant the impartiality and disinterested-
ness with which the author expounds the Confucian doctrines.
CRITICISMS AND DISCUSSIONS. 1 59
Mr. Dawson has no desire to prove that Confucianism is inferior
to any particular ethical or religious system, nor does he wish to
proselyte his readers into Confucianism. He simply presents to us
what the great Confucians thought and taught concerning the
multifarious complexities of life and conduct. He speaks of con-
cubinage with the same calmness with which he discusses the Con-
fucian conception of the state.
It is natural that an undertaking of this kind by one who has
no access to the original texts cannot be entirely free from occa-
sional errors. Numerous unimportant mistakes may be pointed out
at random. For example : ( 1 ) on page xiii, the name of Confucius
appears twice as Kung Chin, which should be Kung Chiu; (2) on
page xiv, Chun Chin should read Chun Chiu ; (3) on page xvi, it
is wrong to include the Hsiao King instead of the Chun Chiu in the
Five Classics ; and (4) on the same page "Pan Ku" and The His-
tory of Han Dynasty are mentioned as two separate works ; whereas,
as a matter of fact, Pan Ku is the author of The History of Han
Dynasty.
Of errors of a more serious nature we find at least three.
In the first place, the title, "The Ethics of Confucius," is not correct.
It is as if a compilation of the ethical theories contained in the
works of Plato, Aristotle and Theophrastus were to be called "The
Ethics of Socrates." Mr. Dawson's book deals with the ethics,
not of Confucius alone, but of what we may call classical Con-
fucianism. For it is almost needless to point out that many of the
Confucian classics, like the Shu King and the Shi King, deal with
historical periods long before Confucius; while others, like the Book
of Mencius and the Li Ki, came long after the death of Confucius.
Book III of the Li Ki, for example, was compiled in the second
century B. C.
In the second place, Mr. Dawson has at times misinterpreted
the meaning of certain passages. Take this illustration:
"The scholar keeps himself free from all stain" (Li Ki,
xxxviii, 15). The Master said, "Refusing to surrender their
wills or to submit to any taint to their persons ; such, I think,
were Pih-E and Shuh-Tse" (Analects, xviii, 8).
"These two passages," says Mr. Dawson, "illustrate the sage's
insistence upon sexual continence, among other virtues." Now the
word "stain" in the first quotation has no reference to sexual rela-
tions. Nor does the phrase "taint to their persons" in the second quo-
l6o THE MONIST.
tation mean sexual immorality. The story of Pih-E and Shuh-Tse (or
Po-I and Shu-Chi), who abandoned their hereditary kingdom and
retired into obscurity, and who, when the Chou Dynasty was
founded, died of hunger rather than live under the new dynasty, —
this story is well known to every Chinese, and is given in a note in
Legge's translation (v. 22).
In the third place, Mr. Dawson has on several occasions taken
a passage quite apart from its immediate and inseparable context,
thus losing the meaning that was intended. An example of this
kind is found on page 248:
"When good government prevails in the empire, cere-
monies, music and punitive military expeditions proceed
from the emperor" (Analects, xvi, 2).
This passage Mr. Dawson takes as "suggesting that wise pa-
tronage and encouragement of art by the government which has
distinguished the most enlightened governments of ancient and
modern times." Now this passage cannot be taken apart from its
context. Here is the context:
"When good government prevails in the empire, cere-
monies, music, and punitive military expeditions proceed
from the emperor. When bad government prevails, these
things proceed from the princes. When these things pro-
ceed from the princes, rarely can the empire maintain itself
more than ten generations."2
Here we can easily see that the point of emphasis in this
passage is from what source these institutions should derive their
authority. The passage no more illustrates the wise patronage of
art than it illustrates the encouragement of punitive expeditions.
It must be pointed out, however, that such errors are very
rare in the entire work. On the whole, Mr. Dawson's book may
be recommended to all students of Chinese philosophy and religion
as an excellent exposition of classical Confucianism.
SUH Hu.
COLUMBIA UNIVERSITY, New York.
2 This is my translation. Legge's rendering is not correct.
VOL. XXVII APRIL, 1917 NO. 2
THE MONIST
THE TEXT OF THE RESURRECTION IN MARK,
AND ITS TESTIMONY TO THE APPA-
RIT1ONAL THEORY.
WITH A PREFACE ON LUKE'S MUTILATION OF MARK.
~^HE greatest literary problem in the New Testament
A is: What is the matter with the Gospel of Mark?
Something happened to the end of it in the first or second
century, and for ages thereafter it was left truncated in
the middle of a sentence or else supplied with a shorter
conclusion than the present one, which scholars long kept
to themselves. Edwin A. Abbott, however, gave it in his
forgotten Gospel analysis of 1884, and the Nonconformist
translators of The Twentieth Century New Testament
have also given it; but it does not appear in any official
translation, though the Revised Version mentions it in a
note at Mark xvi. 8. This is the note :
"The two oldest Greek manuscripts, and some other
authorities, omit from verse 9 to the end. Some other
authorities have a different ending to the Gospel."
Here is the "different ending," translated from a ninth-
century manuscript in the National Library of France,
Codex L, which gives both conclusions, but puts this one
first. (We prefix to it the connecting words of Mark) :
233
II. And they went out and fted from the sepulcher, for
l62 THE MONIST.
trembling and astonishment had come upon them; and they
said nothing to any one, for they were afraid of.
£ . i|i • $ 4* •• j|c j|c •' ^ji ••• ;|t £ • 4t. 4* £ 4* 4(
[Thirteen ornamental marks.]
Where also you must give currency to this :
Now, all things that were commanded, they showed
forth in few words unto those about Peter. And after
these things Jesus himself, also, from the East even unto
the West, sent forth through them the holy and incor-
ruptible preaching of eternal salvation.
But there is also current the following, after the words :
FOR THEY WERE AFRAID OF:
Now, when he was risen early etc. (as in our common
versions, Mark xvi. 9-20).
In their Introduction to the New Testament (Cam-
bridge, 1 88 1, pp. 298, 299) Westcott and Hort remark on
the above "less known alternative supplement" to Mark:
"In style it is unlike the ordinary narrative of the Evan-
gelists, but comparable to the four introductory verses of
St. Luke's Gospel." Conybeare, in his great book, Myth,
Magic and Morals, throws out the suggestion that Luke
mutilated the first edition of Mark because he disagreed
with its contents : viz., an account of apparitions in Galilee,
whereas he expressly limits all these phenomena to Judea,
by making Jesus order the apostles to stay in Jerusalem
until Pentecost. (Luke xxiv. 49; Acts i. 4). If Luke
mutilated Mark, then why not go further and say that he
wrote this smooth-flowing supplement to round him out?
The word ODVTO^ICO^, "in few words," is never found in the
New Testament except in this shorter Mark Appendix and
in Luke's Acts of the Apostles (xxiv. 4).
'E^cr/YE^ro, "to show forth," also occurs only in the
Pauline or Lucan (for Luke was Paul's secretary) Epistle
of Peter (i Pet. ii. 9). 'E|ajioateXXco, "to send forth," is
used seven times in Acts, thrice in Luke's Gospel, and once
THE TEXT OF THE RESURRECTION IN MARK. 163
by his master Paul. "Incorruptible" occurs only in Paul
and the Pauline i Peter.
Luke represents the aristocratic tradition of the capital,
which said: "It all happened here!" Mark represents the
rural tradition of Galilee, which said: "Our poor parish
was the scene of these wonderful things !"So the young man
in white, in Mark, says at the sepulcher: Go, tell his dis-
ciples and Peter: Behold, I am going to Galilee ahead of
you. There shall ye see me. (Thus read some of our best
manuscripts, in the first person.)
Another thing: Luke and John both make the appari-
tions real. In these later Gospels Jesus is objective after the
Resurrection: he eats broiled fish in Luke, while in John
the wounds in his hands and side are felt by Thomas.
Now as the earliest account of the Resurrection in Paul
(i Cor. xv. 4-8) makes the event a series of apparitions,
it is probable that the second earliest account, Mark's, did
the same. Indeed in Matthew xxviii. 17 (under suspicion
of being taken from the lost ending of Mark), "some
doubted." This was because some saw the figure and
others did not. Luke and John leave no room for doubt:
the evidence is sensuous, not subjective.
The first Christian heresy was Docetism, the belief that
Jesus even in life was a phantom. His flesh and blood were
unreal ; he did not really suffer ; his bodily functions were
different from human ones or even non-existent. To fight
this heresy the First Epistle of John was written, and a
curse pronounced upon those who doubted that Jesus had
been actual flesh and blood ( i John iv. 2, 3). Consequently
if Mark repeated Paul's impression that the Galilean ap-
paritions were the same in kind as the one to himself on
the Damascus road, then Mark must go. Who was the
likeliest one to do this work of excision? Answer: Luke.
He was the most literary of all the Evangelists. He is
164 THE MONIST.
the only one of them who says "I." Moreover, as Harnack
has pointed out, he betrays an animus against Mark, ani-
madverting upon his conduct in Acts xv. 36-41. In his
own Gospel Prologue, Luke is undoubtedly thinking of
him as one of the "many" who have "undertaken" to write
the life of Jesus, but who have not begun "accurately from
the first" nor set forth "in order" the sayings and events.
Add to this the Jerusalem tradition of the Resurrection
against the Galilean, and the flesh-and-blood appearances
against the phantom who is only to be "seen" ("there shall
ye see me," in Mark), and we have motive enough for
Luke's high-handed act.
Indeed, we can even surmise the reason why he made
the excision in the middle of a sentence. He would hardly
do this except to get rid of an offensive word. If Mark
had read:
They said nothing to any one, for they were afraid of
the apparition,
this last word would have been the red rag. There must
be no apparition: there must be objective forms. The
young man in white, who, in several MSS., speaks in the
person of Jesus, was indeed he himself in his glorified
being. Thus do I read the texts. Luke too had read some-
thing of this kind, which he reproduces thus :
But they were terrified and affrighted and supposed
that they beheld a spirit. ( Luke xxiv. 37. The Cambridge
MS. and Marcion's edition of Luke both read "apparition"
instead of "spirit")
Let it be understood that I do not deny the possibility
of ectoplastic phantoms, which Myers himself believed in,
though he said he would not press them upon the credence
of the reader, because of the difficulty of correct observa-
tion and the chances of fraud. Dr. Reichel of Germany
has testified to their occurrence here in America. The
difficulty in the New Testament is that they only appear in
THE TEXT OF THE RESURRECTION IN MARK. 1 65
the later accounts. Paul and (I shall show presently)
Mark, our earliest witnesses, know of apparitions alone,
not of materialized forms.
For the fullest account in English of all the problems
the reader should consult The Resurrection in the New
Testament, by Clayton R. Bowen, of Meadville, Pennsyl-
vania (New York, 1911). Professor Bowen is one of a
long series of laymen and liberals, like Griesbach, Lach-
mann, Tischendorf and Tregelles, who have taken the New
Testament out of clerical hands. The three German lay
professors and Tregelles, the English Quaker, were the
ones whose work led directly to the Revised Version of
1 88 1 ; but the task of revision is by no means ended yet.
Bowen was a Unitarian minister, but is now professor
at Meadville. Before reading him, a shorter and clearer
book by Kirsopp Lake should first be mastered.
Kirsopp Lake, of Harvard University, published in
1907 The Historical Evidence for the Resurrection of
Jesus Christ (London, Williams and Norgate). Professor
Lake at that time held the chair of New Testament Exege-
sis in the University of Leiden, to which Rendel Harris
was elected in 1903, but did not serve. The book appeared
in the Crown Theological Library and has been widely
read. It contains a masterly analysis of the Resurrection
narratives in i Corinthians, the Synoptical Gospels, the
Acts of the Apostles, the Mark appendices, the Fourth
Gospel and the apocryphal ones of Peter and the Hebrews.
The conclusion reached is that Paul and Mark's accounts
are historical, and the later ones exaggerated. Babylonian
and other resurrection theories are reviewed, and the book
ends with an allusion to Myers and psychical research.
F. C. Burkitt, of Cambridge, in placing the essay in a
l66 THE MONIST.
bibliography, says : "I introduce this book here as the first
example in original English work of the doctrine of the
priority of Mark being consistently applied throughout an
historical investigation." (The Earliest Sources for the
Life of Jesus, Boston, 1910, p. 129).
The method is that of the Lower Criticism, though the
Higher is also freely used. What I especially wish to
criticize is the following passage (pp. 61-65) which here
we must read in full:
"The young man at the tomb. — The account of what
the women saw at the tomb is contained in Mark xvi. 5.
Dependent narratives are found in Matthew xxviii. 2-5
and in Luke xxiv. 3-5.
"And entering into the tomb, they saw a young man
sitting on the right side, clothed in a white garment; and
they were astonished.
"As it stands in Mark, this account gives rise at once
to two questions: Did they see for themselves that the
grave was empty? and who was the young man who ap-
peared to them? Neither question is answered in Mark,
but before considering the bearing of this fact, it is first
necessary to ask whether the version given above repre-
sents the original text. According to it, the women entered
the tomb and found a young man seated within on the
right hand. No other meaning can be extracted from it,
or ever could have been, in the presence of the word eiaeX-
ftoijaai, 'entering into,' in verse 5 and the reference con-
tained in the corresponding e|eA$oi)aai, 'going out/ in
verse 8. But in case of neither of these words is the text
perfectly certain. The former is in the Vatican MS. weak-
ened to eAftovaai, 'coming/ while the latter is not repre-
sented in the Arabic Diatessaron, and in some MSS. is
altered to axowavrec;, 'having heard/ The weight of tex-
tual evidence is against these alterations, but on the other
hand transcriptional probability is in their favor. It is
THE TEXT OF THE RESURRECTION IN MARK. 167
unlikely that later scribes would have introduced changes
in the text which were calculated to weaken the evidence
for the belief that the women had made a complete exam-
ination of the tomb, and if these changes be made, the text
of Mark would leave it doubtful whether the women saw
the young man on the right hand of the inside or of the
outside of the tomb; for eA-ftouaai eig TO [wjneiov need not
mean more than 'when they came to the tomb.' Is it pos-
sible that this represents the original form of the narrative ?
In the absence of other evidence, it may not be ill-advised
to consider the evidence of a comparison with the two other
gospels, Matthew and Luke, which are closely based on the
Marcan narrative, and of the Fourth Gospel and the Gos-
pel of Peter, which follow it with greater freedom. It has
already been seen, in cases in which the Marcan document
is undoubtedly ambiguous or difficult, that the dependent
narratives adopted divergent methods of elucidating the
points at issue. It may therefore be allowed to reverse
this argument and see whether the dependent narratives
in the present case support the suggestion that the ground
document was ambiguous. They certainly seem to do so.
Matthew represents the angel, who is in his narrative the
equivalent of the young man of Mark, as seated on the
stone which he had just rolled away; he was therefore
regarded by Matthew as outside the tomb. It is equally
plain that Luke regards the two men, who in his narrative
represent the Marcan young man, as appearing within the
tomb. Furthermore, the Fourth Gospel and the Gospel of
Peter narrate that the women did not enter the tomb, but
stooped down and saw an angel or angels sitting within.
These two last accounts may quite well represent an attempt
at conflation between two traditions which differed, or
were not explicit, as to the position of the women and the
angel with regard to the tomb, and so far they support the
suggestion, which is rather strongly made by Matthew and
1 68 THE MONIST.
Luke, that the ground document was ambiguous on this
point. The weak point in this argument is that it does
not take account of the possibility that Matthew altered
the Marcan document owing to the influence of the story
of the watchers. It could be argued that the angel had
to be kept in the presence of the watchers and of the women,
and that the word cbreXftovacci, 'going from,' in verse 8 is
a proof that the ground document of Matthew contained
an account of an actual entry into the tomb. This is per-
haps not a convincing argument, but it may be taken as
practically balancing the previous one. It is impossible
finally to decide between the two. I think that the balance
of probability remains slightly in favor of the view that
the original Marcan document narrated the story of the
vision at the tomb in such a way, as not to state plainly
that the women entered the tomb, but I should not be pre-
pared to put emphasis on the argument."
I hope to show that there is every reason for Professor
Lake to emphasize the argument that the original text of
Mark did actually keep the women outside the tomb. We
may say does actually, for the original text of Mark can be
reconstructed from extant manuscripts and versions, with-
out any appeal to the Higher Criticism. In one case only do
we have to appeal to a lost source, but even this is supported
by a patristic quotation, and therefore belongs to the Lower
Criticism.
Let us begin with this lost source. Eusebius, in his
Questions of Marinus, Question i, which deals with the
absence of the Mark Appendix (Mark xvi. 9-20) from the
oldest manuscripts, says this:
"He who rejects the passage itself might say that the
story does not exist in all the copies of the Gospel according
to Mark; at least, the accurate ones among the copies de-
scribe the end of the story according to Mark in the words
THE TEXT OF THE RESURRECTION IN MARK. 169
of the youth who appears to the women, saying to them:
'Be not astonished; ye seek Jesus the Nazarene,' and so
forth. 'And when they heard, they fled, and said nothing
to anyone, for they were afraid of. . . .' For herein the
end is described in nearly all the copies of the Gospel ac-
cording to Mark, and what follows is seldom found in
any, but would not be superfluous in all, and especially if
they should contain a contradiction to the witness of the
rest of the Evangelists."
We may remark that "afraid of" is Kirsopp Lake's
own translation of the concluding words of the genuine
text of Mark, and it has been adopted by James Moffatt in
his splendid translation of the New Testament (London,
1913). But the words for which we have copied this
famous passage of Eusebius are : "when they heard" (dxoi)-
aaaai). Now it is known that Eusebius had access to the
library collected by Origen in the third century and ex-
tended by Pamphilus. Indeed Conybeare has made use of
this fact to delete the trinitarian formula and the baptismal
charge at the end of the Gospel of Matthew, in the teeth
of all existing manuscripts. He shows that Eusebius read
Matthew xxviii. 18-20 without these theological additions,
and places over against three thousand extant MSS., all
later than the fourth century, that other thousand, now
lost, which went back to the third and the second.
Applying this principle we can put in the forefront of
our textual evidence for dxovaccaai instead of eleXftovaai
the whole weight of the earliest Christian manuscripts.
The ungrammatical dxo-uaavreg quoted by Lake is from a
medieval manuscript in Russia, numbered 565 by Caspar
Rene Gregory in his Prolegomena to Tischendorf's Greek
Testament. Of course Eusebius gives the right reading,
dxovaaacu (feminine). Rallying to the support of this
ancient Greek original are the Washington manuscript
and the Old Syriac and Old Armenian versions, overlooked
I/O THE MONIST.
by Lake. Their testimony is very important; especially
the Armenian, for the Old Syriac and the Washington
Greek betray a transition stage which was tautological.
The "went out" was evidently interpolated before the de-
letion of the "having heard."
The following table will give a view of the process of
corruption. As Eusebius expressly tells us that the most
accurate MSS. omitted the Mark appendix, we need only
deal with those that do so. This gives us a sure criterion.
Six MSS. that omit this can therefore be pitted against
6000 that add it. To the trustworthy ones we may add
those which contain the spurious matter with a caveat, also
those which have a different ending from the current ap-
pendix. To these also must be added a few MSS. that
contain attestations of careful copying from Jerusalem
copies, such as No. 565.
Lost MSS. of the Early Centuries quoted by Eusebius.
[First clause not traced.]
And when they heard they fled and said nothing to any
one, for they were afraid of
(End of Mark.)
Armenian Version.
And entering into the sepulcher.
* * *
And when they heard, they fled from the sepulcher,
because they were terrified; and they said nothing to any
one for they were afraid.
Gospel according to Mark.1
* * #
Introduction to Luke.
1 The colophons here printed in bold-faced type are rubricated in the
original.
THE TEXT OF THE RESURRECTION IN MARK.
Frank Normart, of Glenolden, Pennsylvania, but a
native of Erzerum, has translated for me the passage from
the Old Armenian, as found in his own printed edition ( Con-
stantinople, 1895) and in a valuable manuscript owned by
John P. Peters (Bedrosian) of Philadelphia. (The colo-
phon is from the manuscript, for the Bible Society has
printed the Appendix, as in the King James version, with
a note accusing the Greeks for omitting it, but carefully
suppressing the fact that nearly all Armenian MSS. before
A. D. noo omit it also.)
Both in the Syriac and the Armenian this colophon is
rubricated.
Washington MS.
And entering into the sepulcher
* * *
And when they heard, they went out and fted from the
sepulcher, for fear and astonishment had come upon them,
and they said nothing to any one, for they were afraid of.
Now when he was risen early, on the first day of the week,
he appeared to Mary Magdalene etc.
[This is the earliest MS. that contains the Appen-
dix, which it has in an unusual form, hitherto only
partially known from a fragment in Jerome.]
Old Syriac.
And they entered into the sepulcher
And when they heard, they came forth and went away
and said nothing to any one, for they had been afraid.
ENDETH GOSPEL OF MARK.
172 THE MONIST.
The South Coptic
(Sahidic or Thebaic.)
[First clause wanting.]
* * *
And when they had heard, they came out of the sepulcher,
and they ran, for a trembling was laying hold on them,
and a confusion; and they said not any word to any one,
for they were fearing. But all the things which were
ordered them, to those who followed Peter they said them
openly. After these things also again Jesus was manifested
to them from the place of rising of the sun unto the place
of setting. He sent through them the preaching which is
holy and incorruptible of the eternal salvation. Amen.
But these also belong to them.
[Then follows the Longer Appendix, after a repetition
of the words at the juncture.]
The Vatican MS.
And coming unto the sepulcher
* * *
And they went out and fled from the sepulcher, for trem-
bling and astonishment had come upon them ; and they said
nothing to any one, for they were afraid of
ACCORDING TO MARK.
The Sinaitic MS.
And entering into the sepulcher
* * *
And they went out and fled from the sepulcher, for trem-
bling and astonishment had come upon them ; and they said
nothing to any one, for they were afraid of
Gospel according to Mark.
THE TEXT OF THE RESURRECTION IN MARK. 173
The Old Latin at Turin.
And when they had entered, they saw a youth etc.
* # *
But when they went out from the sepulcher, they fled; for
trembling held them, and awe by reason of fear.
But all things whatsoever that were commanded, those
also who were with the boy briefly explained. And after
these things Jesu himself appeared, and from the East
even unto the East (sic) he sent through them the holy
and uncorrupted [preaching] of eternal salvation. Amen.
Endeth Gospel according to Mark. Beginneth happily
according to Matthew.
The Ethiopic version also omits the Mark appendix,
while medieval MSS. L, and Nos. I and 209 show the doubt
about it; L gives both endings, like the South Coptic,
putting the shorter appendix first as we have seen already.
Nos. i and 209 say at xvi. 8 :
"In some copies, the Evangelist ends here, as far as
Eusebius the [friend] of Pamphilus, has placed his canons;
in others, there are found also these [words] :
"NOW WHEN HE WAS RISEN/' etc.
Several Fathers support the testimony of Eusebius,
so that the proof is overwhelming.
By restoring "when they heard" at xvi. 8, we get rid
of the eleAftooxrai, but this was introduced as correlative
to eiaeTiftovaoci : the two stand or fall together. If we had
not a single manuscript that read eMtowai at xvi. 5, the
Higher Criticism would bid us read it. But our very
oldest Greek authority reads it, plus an eleventh-century
MS numbered 127, together with the fourth-century Gothic
174 THE MONIST.
version. This reads atgaggandeins, "coming at," or. "going
unto." Lower Criticism therefore permits the restoration.
So we have unimpeachable ancient testimony that there
was no e|eA,dxyuaat at Mark xvi. 8 : the women did not flee
out of the tomb, because they had never been in it.
It is vain to protest that we cannot put three authorities
against three thousand that read eiaeA.fto'uaai, "entering
into"; for, by the laws of the Lower Criticism, we can,
not only on the grounds already given, but by a well-known
law of textual criticism. Dean Alford, in the critical ap-
paratus to his Greek Testament, gives us the reason:
"Received text, eioEA,ftovaai, from the parallel in Luke."
(Henry Alford, Greek Testament: New York, 1859, Vol.
I, P- 391-)
Nay, more : the Dean of Canterbury Cathedral did not
hesitate to put eMhwaai into his Greek text and to translate
it in his New Testament for English Readers (London,
1868, Vol. I):
"And when they came to the sepulcher," etc.
The principle upon which Alford did this is perfectly
sound. It is thus expressed by Jerome in his letter to Pope
Damasus, introducing his novel Vulgate edition of the
Gospels in the year 384 :
"Great error, if indeed it be (so), has grown up in our
codices, so long as what one Evangelist has said further
on the same thing, (the scribes), have added in another
because they thought it too little. Or, so long as another
expressed otherwise the same meaning, he who had read
first any one of the Four, considered that the rest ought
also to be amended to the pattern of that one. Whence it
happens that among us everything is mixed, and there are
found in Mark more things of Luke and of Matthew, and
again in Matthew more things of John and of Mark, and in
others of the rest things which are peculiar to others."
Here we have the Protestant reason, stated by the
THE TEXT OF THE RESURRECTION IN MARK. 175
prince of Catholics, for modern revision of the text. Al-
though we have not access to so many ancient manuscripts
as Jerome had, yet we have ancient versions neglected by
him, as well as two Greek codices of his own time and
several Latin ones. We are more justified in ceasing to
regard his work as final because he tells us himself that
he did not do it thoroughly:
"This short preface offers only the four gospels, the
order of which is as follows : Mathew, Mark, Luke, John,
amended by a collation of the Greek codices, but (only) of
old ones. Lest, however, they should differ much from the
accustomed form of the Latin reading, we have so re-
strained the pen that, when such things only were corrected
as seemed to change the sense, we suffered the rest to
remain as they were."
We are now in a position to reconstruct the end of the
Gospel of Mark, and to show that this most historic of all
the Evangelists never told a story about a corpse that got
up and walked off, but simply of some women who came
to a tomb and saw a strange young man. When they saw
him they were astonished, but when he addressed them they
were terrified and ran away. At this point the Gospel ends,
as we now have it. The reason for this abrupt ending requires
a separate discussion, such as briefly outlined in our pro-
logue or such as I attempted in The Open Court at Easter,
1910. Unfortunately I had not then read Kirsopp Lake, or
my attempt would have been better. (At the top of page
133 I made a blunder; line i should read: "Now this note
of doubt is Marcan, not Matthaean," etc.)
In giving the following text, several readings differing
from the common ones have also been given in addition
to those noted. Thus the phrase "on their right" in verse
5 is from the Sinai Syriac, and is all part of the reformed
readings, for it no longer smacks of the inside of the grave
where a youth or an angel was sitting on the right side of
176 THE MONIST.
the corpse. The word "daughter" is also from the Old
Syriac. The Greek has a blank here, which our common
translations supply by "mother." But a second-century
version in the language which Jesus spoke may be pre-
sumed to go back to authentic tradition.
Our reconstruction is guided by the Lower Criticism
alone. If we were to venture upon the dangerous ground
of the Higher, I should strike out the words : "He is risen ;
he is not here." These are lacking in important MSS. of
Luke, and as Luke used the first edition of Mark, he prob-
ably did not find them therein. But, as they appear in all
extant MSS. of Mark, the principles of the Lower Criti-
cism require that they should be retained. Higher Criti-
cism would also query the historicity of the spices and oint-
ment. Matthew says nothing about them, but tells us that
the women came "to see the tomb," and had no errand
inside. John expressly rules out the proceeding by an
elaborate embalming before burial. If Luke's eiaeMhwaai
could be copied into Mark, as Alford following the state-
ment of Jerome and the abundant witness of the manu-
scripts would have us admit, why should not Luke's oint-
ment and spices also have found their way into Mark in
very early times? But here again we are faithful to the
Lower Chriticism and insert the spices and ointment.
Another point. When Dr. Lake says: "eTiftouaai els
TO |j,VTi(i8iov need not mean more than 'when they came to
the tomb,' " does he not understate the case? Can we not
confidently say that they do not mean more? Thayer, in
his lexicon, long since pointed out that elg to |ivr]|ieiov in
John xx, the parallel passage to the one in Mark, means
simply "unto the tomb." In verse I, the Revised Version
renders the phrase : "unto the tomb" ; in verse 3, "toward
the tomb" ; in verses 4 and 8 "to the tomb." So too in John
xi. 31 and 38. In New Testament Greek, therefore, els TO
means "unto the tomb," and in order to introduce
THE TEXT OF THE RESURRECTION IN MARK. 177
the idea of entrance, Luke and the copyists of Mark had
to alter eXftouaai to eiaeXfto'uaai. Then, having gotten the
women into the tomb, they must be gotten out again ; hence
the correlative corruption of axovaaacti ecpvyov mto £§eX-
dovaai eqwyov, which in the Sinai Syriac is even more
tautological :
they went out and went.
This text is a conflation, for it has already given the
original reading known to Eusebius in his Csesarean manu-
scripts :
And when they heard.
In fact, a close study of the documents reveals the fact that
the whole passage has been systematically tampered with.
Mark xvi entire, as in the Oldest Manuscripts.
Revised Text.
230 And when the sabbath was past, Mary Magda-
VIII lene and Mary the daughter of James, and Salome,
brought spices, that they might come and anoint him.
231 And very early in the morning, the first day of
I the week, they came unto the sepulcher at the rising
of the sun. And they said among themselves: Who
shall roll us away the sepulcher stone? (for it was
exceeding great). And when they looked they saw
that the stone was rolled away. And coming unto
the sepulcher, they saw a young man sitting on their
right, clothed in a white robe; and they were bewil-
dered.
232 And he saith unto them: Be not bewildered; ye
II seek Jesus the Nazarene, who was crucified. [He is
risen; he is not here.] Behold, there is his place where
they have laid him. But go your way, tell his dis-
178 THE MONIST.
ciples and Peter : I am going to Galilee ahead of you.
There shall ye see me, as I have said unto you.
233 And when they heard, they fled, and said nothing
II to any one, for they were afraid of
HERE ENDETH THE GOSPEL OF MARK.
The numbered paragraphs are known as the Ammonian
Sections and appear in the Sinaitic manuscript and down
through the Middle Ages. Underneath the Hindu numeral
is the canon ascribed to Eusebius, which is numbered in
Roman. These canons represent an ancient Gospel anal-
ysis of no mean ability: Canon I means that the section is
common to all four Gospels; II, to the three Synoptists;
VIII, to Mark and Luke; X, peculiar to one; the rest to
different pairs.
RECAPITULATION.
From early manuscripts, versions and Fathers we can
reconstruct the text of the Resurrection in Mark, so as to
read, at xvi. 5, "coming unto the sepulcher," instead of
"entering into" it. Comparison of these authorities reveals
the fact that the text has been systematically corrupted in
early times by altering "coming unto" to "entering into,"
and "when they heard," in verse 8, to "going out of." This
has been done so as to make it appear that the women
found the grave empty, implying the doctrine of a fleshly
resurrection. By the Lower Criticism alone we are en-
abled to correct these corruptions and to show that Mark
originally contained no account of a physical resurrection.
The familiar problem of the lost ending of Mark is inci-
dentally dealt with.
ALBERT J. EDMUNDS.
PHILADELPHIA, PA.
BERGSONISM IN ENGLAND.
MBERGSON has told us that on the arena of Europe
• to-day we have a spectacle of "life" in arms against
"matter." He takes the two terms to express respectively
the spirit of his own nation and that of the enemy. We
may take them as expressing at least the alternatives be-
tween which his own philosophy moves. On the arena of
the universe as a whole life and matter are in conflict and
his philosophy seeks to decide between the two. Which is
the ultimate reality? Which is to be explained by the
other? Is life a product of matter or is the truth the other
way, has matter itself been created by life? Bergson's
works may be considered to constitute an elaborate answer
to this question, and to decide it in favor of the latter alter-
native. He thus takes rank as a champion of the living
against the dead. But the main position against which he
argues — what one could call roughly scientific naturalism
— is one with which philosophy in this country has long
been accustomed to deal, and with a like result. Now that
Bergson's philosophy may be said to have struck its first
roots in this country, the time seems opportune for a com-
parison. We might with profit compare what he has to say
with what the philosophy of the past two generations has
been teaching. From some points of view it seems as
though his special followers did not realize sufficiently the
likeness of the two ways of thinking. Nor do they seem
to have seen as yet how much some of Bergson's most
ISO THE MONIST.
valued positions seem to invite and indeed to demand
strengthening by reference to matters which have been
developed at least partially by the older idealism, and which
are absent from him.
The opportunity for making the comparison is ready
to our hand. As is well known, the head and front of
Bergson's English following has come to be pretty much
identified with Dr. H. Wildon Carr. While his work, The
Philosophy of Change,1 shows the influence of Bergson at
every turn, and while Dr. Carr does not conceal his convic-
tion that Bergson's is the most successful attempt yet made
to deal with the questions of metaphysics, still it would be
wrong to say that in this work he has merely been expound-
ing another man's thought. He makes the endeavor to
restate Bergson's central principle and to substantiate it
by fresh applications ; and a glance at the main topics will
serve, we believe, to give a fairly true view of one important
feature of Bergson's thinking which it is necessary to take
account of, if we would bring its relation to English
thought properly into focus. It will show us the kind of
questions which figure in Bergson's pages.
The recurring topic is matter and spirit and the prob-
lems which arise out of their relation. For instance, is
mind produced by the brain? If so, how could mind ever
get at what is outside of this brain? for apparently mind
can. reach not only what is outside of it but what is even
separated from it by immeasurable gulfs of space and time.
Or, what are we to make of the question as to the consti-
tution of matter, now that recent discoveries seem to re-
solve the material atom into something that is not a sub-
stance at all but only so much electricity ? Then there are
the problems connected with the relation of conscious mind
to movement. Why is it that a higher form of conscious-
ness, in the animal world, is so constantly accompanied by
1 Macmillan & Co., 1914.
BERGSONISM IN ENGLAND. l8l
an increased capability on the animal's part of choosing its
movements? There is further the nature of life itself.
How are we to construe the fact that the whole past of
every living being appears to be recorded in its present
structure? And again there is the law of evolution and
its ways of working. How is it that in a material world
supposed to be governed by mechanical law, evolution can
bend and govern the most dissimilar series of conditions
so as to produce a like result — as when (to cull an illustra-
tion from Bergson himself) the series of conditions which
produces a mollusk and the very different series which
produces a vertebrate animal should both alike end in the
one result, the endowment of the creature with an eye?
An outstanding characteristic of the whole way of
thought will be apparent from this cursory survey. Its
leading questions are such as would arise in the course of
the study of natural science. They are questions which
would occur to a scientist with philosophical interests. No
doubt the motive which impels the mind to raise them lies
far back in the perennial human sources of the philoso-
phizing habit. But the actual questions raised come straight
out of modern physics, mathematics, biology and psychol-
ogy. They don't arise out of a study of the history of phil-
osophical disputes. This is characteristic of Bergson. He
is preeminently" one of the writers who attack problems,
not other people's solutions of problems. Hence his philo-
sophical freshness. He does indeed deal with the history
of thought. He deals also with some of the standing con-
troversies of current philosophy. But these are not the
center of his interest. He does not begin with these. His
speculation thus acquires an interest for scientific minds
which most philosophy does not possess. And it is perhaps
not altogether fanciful to say that this feature gives his
thought from the outset a certain advantage with the Eng-
lish mind. Moreover, in Dr. Carr Bergson seems to have
l82 THE MONIST.
found a follower whose interests also are preeminently in
the concrete present. The central matter with him seems
to be, not what can we make of the systems of the past, but
rather what can we make of the report which the sciences
give us now of the world we are in; what is important is
that report and the right interpretation of it.
Fortunately, however, what we have just referred to as
the perennial sources of all philosophy — the great hypoth-
eses of our emotional nature — are touched upon by Dr.
Carr much less lightly and less fleetingly than by his mas-
ter. He even makes bold to adopt as a heading for one of
his later chapters the well-known words in which Kant
summed up the whole demands of our higher emotional
nature, the phrase "God, Freedom and Immortality." It
is a fortunate circumstance for the comparison which we
have in hand. In Kant we have the original source of that
general idealistic view in philosophy which has practically
held the field in English teaching until recently and with
which we wish to compare Bergson's. These postulates
will therefore give us a starting-point. We can compare
what light upon them has been derived from the older
sources, with what has been given to us by Bergson. And
we can compare the two philosophies further as regards
their ability to justify what they have to say on these things,
and give a reason for the hope that is in them.
Before we can consider what light Bergson throws
upon ultimate questions, or compare it with any derived
from elsewhere, we must first try to gain some rough idea
of his general view. It is clearly impossible to go into
detail. We cannot indicate Bergson's actual answers to
the questions we have cited above, still less his answers to
all the questions of which these are only a few taken at
random. But it may be possible to indicate his principle.
It may be possible to point out what in the universe Berg-
BERGSONISM IN ENGLAND. 183
son especially sees and values ; what it is which he believes
to be capable of providing a solution to these and all the
various problems with which he deals. It is in point of
fact nothing else than that elan vital which has figured so
often in the reviews of the new French philosophy, that
vital impulse which we behold forcing itself along the
whole course of evolution and of which in the long run
Bergson holds the universe itself to be the creation.
With this mere hint of the view, let us turn at once to
the question how it stands to the idealism which has been
taught in England and which finds its classical English ex-
pression in the writings of T. H. Green, — how it stands,
especially, as regards an attitude to problems which are
of the last importance for the human mind.
In the first place there are striking general resem-
blances. In Green's doctrine as in Bergson's the funda-
mental reality of the universe is not matter but conscious-
ness. Like the former view, Bergsonism professes never-
theless to be neither "idealism" nor "realism" simpliciter.
And up to a point it adduces the same reason for repu-
diating the former of these titles — for refusing, i. e., to
identify its teaching with any such "idealism" as is usually
associated with the name of Berkeley. The reason in both
cases is this. The reality of the world, though for both
views it is consciousness, is not for either of them anything
constituted simply by our private minds. No thesis is put
forward by either theory to the effect that you and I and
other minds like us are all that exist. The consciousness
referred to is in the literal sense universal. It is over all
the universe, a feature of the whole of it. No such doctrine
is put forward from either quarter as that the universe
which we usually see and know does not exist. What is
said is that it is conscious and is the product of its con-
sciousness. But, for Bergsonism, that self-creative con-
sciousness which the universe is differs from our private
184 THE MONIST.
minds also in another way, a way reminiscent of Schopen-
hauer as well as of the neo-Kantian idealism of Green.
That consciousness is not preeminently representative or
pictorial. It is active. Not the static picture of knowledge
is its characteristic expression, but the energy of will.
In consciousness so conceived, then, Bergsonism finds
the key to the broad facts of life and evolution as science
has revealed them to us. In the evolutionary history of
life on this planet — in the genesis and progress of vegetable
world, animal kingdom and man — what we have is this
active consciousness in the form of life, pushing itself, as
it were, through the surface of matter and seeking free
way. Man's physical organism is the one configuration of
matter through which it finds the free course which it seeks.
The human body is organized for giving outlet to this ac-
tivity. The brain and nervous system are but its cutting-
edge by means of which it thrusts itself forward. The
story of evolution is the story of how the main current of
this vital impulse has worked its devious way through
matter. The different forms of life which we see are the
different channels into which the central stream has split
itself up in process of thrusting itself into matter. The
central stream has not quite dissipated itself as yet into
these branches. The main current is still traceable. It is
found in the life and consciousness of man. It is for this
reason that man is at the head of creation. His life and
mind contain the most complete concentration to be found
anywhere of what was in the original world-impulse. The
fundamental reality of the universe, then, is life; but it is
a life which comes to view best, not in the plant or the
animal but in the conscious life of man. We must note
further that the "matter" through which the stream of life
thrusts itself is in the last resort its own creation, though
we need not go into Bergson's proof of that here. The
vital impulse is thus creative of matter and of all the forms
BERGSONISM IN ENGLAND. l8$
of life in which it finds an outlet, and the whole process
of its advance is named by Bergson "creative evolution."
Some such view is the only one capable, in Bergson's
opinion, of meeting the necessities of the case which nat-
ural science has presented to us. Dr. Carr has endeavored
to go further and show that as a general view it is specially
in harmony with some quite recent scientific discoveries.
The "vital impulse" is nothing if it is not movement. It is,
in fact, pure movement. If it be creative of "things" then
somehow things must be generated out of movement. And
this, Dr. Carr points out, is precisely what science is now
finding.
"The essential principle of the philosophy of change,"
he says, "is that movement is original. Things are derived
from movement, and movement is not a quality or character
that things have added to themselves."2 "A very few years
ago," he says again, "such a doctrine would have sounded
paradoxical and absurd. But now compare the philosoph-
ical doctrine of original movement with the new theories
of science. Let us take first the structure of the atom.
The electrical theory of matter teaches that the atom is
composed of a central mass or core, which is far the larger
part of its substance, and an envelope small in comparison.
The central core is positive electricity, and the outer en-
velope consists of negatively electrified particles held in
position by the electrical relation to the central core. The
atom, in fact, is a solar system in which the positive element
is the sun and the negative element the planets. And all
the qualities of atoms depend upon the arrangement of
these outer negative elements. But what is the ultimate
reality of this atom — something or other that is electrified ?
No, it is electricity, not something electrified, and electricity
is a form of energy, and energy degrades and disperses.
2 Philosophy of Change, p. 11.
l86 THE MONIST.
Reduced to simple everyday concepts it is this, that what
we call matter is a form of movement."8
But it is not merely in the case of the atom that recent
discoveries have tended to resolve into terms of movement
what we had been accustomed to regard under quite other
terms; elsewhere also they have begun to transform the
static into the changing, the resting into the active.
"But now turn to the other side," Dr. Carr continues
(pp. 17-18). "In the last few years it has been possible to
demonstrate that our solar system is not, as was supposed,
at rest in an absolute space or else moving, if it be moving,
without regard to forces outside itself. It belongs to a
larger system, all the parts of which are in movement in
relation to one another. The fifty million stars that our
telescopes reveal are not scattered at random over the
firmament, but are moving along regular courses coordi-
nated to one another. The members of this stellar system
are not, like the planets, revolving round a central mass,
but millions of suns are streaming across an unoccupied
center. The speed of our sun (now about 12% miles a sec-
ond) has been calculated, and its direction and the acceler-
ation it will undergo as it travels across the center and
passes outward again to the periphery. This, however, is
not all. A discovery has been announced that seems likely
to extend indefinitely further than astronomers have yet
imagined the vastness of the spatial universe. Observa-
tions which have been made on the great spiral nebula in
Andromeda show that its spectrum is inconsistent with
the hitherto generally held supposition that it consists of
gaseous matter in a state of extreme tenuity. It is now said
to be a spectrum that is given out by solid glowing masses,
and thus seems to confirm an old view that the nebulae are
star groups immensely distant. This nebula is apparently
not within our stellar system, but itself a vast stellar system
8 Ibid., pp. 16-17.
BERGSONISM IN ENGLAND. 187
lying outside the latter and at an enormous distance away
from it. What other systems lie outside these we do not
know, but all that we discover suggests universal move-
ment. There is no absolute rest. If we conceive an ob-
server placed anywhere in this great universe that we look
out upon from our position on an insignificant planet of an
insignificant sun, whether we suppose him to gather into
one embrace what to us are vast stellar systems or to be
confined to the negatively charged ion of an hydrogen
atom, there will stretch out for him on either side an unlim-
ited expanse of reality of which the ultimate essence is
movement."
And Dr. Carr finds a suggestion of the same point,
viz., that things are not more original than movement but
that movement is more original than things, in the way in
which the recent "principle of relativity" in physics threat-
ens to transform our conceptions of space and time and re-
move the ether from its place as a scientific hypothesis.
All this seems to him to confirm the view expressed by
Bergson in La preception de changement: "Movement
is the reality itself, and what we call rest (immobilite)
is a certain state of things identical with or analogous to
that which is produced when two trains are moving with
the same velocity in the same direction on parallel rails;
each train appears then to be stationary to the travelers
seated in the other." And again : "There are changes, but
there are not things that change; change does not need a
support. There are movements, but there are not neces-
sarily constant objects which are moved; movement does
not imply something that is movable."
The discovery that the whole universe is movement is,
however, very little, if we know nothing about this move-
ment except simply that it moves. Even when we have
brought it so far that we can regard this movement as life,
1 88 THE MONIST.
as creative, and as able at last to burst into consciousness,
we have not even then got very far philosophically unless
we learn something of the inner character of this vast
spiritual force. On its inner character too must rest our
judgment upon Dr. Carr's bold claim for the "new method,"
that it is nothing less than a revolution and that it has
reversed the direction that philosophy has followed
throughout its history of 2500 years.4 It is on the subject
of the inner character of this movement that Bergson's
teaching most directly challenges comparison with that of
Green ; and, we may add, most clearly demands to be sup-
plemented by it.
For Bergson is by no means the only teacher who has
conceived of a universal spiritual energy as sustaining the
universe. Green teaches the same. In the words of by far
the ablest existing short exposition of it, the central concep-
tion of his philosophy "is that the universe is a single,
eternal, activity or energy of which it is the essence to be
self-conscious."5 Nor can we get a distinction for Bergson
out of his repeated claim that the spiritual activity of
which he is thinking is not purely conceptual, because
Green in essentials makes the same claim. A great deal
is made by Bergson of the non-conceptual character of true
philosophical apprehension. You cannot apprehend that
ultimate essence or spiritual force whereby the universe
exists, in the ordinary way; that is, by the intellect. The
reason why, is that the intellect can only apprehend what
is dead, static, given. It cannot grasp living movement.
Now Green has quite as little use as Bergson has for what
can only grasp the given or static. Natural science, for
Bergson, is based on the intellect and that is why it cannot
conduct us into the presence of what verily explains things.
4 See Philosophy of Change, p. 20.
6 R. L. Nettleship in his biography of Green : Green's Works, Vol. Ill, p.
Ixxv.
BERGSONISM IN ENGLAND. 189
Science only sees in the universe what is dead, and there-
fore it cannot exhibit its ultimate spiritual essence. This
is Green's complaint too. Green, indeed, does not say that
we must appeal to something else than the intellect (this
is Bergson's way of putting the matter), but he does say
that we must understand the intellect. We must not be
content to use it uncritically, as natural science and naive
common sense do. We must lay hold of life and activity
with it. But Green is clear that the life and activity of
which the intellect must lay hold, is its own. His philo-
sophic creed, shortly stated, is that this is possible — that
intellect, itself an energy, will reveal a spiritual energy at
the heart of the universe, if it be persevered with and
rightly used. Green does not say so in these words. But
his philosophy says so to any one who has entered into it.
He insists on the one hand that reason and will are one,
in the sense that they are alike expressions of one prin-
ciple;6 and he speaks constantly of that principle as active
and as self-active. His phrase is "self-realizing." On
the other hand, his whole contention against the empirical
school in philosophy was that this self-activity, the essence
of men's minds, was not in men's minds alone. It was the
essence of the universe. The spiritual principle was "in
knowledge," and also it was "in nature," as the most ele-
mentary student of his chief work soon learns to know.
Practically all he had to say about nature in fact was just
this : that it was not inert, dead, merely given ; that it was
a spiritual life, of which our individual minds were the
highest finite manifestation. So far Green and Bergson
are on common grounds.
There is, however, a real divergence between Berg-
son's and the older teaching. They differ in their doctrine
of time. Both agree that what we can see around us with
6 See, inter alia, Works, II, p. 329.
THE MONIST.
the bodily eye is not the ultimate spiritual energy but its
manifestations only. Even with the eye of the mind, they
both hold, the ultimate spirit itself cannot be apprehended
in its whole nature, for only part of its original totality
is contained in the mind of man. But they differ as to what
it is in its own whole nature. With Green it is in itself
already perfect, whereas with Bergson it is still developing
and changing with the course of time and has an immense
and entirely uncertain future ahead of it. With Green the
ultimate spirit is complete, with Bergson it is incomplete.
With Green it is a consciousness, morally and intellectually
all that we could conceive ourselves becoming. With Berg-
son it is a consciousness still always turning into something
different and turning always into something which could
not have been predicted.
At this point also occurs the most marked difference of
the two doctrines in regard to the light which they cast
upon the assumptions of the moral and religious conscious-
ness. And as regards religion, it is not hard to judge
which is in the stronger position. So far as the religious
mind has entertained the belief that behind the phenomena
of the universe and acting as their source, there exists a
mind which is eternal, one who is above time and vicissi-
tude, who is perfect and is not subjected to change, a God
"who was and is and ever will be," in so far it will find its
faith countenanced in Green's teaching but discountenanced
in Bergson's. Dr. Carr himself is clear that the change
Bergson's theory invites us to make in our religious con-
ceptions is profound, though he thinks that it has com-
pensations.
"How is the conception of God affected by the principle
of this new philosophy ? One attribute that has seemed to
attach to this conception can certainly not belong to it —
eternity, in the sense of timelessness. Reality is essentially
movement, movement is duration, duration is change. If
BERGSONISM IN ENGLAND.
we call the original impulse of life God, then God is not
a unity that merely resumes in itself the multiplicity of time
existence, a unity that sums up the given. God has nothing
of the already made. He is not perfect in the sense that He
is eternally complete, that He endures without changing.
He is unceasing life, action, freedom.
"No more profound change can be imagined in the con-
ception of the universe, in the conception of human nature,
in the whole outlook of life, than is involved in this new
conception of God. The conception of God to which we
have been accustomed in philosophy, — the most perfect be-
ing, the ens realissimum, the first cause, the causa sui, the
end or final cause, — is the conception of a reality which
time does not affect. Hence the continual attempt both in
ancient and modern philosophy to conceive two orders or
kinds of existence, the temporal and the eternal, and the
whole problem of philosophy has been to conceive the rela-
tion of these two orders to one another. Time and the
whole order of changing reality must, it has seemed, be of
the nature of an emanation from God, or a manifestation
of God. But however conceived, the time order is regarded
as essentially unreal, appearance and not reality; change
and movement are relative to us."7
Connected with the same difference in regard to time,
there is again a difference of the two theories as regards
the light they throw upon another of the Kantian postu-
lates. So far as the religious consciousness has fixed its
hope on immortality in the sense of a life out of time Berg-
sonism can offer no corroboration, for time and the change
which constitutes it are to this philosophy reality itself,
and to be out of time is ipso facto to be out of existence.
Here again the older view is considerably different. For
it the idea of a life beyond time is at least not contradictory.
Nay, any completing or perfecting of our best life here
7 Philosophy of Change, pp. 187-188.
IQ2 THE MONIST.
would inevitably have this character, since for this view
we are already above time in so far as we think what is
true and do what is unselfish.
As regards the postulates of God and immortality, then,
the effect of Bergsonism is of a negative character. But
these are preeminently religious postulates. The point
upon which Bergsonism claims most confidently to have
substantiated our higher emotional demands is in regard
to the moral postulate, that of freedom. In its clearness
upon this question, indeed, Dr. Carr finds the chief com-
pensation for its attitude upon the others.
"The philosophy of change does not sound any clear
and confident note as to what lies beyond us in the unseen
world. It does not present to us God as the loving father
of the human race, whom He has begotten or created that
intelligent beings may recognize Him and find happiness
in communion with Him. There may be truth in this ideal,
but it is no part of philosophy. Neither does it teach us the
brotherhood of the human race — on the contrary it seems
to insist that strife and conflict are the essential conditions
of activity. Life is a struggle, and the opposing elements
are the nature of life itself, the very principle of it. The
evolution of life is the making explicit of what lies implicit
in the original impulse. Philosophy reveals no ground for
the belief in personal survival, and it shows us that 'how-
ever highly we prize our individuality we are the realiza-
tion of the life-impulse which in producing us has produced
also myriad other forms. What then is the attraction that
this philosophy exercises ? What is there of supreme value
that it assures to us? The answer is freedom."8
Here at length we reach the philosophically important
matter. For here we can interrogate the two views, not
merely as to whether they can corroborate our religious
8 Philosophy of Change, pp. 195-186.
BERGSONISM IN ENGLAND. 193
sense, but as to their grounds for doing so. The whole
question for the critical evaluation of the philosophy of
Bergson, it may be said, is that of the nature of and the
evidence for the freedom which he says characterizes that
ultimate spiritual force of which we are the offspring, and
which by its vast uprush through the universe and through
us creates us and the universe as it goes.
For Green too, as everybody knows, there is freedom.
And he puts the rationale of it thus. Man is free, for him,
both in his knowing and his acting, because in both of these
functions the past is gathered up in the present which is
now before him. Except this were so, says Green, we
could not know. To know, is to know succession. Now if
there were only succession itself — that is, if the past were
not thus gathered up — there could be no consciousness
thereof. This is straightforward reasoning, and at bottom
quite simple. If I am gathering a bunch of flowers, I must
hold the first ones in my hand while I gather the rest. If
I did not do this but dropped each one as I picked it, I
should never have a bunch. Quite similarly, if I hear or
see a succession, say, of strokes upon a knocker, and if I
know that it is a succession of five knocks, my knowing
is evidence sufficient that the earlier strokes have not es-
caped me but have been gathered up in my mind and pre-
sented along with the last one. If each had disappeared
as it occurred there would have been no succession of five
for me. Each one would have been number one; and
when it was over would have been nothing. To perceive
time at all I must not merely have the present before me.
I must have the past along with the present. In Green's
phrase, the various members of the series must be "co-
present" to consciousness.
Bergson has made an analysis of this same experience,
and has given the matter profound attention. He too sees,
that to know succession in the ordinary sense of knowl-
194 THE MONIST.
edge the members of the succession must be somehow co-
present, but he gives the whole matter another turn. He
cannot feel, apparently, that in knowing the successive as
thus co-present we are really knowing the successive at
all. His refrain therefore is, we try to know a time-suc-
cession by the ordinary use of our intellect, but cannot.
We do not, in this fashion, know a time-succession. We
only succeed in knowing space. In counting the strokes
we set out the series of events in a row, along a line, in a
kind of mental space. This we call perceiving their tem-
poral succession. And if one asks, "Why does the intellect
fail? how are we to apprehend time, or what would it be
like if we would apprehend it?" the whole argument of
Bergson's Time and Free Will converges in effect upon
this answer: that the intellect which fails to apprehend
time-succession fails because it can only set out the events
separately along an imaginary spatial line, whereas for
the "intuition" which really apprehends time these events
are not separate, they interpenetrate. This interpenetra-
tion is time. It is fairly easy to see further how, out of
the apprehension of such time, he gets free will. We have
to pull ourselves together in order to grasp this interpene-
tration; and in this attitude, in this tense summoning of
ourselves together, we are free.
We have here the fundamental impeachment of reason
to which Bergson's philosophy seems compelled to have re-
course. To reject the intellect as a means of attaining to
the truth is an obvious weakness, as compared with the
other view, thus far — that it is a species of self-subversion
which the view with which we are contrasting it does not
commit. Both Bergson and Green in philosophizing at all
are endeavoring to settle their account with the problems
of life by thinking them out. Both, in other words, are
making use of the intellect. The difference between them
in regard to the matter before us is that Green trusts the
BERGSONISM IN ENGLAND.
instrument he is using. Having found what the intellect
perceives time and succession to be, he says frankly that
that is what they are. But Bergson, unable to accept the
verdict, will rather make bold to say that our rational mind
is incompetent, that it is incapable of seeing things as they
are, and so has no authority in the case. This is a serious
matter. One cannot feel, after this, that the intellect can
be a very safe instrument to philosophize with. This is
perhaps the rock on which all philosophies eventually split
which attempt to reason the reader into preferring some
supra-rational or sub-rational power before reason itself.
Mere reason may not be fit to see what reality is; but if
not, is it fit to attack itself either ? We cannot endorse this
intellectual abuse of the intellect. If the intellect cannot
justify itself it cannot justify anything. We must accept
the intellect, or our whole attitude is sceptical.
"But the intellect can't allow you free-will," it will be
at once objected. This is an ancient objection, of which,
as we shall see, Bergson himself shows us how to get the
better. What, we have to ask, — what precisely is the free-
dom that Bergson's argument itself will bring us if it is
true? It is easy for Dr. Carr to speak as if Bergson pre-
served for us the privilege of a wide choice in an open
universe. All defenders of freedom have used such lan-
guage. The question is, what evidence has he? What is
there in our own experience that we can fall back upon and
see that the universe is open before us ? What reveals our
identity with a universal principle of freedom which creates
the universe itself, and in whose life we are free?
Whatever answer can be got out of Bergson to the
question must come from the "interpenetration" just men-
tioned. And on inquiry we find that it is a solid answer
enough. We do get evidence of freedom. And it is from
the "interpenetration" that we get it. Bergson is one of
the few people who see where the freedom issue really lies.
196 THE MONIST.
In Time and Free Will he insists that freedom is to be
looked for in the character of an act itself. It is the ques-
tion "what was the act?" that is essential; not the question
"what might it have been?" or "could it have been differ-
ent ?" What we have to ask about two alternative courses
of conduct ahead of us, when we want to know whether we
are free agents, is not "is either equally possible to me
now?" but "what is the inner character of the one chosen
when it does eventuate?" And he indicates, in language
which might have been copied from Green, that our charac-
ter must be in our act. "We are free," he says in Time
and Free Will,9 "when our acts spring from our whole per-
sonality, when they express it, when they have that in-
definable resemblance to it which one sometimes finds be-
tween the artist and his work. It is no use asserting that
we are then yielding to the all-powerful influence of our
character. Our character is still ourselves;" etc., etc. And
what we learn from his lengthy subsequent discussion of
the matter is simply this: that where "interpenetration"
occurs, there our character is; where the multiplicity con-
sciously present in us is made up of items which interpene-
trate, there our personality has its seat. And where the
multiplicity of interpenetrating states is at its maximum
in the great, critical decisions of our life, there our free-
dom is at its maximum because our personality is so. "It is
the whole soul .... which gives rise to the free decision ;
and the act will be so much the freer, the more the dynamic
series with which it is connected tends to be the fundamen-
tal self."10
It takes a great effort, often, to draw the scattered
multiplicity of our conscious states into this interpene-
trating unity. And in his later work, Creative Evolution,
Bergson tries to show that when this concentration of spirit
9 English Translation, p. 172.
10 Time and Free Will, p. 167.
BERGSONISM IN ENGLAND. 197
is relaxed an order of freedom transforms itself into an
external order of necessity. There is no disorder in spirit,
but only these two opposite kinds of order. That is how
he accounts for matter. It is the de-tension of the universal
life-impulse. But the present point is, that an act is free
when our personality is in it, and that happens when it is
one such as gives outlet or utterance to a multiplicity of
states held in an intense interpenetrating unity.
So far, Bergson conducts us along safe and solid
ground. But let us not make a mystery of this interpene-
tration. The highest examples of it are to be found only
rarely, no doubt. We find them in moments when the en-
tire being of a richly endowed mind, all its desires, fears,
hopes, knowledge, emotions, converge in one direction,
meditate one high and hard decision, and that decision is
taken. There you have that contracting together of the
entire soul for the effort, of which Bergson speaks under
so many similes, and which is perhaps the highest act of
a life. But there are simpler examples. The simplest is
the common experience we have already referred to — the
mere watching a series of events go by. The vague im-
pression left by the last "click" of a series to which we have
not been attending will tell us, says Bergson, (if we start
up afterwards and try to count how many we have missed)
when we have counted enough. In such a case the objects
we consciously count are set out in a sort of mental row.
Not so the vague impression which acts as our standard
and says to us when we have counted up, say, four, "that
is enough." This vague impression does itself contain
four. It is an impression of four. But it contains them
in a different way. In it they are not set out in a row, but
interpenetrate. Its "four" character, its quadruplicity if
you will, is a unique quality.
"Whilst I am writing these lines, the hour begins to
strike upon a neighboring clock, but my inattentive ear
198 THE MONIST.
does not perceive it until several strokes have made them-
selves heard. Hence I have not counted them. Yet I only
have to turn my attention backwards to count up the four
strokes which have already sounded and add them to those
which I hear. If, then, I question myself carefully on what
has just taken place, I perceive that the first four sounds
had struck my ear and even affected my consciousness, but
that the sensations produced by each one of them, instead
of being set side by side, had melted into one another in
such a way as to give the whole a peculiar quality, to make
a kind of musical phrase out of it. In order, then, to esti-
mate retrospectively the number of strokes sounded, I tried
to reconstruct this phrase in thought : my imagination made
one stroke, then two, then three, and so long as it did not
reach the exact number four, my feeling, when consulted,
answered that the total effect was qualitatively different.
It had thus ascertained in its own way the succession of
four strokes, but quite otherwise than by a process of addi-
tion, and without bringing in the image of a juxtaposition
of distinct terms. In a word, the number of strokes was
perceived as a quality and not as a quantity; it is thus that
duration is presented to immediate consciousness, and it
retains this form so long as it does not give place to a sym-
bolical representation derived from extensity."1
Now the freedom which Bergson secures, and which
he says cannot be apprehended by the intellect but only by
what he calls "intuition," is this interpenetration. The in-
tellect, he holds, cannot grasp it. But if we put aside his
statement that the intellect cannot grasp this unity of inter-
penetrating items, and attend solely to his description of
what the intellect is alleged not to be able to grasp, we find
that his statement is quite wrong. The intellect can grasp
it, and Green's doctrine is precisely that it can. True,
"interpenetration" is not a favorite word of Green's. He
11 Time and Free Will, Eng. Trans., pp. 127-128.
BERGSONISM IN ENGLAND. 199
speaks of relation. He holds that the members of a suc-
cession in order to be known to our minds as a succession
must be related; so related that they are co-present. But
this interrelation which Bergson says is a misreading of
time and a translation of it into mere "space symbolism"
because the members don't interpenetrate, this intellectual
apprehension of a succession, is already to Green precisely
a complex of interpenetrating elements. True, the items
are connected by relation, but relations are internal for
Green. They are constitutive of the thing's character. The
relations in which each thing stands to the others are what
make its nature. The nature of all the others, therefore,
enters into each, and that of each into all the others. They
must interpenetrate; their natures do so as truly and lit-
erally as two brushes which have been stuck together. The
fact is, it is altogether the same whether we say of certain
elements that their mutual relations are internal to each
of them, or that they penetrate one another.
"But this is not the interpretation that Bergson means,"
it will be replied at once. "This interrelation of Green's
would never yield anything like freedom. What Bergson
means is a vital interpenetration, not any dead static thing
such as could be illustrated by the mere material inter-
penetration of the bristles of two brushes." Entirely so.
The metaphor does not do justice to Bergson's position,
and neither does it to Green's. With Bergson the inter-
penetration of the elements seen by "intuition" is vital, it
is an intense living movement, and he strains language to
express how the elements fuse together, melt into each
other, inter-work and support a real life. But neither,
with Green, are the objects of the intellect in a dead rela-
tion. A relation, with him, is a relating — a living activity,
therefore. He has nothing to teach if he does not teach
this. He has nothing to urge if he does not urge that a
system of relations "implies a relating mind." And surely
2OO THE MONIST.
no one ever took him to mean by that, that the implied
"mind" merely made the system, set it down, and left it for
ever alone to stand there, dead, cold and finished. The re-
lations are alive. They are being kept up. They are a
deed; and not a deed done but a deed ever a-doing. A
relation of two things, with Green, is a supporting of them
in an energy of ceaseless spiritual movement, in precisely
the Bergsonian sense.
"But this movement constitutes the things, with Berg-
son; it is their source, the very stuff of which they are
made." Even so with Green, and much he has been made
to suffer for it ! It is not, says Bergson, things which are
first and which come to interpenetrate afterward. It is
the movement or interpenetration which constitutes the
things. It is not, says Green, things which are first and
which come to be related or interpenetrated afterward. It
is their relation or interpenetration which constitutes them.
A thing is nothing apart from its relations.
So far as regards the tracing of reality to a spiritual
source Bergson indeed uses a language which is different
from that of the older idealists. But in this general matter
his fundamental thought is accurately the same. The only
difference is that the older teaching does not fall back on
any special intuition in order to be assured that reality has
a spiritual source. It relies on the more thorough applica-
tion and the critical use of the intellect itself. It holds that
this most important of truths still is truth, and that by
those who persevere it may be reached by the same meth-
ods through which other convincing truth is reached,
namely, by the exercise of reason.
But this one difference is a difference as of heaven and
earth. By disparaging intellect it puts Bergson in the un-
happy position of constantly needing to discredit that very
faculty of "reasoning" upon which as a philosopher he
BERGSONISM IN ENGLAND. 2OI
must stake his own results; and that is not the whole
of the trouble. It also gives a false cast on the moral side
to the entire physiognomy of his teaching. And with a
glance at this we may close our review.
The significant point is that Bergson does not believe
in the intellect, or in the typical object of the intellect,
namely, space. By not believing in them we mean that he
does not believe in their spirituality. Green does. Green
finds in space itself that very interpenetration or spiritual
movement which Bergson insists cannot be found there. He
finds, that is to say, in the (spatial) object of the intellect
something which fully answers the essentials of Bergson's
description of the interpenetrating, while Bergson con-
stantly speaks of this character in things as though it could
not be seen at all intellectually, but only in glimpses, by the
special power of apprehension which he calls intuition.
Green, in a word, finds in the spatial-intellectual that reality
and truth which Bergson can only find when all "space-
symbolism" has been done away with. This is a serious
difference. For this "space-symbolism," in the wide mean-
ing which Bergson gives to it, is the very stuff and fiber of
the moral life. His teaching therefore means that to be
at the moral point of view is to be out of touch with the real
truth of the world.
And unfortunately his actual ethical teaching bears out
the suggestion. It is quite a mistake, we may note in pass-
ing, to say that Bergson has not written on ethics. It is
true he has not written any book with that name. But he
has a work the real burden of which is an interpretation
of the moral and social life. This is his little treatise On
Laughter. His thesis in that work is that laughter is a
species of social castigation. It is designed to rid society
of the conduct that provokes it. And the question for the
moral implications of Bergson's teaching is, what is it whose
destiny is thus to be socially castigated? Startling as the
2O2 THE MONIST.
answer may seem, it is the moral. It is called the mechan-
ical. In the wide sense in which Bergson eventually uses
the term, it is the intellectual-spatial. But in the concrete
what is it? It is simply faithfulness to principle where
such faithfulness is awkward. In other words it is the very
soul of the moral life, if that is anything at all distinct from
the "esthetic" life. This disbelief in space and the spatial,
this disbelief in the negation which is at the root of these,
is what the present writer has ventured to call the pessi-
mism of Bergson.12
Without repeating here what has been worked out else-
where,13 reference may be permitted to one little point in
elucidation of this view. It concerns Bergson's first illus-
tration in Laughter, his picture of the runner who stumbles
and falls. It is a small matter, of course, but it has always
struck the present writer as a peculiarly significant accident
that Bergson should have opened an essay On Laughter
by taking as his first example of the ridiculous precisely
that figure which has served so many moralists for their
type of the moral life. The runner of Bergson's illustra-
tion, as Bergson describes him, with his eagerness and his
"rigidness," with his omitting to look where he is going,
his stumbling over obstacles and his abundant inability to
adapt his conduct as circumstances require, and follow the
sinuosities of his crooked path, is indeed ridiculous. But it
is only Bergson's light vein that makes him so. There is
nothing essentially ludicrous about such a man. In essen-
tials, he might be Bunyan's pilgrim fleeing toward the
wicket-gate or St. Paul's runner, who also heeds nothing
12 See articles in The Hibbert Journal for October 1912, The International
Journal of Ethics for January 1914 and Mind for July 1913. Compare an article
on "Bergson, Pragmatism and Schopenhauer" by Giinther Jacobi in The
Monist, Vol. XXII, pp. 593ff. The latter article, however, should be read with
caution. The present writer has the best of reasons to believe that the mar-
velous correspondence in detail which exists between Bergson and the prince
of pessimists is largely accidental. Bergson himself learned about it only after
his own principles had been evolved into practically their mature shape.
18 In the article in The International Journal of Ethics referred to.
BERGSONISM IN ENGLAND. 2O3
either right or left, but simply "presses toward the mark."
Of course there would be nothing in a mere illustration
as such, but this one is so absolutely well chosen. This is
the type of man — this steadfast man, this man who just is
not sinuous and yielding and pliable and graceful and free,
this straight-going individual who cannot do anything but
go straight — this is the type whose proper destiny, accord-
ing to the whole tenor of the essay, is to be laughed out of
society ; this is the man for whom society has no use. "Since
when?" some may feel inclined to ask, not without a tinge
of indignation. We confess that to us, hitherto, society
has seemed to have considerable need for him ; nay, to have
had, perhaps, prodigiously little use for the other sort in
comparison.
Moreover it is the discovery of precisely what this social
theory neglects, namely the spirituality in spatiality itself,
that enables the idealist to endorse the religious conscious-
ness of God as eternal and perfect, without losing the other
point, equally important, that the divine nature must also
be movement, activity, freedom. To science the natural-
spatial world is a completed order. If such order implies
spirit, then, there must be a completed mind. As for the
compatibility of such completeness with freedom, the very
reasons which make Bergson to see real, active, free spirit-
ual life except in a present which has the past in it, make it
impossible for the idealist to see the perfection of such free-
dom except in a living present charged not only with the
whole past, but with the whole future as well. The whole
of reality must interpenetrate as Bergson makes the reality
which has so far elapsed do. That interpenetration, with
its inner activity, movement and freedom, makes up the
content of what the religious consciousness has conceived
as the perfect mind of God. Its inward intensity is God's
perfect life, which is also ours so far as we are both good
and great.
2O4 THE MONIST.
With the claim then, which is put forward by most of
Bergson's following here and elsewhere that his philosophy
is both true and "new," we cannot agree. So far as we
have been able to examine it, it differs from other idealism
in an essentially philosophical way only when it has some-
thing to say which is indefensible. Bergson has done im-
portant work in matters which in this paper we have had
to pass over because they are extra-philosophical. He has
done great work in psychology ; and he has also done great
work in the interpretation of the actual story of evolution,
by bringing out new facts there which could easily be
shown to be as compatible with the classical idealistic de-
fense of spirit as with his own. That kind of work is the
limit, it seems to us, of his service; except indeed it be a
service to have presented a great deal of the substance of
idealism from an angle so entirely fresh as almost to trans-
port the reader into the idealistic center of vision, without
his suspecting that he is there. We are not convinced that
this is a small service. Nay, rightly understood, there is
perhaps no greater.
J. W. SCOTT.
UNIVERSITY OF GLASGOW.
THE PRESENT STATUS OF THE UNCONSCIOUS.
^HE unconscious is a topic with which some writers
JL have tried to coquet freely, which others have shunned
scrupulously, and which still others have approached in a
true scientific spirit in the endeavor to find out precisely
what it is and how well it can explain the phenomena that
usually come under its name. The motives that have led
men to write about the unconscious have differed so widely
that it is not surprising to find the works of some fantas-
tical, and those of others useful for practical purposes only
but devoid of scientific information. The interests of the
former type have been purely metaphysical. Their object
being to discover unity and continuity in the universe, they
have postulated the unconscious as the absolute principle.
The interests of the latter, who are chiefly physicians, have
been entirely practical. Naturally they have considered
and still continue to consider the subconscious from the
functional point of view. What its real nature is and how
it is related to consciousness as such, is a problem that does
not fall within their sphere. The third group comprises
the few psychologists who, motivated by a true scientific
and progressive spirit, are seeking to discover the "what"
and the "how" of subconscious activities. As a result of
these diverse attitudes we find that one writer thought
the unconscious a topic sufficiently great and all-embracing
2O6 THE MONIST.
to deserve three large volumes, while another laconically
dismisses it in three monosyllabic words. But the fact
that we still have the problem on our hands shows that
neither the three volumes of von Hartmann nor the three
words of Miinsterberg1 have either brought us any nearer
to its solution or diminished in the least our unavoidable
duty as scientific psychologists to search out the cause of
unconscious activities and, if possible, to bridge the gap
between the conscious and the unconscious.
As an example of the psychologist who avoids all dis-
cussion of the subconscious, I need only mention Titchener,
who, after defining the subconscious as "an extension of
the conscious beyond the limits of observation,"2 goes on to
say that it is always >a matter of inference and therefore
"it can not be a part of the subject matter of psychology."
It is merely employed as an explanatory concept, he de-
clares, but there are two reasons against its use in psy-
chology: First, that the scientific psychologist, like the
scientist in general, is not called upon to explain anything ;
and secondly, the introduction of this inferential concept
may lead to danger inasmuch as it is "impossible to draw
the line between legitimate and illegitimate inference."
As an example of the thinker who interprets all mental
phenomena in terms of the subconscious, I may mention
von Hartmann and Schopenhauer. The former endows
the entire universe with an unconscious mind, declaring
it to be the absolute principle which operates in all things
organic and inorganic. But as James says, "his logic is
so lax and his failure to consider the most obvious alter-
native so complete, that it would .... be a waste of time
to look at his arguments in detail." Nor are the views of
Schopenhauer much more reasonable. According to him
every sense organ unconsciously infers its impinging stim-
1 Psychotherapy, p. 125.
2 A Beginner's Psychology, p. 327.
THE PRESENT STATUS OF THE UNCONSCIOUS. 2O7
ulus "as the only possible cause of some sensation which it
unconsciously feels."3
But the theory of the subconscious dates back farther
than von Hartmann and Schopenhauer. Although Wein-
gartner traces it to Plato and Plotinus, we may say that it
received its first definite formulation at the hands of Leib-
niz in his conception of the petites perceptions which play
the main role in psychic activity. These subliminal per-
ceptions are individually too faint to arouse consciousness,
according to Leibniz, but in their totality they come to a
high degree of consciousness. To use his own words, "the
belief that there are no other perceptions in the soul than
those of which it is conscious, is a great source of error."
Kant's view of the subconscious is somewhat analogous
to that of some modern authors, particularly Lipps. He
declares: "To have sensations and not to be conscious of
them is a contradiction, for how can we know that we have
them, when we are not conscious of them? But we may
infer we have had a sensation or a perception, although
we were not immediately aware of it."4 Such perceptions
Kant calls "vague," and their field he declares is much
broader than that of the clear and definite ones.
Turning to the English school, we find Sir William Ham-
ilton asserting that although he does not wish to maintain
that all consciousness is the product of unconscious percep-
tions, and that knowledge as such is the product of the un-
known and the unknowable, still we must confess, he says,
"that there are things which we neither know nor can know
directly, but which manifest their existence indirectly
through the medium of their effects."6 Hence, since the
mind in its behavior manifests processes of which it is un-
conscious, these processes must have come about through
3 James, Psychology, I, p. 170.
4 Soewenfeld, Bewusstsein und psychisches Geschehen, p. 2.
5 Carpenter, Principles of Physiology, p. 518.
2O8 THE MONIST.
some modification of mind, and this may be called the un-
conscious. Or as Carpenter believes, Hamilton meant by
this "unconscious cerebration."
Maudsley undertakes to interpret the greater part of
conscious behavior in terms of unconscious psychophysical
processes. He observes that almost from the moment of
birth the sensorium receives multifarious impressions which
it assimilates unconsciously, and makes use of them in a
purely mechanical manner even in so-called intelligent ac-
tivity. Even our general and abstract concepts are de-
veloped unconsciously; in short, "the process upon which
our thinking depends," he says, "goes on of its own accord,
without our awareness."
Carpenter devotes a whole chapter to unconscious cere-
bration and declares that since there is reason to believe
that the greater part "of our intellectual activity" — whether
it be reasoning or imagination — is essentially automatic,
it is not unlikely that "the cerebrum may act upon impres-
sions transmitted to it, and may elaborate intellectual re-
sults, such as might have been attained by the intentional
direction of our minds to the subject, without any con-
sciousness on our own part" (p. 515).
This view was subsequently taken up by Huxley and
made to explain all intellectual activity. Noticing that
epileptics can execute complex actions without having any
memory of them upon recovery, and also that somnam-
bulists can write letters and compose original verse while
in their so-called sleeping state, he concluded that since
"these cases are examples of purposive and intelligently
controlled action taking place without consciousness, it
would seem to follow that the mere mechanism of the
nervous system" is all that is needed for the execution of
such actions, independently of all consciousness and con-
scious guidance; and, therefore, we are compelled to as-
sume that when similar actions are accompanied by con-
THE PRESENT STATUS OF THE UNCONSCIOUS. 2OO,
sciousness, the nervous mechanisms are the only essential
conditions, and "consciousness is a superfluous accom-
paniment, so far as the causal sequence is concerned."8
It is obvious from the above that not only do earlier
writers disagree on how the subconscious functions but
they even differ with respect to its essential nature.
Turning to modern authors we find that among them,
too, there are almost as many views as writers on the sub-
ject. On the one hand, men like Freud declare that sub-
conscious phenomena are due to dissociated and suppressed
ideas; that these unconscious ideas are active, though the
individual may not be aware of them while going through
the bodily actions of which those ideas are the prime
causes.7 On the other hand, Sidis retorts that the existence
of unconscious ideas is inconceivable, for "ideas are essen-
tially of a conscious nature" f hence their introduction into
psychology is a self-contradictory concept. The subcon-
scious, according to him, is rather "a diffused consciousness
below the margin of personal consciousness."9 Again,
writers like Prince and James conceive of the subconscious
as the outerlying fringe of consciousness, as dim conscious-
ness, or better still, as the base of a cone, the apex of which
is attentive consciousness. While Irving King refutes this
view, declaring that consciousness either exists or does not
exist, that "it may be more intense at one moment than at
another .... But at any one moment it is .... a unitary
existence without parts which might be thought of as clus-
tering about a center with different degrees of intensity
and adhesion."1 Finally, while Sidis mocks unconscious
cerebration, characterizing nerve currents, nerve-paths and
"McDougal, Body and Mind, pp. 109-110.
7 Freud, "A Note on the Unconscious, "Proc. Soc. Psy. Res., XXVI, 1912,
p. 314.
8 Sidis, "The Theory of the Unconscious," Proc. Soc. Psy. Res., XXVI,
1912, p. 337.
9 Ibid., p. 319.
10 "The Problem of the Subconscious," Psychol. Rev., XIII, 1906, p. 43.
2IO THE MONIST.
neurograms as "figments of imagination/'11 Ribot main-
tains that the psychological aspects of the subconscious
play but a secondary role, that they are a result, an effect
of physiological or neural processes.12
No small part of the above controversy and disagree-
ment is due to the fact that the term subconscious has been
used in widely different senses. Prince gives no less than
six different meanings in which the term has been em-
ployed.
1. The word subconscious has been employed to describe
that portion of our field of consciousness which at any
moment is outside the focus of attention. In this sense it
is equivalent to James's fringe of consciousness.
2. The second meaning asserts that the subconscious is
composed of ideas that are dissociated or split off from
the personal consciousness, i. e., the focus of attention;
that though the subject is unaware of their existence they
are none the less active, and that "they form a conscious-
ness coexisting with the primary consciousness, and thereby
a doubling of conscious results."
3. According to the third meaning of the term, "sub-
conscious states are conceived of as becoming synthesized
among themselves, forming a larger self-conscious personal-
ity, to which the term self is given." These subconscious
states are personified by the people who hold this view,
and referred to as the "subconscious self" or "the hidden
self."
4. The fourth view conceives the subconscious as "in-
cluding all those past conscious states which are either for-
gotten and cannot be recalled, or which may be recalled as
memories," their non-existence being due to the fact that
they are crowded out of consciousness by the bulk of pres-
ent experience.
11 Op. cit., p. 325.
12 "A Symposium on the Subconscious," Journ. Abnorm. Psychol., II, 1907,
p. 37.
THE PRESENT STATUS OF THE UNCONSCIOUS. 211
5. The fifth view, which is that of Frederick Myers,
"declares that subconscious ideas, instead of being mental
states dissociated from the main personality, are the main
reservoir of consciousness, and the personal consciousness
is a subordinate stream flowing out of this great storage."
In short, we have within us a great tank of con-
sciousness, but are aware of only a small portion of it.
6. The sixth and final view asserts that there are no
psychical elements in subconscious phenomena at all, that
automatic writing and speech, the solution of mathematical
problems in sleep, and the carrying out of post-hypnotic
suggestion "are the result of pure neural processes/' un-
accompanied by any mentation whatever.13
Before presenting the detailed arguments in support of
the above theories, let us hastily review some of the more
common subconscious phenomena in order that we may
have freshly before our minds the facts which these theories
endeavor to explain.
The main test that a by-gone experience was accom-
panied by consciousness is memory.14 The ability to recall
an experience without the artificial aid of suggestion or
abstraction, shows that the individual was conscious of
that experience at the time he underwent it. But memory
is composed of three factors : registration, conservation and
reproduction. Something must be impressed on the sen-
sorium in order to be recalled, and it must also be conserved
in some form. The question therefore arises: Does every
impression, however faint it may be, stir up a pulse of con-
sciousness which is immediately forgotten because of its
brevity or faintness, or can reproducible impressions be
made without the least awareness at the time being? And
if so, how are they conserved ? Daily observation and lab-
oratory experiments demonstrate that perceptions of the
18 "A Symposium on the Subconscious," Jour. Abnorm. Psychol., p. 22.
"McDougal, Of. tit., p. 109.
212 THE MONIST.
environment of which the individual did not have the least
awareness, may be conserved. You may pass an acquain-
tance on the street without being aware of him at the time,
but two or three minutes later it will suddenly dawn on
you that you had seen your friend so and so. Again in
hypnosis, by means of automatic writing or abstraction,
people have been able to recall paragraphs in the news-
papers read through casual glances without awareness
thereof. Or the experiment may be put under controlled
conditions, by having the subject take a brief survey of
the room, and then while blindfolded dictate as detailed a
description of it as he can. Thereafter if he is hypnotized
and asked to describe the room once more, "it is often quite
surprising," says Morton Prince, "to note with what detail
the objects which almost entirely escaped conscious ob-
servation are subconsciously perceived and remembered."1
Another method of proving the conservation of uncon-
scious experiences is to have a person concentrate his atten-
tion by giving him something to read or an arithmetical
problem to perform, and while he is so engaged to place
cautiously and surreptitiously objects within his peripheral
field of vision. After their removal he is asked to state
in detail what he has seen. Invariably he is unable to men-
tion any of these surreptitiously introduced objects. On
being hypnotized, however, he mentions them with con-
siderable accuracy and readiness.16
Automatic writing furnishes another group of facts
which presuppose subconscious processes. If into the an-
esthetic hand of an hysterical person a pencil be put the
hand will commence to write mechanically, and the subject
will observe the movements of the hand as if that member
belonged to some other person. Nor will the patient rec-
ognize the written ideas as his, but if he is hypnotized he
15 Prince, The Unconscious, p. 53.
« Ibid.
THE PRESENT STATUS OF THE UNCONSCIOUS. 213
will claim them immediately and explain what he meant
by them. Sometimes the two hands of the same subject
may be made to give written expression to two different
kinds of mental content.
Perhaps the most interesting and common source of
subconscious phenomena is somnambulism. People in this
state have been known to perform the most delicate feats
of physical skill, such as walking across roofs on narrow
planks. Others have been known to perform- events that
in waking life require a great deal of intelligence, — such
as writing letters or verse. Yet they have no memory for
these events. The question arises : Are these highly com-
plex mental activities performed mechanically, without any
mentation, or are they consciously performed, but forgotten
in waking life because dissociated from the personal con-
sciousness ?
Post-hypnotic suggestion is no less a mystery than com-
plete change of personality. An individual is hypnotized
and is told that at a fixed time after he awakens — be it sev-
eral minutes, an hour or a day later — he is to do a certain
deed. He is awakened and asked if he remembers any-
thing that had been said to him during the hypnosis. He
does not. He is permitted to depart and goes about his
business in his customary manner. But precisely at the
fixed time he will carry out the post-hypnotic suggestion,
whether it be to ask for a pail of coal in a jewelry store or
to purchase an overcoat in summer. When he is asked why
he did this he can only reply that something within
prompted him to it, that he felt it was a voluntary deed.
Whether such a case as that of the Rev. Ansel Bourne
would fall into the group of epileptic phenomena or not
matters little. In both instances we know that the subject
will go through many complicated activities, denoting a
high degree of consciousness or the presence of the cus-
tomary kind of intelligence as judged by the adaptation of
214 THE MONIST.
the subject to his environment, yet in neither case does the
normal personality have a memory for these experiences.
How are these phenomena to be explained in the light
of modern psychology?
Two general theories are proposed: the psychical and
the physiological. And it is to these two that the six fore-
going views can be reduced after we eliminate Myers's
metaphysical notion which conceives of the subconscious as
the reservoir of all consciousness, and that other view which
interprets the subconscious as the larger self-conscious per-
sonality.
Freud, Sidis and Janet may be taken as the chief ex-
ponents of the psychological theory of the subconscious,
while Pierce, Jastrow and Ribot, not to mention a host of
others, hold to the physiological view. The former trio in
one form or another declare that the subconscious is dis-
sociated consciousness, or awareness that is dissociated
from the synthesizing personality, and that this aware-
ness exists in consciousness in a latent form all the time.
The latter maintain that not only is it unscientific to speak
of latent ideas and latent feelings, but that there is no
causal relation among psychic elements at all, that the
explanation of unconscious phenomena must be sought in
neural processes.
Let us examine their views individually.
Freud suggests that the term "conscious" should be
applied to the perception which is present to our conscious-
ness and of which we are aware, — while the latent per-
ceptions should be denoted by the term "unconscious."
"Hence an unconscious idea is one of which we are not
aware, but the existence of which we are nevertheless ready
to admit because of other proofs or signs."17 This un-
conscious idea, though latent in the sense that it does not
attain awareness, is by no means inactive while in the
" Op. tit., p. 3i3ff.
THE PRESENT STATUS OF THE UNCONSCIOUS.
mind. That unconscious ideas are active, undergoing com-
bination and recombination among themselves, is demon-
strated by the hysterical patient. "If she is executing the
jerks and movements constituting her fit," says Freud, "she
does not consciously represent to herself the intended ac-
tions, she may perceive those actions with the detached
feelings of an onlooker. Nevertheless, analysis will show
that she is acting her part in the characteristic reproduc-
tion of some incident in her life, the memory of which was
unconsciously present during the attack."
Freud distinguishes two kinds of latent ideas: those
which enter consciousness with no difficulty whatever, and
those which do not penetrate into consciousness however
strong they may be. The first type constitute the fore-
conscious, the second type the unconscious. "The term
unconscious," he says, "now designates not only latent
ideas in general, but especially ideas with a certain dynamic
character, ideas keeping apart from consciousness, in spite
of their intensity and activity." In explaining the phenom-
enon of double personality Freud would say that it is a
shifting of consciousness, an oscillation between two dif-
ferent psychical complexes which become conscious and
unconscious alternately.18
But the question still remains : Why does foreconscious
activity pass into consciousness with no difficulty, while an
unconscious activity is cut off from consciousness? (It is
to be noticed here that he no longer speaks of foreconscious
and unconscious ideas, but replaces the word idea by the
term activity.) In answering this question he says that
frequently when we try to represent an idea or a situation
to ourselves we become aware of a distinct feeling of re-
pulsion which must be overcome ; and when we try to inject
such an idea into a patient, we get the signs of what may
be called his resistance to it. "So we learn that the un-
i*Ibid., p. 315.
2l6 THE MONIST.
conscious idea is excluded from consciousness by living
forces, which oppose themselves to its reception ; while they
do not object to other ideas, — the foreconscious ones." At
the present state of our knowledge, therefore, he suggests
the following as the most probable theory that can be
formulated: "The unconscious is a regular and inevitable
phase in the processes constituting our psychical activity;
every psychical act begins as an unconscious one, and it
may either remain so, or go on developing into conscious-
ness, according as it meets with resistance or not." Freud
illustrates this view by referring to ordinary photography.
The first stage of the photograph is the "negative" ; every
picture has to pass through the negative process ; and those
negatives which on examination prove to be satisfactory
are admitted to the positive process, ending in the picture ;
those which do not are rejected. Such is the distinction
between the foreconscious and unconscious ideas or activ-
ities. In reply to his critics that an unconscious idea is
inconceivable, he declares that "the existence of an un-
conscious consciousness is still more objectionable."
Sidis gives three definitions of the subconscious which
may be called the medico-popular, the metaphysical and
the scientific, respectively. In one place he defines the sub-
conscious "as mental processes of which the individual is
not directly aware." In another place he refers to it "as a
diffused consciousness below the margin of personal con-
sciousness"; and on a third occasion he defines it "as con-
sciousness below the threshold of attentive personal con-
sciousness."
The subconscious like the conscious may be, according to
Sidis, of three types : desultory, synthetic, or recognitive.20
Sidis would almost banish the term subconscious from
literature, and what is commonly called subconscious he
19 The Theory of the Unconscious, p. 319.
20 Psychology of Suggestion, p. 201.
THE PRESENT STATUS OF THE UNCONSCIOUS. 217
would call conscious, while that which is commonly known
as the conscious he would call the self-conscious. The self-
conscious is that form of mentation which is aware of itself ;
it is "the knowledge of consciousness within the same mo-
ment of consciousness." and in that sense it is identical
with personality,21 On the other hand, the secondary or
subconscious self must not be regarded as an individual;
"it is only a form of mental life"; it is a coordination of
many series of moments-consciousness," — i. e., pulses of
consciousness. And it is these moments-consciousness that
are at the heart of the subconscious. Therefore, subcon-
scious experience is not wn-conscious experience. The proof
is this: Normal memory is a reproduction of conscious
states. Now, when a subject is hypnotized he can be made
to recall an experience which he does not remember in his
waking state; and in this he displays memory like normal
memory. Therefore, we have proof that his experience
was accompanied by consciousness at the time it occurred.
Or, to use Sidis's own words, "that in subconscious states
there is really present a subconscious consciousness."22
It is to be noticed that this is not the same thing as
saying that the ego or the personality was aware of that
experience, but on the contrary, there was an awareness
of which the attending self had no consciousness.
Having eliminated the subconscious from literature,
therefore, there are only two forms of awareness to be
considered, according to Sidis: consciousness as such and
self-consciousness. The difference between these two states
may be made clear in the words of Hoffding. "Many feel-
ings and impulses stir within us, without our clearly ap-
prehending their nature and direction. A man who has
this feeling does not know what is astir in him; perhaps
others see it, or he himself gradually discovers it; but he
« ibid., P. 198.
22 The Theory of the Unconscious, p. 331.
2l8 THE MONIST.
has the feeling that his conscious life is determined in a
particular way."23 What Hoffding means is that there are
"mental states of which we have consciousness, but which
do not reach the personal consciousness." This is the dis-
tinction that Sidis makes between the subconscious and the
self-conscious.
It naturally follows from the above, and there are many
facts in support of the conclusion, that "the stream of sub-
waking consciousness is broader than that of the waking
consciousness, so that the submerged subwaking self knows
the life of the upper, primary self, but the latter does not
know the former." He admits, however, that there are cases
on record showing that the two streams may flow in sep-
arate channels ; that the two selves may be ignorant of each
other.24
On the basis of the foregoing view, the phenomenon of
double personality is not difficult to explain, thinks the
author. When a sufficient number of the submerged mo-
ments of consciousness have accumulated they tend to be-
come synthesized, to group themselves in constellations and
break forth into attentive consciousness, as do hallucina-
tions, for example. In this manner the secondary con-
sciousness attains self-consciousness, and appears as a new
and independent personality. Now and then it "rises to
the surface and assumes control over the current of life."
This secondary self is aware of and passes judgment on the
primary self, while the latter, when it returns, has not the
least knowledge of the intruding ego.
It is apparent that the views of Freud and Sidis are
essentially the same. The argument, therefore, that exists
between these two writers is purely verbal and meaningless.
There is no fundamental difference between an unconscious
idea and an unconscious moment-consciousness, or even an
23 Quoted by Sidis in The Theory of the Unconscious, p. 339.
24 Psychology of Suggestion, p. 198.
THE PRESENT STATUS OF THE UNCONSCIOUS. 219
unconscious consciousness. There may be a difference in
quantity but not in quality. Yet we find Freud declaring
that if philosophers find it difficult to accept the existence
of unconscious ideas, the existence of unconscious con-
sciousness is still more objectionable. To which Sidis re-
torts: "An idea is essentially of a conscious nature. To
speak, therefore, of unconscious ideas is self-contradictory,
— it is equivalent to the assumption of an unconscious con-
sciousness."2 I do not see why Sidis should find fault with
this conclusion, since it is the very assumption with which
he opens his own discussion on the theory of the subcon-
scious. There he defines the subconscious as mental proc-
esses of which the individual is not aware. But what are
mental processes if not ideas, images and perceptions?
His definition, therefore, turns out to be precisely the same
as Freud's.
Though the views of neither of these men lend them-
selves to acceptance in the light of the fundamental postu-
late of psychology, — namely, that every psychosis has its
neurosis (but not the reverse), still Freud's doctrine of the
subconscious is somewhat more palatable than that of Sidis.
At least it is capable of interpretation in terms of our
existing knowledge of neurology; it does not assume too
much and does not pretend to offer a solution of all mental
phenomena. The view of Sidis, on the other hand, is en-
tirely out of harmony with the fundamental postulate of
psychology, and it is so all-embracing and metaphysical
in nature as almost to remind one of the teachings of von
Hartmann.
This is demonstrated by the vigorous but wholly un-
justifiable attack that Sidis launches against the theory of
unconscious cerebration. This doctrine, it will be recalled,
states that physiological processes may go on in the sen-
sorium which enable the organism to adapt itself to its
28 The Theory of the Unconscious, p. 337.
22O THE MONIST.
environment without any consciousness on its part. If this
is so, says Sidis, then there is no reason why similar adap-
tations which are accompanied by consciousness should not
also be purely mechanical and automatic. If the writing
of letters during somnambulism is automatic, then the cor-
respondence of waking life must be carried on in the same
manner. But, he asks, "Can unconscious physiological
processes write rational discourse? It is simply wonderful,
incomprehensible." Assuming that every sense impression
leaves behind it a trace, or a slight modification of nerve
tissue, he says, still this does not explain why it is that a
series of sensations, ideas, and images experienced at dif-
ferent times "should become combined, brought into a unity,
felt .... like copies of one original experience."2 Conse-
quently the subconscious must be considered not as "an
unconscious physiological automatism," but as "a secondary
consciousness," as a secondary self.27
It is doubtful whether the theory of unconscious cere-
bration can account for the whole of unconscious phenom-
ena, but there is no doubt that Sidis's notion does not
account for even a fraction of it. The weakness of his
logic is seen in such passages as the following : "Reactions
to environment accompanied by intelligence in us are rightly
judged to have the same accompaniment in others." From
which, of course, he would have us draw the conclusion that
since we guide our footsteps on the crowded street, or
build a fire, with some degree of waking consciousness or
intelligence, therefore the stroller who is absorbed in his
newspaper or the somnambulist who builds a fire is also
guided by awareness. This conclusion would be correct,
provided the proposition on which it is based were not re-
versible. But it is reversible. It is precisely because we
perform many so-called intelligent actions (as judged by
26 Psychology of Suggestion, p. 125.
27 Ibid., p. 128.
THE PRESENT STATUS OF THE UNCONSCIOUS. 221
their end product) without any consciousness in our normal
life, that we rightly claim such actions to be devoid of in-
telligence or active consciousness in other beings when
performed under the same conditions, or when those beings
are abnormal. The above proposition, therefore, stands
incomplete without its complement, which says with equal
right: Reactions to environment not accompanied by in-
telligence and attentive consciousness in us are rightly
judged to be devoid of these accompaniments in others,
especially when those others can give no direct testimony
as to the presence of consciousness.
Let us take an instance of so-called intelligent action
which is accompanied without consciousness so far as mem-
ory can testify, and see whether it must be explained only
on the basis of unconscious-consciousness, or whether a
better explanation cannot be found. The case of the per-
son who, though absorbed in his magazine, still picks his
way through the crowded thoroughfare will do quite well.
Now two wholly unrelated streams of thought cannot oc-
cupy the same mind at the same time. To be sure, we may
dream and know that we are dreaming, or dream and ex-
perience a desire to wake up ; or experience both the music
and the color effect of an opera at the same time ; but these
are somewhat related mental complexes: at least they are
logically related. We certainly can not solve mathematical
problems and at the same time think of our social engage-
ments. Suppose, then, we assume that our hypothetical
person is strongly conscious of his reading material only,
and is oblivious to the people on the sidewalk. How shall
we explain his ability to pick his way through the crowd?
The process may be described thus : Two sorts of stimuli,
diverse in nature, impinge on a single sensory organ, the
eye. The one stimulus is the words on the printed page,
which falls in the center of visual regard ; the other stim-
ulus is the people on the sidewalk, perceived in the periph-
222 THE MONIST.
ery of vision. Tracing these diverse impressions it seems
reasonable to assume that the impression of the printed page
is conducted to the occipital lobes, from there to the associa-
tion centers, and from these the nerve energy is distributed
to the other centers, including the motor center, so that
when the individual reaches the bottom of the page he
makes a conscious and coordinated movement with the hand
to turn over a new page. The other vague impressions
which fall on the periphery of vision are also conducted to
the occipital lobes, but the path to the association centers
is already blocked. Naturally the nerve energy seeks an
outlet in some other direction. Now in the course of the
individual's life, strong association bonds had been formed
between visual perceptions of the kind that now impinge
on the periphery of his vision and specific organic reactions,
i. e., seeing a body coming toward him and moving out of
its way. Psychophysically speaking, these strong associa-
tion bonds are smoothly working conduction-paths between
the visual and motor centers. Consequently when now a
visual impression of the same kind reaches the visual cen-
ter, it immediately discharges itself through the path of
least resistance, and upon reaching the motor center re-
leases the customary response which, of course, is an adap-
tation to the external situation. Since all this takes place
without reaching the association centers, we have uncon-
scious "intelligent" action.
But it will be asked: How does this view account for
the fact that if the individual is hypnotized he can be made
to give an account of persons and places he had passed
though wholly oblivious of them at the time ? The answer
to this question involves the physiological theory of the
unconscious, and it is to this that we turn next.
Generally stated, this theory means that the subcon-
scious is not psychical at all, but purely physiological ; that
the presence of awareness cannot be measured by adaptive-
THE PRESENT STATUS OF THE UNCONSCIOUS. 223
ness of action, for there are many glands and thousands of
cells in the human body performing very complex adaptive
acts, or acts designed for the preservation of the organism ;
yet we do not say that these are mental. Why should we
expect less from the tissue of the central nervous system
than we do of all other tissue ? Or in the words of Miinster-
berg, "Why cannot they, too, produce physiological proc-
esses that yield to well-adjusted results?," i. e., to pur-
posive sensorial excitements and motor impulses.28
The same view is advanced by Ribot, who declares that
the psychological solution of the unconscious rests on the
assumption that consciousness is a quantity which may
decrease indefinitely without ever reaching zero. But there
is no justification for this postulate ; he says : "The results
of psychophysists with regard to the threshold of con-
sciousness seem to justify the opposite view, namely, the
perceptible minimum appears and disappears instantane-
ously, and this fact is unfavorable to the hypothesis of an
increasing and decreasing continuity of consciousness."
The physiologic solution, moreover, is simple, inasmuch as
it maintains that subconscious activity is purely cerebral.29
The same theory is shared by Jastrow. He deems it a
fundamental requisite of any adequate conception of the
subconscious that it make a vital connection with normal
mental activity; it must find a natural place in an evolu-
tionary interpretation of psychic functions, and like normal
activity it must be interpreted in terms of neural disposi-
tions. He proposes a criterion, therefore, for the measure
of awareness. "The measure of awareness that shall ac-
crue to any given nervous structure to an environmental
situation, in order to render the response advantageous. . .
will be determined by the status of the need thus satisfied
in the organic life of the individual. The simplest, recur-
2SJourn. Abnorm. Psychol., II, 1907, p. 30.
29 Op. cit., p. 35.
224 THE MONIST.
rent and constant needs will be sufficiently met by neural
dispositions without conscious states, or with the lowest
type thereof."30
Irving King advances the same view and almost in the
same words. "Neural processes," he says, "are accom-
panied by psychical processes only when there is some need
for them."3 According to him, consciousness is definitely
related to the facilitation of reactions and adjustments re-
quired by the life process, but which the automatic arrange-
ments of the organism cannot meet. Consciousness either
is or is not. It may be more intense at one moment than
at another, but it does not consist of different degrees of
intensity, as James's theory of the "fringe" would imply.
On the neural side, however, we do have a system which
may be spatially represented. In terms of this system con-
sciousness is not "the sum of the organization of psychic
elements, but rather the unique and single accompaniment
of a peculiar organization of neural processes." From this
definition it follows that each neural element will determine
the complexion of consciousness. If it is in the center
of the system, it has dynamic conscious value ; if outside of
that system, it has potential value only. The subconscious,
therefore, is not to be conceived as dim consciousness, but
rather as a "physical mass of neural dispositions, tensions
and actual processes which are in some degree, perhaps
organized, the remnants of habits and experiences, both
those which have lapsed from consciousness and those which
have never penetrated the central plexus."
On the basis of these definitions it becomes fairly easy
to understand most of the phenomena that come under the
heads of the conscious and the unconscious. "When con-
sciousness is present," say King, "the neural processes in-
volved are much more intense than otherwise." The dream
consciousness is a condition in which the central activity
80 The Unconscious, p. 411. 31 Op. cit., p. 42.
THE PRESENT STATUS OF THE UNCONSCIOUS. 22$
is so subdued that more or less fragmentary neural dispo-
sitions are aroused. In hypnosis, again, the center of activ-
ity is shifted in more or less degree, resulting in the tem-
porary lapse from consciousness of some processes and the
incorporation of others which were previously mere neural
dispositions. While in multiple personality there are one
or more strongly organized potential systems of neural
elements which, under appropriate conditions, can sep-
arately become sufficiently active to be conscious.32
It is to be noted that the chief characteristic of the ex-
ponents of the physiologic theory is that they do not endow
the subconscious with any mysterious powers, they do not
regard it as the reservoir of consciousness, but on the con-
trary, they consider subconscious events as very much like
the ordinary facts of waking consciousness; and their
method of explanation is to proceed in a true scientific man-
ner from the known to the unknown, from the facts of the
conscious to those of the unconscious. And although Mor-
ton Prince does not hold this view in its entirety, it is
nevertheless in this fashion that he commences the presen-
tation of what is without doubt the most able and most
cogent theory of the unconscious that has appeared in re-
cent years.
The problem of the subconscious, according to him, is
the problem of memory. Whoever solves the latter will
also have solved the former. Memory should be considered
from two points of view : as a process and as an end result.
As a process it is composed of three factors, — registration,
conservation and reproduction. The last is the end result,
but to understand this we must know something of, or at
least have a plausible theory concerning, impression and
conservation.
Instances of the conservation of forgotten experiences
abound both in normal and pathological life. They are
32 Ibid., pp. 4Sff.
226 THE MONIST.
such as lapses of memory, forgotten acts, failure to recog-
nize, or in abnormal cases they become manifest in auto-
matic writing and speech, in post-hypnotic suggestions, and
so forth. After examining the facts in great detail, Prince
comes to the conclusion that it does not matter at what
period of life or in what state experiences have occurred,
"or how long a time has intervened since their occurrence,
they may still be conserved. They become dormant, but
under favorable conditions, they may be awakened and
may enter conscious life."3 Naturally these experiences
must be conserved in some form ; and whatever the nature
of this form may be it is obvious that the experiences them-
selves must have "a very specific and independent existence,
somewhere and somehow, outside of the awareness of con-
sciousness."
Now in order to account for normal memory we must
posit that ideas which have passed through the mind have
been conserved through some residuum left by the original
experience. This residuum must be either psychological
or physiological. Suppose we consider the former alterna-
tive first. We shall have to assume that sensations, per-
ceptions, emotions and even complex systems of ideas are
capable of pursuing "autonomous and contemporaneous
activity outside of the various systems of ideas that make
up the personal consciousness." This is an untenable view,
for it would necessitate the storing up of millions of ideas
and infinite forms of associations. Let us, therefore, con-
sider the other alternative, namely, conservation as phys-
ical residua. This view is based on the assumption that
whenever we have a mental experience of any sort some
change or trace is left in the neurones of the brain. This
does not necessarily mean that the neural modification is
the cause of the conscious process. On the contrary, it
assumes the postulate of psychophysical parallelism and
83 The Unconscious, pp. 82ff.
THE PRESENT STATUS OF THE UNCONSCIOUS. 227
declares that with every passing state of conscious ex-
perience, with every idea, emotion and perception, the brain
process that is functioning leaves some trace, some residua
of itself within the neurones and in the functional arrange-
ments among them. This physiological conception is at the
basis of the association theory, wherein it is assumed "that
whenever a number of neurones involved in a coordinated
sensory-motor act are stimulated into functional activity,
they become so associated and the paths between them be-
come so opened or sensitized, that a disposition becomes
established for the whole group to function together and to
reproduce the original reaction when either one or the
other is afterward stimulated into activity. This 'dispo-
sition' is spoken of in physiological language as a lowering
of the threshold of excitability. This change we may speak
of as a residuum,"34 says Prince.
We are now in a position to answer the question raised
a while ago concerning the ability of a hypnotized person
to recall a forgotten experience or one that he was not
aware of at the time of its occurrence.
The neurones in retaining the residua of the original
process have become organized into a functioning system
corresponding to the system of mental states — whether
ideas, perceptions or emotions — which accompanied that
original experience and are now capable of reproducing it.
Hence when we reproduce the original ideas in the form of
memory it is because there is a refunctioning of the physio-
logical neural process. On hypnotizing a person, therefore,
and asking him to recall a forgotten event, we simply start
that process by introducing what may be called a catalytic
agent, i. e., we stir one neurone or one brain cell, or one part
of the system, and that sets the entire system working pre-
cisely as it did on the original occasion. This physiological
functioning now reaches consciousness or motor expression,
**Ibid.t pp. 119-120.
228 THE MONIST.
because all other mental processes are arrested for the time
being, thus facilitating a greater discharge of nerve energy
in this one direction.
The same is true of crystal gazing and automatic writ-
ing. In the former occurrence there is an intense concen-
tration of primary attention. That is, the subject does not
attend to any idea or to a situation from which he tries to
derive meaning, but merely to a visual stimulus. In this
manner all distracting influences and mental processes
which do not harmonize with the original experience, of
which it must be said the individual has some intimation
to begin with, are arrested. Thus the resumption of the
original neural process is facilitated and with it, of course,
the psychical accompaniment. Anything that will hold the
attention will do as well as a crystal. A soft light will
work just as well.
Equally well can automatic writing be explained on the
basis of this theory. The writing habit is very highly and
delicately "developed in us writing mortals," to use a
phrase of Pierce, and it is no wonder that it may operate
mechanically, when for some reason its neural system has
become detached from that other system which constitutes
self-consciousness. Nor do the specimens of automatic
writing show this phenomenon to be essentially different
from the uncontrolled movements of the hands and bodily
twitching that most of us have at times ; and by no means
is it different from such nervous troubles as chorea and
locomotor ataxia. The hand has been observed to write
backwards, to write mirror script, to follow indefinitely a
direction given to it by the experimenter such as moving
in a circle, it misplaces and omits letters. "Surely," says
Pierce, "such occurrences point clearly to a disordered
neural mechanism, rather than to a perverse or humorously
inclined secondary consciousness."3
35 Carman, Studies in Phil, and Psychol, 1906, pp. 327-328.
THE PRESENT STATUS OF THE UNCONSCIOUS.
We see, then, that most if not all subconscious phenom-
ena can best be explained in terms of cerebration. Now it
is necessary to have some term to designate the separate
neurological modifications, and Prince calls these "neuro-
grams." A neurogram, therefore, is a brain record; and,
just as a phonogram characterizes the form in which the
physical aspect of spoken thought is recorded, so a neuro-
gram characterizes the form in which thoughts and other
mental experiences are recorded in the brain tissue. Of
course this is merely a theoretical concept, like atoms and
moments of force.
Though memory is regarded in psychology as a con-
scious process, it is evident that on the basis of the fore-
going view any process that consists of the three factors,
registration, conservation and reproduction of experiences,
must be considered as memory, "whether the final result
be the production of a conscious experience or of one to
which no consciousness was ever attached."3
That memory is ultimately a physiological phenomenon
was demonstrated by the experiments of Rothmann who
showed that decorticated animals can be educated, i. e.,
new dispositions and new associations may be established
in the lower centers "without the intervention of the in-
tegrating influence of the cortex or conscious intelligence."37
The bearing of this fact is that unconscious processes are
capable of being conserved in the form of physiological
memory.
If we accept the psychophysiological theory of memory,
then, we may define the unconscious as the brain residua,
the physiological dispositions or neurograms in which the
experiences of life are conserved. The co-conscious, on the
other hand, means "a coexisting consciousness of which
the personal consciousness is not aware. And since these
36 Prince, The Unconscious, p. 135.
87 The Unconscious, p. 238.
23O THE MONIST.
two function together we need an inclusive term, one that
will embrace them both, and that is the subconscious."3
Here the truly scientific discussion ends. The rest that
Prince has to say about the subconscious is metaphysical,
and not unlike the views of Sidis, von Hartmann and My-
ers. He declares, for instance, that the subconscious, rather
than the conscious, is the important factor in personality
and intelligence; that the subconscious furnishes the ma-
terial out of which our judgments and beliefs, our ideals
and characters, are shaped. Yet I can hardly see how he
squares this statement with the next in which he says that
the unconscious complexes are kept in check by the normal
inhibitions and the counterbalancing influences of the nor-
mal mental mechanism.39 Evidently, then, it is the normal
mental mechanism, by which I suppose he means attentive
consciousness or intelligence, which exercises a determining
control over the unconscious complexes. Hence it is the
conscious and not the unconscious which is at the basis
of our beliefs, our ideals and character.
Be this as it may, Prince's metaphysical interpretation
does not change the facts nor the value of his scientific
concepts which so excellently explain those facts. For by
resolving the subconscious into unconscious physiological
dispositions on the one hand, and coactive conscious states
on the other, we are able to understand more clearly the
nature of lapsed memory, absent-mindedness, post-hypnotic
suggestion, artificial hallucinations, hysteria, psychoneu-
rosis and multiple personality.
With respect to bridging the gap between the conscious
and the subconscious, Prince declares that no gap exists.
What belongs to one at times passes into the other, and
vice versa. Consciousness may be conceived of as a round
disk with attention or the focus of awareness at the center.
38 Ibid., p. 253.
39 Ibid., p. 262.
THE PRESENT STATUS OF THE UNCONSCIOUS. 23!
Surrounding this is a zone which constitutes the fringe of
awareness. Embracing that is the co-conscious, i. e., un-
conscious mentation, while the outermost zone comprises
the unconscious processes. There is a gradual shading
from the center to the edge of this figurative disk or sphere
of consciousness. But here again Prince treads on the
metaphysical, and we have not the time to follow him.
The space at our disposal only permits us to suggest
that more original work ought to be done in this field. Too
many writers weave their theories around the same cases
of somnambulism and double personality. The cases ex-
amined by Morton Prince, Janet and Bernheim constitute
a sort of stock-in-trade making their rounds in the litera-
ture on the subconscious. But a theory does not gain
credence by hopping about on the same crutches; it must
gather new facts if it would increase in strength. In this
respect Professor Lillien Martin has shown a good way
in her experimental investigation of the subconscious. She
does not add anything new, but her method of investigation
which consisted in having normal subjects permit images
to arise of themselves and then introspect on them, is more
reliable than the questionnaire method used by some au-
thors, or the observation of pathological cases upon which
still others have built their concepts. Experimental research
under strictly controlled conditions, should be the slogan
of psychologists in the field of the subconscious as it is in
that of the conscious.
Thus we stand at the present moment with three theories
of the unconscious before us. The psychometaphysical,
the psychophysiological with metaphysical leanings, and
the psychoneurological with scientific leanings. The first
declares that the whole universe is permeated with con-
sciousness, that there is intelligence in all animals, in plants
and even in inorganic matter. This notion is held by writ-
ers like von Hartmann, Myers, Delboeuf and persons in-
232 THE MONIST.
terested in psychical research. It is obvious, however, that
this view will not bring us anywhere.
The psychophysiological theory with metaphysical lean-
ings also maintains that there is consciousness in all organ-
isms, only it is not conscious of itself. That in living organ-
isms this consciousness is accompanied by physiological
changes, but these changes are not necessarily the cause
of conscious phenomena. Neurological modifications are
only conceptions assumed for the purpose of explaining
unconscious activity. But the psyche is the fundamental
principle. The unconscious is the source of all intelligence.
This view is held explicitly or implicitly by Freud, Sidis,
Prince, Lloyd Morgan and Janet.
Finally, the psychophysiological theory with scientific
leanings asserts that neurological modifications are the es-
sential factors of conscious and unconscious phenomena.
That consciousness appears only when the neurones attain
a certain tension, or function in a certain relation; that
consciousness may or may not accompany so-called intelli-
gent actions performed under pathological conditions ; that
it is certainly not present in instinctive functioning which
characterizes the life of lower animals ; that the unconscious
is not the storehouse of the conscious, that there is nothing
mysterious or wonderful about it, and that with further
investigation its precise nature and place in the scale of
psychogenesis will be at the command of psychologists.
This view is held by writers like Ribot, Pierce, King and
Jastrow.
Such are the three views that present themselves for
our consideration. There is no doubt about the one that
scientists will adopt as leading to a greater extension of
human knowledge.
GUSTAVE A. FEINGOLD.
HARTFORD, CONN.
NIRVANA
THE BUDDHIST'S FINAL GOAL.
NIRVANA, state of rest unbroken, where
Benign extinction of all 'passion rules —
A rest so deep that in eternity
It shall not be disturbed — I long for thee!
After life's storm and stress thou grantest peace.
Weary of this world's wild anxieties,
Its pains and empty pleasures, I will seek
The everlasting in blank vacancy,
Thus to attain the boon of dreamless sleep
From which nor rancor of a villainous
Intrigue, planned by malevolence or hate,
Nor the misfortune of a sorry slip
Of my misguided feet, will waken me,
But unconcerned and calm I shall remain
In perfect quietude. For I'll be safe
From all the worry and from all the trouble
That restlessly stirs life and keeps it moving.
The bustle of the world, its vulgar noise
With its deplorable afflictions, trials
And eke malicious slander, will be hushed.
There is a refuge, vainly sought for here,
And in its sanctuary I'll find shelter
From life's great woes and small annoyances.
There paltry problems will no longer vex
Nor will demand immediate solution.
234 THE MONIST.
I shall no longer be disquieted
By urgent needs to be responded to
In energetic action. E'en my ego
With its ambitions, wants and vanities;
Its recollection of the past with all
Its sweet and bitter memories — all that,
My very consciousness, will be extinct.
I shall be left in tranquil emptiness
And in a soothing void of non-existence,
A clean, pure state of rest most absolute,
Without the slightest ripple of disturbance,
A panacea for all earthly ills,
An anodyne for any pang or pain.
In former ages mankind felt assured
Of a survival of the soul. The savage
Met his dead parents and his friends in dream.
He saw them, he conversed with them, and dreams
Were real to him just as actual life.
When man grew wiser, he began to doubt
And he grew anxious for convincing proof.
Though proofs were negative, yet he still clung
To hope expressing his desire to live
And to prolong his life beyond the grave.
Oh foolish man, why dost thou shrink from death
And yearnest greedily for prolongation
Of thy ill-favored self? Thy selfishness
Thou wishest to preserve, thy abject foibles,
Instead of gladly hiding them into
The darkness of a taciturn forgetting,
As in wise justice Nature has intended,
Thou wouldst perpetuate with petulance
And peevish childishness that of thyself,
Exactly that, whose riddance should be welcomed
NIRVANA. 235
As a great boon, a seemly liberation
From slavery, its drudgery and curse.
Why should we cling to chains that burden us
When we might cast them off and free ourselves ?
Why should we serve new terms as sentenced convicts
When duly our acquittal is pronounced
And a reprieve has graciously been granted?
Mara, the Evil One will envy me
In my benign repose; he will continue
His vicious persecution. Shall I help him
And do the wrong myself unto myself
By pressing from a place of safety into
My prison with its ugly bars ? Oh no !
No, I shall not! For I prefer my freedom!
There I shall be where most malignant foes
Shall not be able to do any harm.
And if they should go on abusing me
I shall no longer heed their defamation;
I'll leave them to their fate which in full justice
Will come to them without my interference.
Their lies no longer touch me. In Nirvana
I shall be free; the vicious will remain
In a gehenna builded by themselves.
The wild desires of my hot pulsing heart
Will then be calmed, all hunger will be stilled,
All thirst be quenched in deepest satisfaction.
And mine shall be the glory of Nirvana ;
Having achieved the conquest of all pain,
Having attained final emancipation,
It will be mine, Nirvana will be mine.
I shall be free when I have closed mine eyes,
To enter death, life's solemn grand finale,
Its fruitage, benison and consummation.
Nirvana's holy peace shall then be mine.
236 THE MONIST.
Indeed it is mine here ; I live it now
If I but understand the art of living
The truth : It is and will be mine, when I
Surrender transient things to transiency
And live in that alone which will endure.
Oh, the inanities of self! how puny,
How paltry are they; and how kind is death
To brush them off with gently sweeping stroke
Like spider webs out of a gloomy corner,
Together with the spider who has built them.
O let them go without regret and sorrow,
The ego with its portion is not worthy
Of preservation. It is but the burden
Of our existence, the receptacle
In which the weaknesses and faults of life
Are bred, in which its plagues are caught and stored.
So let them go and bless their disappearance.
They are like painful sores that should be healed,
And when our ego passes they are cured.
The right ideas only which we've thought,
The good deeds too which we have done and things
Of beauty we have shaped, they shall survive.
They are our better selves ; they will be helpful,
Helpful to others, to the generations
That are to come, helpful like gifts of God,
Like rain or sunbeams, showered down on earth,
Profuse, unstinted, and with utter lack
Of egotism. But do not cling to self,
Nor yearn for any undue preservation
Of personality. Our ego's life
And all that's of an accidental nature
Be handed over to its destination
Which is a dissolution into naught.
NIRVANA. 237
Our conscious ego has originated,
It has been growing, and 'twill pass away.
Such is its destiny and so 'tis best.
But I will glory in my future lot —
Nirvana's boon, the state of perfect peace.
Yea, I can enter even now into
Nirvana's hallowed temple where my soul
Is liberated from all transiency
And will be ready for a final exit
Out of existence with its narrowness
Into the better and superior realm,
The realm of bliss, Nirvana's noble bliss.
Praised be Nirvana, glorious radiant state
Of biding peace, hope of all living creatures
And comfort of the dead. Holy asylum
Which grander is than highest joy in heaven
And more divine than the divinity
Of Brahma and his gods in all their splendor.
Praised be Nirvana, goal of all the Buddhas!
And blest is he who enters there, who lives
There in Nirvana; lives there in the truth
Which therein is revealed; he who is free
From vain attachment, who's above temptation.
'Tis he in whom all passion is extinct ;
Who has attained life's final aim Nirvana,
Goal of the wise, and of the blessed Buddhas.
He who has reached it is the Conqueror,
The conqueror of Evil, the great Jina,
He's the Enlightened One : he is the Buddha !
And he is blest ; the Buddha, yea ! is blest.
Pathfinder to Nirvana ! Praised be he !
Namo tassa Bhagavato Buddhassa.
PAUL CARUS.
THE MANUSCRIPTS OF LEIBNIZ ON HIS DIS-
COVERY OF THE DIFFERENTIAL CALCULUS.*
PART II.
§m.
The following notes, on certain MSS. which Gerhardt does not
give in full, are taken from G. 1848, p. 20 et seq. (see also G. 1855,
p. 55 et seq.)
In a manuscript of August, 1673, bearing the title Methodus
nova investigandi Tangentes linearum curvarum ex datis applicatis,
vel contra Applicatis ex datis productis, reductis, tangentibus, per-
pendicularibus, secantibus, Leibniz begins at once with an attempt
to find a method that is applicable to any curve for the determination
of its tangent. "But if," says Leibniz with regard to the classifica-
tion of curves which Descartes laid down as fundamental for his
method of tangents, "the figure is not geometrical — such as the
cycloid — it does not matter; for it will be treated as an example
of a geometrical curve, by supposing that there is a relation between
the straight lines and curves by which they are made known to us ;
in this way, tangents can be drawn just as well to either geometrical
or ageometrical curves, as far as the nature of the figure allows."
He considers the curve as a polygon with an infinite number of
sides, and here already he constructs what he calls the "Character-
istic Triangle," whose sides are an infinitely small arc of the curve,
and the differences between the ordinates and between the abscissae ;
this is similar to the triangle whose sides are the tangent, the sub-
tangent and the ordinate for the point of contact. In just the same
manner as used by Descartes, Leibniz seeks the tangent by means
* Part I appeared in The Monist of October, 1916.
THE MANUSCRIPTS OF LEIBNIZ. 239
of the subtangent ; he denotes the infinitely small differences of the
abscissae by b, and verifies for the parabola, that his method works
out correctly, when the terms of the equation that contain the in-
finitely small quantities are neglected. The omission of these terms,
however, does not appear to Leibniz to be a method to be relied
upon. In fact, he says: "It is not safe to reject multiples of the
infinitely small part b, and other things; for it may happen that
through the compensation of these with others,1 the equation may
come to a totally different condition." So he seeks to obtain the
determination of the subtangent in some other way. "The whole
question is, how the applied lines can be found from the differences
of two applied lines," are his own words. He then finds that the
solution of this problem reduces to the summation of a series, of
which the terms are the differences of consecutive abscissae.
At the end of the manuscript Leibniz proceeds to speak of the
inverse problem: "It is an important subject for investigation,
whether it is possible, by retracing our steps, to proceed from tan-
gents and other functions to ordinates. The matter will be most
accurately investigated by tables2 of equations ; in this way we may
find out in how many ways some one equation may be produced
from others, and from that which of them should be chosen in any
case. This is, as it were, an analysis of the analysis itself, but if
that is done it forms the fundamental of human science, as far as
this kind of things is concerned." Ultimately Leibniz obtains the
following result: "The two questions, the first that of finding the
description of the curve from its elements, the second that of find-
ing the figure from the given differences, both reduce to the same
thing. From this fact it can be taken that almost the whole of the
theory of the inverse method of tangents is reducible to quadra-
tures."
According to this, Leibniz has in the middle of the year 1673
already attained to the knowledge that the direct and the so-called
inverse tangent-problem have an undoubted connection with one
another ; he has an idea that the latter may be ..capable of reduction
tc a quadrature (i. e., to a summation).
Again, in a manuscript dated October 1674, i. e., fourteen
months later, which bears the title Schediasma de Methodo Tan-
1 It is impossible to see, without a fuller knowledge of the context.whether
this refers to "compensation of errors," or whether Leibniz is alluding to the
possibility of all the finite terms cancelling one another.
2 Leibniz comes back to this point later ; see § IV.
240 THE MONIST.
gentium inversa ad circulum applicata, he is able to say for certain
that "the quadratures of all figures follow from the inverse method
of tangents, and thus the whole science of sums and quadratures
can be reduced to analysis, a thing that nobody even had any hopes
of before."
After Leibniz thus recognized the identity between the inverse
tangent-problem, of which the general solution had not been found
by Descartes, and the quadrature of curves, he applied himself to
the investigation of series by the summation of which quadratures
were then obtained. In a very extensive discussion, bearing the
date of October, 1674, and the title Schediasma de serierum summis,
et seriebus quadraticibus, Leibniz starts from the series
and obtains the following general rule: "By calling the variable
ordinates x, and the variable abscissae y, and b the abscissa of the
greatest ordinate e, and d the abscissa of the least ordinate h," are
Leibniz's own words, "we have the following rules :
hzw dzh x
—- + —- = xy--,e_k = w,
* w2
f7/l — - _ -
2 2'
yw = x in decreasing values, for in ascending or increasing values
yw = eb - x"s
Leibniz then goes on to remark : "These rules are to be altered
slightly according as the series increase or decrease; also mention
of the least ordinate may be omitted, if it is always understood to
be the last ordinate; on the other hand, w can always be inserted
wherever mention is made of w. All series hitherto found are con-
tained in the one by means of these rules, except the series of
powers, which is to be obtained by taking differences."
3 This, without either proof or figure, is a hopeless muddle ; and yet it is
repeated word for word, without any addition or remark, in Gerhardt's 1855
publication. Goodness knows what the use of it was supposed to be in this
form! Unless Leibniz has omitted some length, which he has supposed to be
unity, the dimensions are all wrong.
THE MANUSCRIPTS OF LEIBNIZ. 24!
In the same essay, Leibniz makes use of a theorem, which he
has probably found to be general at an earlier date, namely:
"Since BC is to BD as WL to SW, there-
fore BC^SW,4 that is, the sum of every BC
[applied to AC], is equal to BD^WL, that is,
the sum of every BD applied to the base ; more-
over, the sum of every BD applied to the base is
equal to half the square on the greatest BD.
Further, it is evident that the sum of every WL
is equal to the greatest BD."
Accordingly, Leibniz comes to the further
conclusion that the method of Descartes, which
uses a subsidiary equation with two equal roots, to solve the general
inverse-tangent problem, is unsatisfactory. In a manuscript of
January, 1675, Leibniz says : "Thus at last I am free from the un-
profitable hope of finding sums of series and quadratures of figures
by means of a pair of equal roots, and I have discovered the reason
why this argument cannot be used; this has worried me for quite
long enough."5
§ IV.
The manuscript that comes next in da'te is one that is
given in G. 1855. It really consists of three short notes,
(i) a theorem on moments, (2) a continuation of the idea
started at the end of the manuscript of August, 1673
(§ III), namely the formation of tables of equations that
are derivable from certain standard equations, with the
appropriate substitutions for each case, (3) a return to
the consideration of moments.
This is the first appearance of the word "moment," but
from the context it is evident that Leibniz has done some
considerable amount of work upon the idea before. If the
theorem that is first given is written in modern notation,
4 The sign ° signifies multiplication.
6 Observe that as yet nothing has been said about the area of surfaces of
revolution or moments about the axis, although we should expect them to be
mentioned in connection with the figure that is given ; for the next manuscript
shows that in October 1675, Leibniz has already done a considerable amount
of work on moments.
242
THE MONIST.
it takes the form of an "integration by parts" and serves
to change the independent variable. Thus we have
and it is readily seen that if x can be expressed as a square
root of a simple function of y, as for the circle and the
conic sections, then the integral on the right-hand side
has no irrationality. This, I take it, is the connection
between this theorem and those which follow.
The proof is not so clear as it might be on account of
two errors, both I think errors of transcription or mis-
prints. The first a should be an x, and the second a should
be the preposition a (= from) ; also, for modern readers
the figure might be improved by showing the variable lines
AB (=x), BC (=30 as in the accompanying diagram.
The argument then is as follows:
Moment of BC(=30 about AD is xy, when it is applied
to AB for the summation ; for this brings in the infinitesi-
mal breadth of the line.
B
Moment of DE (= x) about AD is xz/2, when applied
to AD, so as to include the infinitesimal breadth of the
line, and assuming that the line may be considered to be
condensed at its center of gravity. The theorem follows
at once.
Note the use of the sign n as a symbol of equality,
which I have allowed to stand in the opening paragraph.
Leibniz adopts the ordinary sign two months later, or Ger-
THE MANUSCRIPTS OF LEIBNIZ. 243
hardt makes the change,8 so I have not thought it necessary
to adhere to it, but only to show it in the opening para-
graph.
The only remark that seems to be necessary with regard
to the second part of this manuscript is that Weissenborn7
argues from the continued allusion by Leibniz to the de-
sirability of forming tables of curves whose quadratures
may be derived from those of others, especially the conic
sections, (starting with the manuscript of November, 1675,
where Weissenborn states that it is first hinted), that
Leibniz had probably either seen or heard of the Cata-
logus curvarum ad conicas sectiones relatarum of Newton.
The point is that Weissenborn seems to have missed the
clear reference to the reduction of curves to those of the
second degree, in this manuscript of October, 1675. It
may of course be just possible that G. 1855, m which this
MS. appears, was not at Weissenborn's hand at the time
that he wrote, for Weissenborn's book was published in
1856.
With regard to the third part, it will be found in the
original Latin- that Leibniz, after apparently starting with
perfect clearness, gets rather into a muddle toward the end.
This is however only apparent, being partly due to an in-
accurate figure, and partly to what I am convinced is an
error of transcription. This incorrect sentence makes Leib-
niz write apparently absolute nonsense ; but if a correction
is made according to the suggestion in the footnote, and
reference is made to the corrected diagram that I have
added on the right of the figure of Leibniz, as given by
Gerhardt, then the proof given by Leibniz reads perfectly
smoothly and sensibly.
6 Gerhardt has a footnote to the effect that, as nearly as possible he has
retained the exact form of this and the manuscripts that immediately follow ;
except in the matter of this one sign I have adhered to the form given by
Leibniz.
7 Weissenborn, Principien der hoheren Analysis, Halle, 1856.
244
THE MONIST.
25 October, 1675.
Analysis Tetragonistica Ex Centrobarycis.
Analytical quadrature by means of centers of gravity.
Let any curve AEC be referred to a right angle BAD ; let AB n
DCna,8 and let the last xnb; also let BCnADn^, and the last
ync. Then it is plain that
ornn. yx to x = — — omn. ~^~ to y.
(1)
For, the moment of the space ABCEA about AD is made up
of rectangles contained by BC (= y) and AB (= x} ; also the moment
B
about AD of the space ADCEA, the complement of the former
/ x2\
is made up of the sum of the squares on DC halved ( = — ) ; and if
this moment is taken away from the whole moment of the rectangle
i2
ABCD about AD, i. e., from c into omn. x? or from — , there will
remain the moment of the space ABCEA. Hence the equation that
I gave is obtained ; and, by rearranging it, it follows that
omn. yx to x + omn. — to y =
(2)
In this way we obtain the quadrature of the two joined in one
in every case ; and this is the fundamental theorem in the center of
gravity method.
Let the equation expressing the nature of the curve be
ay*+6x*+cxy+dx+ey+/=0, (3)
and suppose that xy=z, - • • • (4), then y = — .
x
(5)
Substituting this value in equation (3), we have
8 This a should be x.
9 Here, in the Latin, "ac in omn.*-" should be "a c in omn.*."
THE MANUSCRIPTS OF LEIBNIZ. 245
*+/-0. (6)
X" X
and, on removing the fractions,
az2 + bx* + cx2z + dxs + exz + fx2 = 0 (7)
Again, let x2 = 2w (8) ; then, substituting this value in
equation (3), we have
and therefore
x_-a*-2bw -ey-f (10)
ey+d
= V2w; (11)
and, squaring each side, we have10
azy2 + 4aby2w + 2aeya + 2afy2 + 4b2w2 + 4bewy + 4bfw
+ ezy2 + 2fey + f2- 2c2y2w - 4cdyw - 2d2w = 0. . . ( 12)
Now, if a curve is described according to equation (7), and
also another according to equation (12), I say that the quadrature
of the figure of the one will depend on the quadrature of the figure
of the other, and vice versa.
If, however, in place of equation (3), we took another of
higher degree, the third say, we should again have two equations
in place of (7) and (12) ; and continuing in this manner, there is
no doubt that a certain definite progression of equations (7) and
(12) would be obtained, so that without calculation it could be
continued to infinity without much trouble. Moreover, from one
given equation to any curve, all others can be expressed by a general
form, and from these the most convenient can be selected.
If we are given the moment of any figure about any two
straight lines, and also the area of the figure, then we have its
center of gravity. Also, given the center of gravity of any figure
(or line) and its magnitude, then we have its moment about any
line whatever. So also, given the magnitude of a figure, and its
moments about any two given straight lines, we have its moment
about any straight line. Hence also we can get many quadratures
from a few given ones. Moreover, the moment of any figure about
any straight line can be expressed by a general calculation.
The moment divided by the magnitude gives the distance of the
center of gravity from the axis of libration.
10 In view of this accurate bit of algebra, the faulty work in subsequent
manuscripts seems very unaccountable.
246
THE MONIST.
Suppose then that there are two straight lines in a plane, given
in position, and let them either be parallel or meet, when produced
in F. Suppose that the moment about BC is found to be equal to
ba?, and the moment about DE is found to be ca2. Call the area
of the figure v\ then the distance of the center of gravity from the
ba2
straight line BC, namely CG, is equal to — , and its distance from
v
2
the straight line DE, namely EH, is equal to - — ; therefore CG is to
v
EH as b is to c, or they are in a given ratio.11
GERHARDT'S DIAGRAM.
SUGGESTED CORRECTION.
Now suppose that the straight line EH, remaining in the plane,
traverses the straight line DE, always being perpendicular to it, and
that the straight line CG traverses the straight line BC, always per-
pendicular to it, and that the end G leaves as it were its trace, the
straight line G(N), and the end H the straight HN. Then, if BC
and DE meet anywhere, G(N) and HN must also meet somewhere,
either within or without the angle at F. Let them meet at L ; then
the angle HLG is equal to the angle EFC, and PLQ (supposing
that PL = EH and LQ = CG) will be the supplement of the angle
EFC between the two straight lines, and will thus be a given angle.
If then PQ is joined, the triangle PQL is obtained, having a given
vertical angle, and the ratio of the sides forming the vertex, QL : LP,
also given.
When then BL is taken, or (B)(L), of any length whatever,
since the angle BLP always remains the same, and in addition we
have BL to LP as (B) (L) to (L) (P), therefore also BLto (B) (L)
as LP to (L) (P) ; and this plainly happens when FL is also propor-
11 This proves the fundamental theorem given lower down, with regard
to a pair of parallel straight lines; and he now goes on to discuss the case
of non-parallel straight lines.
THE MANUSCRIPTS OF LEIBNIZ. 247
tional to these, that is, when a straight line passes through F, L,
(L),
Hence, since we are not here given several regions, it follows
that the locus is a straight line. Therefore, given the two moments
of a figure about two straight lines that are not parallel, ,
the area of the figure will be given, and also its center of gravity.12
Behold then the fundamental theorem on centers of gravity. If
two moments of the same figure about two parallel straight lines
are given, then the area of the figure is given, but not its center of
gravity.
Since it is the aim of the center of gravity method to find
dimensions from given moments, we have hence two general the-
orems :
If we are given two moments of the same figure about two
straight lines, or axes of libration, that are parallel to one another,
then its magnitude is given; also when the moments about three
non-parallel straight lines are given. From this it is seen that a
method for finding elliptic and hyperbolic curves from given quad-
ratures of the circle and the hyperbola is evident.13 But of this in
a special note.
§ V.
The next manuscript to be considered is a continuation
of the preceding, and is dated the next day. Its character
is of the nature of disjointed notes, set down for further
consideration.
12 The passage in Gerhardt reads :
Datis ergo duobus momentis figurae ex duabus rectis non parallelis, dabi-
tur figurae momentis tribus axibus librationis, qui non sint omnes parallel!
inter se, dabitur figurae area, et centrum gravitatis.
For this I suggest :
Datis ergo tribus momentis figurae ex tribus rectis non parallelis, aliter
figurae momentis tribus axibus librationis, qui non sunt omnes paralleli inter
se
The passage would then read:
Given three moments of a figure about three straight lines that are not
parallel, in other words, the moments of the figure about three axes of libra-
tion, which are not all parallel to one another, then the area of the figure will
be given and also the center of gravity.
If the alternative words are written down, one under the other, and not
too carefully, I think the suggested corrections will appear to be reasonable.
18 Apparently, here Leibniz is referring back to the theorem at the beginning
of the article.
248 THE MONIST.
26 October, 1675.
Another tetragonistic analysis can be obtained by the aid of
curves. Thus, let the same curve be resolved into different elements,
according as the ordinates are referred to different straight lines.
Hence also arise diverse plane figures, consisting of elements similar
to the given curve ; and since all of these are to be found from the
given dimension of the curve, it follows that from the dimension
of any one of the curves of this kind the rest are obtained.
In other ways it is possible to obtain curves that depend on
others, if to the given curve are added the ordinates of figures of
which the quadrature is either known or can be obtained from the
quadrature of the given one.
Just as areas are more easily dealt with than curves, because
they can be cut up and resolved in more ways, so solids are more
manageable than planes and surfaces in general. Therefore, when-
ever we divert the method for investigating surfaces to the con-
sideration of solids, we discover many new properties; and often
we may give demonstrations for surfaces by means of solids when
they are with difficulty obtained from the surfaces themselves.
Tschirnhaus observed in a delightful manner that most of the proofs
given by Archimedes, such as the quadrature of the parabola, and
dependent theorems on the sphere, cone, and cylinder, can be re-
duced to sections of rectilinear solids only, and to a composition that
is easily seen and readily handled.
Various -ways of describing new solids.
If from a point above a plane a rigid descending straight line
is moved round an area, of any shape whatever, diverse kinds of
conical bodies are produced. Thus if the plane area is bounded
by the circumference of a circle, a right or scalene cone is produced.
Also if the figure used for the base, or the plane area, has a center —
an ellipse for example — then we get an elliptic cone, which is a right
cone if the given point is directly above the center, and if not it is
scalene. Another conic gives another elliptic cone.
If the rigid line drawn down from the point is circular or some
other curve, at one time it is so fixed to the point or pole that it has
freedom to move in one way only, say round an axis, in which case
it is necessary that the base should be a circle and that the fixed
point or pole should be directly over the center. At another time
it is necessary that the rigid line should have freedom for other
motions, such as an up and down motion, or some other motion,
THE MANUSCRIPTS OF LEIBNIZ. 249
controlled by some straight line; and then it will always ascend
or descend when necessary, so that it ever touches the given plane
area by its rotation round the axis ; and this is the second class of
cones. A third class consists of those in which, besides the double
motion of a rotation round an axis and an up and down motion,
the curve alone, or the axis alone, or even both the curve and the
axis, also perform other motions meanwhile, or even the point itself
moves.
Here is another consideration.
The moments of the differences about a straight line perpen-
dicular to the axis are equal to the complement of the sum of the
terms ; and the moments of the terms are equal to the complement
of the sum of the sums, i. e.,
n ult.;r, omn.w,, — omn. omn.w
OTun. ur
az
Let xw n az, then w ""> — , and we have
x
az az
omn.as " ult..r. omn. omn.omn. — :
x x '
az az az
hence omn. — ""• ult.^r omn. — = - omn.omn. —5 ;
X X2 X2 '
inserting this value in the preceding equation, we have
10 az i, az
omn.a^r *~> ult.-r2 omn. -=• - ult.^r. omn.omn. -=• ,
x2 x2
14 I have given this equation, and those that immediately follow it, in
facsimile, in order to bring out the necessity that drove Leibniz to simplify
the notation.
We have here a very important bit of work. Arguing in the first instance
from a single figure, Leibniz gives two general theorems in the form of moment
theorems. The first is obvious on completing the rectangle in his diagram,
and this is the one to which the given equation applies. In the other the whole,
of which the two parts are the complements, is the moment of the completed
rectangle ; its equivalent is the equation
omn^ey = ult.jr omn.y — omn. omn.y.
Now, although Leibniz does not give this equation, it is evident that he rec-
ognized the analogy between this and the one that is given ; for he immediately
accepts the relation as a general analytical theorem that he can use without
any reference to any -figure whatever, and proceeds to develop it further.
This would therefore seem to be the point of departure that led to the Leib-
nizian calculus.
250 THE MONIST.
ii: az
— omn. ult.jr. own.—: — omn.omn.— 7 ;
x2 x2
and this can proceed in this manner indefinitely.
a a a
Again, omn. — •"» x omn. -=- — omn.omn. — z-
x x* xf
and omn.a n ult JT omn. — omn. omn. — ;
x x
the last theorem expresses the sum of logarithms in terms of the
known quadrature of the hyperbola.15
The numbers that represent the abscissae I usually call ordinals,
because they express the order of the terms or ordinates. If to the
square of any ordinate of a figure whose quadrature can be found,
you add the square of a constant, the roots of the sum of the two
squares will represent the curve of the quadratrix. Now if these
roots of the sum of the two squares can also give an area that has
a known quadrature, then also the curve can be rectified.16
15 Having freed the matter from any reference to figures, he is able to
take any value he pleases for the letters. He supposes that s=l, and thus
obtains the last pair of equations. He then considers x and w as the abscissa
and ordinate of the rectangular hyperbola JTW = a (constant) ; hence omn.a/T
or omn. w is the area under the hyperbola between two given ordinates, and
therefore a logarithm; and thus omn. omn.o/jr is the sum of logarithms, as
he states.
*• There only seem to be two possible sources for this paragraph, (1)
original work on the part of Leibniz, and (2) from Barrow. For we know
that Neil's methods was that of Walk's, and the method of Van Huraet used
an ordinate that was proportional to the quotient of the normal by the ordinate
in the original curve.
Now Barrow, in Lect XII, §20, has the following: "Take as you may
any right-angled trapezial area (of which you have sufficient knowledge),
bounded by two parallel straight lines AK, DL, a straight line AD, and any
Hue KL whatever; to mis let another such area be so related that when any
straight line FH is drawn parallel to DL, cutting the lines AD, CE, KL in the
points F, G, H, and some determinate line Z is taken, the square on FH is
equal to the squares on FG and Z. Moreover, let the curve AIB be such that,
if the straight line GFI is produced to meet it, the rectangle contained by Z
and FI is equal to the space AFGC; then the rectangle contained by Z and
the curve AB is equal to the space ADLK. The method is just the same,
even if the straight line AK is supposed to be infinite.
This striking resemblance, backed by the fact that there seems to be no
connection between this theorem and the rest of the paper, that Leibniz gives
THE MANUSCRIPTS OF LEIBNIZ. 25!
To describe a curve to represent a given progression.
From the square of a term of the progression, take away the
square of a constant quantity; if the figure that is the quadratrix
of the roots formed from the two squares is described, it will give
the curve required ; it does not follow that a rectifiable curve can
be described.
The elements of the curve described can be expressed in many
different ways. Different methods of expressing the elements of
a curve may be compared with different methods of expressing
a figure having similar parts with it, according as it is referred in
different ways. Lastly, a solid having similar parts with a curve
can thus far be expressed in many ways, and so also for a surface
or figure having similar parts with the curve.
§ VL
Three days later, Leibniz considers the possibility of
being able to find the quadratrix in all cases, or when that
is impossible, some curve which will serve for the quadra-
trix very approximately. He makes an examination of the
difficulties that are likely to be met with and the means to
overcome them, and he seems to be satisfied that the method
can be made to do in all cases. But in the absence of an
example of the method he proposes to adopt, he seems only
to have been wasting his time. But this may be dismissed,
for it is not here that the importance of this essay lies; it
is altogether in what follows.
The rest of the essay is in the form of disjointed notes:
it is just the kind of thing that any one would write as
notes while reading the works of others. This is what I
take it to be ; and the works he is considering are those of
no attempt at a proof, (indeed I very much doubt whether I could have made
out his meaning from the original unless I had recognized Barrow's theorem)
and that Leibniz gives 1675 as the date of his reading Barrow, almost forces
one to conclude that this is a note on a theorem (together with an original
deduction therefrom by himself) which Leibniz has come across in a book
that is lying before him, and that that book is Barrow's. Against it, we have
the facts of the use of the word "quadratrix," not in the sense that Barrow
uses it, namely as a special curve connected with the circle; that the quad-
ratrix is one of the special curves that Barrow considers in the five examples
he gives of the Differential Triangle method; and that another example of
this method is the differentiation of a trigonometrical function which seems
to be unknown to Leibniz.
252 THE MONIST.
Descartes, Sluse, Gregory St. Vincent, James Gregory and
Barrow. Descartes he has already dismissed as imprac-
ticable in the manuscript of January, 1675; but there are
indications that the former's method has still some influ-
ence. An incidental remark leads to the consideration of
the ductus of Gregory St. Vincent ; but these too are soon
cast aside, truly because Leibniz does not quite grasp the
exact meaning of Gregory. He then either remembers
what he has seen in Barrow or refers to it again, for the
next thing he gives is some work in connection with which
he draws the characteristic triangle, which is here for the
first time, as far as these manuscripts go, the Barrow form
and not the Pascal form. He immediately obtains some-
thing important, namely,
omn. I2
= omn. omn. /— .
a
Noting that, in modern notation, / is dy, and a is dx,
and also, since a is also supposed to be unity, that the
final summation on the right-hand side is performed by
"applying the successive values to the axis of x, while the
summation denoted by omn./ is a straightforward summa-
tion, it follows that the equivalent of the result obtained
dy
by Leibniz is %y* = fy -r dx.
However, in attempting to put this theorem into words
as a general theorem he makes an error ; he quotes omn./2 as
the "sum of the squares" instead of the "square of the
final y." This I think is simply a slip on the part of Leib-
niz, and not, as suggested by Gerhardt and Weissenborn,
- an indication that Leibniz confused omn./2 with omn./2, and
considered them as equivalent. Neither of these authori-
, ties appears to have noticed the fact that when Leibniz
has invented the sign / (which he immediately proceeds
to do) he carefully makes the distinction between the
THE MANUSCRIPTS OF LEIBNIZ. 253
equivalents to the square of a sum and the sum of the
squares. Thus we find that his equation is written as
J 4 = J Jl - , (note the vinculum)
while later in the essay we have j*/3 to stand for the sum
of the cubes. Further, apart from this. I do not think that
any one can impute such confusion of ideas to Leibniz, if
it is noted that so far this is not the differential calculus,
but the calculus of differences, i. e., / is still a very small
but finite line and not an infinitesimal ; for in § IV, Leibniz
had squared a trinomial successfully, and must have known
that the sum of the squares could not be equal to the square
of the sum. Both these above-named authorities seem to
find some difficulty over the introduction of the letter a,
apparently haphazard. This difficulty becomes non-exis-
tent, if it is remembered that a is taken to be unity, and
the remarks made about dimensions by Leibniz are care-
fully considered; it will then be found that the a is in-
troduced to keep the equations homogeneous! Weissen-
born also remarks that Leibniz jots down the integral of
x2 without giving a proof, and appears to be in doubt how
he reached it. If this is so, it confirms the opinion that I
have already formed, namely, that neither Gerhardt nor
Weissenborn tried to get to the bottom of these manu-
scripts, being content with simply "skimming the cream."
I suggest that Barrow, Gregory St. Vincent, and even
Sluse, now join Descartes on the shelf or the floor, and that
the rest of the essay is all Leibniz. He writes the two
equations he has found, the equivalents to two theorems
obtained geometrically, notes the fact that these are true
for infinitely small differences (without, however, men-
tioning that they are only true in such a case), discards
diagrams, and proceeds analytically; that is, the y's are
successive values of some function of x, where the values
254 THE MONIST.
of x are in arithmetical progression; hence, substituting
x for / in the equation
omn.xl = omn./ — omn. omn./,
and remembering that omn.jir == x2/?, as he has proved,
we have
_2T _2T 2T
omn. x* = x — - — omn. — , or omn. x* = — - .
22 3
/.j-3 jj/4
— = — correctly (although
O T"
there is an obvious slip or, as I think, a misprint of / for x) ;
this could have been obtained in the same way.
x^ x^ x^
omn. x* = x — — omn. — , or omn. x* = — .
Similarly, Leibniz could have gone on indefinitely, and
thus obtained the integrals of all the powers of x. But
his brain is too active ; as Weissenborn says, his soul is in
the throes of creation. He merely alludes in passing to the
inverse operation to / as being represented by d, which
he for some reason writes in the denominator (probably
erroneously because he has noted that / increases the di-
mensions) ; and then he harks back to the opening idea of
the essay, the obtaining of the quadratrix by means of
transformation of equations, an idea truly as hopeless as
the method of Descartes which he has discarded. Never-
theless, even then he obtains something remarkable, noth-
ing more or less than the inverse of the differentiation of a
product. This fundamental theorem is obtained geomet-
rically; the proof of the little theorem on which the final
result is founded is not given, neither is there a diagram.
It cannot therefore be supposed but that Leibniz is work-
ing from a diagram already drawn, and I suggest he was
referring to one of those theorems, with which he had
filled "hundreds of pages" between 1673 and 1675. The
THE MANUSCRIPTS OF LEIBNIZ. 255
proof follows quite easily by the use of the characteristic
triangle, and is given in a footnote. This theorem is not
in Barrow, nor can I remember seeing it in Cavalieri;
I have not yet been able to procure a Gregory St. Vincent ;
it may be in James Gregory.
The benefits of this discovery are lost as before, for
Leibniz once more alludes to the transformation of equa-
tions for the purpose of obtaining the quadratrix.
Summing the whole essay, we can say that in it is the
beginning of the Leibnizian analytical calculus.
29 October, 1675.
Analyseos Tetragonisticae pars secunda.
(Second part of analytical quadrature.)
I think that now at last we can give a method, by which the
analytical quadratrix may be found for any analytical figure, when-
ever that is possible ; and, when it can not be done, it will yet always
be possible that an analytical figure may be described, which will
act as the quadratrix as nearly as is required. This is how I look
at it:
Suppose the equation of the curve, of which the quadratrix
is required, is given, and that the unknowns in it are x and v. Let
the equation to the curve required be17
v = b + cx + dy + ex2 + fy2 + gyx + hys + lx* + mxyy + yxx + etc. ; ... (i)
let it be set in order for tangents, as follows :
-dy- 2fy2 - gyx - 3hy3 - 2mxy2 - mxzy - etc.
(ii)
17 This is either a misprint, v instead of O, or else Leibniz is in error.
For Slusius's method there must be only two variables in the equation. In the
Phil. Trans, for 1672 (No. 90), Sluse gives his method thus:
If y5 -f- by* = 2qqv3 — yyv3, then the equation must be written y5 -f- by4 -f-
yy3 = 2qqv3 — yyv3 ; then multiply each term on the left-hand side by the
number of y's in the term, and substitute t in place of one y in each ; similarly
multiply each term on the left-hand side by the exponent of v; the equation
obtained will give the value of t.
The use of the letters v and y is to be noted in connection with Leibniz's
use of the same letters ; it does not seem at all necessary that Leibniz should
have seen Newton's work, with this ready to the former's hand, as a member
of the Royal Society. I suggest that Sluse obtained his rule by the use of a
and e, as given in Barrow. Can Barrow's words usitatum a nobis (in the
midst of a passage written in the first person singular) have meant that the
method was common property to himself and several other mathematicians
that were contemporary with him? This would explain a great deal.
256 THE MONIST.
Now, t/y = a/v; hence, if from the equation t/y = a/v, we elim-
inate t and y by the help of equations (i) and (ii), that equation
should be produced which represents the figure that has to be
quadratured ; and by comparing the terms of the equation thus ob-
tained with the given equation, unless indeed there is no possibility
of comparing them, we shall obtain the quadrature.
But if an impossibility arises, it is then known that the given
analytical figure has no analytical quadratrix. But it is quite clear
that if we add to it such as will change it almost imperceptibly, then
a quadrible figure may be obtained, since this plainly produces an-
other equation. However, as an impossible case may arise, we must
consider the difficulties.
Say that the equation that is obtained is of infinite prolixity,
while the given one is finite. My answer is, that in comparing the
one with the other it will be seen how far at most the powers of
the unknowns in the indefinite equation can go. The retort may
be made, that it may happen that the indefinite equation obtained
may have more terms than the finite equation that is given and yet
may be reduced to it, for it may be divided by something else that
is either finite or indefinite. This difficulty hindered me for a long
time a year ago, but now I see that we should not be stopped by it.
For it may happen that from a certain determinate figure (whose
equation is not divisible by a rational) by the method of tangents
there may arise an ambiguous figure ; for it is impossible to say
that, for any figure, there shall be only one tangent at any one point.
Hence the produced equation can neither be divided by a finite nor
by an indefinite quantity ; for in truth indefinite figures, or those
whose ordinates are represented by an infinite equation, have some-
times these very ordinates finite, and these ought to satisfy the
equation. Notwithstanding that, I foresee another difficulty ; for
indeed it seems that sometimes it may happen that all the roots of
the equation will not serve for the solution of the problem. Yet,
to tell the truth, I believe they will do so.
Now here is a difficulty that really is great. It may happen
that a finite equation may also be expressed as an indefinite one,
so that the equation obtained may really be the same as the given
equation although it does not appear to be. For example,
<y2 = x/( 1 + x*) = x - xz + x3 - x* + x* - x6 + etc. ;
and in the same way others can be formed by various compositions
and divisions. This I confess is truly a difficult point, but it can be
THE MANUSCRIPTS OF LEIBNIZ.
257
answered thus : If a figure has an analytical quadratrix of any sort,
in all cases it may be assumed to be an indefinite one ; and then it
will not in all cases give an indefinite, but sometimes a finite, equa-
tion that is equivalent to the given equation. In the same way,
it is certain that the quadratrix of a given curve as it is usually
investigated, whenever there is one, will also be determined ; and
that too given uniquely and not ambiguously, so that any that differs
from it, differs only in name. There is still one difficulty left; it
seems impossible to determine which is the end or first term of the
indefinite equation that is obtained ; for it may happen that the terms
of lower degree are cut out, and then it is divisible by y or x
or yx or powers of these; nor do I see that there is anything to
prevent this. There is the same difficulty whether you start from
the lowest or the highest degree in the equation assumed to begin
with as indefinite. Suppose then that in the equation obtained this
division is possible, then it is necessary that the constant term
should be absent, and also all those terms in which x alone or, if
you like, all the terms in which y alone is absent ; and if we examine
this continuously we may light upon an impossibility.
In this general calculus then, we may take it as certain that
this difficulty is solved, and that such a division after the calcula-
tion can never happen; or if it is possible for it to happen, then
the terms will go out, one after the other, so that the equation can
be depressed and the comparison be made ; and then it is to be seen
whether this difficulty cannot be overcome in general, and the com-
parison proceed as we proceed with the elimination. Perhaps if
the figure to be quadratured is reduced beforehand to its simplest
equation possible, impossibilities will the more readily be detected.
For then presumably the quadratrix must become more simplified.
In addition we have another source of assistance ; for various cal-
258 THE MONIST.
culations leading to the same thing, though obviously differing from
one another, can be contrived, from which equations are comparable.
Let BL = WL = /, EP = , TB = t, GW = a, then = omn.l.
Incidentally I may remark that there are composite numbers
that cannot be added or subtracted from one another by parts,
namely those denominated by powers, or by sub-powers or surds.
There are also other denominate numbers which cannot be multi-
plied together by parts, such as numbers representing sums ; for
instance, omn.l cannot be multiplied by omn.p, nor can we have
y = 2omn. omn.pl. However, as such a multiplication may be im-
agined to occur under certain conditions, we must consider it as
follows :
We require the space that represents the product of all the
p's into all the I's; we cannot make use of the ductions of Gregory
St. Vincent, where figures are multiplied by figures, for by this
method one ordinate is not multiplied by all the others, but one into
one. You may say that if one ordinate is multiplied by all the rest
it will produce a sursolid space, namely, the sum of an infinite num-
ber of solids. For this difficulty I have found a remedy that is
really admirable. Let every / be represented by an infinitely short
straight line WL, that is, we want the quadratrix line representing
omn. / ; well, the line BL = omn. / ; and if this is multiplied by every
p, each represented by a plane figure, then a solid is produced.
If all the I's are straight lines and all the p's are curves, a curved
surface is produced by a duction of the same sort. But these things
are all old; now, here is something new.
If upon WL, MG, or every single /, is superimposed the same
curve representing all the p's, where the curve p is originally all in
the same plane and is carried along the curve AGL while its plane
always moves parallel to itself, then what we require will be ob-
tained. In place of a curve a plane may be carried along the curve
in the same manner, and a solid will be obtained, whereas by the
former method it was a curvilinear surface; and both for the sur-
face and for the solid the section always remains the same. It
remains to be seen whether a number of analytical surfaces cannot
be ascertained, as in the case of analytical lines ; but this is men-
tioned only incidentally.
N. B. The curvilinear surface formed by the motion of a
curve parallel to itself along the curve will be equal to the cylinder
THE MANUSCRIPTS OF LEIBNIZ. 259
of the curve under BL, the sum of all the I's but this is also men-
tioned incidentally.
To resume, — = — — — = y, therefore p=~^— /. Hence,
a omn. / a
omn. y - does not mean the same thing as omn.y into omn./, nor yet
a
y into omn./ ; for, since p = — / or - ™— /, it means the same thing
a a
as omn./ multiplied by that one / that corresponds with a certain
p; hence, omn./> = omn. — /. Now I have otherwise proved
a
omn.p= ¥-, i. e., = — ir~'> therefore we have a theorem that to me
Lt £
seems admirable, and one that will be of great service to this new
calculus, namely,
omn. I2 - , / , ,
- — = omn. omn./—, whatever / may be;
2 a
that is, if all the I's are multiplied by their last, and so on as often
as it can be done, the sum of all these products will be equal to half
the sum of the squares, of which the sides are the sum of the /'s
or all the I's. This is a very fine theorem, and one that is not at all
obvious.
Another theorem of the same kind is:
omn.^r/ = x omn./ - omn.omn./ ,
where / is taken to be a term of a progression, and x is the number
which expresses the position or order of the / corresponding to it ;
or x is the ordinal number and / is the ordered thing.
N. B. In these calculations a law governing things of the same
kind can be noted ; for, if omn. is prefixed to a number or ratio, or
to something indefinitely small, then a line is produced, also if to
a line, then a surface, or if to a surface, then a solid ; and so on
to infinity for higher dimensions.
It will be useful to write J for omn., so that
j*/ = omn./, or the sum of the I's.
Thus,
From this it will appear that a law of things of the same kind
26O THE MONIST.
should always be noted, as it is useful in obviating errors of cal-
culation.
N. B. If (I is given analytically, then / is also given ; therefore
if j* J7 is given, so also is /; but if / is given, J*/ is not given as well.
In all cases (x = x'*/2.
N. B. All these theorems are true for series in which the
differences of the terms bear to the terms themselves a ratio that is
less than any assignable quantity.
/& _ •*_
" 3
Now note that if the terms are affected, the sum is also
affected in the same way, such being a general rule ; for example,
I — I = ^- x I / , that is to say, if ^ is a constant term, it is to be
^ b b J b
multiplied by the maximum ordinal ; but if it is not a constant term,
then it is impossible to deal with it, unless it can be reduced to terms
in /, or whenever it can be reduced to a common quantity, such as
an ordinal.
N. B. As often as in the tetragonistic equation, only one letter,
say I, varies, it can be considered to be a constant term, and J/ will
equal x. Also on this fundamental there depends the theorem:
/ -/?
//, that
Hence, in the same way we can immediately solve innumerable
things like this ; thus, we require to know what e is, where
c
1/
J
/3 _f_ I /3 _ ^3.
S a~ J
we have
«j CJC^ j 9 ^ o
ar<? = — + barx + T~ + **•
«5 TC •
For indeed Cl3 = x, because / is supposed to be equal19 to a for the
purpose of the calculation ; — = x.
J a
18 There is evidently a slip here ; / should be x.
19 This is an instance of the care which Leibniz takes ; in the work above
/ has been the difference for y, and a the difference for x ; he is now integrating
an algebraical expression, and not considering a figure at all.; hence /==o, and
o is equal to unity, and therefore / I3 = fix — azx = x \ Thus what is gen-
erally considered to be a muddle turns out to be quite correct. The muddle
is not with Leibniz, it is with the transcriber. It is certain that these manu-
scripts want careful republishing from the originals ; won't some millionaire
pay to have them reproduced photographically in an edition de luxe?
THE MANUSCRIPTS OF LEIBNIZ. 26l
da.
Also fcJp = C-?~, that is = CJJ- f ba2 = f /
3a? ' ^
Also it is understood that a is unity. These are sufficiently new and
notable, since they will lead to a new calculus.
I propose to return to former considerations.
Given /, and its relation to x, to find j*/.
This is to be obtained from the contrary calculus, that is to say,
suppose that fl = ya. Let l = ya/d ; then just as J will increase, so d
will diminish the dimensions. But J* means a sum, and d a differ-
ence. From the given y, we can always find y/d or /, that is, the
difference of the y's. Hence one equation may be transformed into
the other; just as from the equation I c J I2 = c , we can ob-
J 3a3
tain the equation c (~ft— ^
" 3a*d
N-B . f*? + f^=: (*?+£?«. Andinthe
J b J e J b e
, x* . x^a
X Xrd - H --
same manner, — 4- -; — — b e .
do de
d
But to return to what has been done above. We can investi-
gate J/ in two ways; one, by summing y and seeking ya/d = l;
the other, by summing z2/2a = y, or by summing ^/2ay = z, and
then zz/t = p = l=ya/d. Hence, if in an indefinite equation, we
eliminate y by substituting in its place zz/2a, and investigate the
t of this new equation which is indefinite like the first, and
then by the help of the value zz/t = l, and after that by the help of
the new value of t, eliminate z from the indefinite equation con-
taining z and t, there will remain out of the (three) letters x,z,t,l,
the letter / alone ; and again we ought to get an equation which
should be the same not only as the given one, but also the same as
the one that was obtained a little while ago. Hence, since we have
two indefinite equations, containing not only the principle quanti-
ties, but also arbitrary ones, yet not altogether unlike the former;
and these ought to be identical ; it will appear to show whether
certain terms cannot be eliminated, whether it is not possible that a
comparison should be made, and other things of the sort ; and, what
is really the most important thing, which terms are really the
greatest and the least, or the number of terms of the equation.
Moreover, since in the similar triangles TBL, GWL, LBP, no
262
THE MONIST.
mention has yet been made of the abscissa x or of the fixed point A,
let us then suppose that through the fixed point A there is drawn
an unlimited straight line AIQ, parallel to LB, meeting the tangent
LT in I; and let AQ = BL; bisect AI in N; then I say that the
sum. of every QN will always be equal to the triangle ABL, as can
easily be shown by what I have said in another place.20
N \l
B
(B)
These considerations give once more a fresh fundamental theo-
rem for the calculus. For xv/2 = y, where we suppose that ~BL, = v
and QN = /, and y= J7;
t—x
, AI t-x
but — = —
v t
therefore AI =
xv
and QI = v — AI = v — v + — , i. e. QI = — ,
* 2* t
(21)
2 / ' 2 2/ 2t
Now, by the help of the equation (xv+tv)/2t = l, and of the former
equation y=xv/2, and taking once more the first indefinite or gen-
eral equation as a third, and eliminating first of all y, then t by
means of the value found for the ratio of t to x from the indefinite
equation containing x and v, and lastly v by the help of the equation
(xv+tv)/2t = l, in which the principal quantities x and / alone re-
main, as before; and this again should be identical with the given
equation.
Thus we have found three equations obtained in different ways,
which should all be identical with one another and with the given
equation ; and these three are not only identical but should also
20 Since the triangles QLI, WL(L) are similar, QI.B(B) = QL.Q(Q),
hence omn.QI (applied to AB)=omn. QL (to AQ) = figure AQLA, hence
omn.(QI + QA) = rect. ABLQ = 2AABL.
21 Since / is the difference for y, therefore 21 is the difference for xv ;
this is shown to be {xv + tv)/t or x(v/t) +v; and this is the equivalent to
(since v/t = dv/dx — dv)
d(xv) = xdv + v = xdv + vdx.
THE MANUSCRIPTS OF LEIBNIZ.
263
consist of the same letters and signs; and whether this is possible,
will immediately appear on being worked out analytically.
§ VII.
The next manuscript is a further continuation of the
preceding, written two days later. In this Leibniz returns
to the idea that he has found so prolific, namely, the mo-
ments of a figure. It is to be observed that he speaks of
the method of breaking up an area into segments as some-
thing that he has already worked out ; this will be remarked
upon in a note on a later manuscript, where it will help to
clear up a small difficulty. The accuracy of the rather in-
volved algebraical work is also a point to be noticed.
1 November, 1675.
Analyseos Tetragonisticae pars tertia.
(Third part of Analytical Quadrature.)
It was some time ago that I observed that, being given the
moment of a curve ABC, or of a curvilinear figure DABCE, about
two straight lines parallel to one another, such as GF, LH (or MN,
M
0
n
L
G,
r
XB
e
\
\
L
H
Q
N
PQ), then the area of the figure could be obtained; because the
two moments differed from one another by the cylinder of the
figure, where the altitude was the distance between the parallels.
Now, this is true of every progression, whether of numbers
or of lines ; that is, even if we do not use curvilinear figures but
ordinated polygons; in other words, where the differences between
the terms are not infinitely small. Suppose we have any such
ordinated quantity z, and let the ordinal number be x, then
b omn.z •"" ± omn.zx q= omn.zx+ b
and this is evident by the calculus alone.
By the help of this rule, the sums of terms of an arithmetical
264
THE MONIST.
progression refolded reciprocally ;22 and this multiplication takes
place when it is required to find the moment of the ordinates about
a straight line perpendicular to the axis. But if the moment about
any other straight line is required, there is the following general
rule:
From the center of gravity of each of the quantities of which
the moment is required, a perpendicular is drawn to the axis of
libration ; then the sum of the rectangles contained by the distances
or perpendiculars and the quantities will be equal to the moment
about the given straight line.
Hence, if the given straight line is the axis of equilibrium,
it immediately follows that the moment of the figure about the axis
is equal to the sum of the half-squares. Also when it is parallel
to that, it will differ from the foregoing by a known quantity.
Now, let us take another straight line : for the circle for instance,
let ABCD be a quadrant, vertex B, and center D ; let another straight
line be given, that is to say, let the prependicular DF be given and
also EF where it meets the diameter, and thus also DE; let HB
be the general ordinate to the circle, and L its middle point; let
LM be drawn perpendicular to EF.
Then it is clear that the triangles EFD, EMN (where N is
the intersection of ML and AD), and LHN are similar.
y
Let
= *, then HL = - =
NH
But, on account of
the similar triangles, ^^r = ^^7 — T\ >
ri L> r r/ (^ =/ )
therefore
22 The meaning of this is probably a series such as that considered by
Wallis. If a, a -f d, a -\- 2d, etc. is the arithmetical progression, and /, / — d,
I — 2d, etc. is the series reversed, then the series refolded reciprocally is al,
(o + rf)(/ — d), (a-\-2d)(l — 2d), etc. It may however mean the sum of the
squares of the arithmetical progression. B.ut the point is not very important.
THE MANUSCRIPTS OF LEIBNIZ. 265
Hence, EN = DE(=0 -HD(=*) -NH ( = -^ =<-*-^.
Now NL=
MN NH NH.EN
and =77- = TTT » or MN = — rrj ; thus we have
and
'-D*
hence, since ^= -<ffi—d2, we have 23
and this calculation is general for any curve, so long as x is always
taken as the abscissa and y as the ordinate.
Therefore the rectangle contained by ML and HB (=y), or the
moment of each ordinate taken with regard to the straight line EF,
or wa, will be equal to
Hence, omn.w will be obtained from the known values of
omn.jr, omn.xy, and omn.y2 ; also, if any three of these four are
given, the fourth is also known.
Now, omn..ry will be equal to the moment of the figure about
the vertex, omn. y2 will be equal to the moment of the figure about
the axis ; hence, given three moments of the figure, that is to say,
the moments about two straight lines at right angles and any third,
the area is given.
This theorem, however, is less general than the one that was
given before, in the first part of this essay, where it does not matter
28 The accuracy of the algebra is noteworthy in comparison with the in-
accuracies that occur later. There is however a slip : ez = fj -\- d2 and not
f2 — d2 ; this must be a slip and not a misprint, because it persists throughout.
It should be noted that the figure given by Gerhardt is careless in that LM is
made to pass through A.
266 THE MONIST.
what the angle between the straight lines may be, if only we are
given three moments; but it is always understood that they are in
the same plane. ( Meanwhile, however, this theorem will suffice for
the curve of the primary hyperbola ; for, if / is infinite, or if FE
and ED are parallel, dy + y2/2 = wa, as has already been proved.)
It is to be observed that by other calculation the area of a
quantity, whose center of gravity lies in a given plane (even though
the whole quantity does not), can be found from three given
moments about three straight lines in that plane. From this it is
to be seen whether the results obtained, when compared with one
another, will not produce something new.
If instead of the moment of a figure we require the moment
of all the arcs BP, PC, etc., the perpendiculars are to be drawn
from the points B, P, C, etc. only, to the straight line; for it will
make no difference whether they are drawn from the end or from
the middle of BP, for instance, for the difference between two such
perpendiculars is infinitely small. Hence, calling the element of
the curve z, the moment of the curve about the straight line EF is
d V/2 - dzz - dxz + fyz
~
Most of the theorems of the geometry of indivisibles which
are to be found in the works of Cavalieri, Vincent, Wallis, Gregory
and Barrow, are immediately evident from the calculus; as, for
instance, that the perpendiculars to the axis are equal to the surface
or moment of the curve about the axis, for you find that a perpen-
dicular is equal to the rectangle contained by an element of the
curve and the ordinate. Therefore I do not set any value on such
theorems, or on those about applications of intercepts on the axis
(intercepted between the tangents and the ordinates) to the base.
Such theorems bring forth nothing new, except maybe they afford
formulas for the calculus.
But my theorem about the dimensions of the segments does
bring out a new thing, because the space whose dimension is sought
is broken up in a different way, that is to say, not only into ordi-
nates but into triangles. Also perhaps the Centrobaric method
yields something new. Maybe an easy method can be obtained, by
which without diagrams those things which depend on a figure can
be derived by calculus. Gregory's theorem, on ductions of two
THE MANUSCRIPTS OF LEIBNIZ. 267
parabolas,24 one under the other, equal to a cylinder, is immediately
evident by calculus; for the ordinate of a circle y=^/a2-x2, that is,
the product of -\/a + x and ^a-x; and in the same way, -\/2av-v2
= y, which gives y=V^ mto ^2a-v; and these come to the same
thing.
If the same ordinate y is multiplied by some quantity z, and
afterward by the same z ± some known or constant number b, the
difference between the sums produced will be equal to the cylinder
of the figure; so that
zy,,-zy + by """ by.
Although this is evident in general by itself, yet applications of it
are not always evident. For instance, let
x2 x2
y —
ax-b* ,
x2
then, multiplying by ifax + bt we have
— o
and, multiplying by ^ox — b, we have 1= — ~ — 5
•\ax + b
ax2 bzx
but, since instead of — — , we can have x + TT — ,
ax — e>2 ax — b£
which depends on the quadrature of the hyperbola ; and thus if one
of the two things, (A) or (B), is given, then the other is also
known, supposing that the quadrature of the hyperbola is known.
Suppose that at the points C, D, E of a curve situated in any
plane there are imposed, perpendicular to the plane, the ordinates
of another curve FGH (not necessarily of the same constitution),
in such a manner that the middle point of each of these ordinates
lies in the plane; then it is evident that LG, MD, NE, multiplied
by FL, GM, HN, (that is, the lines imposed at C, D, E of the curve
BCDE) or the rectangles FLG, GMD, HNE, or the duction of
these two planes into one another, will be equal to the moment of
every LC, MD, NE, etc. Hence, if PR is another axis, and the
interval between it and QL is the straight line PQ, the moment
24 Such theorems are also considered in Wallis, where it is shown that
the products for two equal parabolas are the squares on the ordinates of a
semicircle ; the axes of the parabolas being coincident, but set in opposite sense.
268
THE MONIST.
about PR differs from that about QL by the cylinder whose base
is LC, MD, etc., and whose height is PQ.25
But, if the moment about the straight line PQ, and also that
about some other straight line in another position, as TS, of all the
ordinates LF of the same figure, imposed at the points C, then we
shall have the cylinder corresponding to all the LF's, as I will now
prove.
/ g
If we call QL, x, and CL, y, then TC= — x+ -y + h\ and this
multiplied by z, where FL or MG = z, will give
/ g
— xz + — yz + hz .
a a
Now xz is given, being the supposed moment about PQ, which is
the same whether the s's are placed where they were in the lines
LF, MG, etc., or at the points C, D, E. Also yz is given, either
as the rectangle FLC or as the duction, by hypothesis. Hence, if in
addition there is given one moment of the ordinates imposed upon
25 This is obviously wrong ; the base of the cylinder is the area made up
of FL, GM, HN, etc. The whole of this last passage proved to be difficult to
make out ; Leibniz has not completed his figure, by showing the surface formed
by placing the ordinates FL, GM, HN with their middle points at C, D, E,
and the ordinates themselves perpendicular to the plane of the curve BCDE,
which figure I have added on the right-hand side of Leibniz's figure. Even
when this is given, there is another difficulty added because as given by Ger-
hardt, CS is the tangent at D instead of the proper line, namely, the perpen-
dicular from C to TS ; in addition through a misprint, this line is afterward
referred to as TC. Lastly, "the rectangle FLG" is a misprint for FLC, which
with Leibniz stands for FL.LC; this notation for a rectangle is, as far as I
can remember, used by Wallis and Cavalieri.
When all these errors are revised, what at first sight seemed to be rather
a muddle turns out to be an exceedingly neat idea in connection with the
moments of a figure, and their use to find an area, although mostly imprac-
ticable.
Note. The values f, g, a, h, are the lengths of TQ, QP, PT, and the per-
pendicular from Q on PT.
THE MANUSCRIPTS OF LEIBNIZ. 269
the curve at the points C, D, E, and this is taken to be equal to
/ K
— xz + -ys + hz, then we have hz or the cylinder required.
Hence, the curve BCDE is to be chosen such that the ordi-
nates of the given curve can be multiplied by different ordinates of
the former, drawn either to the axis QL or to the axis TS, with
some advantage of simplicity; and the curves that are suitable for
this are those that have several suitable axes, such as the circular
or primary hyperbola, which has a pair of asymptotes, or an axis
and a conjugate axis.
§ VIII.
Much comment has been made on the fact that the date
of the next manuscript was originally "n November
1675"; that the 5 had been altered to a 3, the ink being
of a darker shade; and that it is almost certain that this
alteration in date was made for some ulterior motive by
Leibniz himself. Hence, if he was capable of falsifying a
date in one particular case, then he is not to be trusted in
others, . . . . , and so on. Instead of trying to explain away
this alteration, let us try to find an explanation as to the
reason of its having been made by Leibniz; I offer the
following as at least feasible.
The essay starts with the words, '7am superiore anno
mihi proposueram qucstionem, . . . . " I suppose that by this
Leibniz intended: "A year or two ago, I set myself the
question, " This conforms with what follows ; the
theorem that he sets down is one such as those that were
suggested to him by Huygens, and further theorems that
came to him as deductions during his first intercourse with
Huygens. Years later, I therefore suggest, Leibniz refers
to this manuscript, reads his own Latin, superiore anno, as
"in the above year," gets no further, recognizes the theo-
rem by its figure as one of the Huygens-time batch, and
says to himself "1675 ? No, that's wrong, should be 1673,"
2/O THE MONIST.
and proceeds to alter it to what he remembers was the
date for the first consideration of the theorem.
N. B. Gerhardt himself has remarked on the darker
tint of the ink used in the alteration ; hence my argument,
made at a later date.
The date 1675 is incontestable; for this composition is
quite glaringly a development of the work that has been
so efficiently started in that of November i, 1675. Progress
is still delayed by the idea that has obsessed Leibniz up till
now, that of the transformation of equations, so as to be
able to eliminate more unknowns than the original number
of his equations warrant. He sets himself the problem:
"To determine the curve in which the distance between the
vertex and the foot of the normal is reciprocally propor-
tional to the ordinate," i. e., the solution of the equation
x + y dy/dx = a?/y, in modern notation. This is a very
unlucky choice for him: for I have it on the authority of
Prof. A. R. Forsyth that this is incapable of solution in
ordinary functions or even by a series in which the law
of the series is easily and simply expressible — at least he
confesses that he is unable to obtain such a solution, which
I take it comes to the same thing.
Leibniz professes to have found the solution and gives
(y2 -(- xz) (a* — yx) = 2y2 logy, and unfortunately this
false success but enhances the value in his eyes of the
method mentioned above. But from the equation given as
the solution we may draw an incontestable conclusion ; for
in a previous problem Leibniz verifies his solution by the
method of tangents, i. e., by differentiation, although the
method does not as yet convey that idea to him ; but he does
not verify the solution in this case, because he is unable at
this date to differentiate the product y2 logy.
The introduction of dx instead of x/d marks a further
advance, more important perhaps than the use of fy dy ;
THE MANUSCRIPTS OF LEIBNIZ,
for he still writes $x, considering dx to be constant and
equal to unity. He is beginning to grasp the infinitesimal
nature of his calculus, and that infinitesimals are not to be
neglected because of their intrinsic smallness, but because
of their smallness with respect to other quantities which
come into the same equations and are finite; but he is far
from being certain about it as yet, as is evidenced by the
discussion as to whether d(v/ty} = dv/dty or not. How-
ever, the whole manuscript marks a distinct advance on
anything that has gone before. From now on he probably
discards geometry, and only refers to Descartes, Gregory
and Barrow for examples to show how much superior is
his method to theirs. I put his final reading of Barrow
down to the interval between the date of this manuscript,
ii November, 1675, and November, 1676; it is at this
time that he inserts his sign of integration in the margins
of the theorems. The next person that examines the orig-
inals of these manuscripts (I am convinced that this is
very necessary), should carefully see whether the ink used
for the note "novi dudum" (which I have mentioned) is
the same as that used for the sign of integration ; also the
other books that were used by Leibniz in his self-education
should be searchingly scrutinized for clues.
The last remark I have to make is one of astonishment
at the errors in the algebraical work which brings this
essay to a close, and to a less degree throughout the essay ;
for we have seen the accuracy to which Leibniz has at-
tained in a previous manuscript ; of course, a great deal of
erroneous work can be explained by supposing none too
careful transcription ; but a re-examination of the whole of
the Leibnizian remains should include a careful scrutiny
on the point as to whether some of the extracts given by
Gerhardt are not the work of pupils of Leibniz, whose
writing would naturally be somewhat similar. Perhaps
too some of those early geometrical theorems might be un-
2/2
THE MONIST.
earthed ; and this would well reward the most painstaking
search. Nobody can assert that anything like an adequate
tale of the progress of the Leibnizian genius has so far
been told.
11 November, 1673.28
Methodi tangentium inversae exempla.
(Examples of the inverse method of tangents.)
A year or two ago I asked myself the question, what can be
considered one of the most difficult things in the whole of geometry,
or, in other words, what was there for which the ordinary methods
had contributed nothing profitable. To-day I found the answer
to it, and I now give the analysis of it.
Find the curve C(C), in which BP, the interval between the
ordinate BC and PC the normal to the curve, taken along the axis
AB(B), is reciprocally proportional to the ordinate BC.
Let AD(D) be another straight line perpendicular to the axis
AB(B), and let ordinates CD be drawn to it, so that the abscissae
AD along the axis AD(D) are equal to the ordinates BC to the
axis AB(B), and the ordinates CD to the axis AD(D) are equal
to the abscissae AB along the axis AB ( B ) . Let us call AD = BC = y,
and AD = BC = *; also let BP = w and B(B)=*. Then it follows
from what I have proved in another place that
29 See Cantor, III, p. 183 ; but neither Cantor nor Gerhardt appears to
offer any suggestion as to why this date should have been altered.
THE MANUSCRIPTS OF LEIBNIZ. 273
y y2 27
Jwz= 2' °rWZ2d
yZ
But from the quadrature of a triangle it is evident that— 3=.?;
and therefore wz = y.
Now, from the hypothesis, w = b/y, for thus w and 3; will be
reciprocally proportional to one another. Hence we have
bz y2
— =y, and thus z = ^ .
/y2
-r ; and from the quadrature of the
/y2 y3 y3
-r = —r- ', hence, x= -=^— ; and this is the required
equation expressing the relation between the ordinates 3; and the
abscissae x of the curve C(C), which was to be found. Therefore
we consider that the curve has been found and it is analytical; in
short, it is the cubical parabola whose vertex is A.
We will therefore see whether the truly remarkable theorem is
not true, namely, in the cubical parabola C(C), the intervals BP
between the normals to the curve, PC, and the ordinates to the
axis, BC, taken along the axis ABP, are reciprocally proportional
to the ordinates, BC.
The truth of this is easily shown by the calculus of tangents.
For the equation to the cubical parabola is xc* = y3 ; taking c to be
the latus rectum, and supposing that for c2 we put 3ba, or c=^3ba,
we have 3xba = y3.
Now, by Slusius's method of tangents, we have t = y*/3ba,
where t is put for BT, the interval along the axis between the
tangent and the ordinate.
v2
y9 — ba
But BP=w=— , and therefore w= y^ == — ; hence, the w's
1 ba y
and the y's are reciprocally proportional as was to be proved.
27 This was obtained in the form omn./> = y*/2, previous to October, 1674,
from the Pascal form of the characteristic triangle ; it is quoted as a known
theorem in the essay dated 29 October, 1675. See §§ III, VI.
It is probably at this date that he began to revise his ideas as to d dimin-
ishing the dimensions ; being forced to reconsider them by the occurrence of
such equations as wz = y. It is seen in the next paragraph how careful he is
to keep his dimensions equal; for he introduces an apparently irrelevant
a(= 1) for this purpose. It gradually dawns on him that neither / nor d alter
the dimensions, but that a "sum of lines" is really a sum of rectangles, on
account of the fact that they are applied in a certain fixed way to an axis;
he is not quite certain of this however until well on in the next year, when
we find him using fdx y.
2/4 THE MONIST.
The artifice of this analysis28 consisted in obtaining the abscissa
from the ordinate; and this idea was never previously thought of.
It is not a more difficult question either, if the curve is required
in which BP, the interval between the normals and the ordinates,
is reciprocally proportional to the abscissae AB. Indeed, iv=az/x;
but w = yz/2 ; hence, we have
— 12 C w or /2 i a2
~V J \ J *•
Now fw cannot be found except by the help of the logarithmic
curve.29 Hence, the figure that is required is that in which the
ordinates are in the subduplicate ratio of the logarithms of the
abscissae; and this curve is one of the transcendental curves.
Now, in truth, it is a much harder question,30 if the curve, in
which AP is reciprocally proportional to the ordinate BC is re-
quired.
a2 y2 f
For then x + w= — and wz=—-. also \z = x,
y 2<t J
, = %', thus, «^ = |1, and «^w£;
y2 x a?
hence, *+_w_=_.
If we suppose that the x's are in arithmetical progression then
x/d-z will be constant, and we shall have
/ a2 r Cc? /
±— — — or I #= I — ,
2d y J J y 2
y
therefore
*2 ^ (* v
- + — = I - or d
J
2 2 y y
28 It is difficult to see exactly what Leibniz means by this statement ; I
can only guess at substitution by means of the theorem ws = y, the equivalent
to the recognition of the fact that y dy/dx . dx = ydy. The wording is however
impersonal, and may mean that he himself had never thought of the idea
before.
29 Required y = /(*), such that y dy/dx = a2/x; the solution is y2 = 2a2
logeA^r. Weissenborn remarks on the omission of the o as being incorrect;
from Leibniz's standpoint I cannot agree with him. Leibniz, from Mercator's
work, connects az/x with the ordinate of the equilateral hyperbola xy = a2,
and its integral with the quadrature of this curve. The omission of the a2
only alters the base of the logarithm, and Leibniz merely states that the solu-
tion is of a logarithmic nature without attempting to give it exactly.
30 How does he know until he has tried it? This rather combats the idea
that these were mere exercises ; it gives this essay the appearance of being a
fair copy intended either for publication or for one of his correspondents. If
this were the case, the errors later in algebraical work are all the more un-
intelligible. The idea that Leibniz was a man who was accustomed to writing
down his thoughts as he went along does not appeal to me at all ; this is the
method of the slow-working mind, rather than that of genius.
THE MANUSCRIPTS OF LEIBNIZ. 275
but, if we join AC, A(C), then these are equal to ^/*2 + y2; and if
with center A and radius AC we describe an arc CE to cut the
straight line AE(C) in E, then E(C) will be the difference between
AC and A(C) ; that is, E(C) =e = dxTTf
•'• e = 2a?/y.
If then it were allowable to assume that the y's were also in
arithmetical progression, we should have what was required; yet
it seems that it does not make any difference even if the x's have
been assumed to be in arithmetical progression. For if we do
assume that the JF'S are in arithmetical progression, it follows that
the AD's, or the y's are the reciprocals of the E(C)'s or the e's.
Moreover, if they are so at any one time they are so at all times. Also,
the sums of an infinite number of reciprocal proportionals, no matter
what the progression may be of which they are taken as the recip-
rocal proportionals ; for in this case there is not any consideration
of rectangles, where there is need of equal altitudes, but a sum of
lines is calculated, that of all the E(C)'s.31 Hence I see the difficulty
arise from the fact that the sum of every e, or every 2az/y, or every
E(C), cannot be obtained, unless we know to what progression the
y's belong. In this case, that information is not given ; for it is
necessary that the .r's should be in arithmetical progression, and
hence that the y's are not so.
On the other hand, if we suppose in the above equation,
y* x a?
+ 2d^d = J'
that the y's are in arithmetical progression, then we have
y a2 /
x + — = — or xy + — =a2 ;
dx y dx
and, finally, by assigning the progression to neither x nor y, we have
in general
xy+y =
But we have not as yet really obtained anything. Let us
therefore consider it from the standpoint of "indivisibles" ; let PCS
produced meet AD in S ; then the sum of every AP applied to AB
31 This seems to be the root of the error into which he falls ; he has not
yet perceived that the e's have to be applied to some axis, before he can sum
them ; and this is to a great extent due to the omission of the dx, taken as
constant and equal to unity. He is thus bound to fall back on the algebraical
summation of a series.
2/6 THE MONIST.
is equal to the sum of every AS applied to AD ;82 or calling DS, v,
we have
dy fy + dy $v = dx §x + dx j* w,
or
by the hypothesis of the question.
Now, if we take the y's to be in arithmetical progression, we
have
r2 .x2
~ + y = </*Logy. 33
But just above, making the same supposition that the y's were in
arithmetical progression, we had
xy+ — = a or dx= -»
»
dx a—xy
and now we have
£,:*+£,
dx- - ---
* Logy
Hence at length we obtain an equation, in which x and y alone
remain, and unshackled, namely
yz + xz, a?-yx=2y2L,ogy ;
and this equation, since it is determinate, will give the required
locus.
This then is an exceedingly remarkable method, for the reason
that when it is not in our power to have as many equations as there
are unknowns, yet often we shall be able to obtain some more
equations, by the help of which we shall be able to eliminate certain
terms, as the term d.x in this case, which alone stood in our way.
Either of the two equations, by itself, contained the whole nature
of the locus, although from neither of them could the solution be
derived, because so far easy means were lacking; yet the combina-
tion of the two equations gave the solution at once.
I see that the same thing could be otherwise obtained by
moments ; and here there comes to my mind a new consideration
that is not altogether inelegant.
32 From the characteristic triangle, AS : AP = dx : dy.
33 This is of course nonsense. The error seems to arise from the dx being
placed outside the integral sign ; thus he assumes that dx is constant, while, for
the integration, he also assumes that the dy is constant.
We cannot argue from this equation that Leibniz did not at this date
appreciate what an infinitesimal was, on account of the infinitesimal being
equated to a finite ratio ; for since he is assuming that dy is an infinitely small
unit, dx really stands for dx/dy.
THE MANUSCRIPTS OF LEIBNIZ.
277
In the attached figure, let EC = y, FC = dy; let S be the middle
point of FC; then it is evident that the moment of FC is the
urn «s»
rectangle contained by FC and BS, i. e., the rectangle BFC; this
follows from the fact that it is equal to BFC+SFC, and the latter
can be neglected as being infinitely small compared to the former.34
Hence fydy = y2/2, or the moment of all the differences FC
will be equal to the moment of the last term, and ydy = d(yz/2), or
y*dy = y dy2/2.
Now, just above, in equation (A), by making x arithmetical,
we had
y d— = a*-xy , or d— =
a —xy
a —xy
but this is the same thing as y dy ; hence y dy = -- - , and therefore
— — f~* x*
y dy — I ~ — —
J V i
But we have already found that
C~=L y*
I y dy=^\
J i
therefore y2 + x* = 2 I — , as before; i. e., dx3 + y* =
From this there follows something to be noted about these equa-
tions, in which occur j* and d, where one quantity, in this case for
instance the x, is taken to proceed arithmetically, namely, that we
cannot make a change, nor say that the value of x is found, thus,
x=2(az/y) -dy2; for dy2 cannot be understood unless the nature
of the progression of the y's is determinate. But the progression of
the y's, in order that it may be used for d y2, must be such that the
xfs are in arithmetical progression ; hence the dy's depend on the
^r's, and therefore the x's cannot be found from the dy's. For the
rest, by this artifice many excellent theorems with regard to curves
that are otherwise intractable will be capable of being investigated,
namely, by combining several equations of the same kind.
In order that we may be better trained for really very difficult
34 Note the advance in ideas suggested by the words "infinitely small
compared with the former." Here, of course, the notation BFC is the usual
notation of the period for BF.FC, the rectangle contained by BF and FC.
278 THE MONIST.
considerations of this kind, it will be a good thing to attempt just
one more, as for instance when the AP's are reciprocally propor-
tional to the AB's.
Here x + a/= — , and2w = a—t and z=dx; and so we obtain
X £ •
d~
w— — = — , hence x + — = x '
z dx dx
The solution of this is not now difficult ; for if we suppose that
the x's are arithmetical,35 we have
(36)
J
x+ =
Hence, ~\/x2 + yz = AC= \/2LogAD; and this is a simple enough
expression for the curve. In this however the AP's are required
to be in arithmetical progression ; but on the other hand, if the y's
are taken to be in arithmetical progression, we have x + y/dx = a?/x ;
and from this latter the nature of the curve is not easily obtained.
Let us see whether there can be a curve in which AC is always
equal to BP; in this case y/xz + yz = w} and w = dyz/2dx. Let the
^s be in arithmetical progression then (f^/*2 + y2=) JAC = y2;
this, however, is not sufficient to describe the curve practically,
that is to say, by points following one another consecutively. When
*=1, let BC=(y); then V1+ (/) = (y2), or 1 + (y2) = (?*).
Whence (y) may be obtained ; thus, from the equation
,yc (37)
, or (y) = .
Further, in the same way,
AC A(C)
and thus again ((y)) can be found. By the help of this a third
35 Note in general that this is Leibniz's equivalent of the modern phrase,
"integrate with respect to x."
36 This I think is more likely to be a slip on the part of Leibniz, than a
misprint; for in the next line he has AD, which is the correct equivalent of y.
Further, AP varies inversely as x, hence the AP's have to be in harmonical
progression, not arithmetical, otherwise x is not equal to x*/2. If on the
other hand, we assume three errors of transcription, and replace x for y, AB
for AD, AB for AP, the whole thing is correct with an arbitrary base.
37 It is hardly necessary to point out the error in the arithmetical solution
of the quadratic ; nor is it important. It is however to be noted that if AC = v,
the equation reduces to vz=.x(x -}-v), and the solution is a pair of straight
lines.
THE MANUSCRIPTS OF LEIBNIZ. 279
AC can be found, and some sort of polygon can be found, which
is more and more like the curve that is required, in proportion as
the thing taken for unity is less and less.
That the x's are in arithmetical progression signifies that the
motion (in describing it) along the axis AB is uniform. But
descriptions that suppose any motion to be uniform are not within
our power.38 For we cannot produce any uniform motion, except a
continually interrupted one.
Let us now examine whether dxdy is the same thing as dxy,
1C
and whether dx/dy is the same thing as d—\ it may be seen that
if y = z2 + bz, and x = cz + d; then
dy = z2 + 2pz + p2, + bz + bp, -z2-bz,
and this becomes dy = 2z + b/3.
In the same way dx = + cf$, and hence
dx dy = 2z + b eft2,
But you get the same thing if you work out dxy in a straight-
forward manner. For in each of the several factors there is a
separate destruction, the one not influencing the other; and it is
the same thing in the case of divisors.
Now let us see if there is any distinction when we seek the
sums of these things. We have ^dx-x, fdy = y, and $dxy = xy.
If then we have an equation, dxdy-x say, then §dxdy = §x. But
J*JT = xz/2, hence xy = xz/2, or x/2 = y ; and this satisfies the equation
dx x* (39)
dxdy = x; for substituting for y its value, ax— =x, or a— =x,
£ £
which is known to be true.
In sums these results do not hold good ; for §x j*y is not the
same thing as $xy, the reason is that a difference is a single
quantity, while a sum is the aggregation of many quantities. The
sum of the differences is the latest term obtained. However, from
the sums of the factors we can find the sums of products, not indeed
as yet analytically, but by a certain method of reasoning; such as
Wallis has done in this class of thing, not by proving them, but by
a happy method of induction. Nevertheless to find proofs for them
would be a matter of great importance.
38 This is strongly reminiscent of Barrow, Lect. I (near the beginning)
and Lect. Ill (near the end).
39 Leibniz, as a logician, should have known better than to trust a single
example as a verification of an affirmative rule.
With regard to infinitesimals note the equation dx dy = x !
28O THE MONIST.
Suppose J zy to be the sum that is required. Let J zy = wt
dw r [ dw 0. r [ dw
then zy = dw, and y= — , and J y= J — . Similarly, J z = J — .
& *5 jr
Suppose that Jy is known, = v, and that Jz is known, =<A; then y
dw ,. dw dv 2 „.
= dv= — , and z=dy= — , and -77 =— . From this it would seem
z y dy y
1) 2 1) ( 2 (2
to follow that d-r = -, and therefore that 7 = I -. Therefore I - =
y y T ~ y •* y
-j- , which is obviously incorrect. (40) Hence it follows that I -r:
v
cannot be equal to r .
What then can it be? We have to sum the difference for v
divided by the difference for y. That is, not every one of the
differences for, or the whole of, v is to be divided by each single
difference for the y; this is not so, I say, because each single one
of the first set is only divided by the single one of the other set
that corresponds to it, and not by all of them. Therefore
7 is not the same as 777, or -7. Will not then d— be something
J dy fdy v y
, r r j
different from -77 ? If it is the same, then also I d-r = I -jr , that is
-= C— =
y ~ J dy ~
dy
which is absurd.
Similarly, if we can suppose that dv$ = dv d$, then J dvty, or
Jdvdty. Now -v^—jdv Jdty; hence, jdvdfy = jd
which is absurd.
Hence it appears that it is incorrect to say that dvd\Jf is the
same thing as dvti, or that -^=d^r ; although just above I stated
ay v
that this was the case, and it appeared to be proved. This is a
difficult point. But now I see how this is to be settled.
If we have v and \f/, and they form some quantity, say <f> = v^/
or v/\j/, and if the values of v and ^ are expressed as rationals in
terms of some one thing, for instance, in terms of the abscissa x,
then the calculus will always show that the same difference is pro-
duced, and that d<f> is the same as dvd$ or dv/dy. But now I see
40 If Leibniz can see that this equality is "obviously incorrect," what is the
use of the argument that has preceded this sentence; for the final result must
also be obviously incorrect.
THE MANUSCRIPTS OF LEIBNIZ. 28l
the former can never happen, nor can it come to the latter by
separation of parts ; for example,
x + /?, ^ x + ft,, -, x, x, becomes 2$x,
which is quite a different thing from
x + p,-x,,r^x + p,-x which gives p2.
Hence it must be concluded that dv$ is not the same as dvdifr, and
.v . dv (41)
a T is not the same as -77 .
Take an equation of the first degree, a + bx + cy = 0. Let DV = 6,
AB = x} BC = ;y, and TB = t. Then, by making use of the method
of tangents,42 we have bt = -cy, or t=-cy/b. In the same way,
0=-bx/c.
T
8
Let WC = w, and WS = /3, then it is evident that t/y = f$/w, and
therefore w=—p-, and in the same way, fi= —7— .
Second degree. a + bx + cy + dx2 + ey2 + fyx = 0. Making use of
'the method of tangents, we have
bt + 2dxt + fyt = -cy - 2ey2 - fyx ;
41 Leibniz here justifiably verifies the falsity of his supposition being a
general rule by a single breach of it. He uses v =• ^ = x, and changes x into
* + /3; thus,
d(xx) = (* + /8)(jr + /B)-. xx - 2£x
dx dx = (JF + /J — *)(* + £ — *) = 02.
Here we see the first idea of the method that is the same as that used by
Fermat and, afterward by Newton and Barrow ; this consideration, whatever
the source, is that which leads him later to the substitution x -\- dx, y-\-dy in
those cases in which Barrow uses a and e.
42 "ordinando et accommodando," literally setting in order and adapting.
It is to be remembered that Sluse gave only a rule, and not a demonstration
of the rule. Part of the rule was that, if the equation in two variables con-
tained terms containing both the variables, these terms had to be set down
on each side of the equation. Thus, for the equation y3 = bw — yw would
first of all be written
y3-\-yw = bw — yw ordinando (?)
then each term on the left is multiplied by the exponent of y, and each term on
the right by that of v, thus,
3ya -}- yw = 2bw — 2yw accommodando ( ?)
and finally one y on the left, in each term, is changed into a t, where t is
the subtangent measured along the y axis.
282 THE MONIST.
hence t= — 7- — ^-r- — P— • From this it is quite evident that t can
b + 2dx+fy
always be divided by y (and 0 by x), and since w = fiy/t, therefore
we have
$b + 2dx +fy -w c+fx,^Pb + 2 dx
w = - 5 -- ,- , and y = -- f 0 - — ,
— c — 2ey —jx j+2e
but from just above \= - , — . hence we have
c + ey +/x
(43)
-w,
— we +/x, —P6 + 2dx
f+2e
Hence we have an equation in which there is no longer any
y;44 and all figures that can be formed from this equation by a
variation of the letters that stand for the constants can be squared ;
and also all others that by other methods can be shown to be con-
nected with it.
§ IX.
In the manuscript that follows we must refrain from
being critical; for, as suggested by the opening remark,
it contains nothing more than random notes, jotted down
as they came into Leibniz's mind, as materials for further
investigation. In the ten days that have intervened since
the date of the last MS., he has either had no spare time
for further work on the lines of this last manuscript, or
else he has found that he cannot proceed any further use-
43 This is hopelessly inaccurate ; all except one error, namely, f -\- 2e,
which should be Pf + 2ew, may be put down to bad transcription. Even if
Leibniz's writing were execrable, the correct version of an ambiguous sign
(through bad writing) could easily have been settled, by working through the
algebra. Thus the first of the last pair of values, in Leibnizian symbols
should be
.._ —w,c + fx,—ft, b + 2dx,,
— •w, c + fx, —ft, b + 2dx,, ^>—e,
with a similar correction in the second value.
44 Even if Leibniz had worked out the correct result, and obtained what
he was trying for, namely, w/P in terms of x, he would have got a very
lengthy quadratic, and the roots would be quite beyond his power to use at any
time. But he convinces himself that he can thus find the quadrature of any
conic, or figures that can be reduces to them.
THE MANUSCRIPTS OF LEIBNIZ. 283
fully until he has perfected the method he had in hand.
He therefore reverts to the method of breaking up the
figure into triangles by means of a set of lines meeting in
a point, coupled with the ideas of the moment and the
center of gravity, in order to try to obtain further general
theorems for analytical use. In this way, he again comes
across the differentiation of a product in the form of an
"integration by parts" ; but he does not recognize in it the
differentiation of a product, for he says that as he has
obtained this before he can get nothing new from it. He
is still wasting his energies over the idea of obtaining
dy/dx as an explicit function of x, for the purposes of
integration or quadratures. The fact that he can use the
method of Slusius as an unproved rule seems to have hid-
den from him the necessity of pushing on his investigations
with regard to the laws of differentiation, or the direct
tangent method.
21 November 1675.
Pro methodo tangentium inversa et aliis tetragonisticis spe-
cimina et inventa. Trigonometria indivisibilium. Aequa-
tiones inadaequatae. ordinatae convergentes. Usus singu-
laris Centri gravitatis.
[Examples and discoveries by means of the inverse method of
tangents and other quadratures. Trigonometry of indivi-
sibles. Inadequate equations. Converging ordinates. Spe-
cial use of the Center of Gravity.]
Subject-matter for a new consideration of the Center of Grav-
ity method, as follows:
A segment AECD having been broken up into infinite tri-
angles, AEC, ACF, etc., let the center of gravity of each of these
triangles be found ; this is a simple matter, for the center of gravity
is always distant from the base a third of the altitude. Then, since
the path of the center of gravity multiplied by the area of the
triangle is equal to the solid formed by its rotation, and also since
the products of the AH's and the infinitesimal parts of the axis are
twice the areas of the triangle, also it is plain that the AG's multi-
284
THE MONIST.
plied by the distances of the centers of gravity of the triangles AEC
from the axis are equal to the moment of the segment about the
axis; by the help of this idea a number of things can be at once
obtained in two ways : first, by taking some general figure and mak-
ing a general calculation, and then so expressing it that the center
of gravity can be easily found; in this way we may obtain the
moments of spaces which would be a matter of difficulty otherwise,
if they were investigated by the ordinary method of ordinates.
Secondly, on the other hand, if figures of which the moments are
easily obtained in the ordinary way are treated by this method, we
shall arrive at certain very difficult curves, the dimensions of which can
always be deduced from some that are easier. Here then we have
a remarkable rule, by the help of which useful properties can always
be obtained from any method however complicated. It is often
useful when problems arise that we know are naturally simple, and
from other reasons are soluble; for thus many notable cases are
discovered. See what Tschirnhaus noted about the Hastarian line.
In irregular problems, such as cannot be treated in a straight-
THE MANUSCRIPTS OF LEIBNIZ. 285
forward manner or reduced to an equation that is sufficiently de-
terminate, because, say, something has to be done inversely, it is
useful to compare several ways with one another, of which the
results should be identical. This seems to be useful for the inverse
tangent method. Here is a case in point.
The figure, in which BP and AT are reciprocally proportional,
is required.
Let TB = f, then AT = t-x, andBP = a2/(*-*). If this is multi-
plied by t, we have
hence. ta2 = ty2 - xy2,
or t = xyz/(az-y2) ;45 and therefore t/x=y2/(a2-y2), or all the f's
together equal the moment about the vertex of every y2/(a2-y2).
But from other reasons, all the TP's applied to the axis are
equal to the TC's applied to the curve.
0y _ Pa? —y*
Now t/y={i/w, and therefore w= y*x xy .
* == 2 2
But fw = y, therefore
fW-£=y (A)
xy
Further, wx= — — — , and J wx=-yx— J yft,
/Par—~y*
m§
n 2 _ 2
Also w=dy, dy— - •—*-, and therefore
y
Now if we suppose that the y's are in arithmetical progression,
then w = dy is constant and ft is variable;
hence, P= - 3 - 2
a* -y'
2 2
But from equation (B), /?^-— —
y
2
hence, ft— =dyx.
40 There is a mistake in sign ; a2 — yz should be yz — a2 ; hence the work
that follows is also wrong.
286 THE MONIST.
We have thus obtained two equations that are mutually inde-
d* vie (46'
pendent, the first f = - ............... (1)
ay a + y, a—y
and the second dyx= — — ....................... (2)
V
Let us seek to obtain others in addition, such as
J t dy = fy dx.
Now this furnishes us with nothing new; but Ctw+ Cxw = xy
or t dy + x dy = dxy. and t= -- y; hence the latter = — *— ^—^,
dy dy
Therefore dx y = dxy - x dy.
Now this is a really noteworthy theorem and a general one
for all curves. But nothing new can be deduced from it, because
we had already obtained it.
However, from another principle we shall obtain a new theo-
rem; for it is known that the sum of every BP = BC2/2; that is to
say, BP= — , /= & = — y, and therefore
t—x w dy '
dxy-dyx 2
We therefore have two equations, in which dx occurs, namely,
the first and the third ; by the help of these, by eliminating dx, we
shall have an equation in which only one of the unknowns remains
shackled; thus from equation (1), we have dx= % yx<> , and now
a —y
from equation (3), we get dxydy2-dydy2x=2a?dy. Hence,
2c?dy + dy dy^x
y dy2
We have therefore an equation between the two values of dx,
in which only the y remains shackled. From this, by assuming
46 Although the variables are separable, Leibniz does not recognize the fact
that he can make use of this. For later he states that the solution of a prob-
lem cannot be obtained from a single equation. In this case we have
dx y dy dv .
- = -£ - =5 = — , if yz — a2 = ± v2.
x y2— a? v '
Supposing this substitution to have been effected, Leibniz would have concluded
that x = v, and would have stated that he had solved the problem.
But here again he has made an unfortunate choice, for the origin (A)
cannot fall on any of the curves Cx = v or Cxz ±yz = ± a2, which is the gen-
eral solution of the equation. Hence the problem is impossible.
THE MANUSCRIPTS OF LEIBNIZ. 287
the y's to be in arithmetical progression, that is that dy = fi a con-
stant, and dy2 = z, and z = s*/2 - y- ; z = V 2 y = df." Thus we have
obtained what was required.
We have here a most elegant example of the way in which
problems on the inverse method of tangents are solved, or rather
are reduced to quadratures. That is to say that the result is obtained
by combining, if possible, several different equations, so as to leave
one only of the unknowns in the tetragonistic shackle. This can
be done by summing ordinates in various ways, or on the other
hand, instead of ordinates, converging or other lines.
Note. If, instead of x or y, some other straight line can be
found, either one that is oblique, or one of a number converging to
the same point, by the employment of which one only of the un-
knowns is left in bonds, it may be employed with safety. Take
for instance the case of finding the relation for the AP's ; here the
sum of AP's applied to the axis is half the square on AC. When-
ever the formula for the one unknown that is left in shackles is
such that the unknown is not contained in an irrational form or as
a denominator,48 the problems can always be solved completely ;
for it may be reduced to a quadrature, which we are able to work
out ; the same thing happens in the case of simple irrationals or
denominators. But in complex cases, it may happen that we obtain
a quadrature that we are unable to do. Yet, whatever it may come
to, when we have reduced the problem to a quadrature, it is always
possible to describe the curve by a geometrical motion ; and this is
perfectly within our power, and does not depend on the curve in
question. Further, this method will exhibit the mutual dependence
of quadratures upon one another, and will smooth the way to the
method of solving quadratures. Meanwhile I confess that it may
happen that there may be need for a very great number of inade-
quate equations (for so I call them, when there is need for many
to solve the problem, although each alone would suffice provided
it could be worked out by itself), in order to completely free one of
the unknowns from its shackles. For, unfortunately, a solution
cannot be obtained from a single equation, unless one of the terms
is free from shackles ; and if this term appears oftener, then not
unless it is freed at least once. Thus there may be a great number
47 This is quite unintelligible to me as it stands ; query, is it an accurate
transcription?
48 This is tantamount to a confession by Leibniz that he cannot explicitly
integrate fa2/y, although he knows that it is logarithmic or reduces to the area
under the hyperbola; for he has given this in the MS. for Nov. 11.
288
THE MONIST.
of inadequate equations to be found ; and we have to examine
which of them are in some way independent of the others, i. e.,
such as cannot be derived from one another by a simple manipula-
tion; for instance, the sum of all the AP's and the sum of all the
AE's.
A new kind of Trigonometry of- indivisibles, by the help of
ordinates that are not parallel but converge.
Let B be a fixed point ; let BDC be a very narrow triangle stand-
ing upon a curve ; let DE be the perpendicular to BC ; from the point
B let BA, perpendicular to BC or parallel to DE, be drawn to meet
the tangent AHDC, and let BH be the perpendicular to the tangent
DC produced.
Then the triangles CED, CHB, BHA are similar; hence we
have BH/CE = HA/DE = B A/CD, and therefore BH, DE = CE, HA,
and BH,CD = CE,BH. Hence it follows that the sum of the tri-
angles or the area of the figure is equal to the products of the AB's
into the CE's, or the differences of the ED's and lastly AH,CD =
DE, BH.49
Further, CH/CE = HB/DE = CB/CD; hence, again, CH, DE =
CE,HB, and HB,CD = DE,CB; i. e., the area of the triangle, as is
in itself evident, is equal to itself. Lastly, CH, CD = CE, CB; and
this last result seems to be worth noting for the case of a Trochoid.
For, if by the rolling of a curve DC on a fixed plane CA, a
trochoid curve is described by the point B fixed in DC, and it is
given that the ordinate of the trochoid drawn to the fixed plane CA
48 There are several errors in the letters in this paragraph, which are
probably due to transcription; thus, an E for a (? badly written) B, an H
for an A, etc., would be quite an easily-imagined error, provided the work was
not verified during transcription.
THE MANUSCRIPTS OF LEIBNIZ. 289
is BH, then the sum of the intercepts CH applied to DC will be equal
to the sum of the CB's applied to their own differences. Now if
any ordinates are applied to their own differences, the same thing
is always produced as in the case where we try to find the moment
of the differences about the axis, which is the same as the moment
when we take the sum of each, or the maximum ordinate, into the
\
distance of its center of gravity from the axis, i. e., its middle point,
that is to say into half itself. Finally this is equal to half the square
on the maximum ordinate. Therefore we can always obtain the
sum of all the rectangles BC, CE, which is always equal to half
the square on BC, or to the sum of all the BP's applied to the axis
in F, where CP is the normal to the curve DC.
§ X.
Leibniz now directs his attention to the direct method
of tangents, and proceeds to generalize the methods of
Descartes. Is it only a coincidence that Barrow uses this
method regularly, the curve that he is especially partial to
being the rectangular hyperbola? Weissenborn suggests
the same coincidence occurs with respect to the method of
Newton, who uses analytical approximations; but if there
is anything in either of these suggestions. I think that the
Harrovian idea, which is purely for the construction of
tangents, is much nearer to that of Leibniz in this manu-
script than is the Newtonian.
However this may be, Leibniz is at last beginning to
consider the point as to the method by which the principle
of Sluse is obtained. He ascribes it to a development of the
method of Descartes; but in this connection I cannot get
out of my head the suggestion raised by Barrow's use of
the first person plural, "frequently used by us," in the
29O THE MONIST.
midst of a passage that is written, contrary to his usual
custom, in the first person singular throughout, where he
describes the differential triangle and the "a and e" method.
I consider that Sluse has enunciated a working rule for
tangents, which he has generalized by observation of the
results obtained by the use of the "a and e" method; and
that this method had been circulated by Barrow some time
before the publication of the Lectiones Geometricae, al-
though I confess that I have not found any record of this,
nor any distinct evidence of a correspondence between
Barrow and Sluse ; but there is more than a suggestion of
this in the fact that Sluse's article was published in the
Phil. Trans, for 1672.
It seems more than strange to me that there should be
such a prolific crop of differential calculus methods within
a couple of years of the work of Barrow in all sorts of
places, raised by many different people, and that none of
them allude to the general seed-merchant, as I consider
Barrow to have been.
22 Nov. 1675.
Methodi tangentium directae compendium calculi, dum jam
inventis aliarum curvarum tangentibus utimur. Quaedam
et de inversa methodo.
[Compendium of the calculus of the direct method of tangents,
together with its use for finding tangents to other curves.
Also some observations on the inverse method.]
In that which I wrote on Nov. 21, I noted down those things
which came to my mind concerning the method of tangents. Re-
turning to the subject, let ACCR and QCCS be two curves that cut
one another in one, two, or more points C, C; let AB(B) be the
axis; let AB = jr be the ordinates, and BC = 3/ the abscissae; then
we shall have two equations to the two lines, each in terms of these
two principal unknowns. Now if these two equations have equal
roots, or the equations have equal values, then the lines will touch
one another. Instead of the line QCCS, Descartes chooses the arc
of a circle VCCD, whose center is P, so that PC is the least of all
THE MANUSCRIPTS OF LEIBNIZ.
291
the lines that can be drawn from the point P. It will come to the
same thing, and often more simply, if we take not the arc of a
circle but the tangent line TC(C), that is the greatest of all those
that can be drawn from a given point T to the curve. Let TA = b,
AE = £, be assumed as given; required to find AB, BC. The two
equations are, the one for the curve AC(C), namely, ax- + cy2 + etc.
= 0, and the other to the straight HneTC(C) which, on account of the
relation TA/AE = TB/BC. will beb/e=(b±x)/y or ±x=(b/e}y-b
or y=± e
Thus the value of either one or other of the unknowns can always
be obtained explicitly, and thus can be worked out immediately
without raising the degree of the equation of the given curve
AC(C) ; and then at once we shall obtain an equation giving the
unknown that alone remains, so that we may determine the condition
for equal roots. Doubtless this is the principle of Sluse's method.
If however the arc of the circle whose center is P is used,
following Descartes, then the new equation, for the circle, will be
as follows: let the radius PC = .y, and PB = z/-x, and we have
s2 = y- + v2 + A'2 - 2vx. Hence it is clear that we have the choice of
either a circle or a straight line ; and when, in the equation to the
given curve, only an even power of y appears (as can always be
made to happen in the case of the conies), then it will be more con-
venient to use equations to circles ; for thus, by the help of the two
values of y2, the unknown x can be immediately worked out; but,
in general for all equations to curves expressed by a rational rela-
tion, the method of the straight line may be usefully employed.
Hence I go on to say that not only can a straight line or a
circle, but any curve you please, chosen at random, be taken, so
long as the method for drawing tangents to the assumed curve is
292 THE MONIST.
known ; for thus, by the help of it, the equations for the tangents
to the given curve can be found. The employment of this method
will yield elegant geometrical results that are remarkable for the
manner in which long calculation is either avoided or shortened,
and also the demonstrations and constructions. For in this way
we proceed from easy curves to more difficult cases, and an equa-
tion to a curve being supposed known, it is always possible to choose
an equation to some other curve whose tangents are known, by the
help of which one of the unknowns can be worked out very easily.
Thus, if it is given that hy2 + y3 = cxs + dx* + ex + f is the equa-
tion to a curve of which the tangents are required, assume a curve
of which the equation is hy2 + yz = gx + q, for that of which the
tangents are already known ; eliminating y, we have an equation
such as gx + q = cx3 + dx2 + ex + f. This can be determined for two
equal roots, either by Descartes's method of comparisons, or Hudde's
by means of an arithmetical progression ; and thus by working out
the value of x, the value of either g or q may be found ; and one of
the two letters q or g can be chosen arbitrarily.50 Hence, a way of
describing that other curve that touches the given curve is obtained ;
now, when this is described, let the tangent be drawn at the point
which is common to it and the proposed curve, which tangents we
have supposed to be already known; then this tangent will touch
the given curve.
I think that, in general, the calculation will be possible by this
method of assuming a second curve, as we have done in this case,
which evidently works out one of the unknowns. Hence I fully
believe that we shall derive an elegant calculus for a new rule of
tangents, which in addition may be better than that of Sluse, in that
it evidently works out immediately one of the two unknowns, a thing
that the method of Sluse did not do. Now this very general and
extensive power of assuming any curve at will makes it possible,
I am almost sure, to reduce any problem to the inverse method of
tangents or to quadratures. Indeed let any property of the tan-
gents to a curve be given, and let the relation between the ordinates
50 The method of Hudde appears to be similar in principle to that of Sluse,
while that of Descartes was the construction of the derived function by assum-
ing roots, forming the sum of the quotients of the function divided by each of
the assumed root-factors in turn, and comparison with the original function.
Both therefore reduce to finding the common measure of the equation to the
curve (where the right-hand side is zero) and the differential of it.
Leibniz, however, strange to say, does not note that by taking one of his
arbitrary constants, q, equal to f, the equation has its degree lowered in the
particular case he has chosen.
THE MANUSCRIPTS OF LEIBNIZ. 293
and the abscissae be required. Then an equation can be derived,
which will contain the principal unknowns, x, y, and always two
others as incidentals, such as s and v, or b and e, or the like ; now,
as the equation contains the property of the tangents, by which s
and b may be expressed so as to have a relation to the tangents,
assume in this case any new curve chosen arbitrarily, and then s
and v will also have some known relation to this curve. By means
of the equation to the arbitrarily chosen curve, we shall be able
to replace the given property of tangents in favor of the curve re-
quired, namely, by removing one or other of the unknowns ; and
by thus reducing the problem to such a state the inverse calculation
will come out the more easily.
The whole thing, then, comes to this; that, being given the
property of the tangents of any figure, we examine the relations
which these tangents have to some other figure that is assumed as
given, and thus the ordinates or the tangents to it are known. The
method will also serve for quadratures of figures, deducing them
one from another ; but there is need of an example to make things
of this sort more evident ; for indeed it is a matter of most subtle
intricacy.
The manuscripts mentioned above seem to be all that
were found by Gerhardt belonging to the period 1673-5.
I feel that it is a great pity that they were not given in
full, or at least a little more fully. For instance, Gerhardt
mentions that Leibniz in the MS. of August 1673 con-
structs the so-called characteristic triangle, but does not
give Leibniz's figure in connection. This figure should
have been given; for the figure given in October 1674 is
not the characteristic triangle as given by Leibniz in the
"postscript" (§1), or the Historia (§11), but it is the
Pascal diagram (assuming that the figure given by Cantor
is the correct one). It would be useful to know the date
at which Leibniz drops the Pascal diagram in favor of
one or other of the Barrow diagrams.
It is to be noticed at this date that Leibniz uses one
infinitesimal only, and verifies that the method of Des-
cartes comes out correctly in the simple case of the parab-
294 THE MONIST.
ola; but he is not satisfied with the generality of the
method of neglecting the vanishing quantities.
Again, the second manuscript of October 1674 appears
to be immensely important; especially as it contains the
groundwork of some of the later manuscripts. Judging
by the little that is given of it, it would seem to be most
desirable that fuller extracts, at least, should have been
given. It is a matter for remark that this manuscript is
a long essay on series. Can this possibly have had any-
thing to do with the fact that it is not given in full?
(TO BE CONTINUED.)
J. M. CHILD.
DERBY, ENGLAND.
CRITICISMS AND DISCUSSIONS.
MECHANISM AND THE PROBLEM OF FREEDOM.
Men's opinions are far more commonly the result of the gen-
eral presuppositions and prejudices of the age in which they live
than the outcome of a rational process. Thus men believe whatever
fits in with their general view of life and dismiss without a hearing
anything which conflicts with it. In this age of science the scientist
has become the arbiter of all questions, and his view is commonly
accepted as authoritative. Hence problems which he refuses to
examine, e. g., the question of the existence of ghosts, are at once
relegated to the realm of superstition. Now there is some danger
of freedom being placed among such problems. An indication of
this is found in the following words of Haeckel, which represent
the attitude of many contemporary scientists and psychologists
toward the question of freedom: "The great struggle between the
determinist and the >indeterminist, between the opponent and the
sustainer of the freedom of the will, has ended to-day, after more than
two thousand years, completely in favor of the determinist. The
human will has no more freedom than that of the higher animals,
from which it differs only in degree, not in kind .... We now know
that each act of the will is as fatally determined by the organization
of the individual and as dependent on the momentary condition of
his environment as every other psychic activity."1 This view has
won its way by its scientific prestige, and has been eagerly accepted
by many who have never examined the evidence for freedom, but
who nevertheless smile indulgently at those who are still so benighted
as to believe in it. Therefore, in view of the great popularity and
influence that Haeckel has enjoyed it seems profitable to examine
the question of the relation of mechanism and freedom with ref-
erence to his specific teaching.
1 The Riddle of the Universe, p. 130f.
296 THE MONIST.
There are two main principles on which Haeckel bases his sys-
tem: the doctrine of evolution, and the "law of substance." The
doctrine of evolution furnishes the principal evidence for his denial
of a spiritual principle in man and together with the "law of sub-
stance" leads to his mechanistic determinism. We shall therefore
first consider the question whether the existence of a spiritual prin-
ciple is precluded by evolution or by any other arguments suggested
by Haeckel. We shall then proceed to the question of the univer-
sality of the "law of substance" and to the problem of its relation
to mechanism, and finally examine briefly the adequacy of mechan-
ism itself as a philosophic explanation of the universe.
ii.
Haeckel never tires of citing facts in support of the doctrine
of evolution. Since this doctrine in some form or other is now
almost universally accepted as true we need not stop for an instant
to inquire concerning the adequacy of this evidence. The only
question for us to consider is whether the doctrine of evolution
inevitably leads to the reduction of mind to matter. Put very
simply Haeckel's argument for this conclusion is that since man
developed from the lowest forms of life there is no reason to at-
tribute to him a separate immaterial or spiritual principle not found
in the forms of life from which he originated. Many objections
to this argument at once suggest themselves to the thoughtful reader.
In the first place it takes for granted the old scholastic idea of rigid
continuity, according to which nothing new can ever arise. Now
there are grave difficulties in this view, but even waiving these for
the moment, Haeckel's conclusion by no means follows. Rather,
the doctrine of continuity, if strictly held, would force him to read
into the life of the lowest organism all the complex processes and
meanings which have been evolved in the highest forms of life.
For if evolution were rigidly continuous the very fact that certain
phenomena, such as sensation and will, have developed in the later
stages of the evolutionary process would show that these phenomena
were implicit in the earlier forms. Thus Haeckel would be com-
pelled to understand the protozoon in the light of man, rather than
to reduce man to the level of the protozoon. He indeed seems some-
times to do this, and with an extraordinary anthropomorphism
bestows elementary will and elementary emotion upon even inani-
mate matter.2 If he held consistently to this view his final system
2 Cf . infra, p. 304.
CRITICISMS AND DISCUSSIONS. 297
would be in the nature of a theological hylozoism rather than a
strictly mechanistic determinism. However, this reading of con-
tinuity, this attribution of man's processes to the lower forms of
life, is misleading, as it involves what Baldwin calls "the fallacy
of the implicit." As a matter of fact, if progress is genuine, new
processes and new meanings must arise which cannot be interpreted
in terms of the lower stages. Thus even though life has arisen
from the inanimate, and consciousness from the unconscious, yet
they involve meanings and processes which cannot be expressed in
terms of the stages from which they arose. It is fallacious either
to deny these new meanings and attempt to reduce them to earlier
stages, or to read them back as implicit in the earlier stages. Hence
the doctrine of evolution in no wise militates against the spiritual
nature of man. Rather it leads us to expect man's nature to be
higher or more developed than the merely physical or the merely
biological.
In addition to the argument drawn from evolution, Haeckel
adduces several other considerations in support of his denial of the
spiritual nature of man. He brings forward the evidence of ex-
periments which have shown that various functions of the soul,
such as speech and sense images, are connected with definite areas
of the cortex of the brain and disappear when these areas are
diseased or destroyed.3 Again he calls our attention to the close
connection between man's higher cerebral functions and purely phy-
siological processes — a connection especially plain in the case of
emotions.* He also emphasizes facts concerning the individual's
development which, in his opinion, indicate that the soul originates,
grows, and decays with the body.5 Finally he points out that we
never find a single instance of a spiritual principle unconnected with
a physical substrate.6
Now although all the above arguments show clearly that there
is some relation between mind and body, yet they do not succeed
in reducing mind to body. The facts can be read as easily the
other way. As a matter of fact, we are as conscious of the influence
of mind on body as of body on mind. It is true that illness or
various physical causes affect man's mental processes, but it is also
true that man's mental processes affect his physical condition. In-
3 Cf. Last Words on Evolution, 98f ; The Riddle of the Universe, p. 204.
4 Cf. The Riddle of the Universe, pp. 127, 204.
5 Cf. The Riddle of the Universe, Chap. VIII.
« Cf. Ibid., pp. 90, 91.
298 THE MONIST.
deed every act of will is evidence of the power of mind over matter.
Now to this Haeckel might retort: "What you call will is merely
a certain functioning of a physiological organism. I can even dis-
close to you, with my microscope, the minute structures in the brain
by which willing takes place." But even if this last contention
were granted it would not prove Haeckel's point. The brain with
its various structures may be the instrument of mind's expres-
sion without being the cause of mind.7 Moreover the very facts
of pathology which are cited by Haeckel to show the dependence
of mind on matter are used by Bergson to prove that mind cannot
be located in the brain nor determined by it.8 Furthermore in this
controversy concerning the relation of mind and body, the idealist
can always go back to Berkeley's position and retort, "The brain,
the nervous system, etc., to which you attempt to reduce mind, are
known only as ideas of mind, and cannot be proved to exist apart
from mind."
A further weakness in Haeckel's arguments is that they often
betray a total misunderstanding of his opponents' position. They
are all directed against the existence of a separate, immaterial sub-
stance or soul. Most idealists, however, regard the soul as activity
or functioning, rather than as substance. They do not insist on
the separateness of the psychic principle or on the existence of any
disembodied spirit, but rather on the fact that man's activity cannot
be explained in purely physical or physiological terms. Suppose
it be granted to Haeckel that the soul is but the "sum total of phy-
siological functions," yet the problem of the activity of the soul is
not thereby solved. Consciousness is a fundamental fact of ex-
perience, and it cannot be explained by being set aside or labeled
an epiphenomenon. Therefore the materialist must explain not only
how the body reacts, but how it is conscious, how it thinks, evalu-
ates, loves, struggles, and sacrifices. It is indeed questionable
whether this activity can be interpreted in purely biological or phy-
siological terms. The so-called body becomes equivalent to the
mind and demands the same sort of an explanation.
We conclude then from this discussion that Haeckel's reduction
of the psychical to the physical is not valid, and we turn to an
7 Cf. Schiller, Riddles of the Sphinx, pp. 293ff ; Bergson, Matter and Mem-
ory, PP- 299ff; James, Human Immortality, pp. 7-29.
8 Cf. Bergson, Matter and Memory, Chapter II.
CRITICISMS AND DISCUSSIONS. 299
examination of the "law of substance" — the second main support
of Haeckel's system.
in.
The "law of substance" is a combination of the well-known
scientific laws of the "conservation of matter" and "the conservation
of energy." According to Haeckel, these laws are but two aspects
of one great cosmic law, since they relate to the two inseparable
attributes of substance. In passing it may be noted that little is
gained by this combination of the two laws, since Haeckel's unknown
substance is incapable of showing concretely the relation between
matter and energy. Haeckel regards this law as the one great
eternal cosmic law. On what evidence then can he base its validity ?
The evidence for the law adduced by Haeckel lies in the realm
of scientific experiment. Thus the law of "conservation of energy"
rests on the fact that many experiments have shown that when one
form of energy is changed into another, it may be reconverted into
the original form of energy with only a slight loss due to the escape
of part of the energy into an unavailable form. Similarly the law
of the "conservation of matter" rests on experiments which have
demonstrated that the weight of a substance does not change
throughout a series of chemical transformations. Moreover no ex-
periments have given any indication of the creation or destruction
of matter or of energy, and the generalization of a great number
of phenomena under these laws has been indeed a great achievement
of science. Yet it is one thing to regard these laws as useful gen-
eralizations for the purposes of science, and quite another to erect
them into ontological and absolutely universal laws. Against this
latter proceeding, which is that of Haeckel, an emphatic protest
must be made. There are three grounds for this protest: (1) the
laws have never been proved to hold exactly in any field; (2) the
fact that the laws appear to hold in one or two fields is no justi-
fication for the assertion that they must hold in all fields; (3) ex-
perience can never prove the absolute universality of any law.
In the first place, it is manifestly impossible to prove that the
laws hold exactly in any field, since the inaccuracy of scientific instru-
ments is such that small differences might pass unnoted. Further-
more there are always extraneous circumstances which must be
taken into account in an appraisal of the results of an experiment.
A scientific result is always an approximation, and the scientific
3OO THE MONIST.
law states what would take place under ideal circumstances rather
than what occurs in any concrete situation.
In the second place, the fact that the laws appear to hold
true within certain fields of our experience does not show that
they must hold in all fields. Thus the demonstration of the laws in
the case of physical and chemical changes would furnish no proof
of their applicability to the relation between the physical and the
psychical. It is at this point that Haeckel's assertion of the uni-
versality of the law depends upon his reduction of the psychical
to the physical. Since this is not valid, he is not entitled without
more ado to extend the application of the law to the psychical
realm. The application of the law here must rest upon experi-
ments showing that a certain amount of physical energy can be
transformed into a definite amount of psychical energy and re-
converted into the original amount of physical energy. Manifest
difficulties stand in the way of such experimentation, but until
something of the sort is carried out there is little significance in
speaking of the psychical life as a form of energy. To do so
merely covers up the fact of our ignorance concerning the relation
between the psychical and the physical. Now apparently Haeckel
himself is aware of some of the difficulties in the way of regarding
the psychic as a form of energy since in his last work, contrary to
many of his previous assertions,9 he explicitly teaches that the
psychic is a separate attribute of substance, coordinate with matter
and energy.10 If this be admitted, however, the psychic must de-
mand its own law of the "conservation of the psychic," if it is to
come under the "law of substance." In any case, the important
point for our purpose is that until the law is proved to be valid in
the psychical field it furnishes no ground for a denial of freedom.
Our last objection needs no justification, as it is a philosophic
commonplace that laws resting on experience can be universalized
only by means of the supposition of the uniformity of nature. This
uniformity, however, cannot be proved by experience without the
assumption of its own existence in the attempted proof. Thus the
observation that the laws apparently hold in a comparatively few
instances within the narrow range of our experience is no proof
that they have always held and will always hold throughout the
length and breadth of the universe.
We have seen reason to question the dogmatic assertion of the
9 Cf. The Riddle of the Universe, p. 220; Anthropogenic, p. 941.
10 Cf. Die Lebenswunder, p. 185.
CRITICISMS AND DISCUSSIONS. 3OI
universality of the "law of substance." Yet if it be admitted for
the sake of argument that the law is universal and necessary, it by
no means follows that this law alone gives an adequate account of
reality and a solution of all its riddles. The law is an abstraction ;
it is purely quantitative, and as such leaves out of account the
qualitative aspects of the universe. Thus although the amount of
matter and energy in the universe remain constant, changes in their
form or in their combination bring about new qualities not reducible
to mere quantity. Take, for example, the case of a chemist who
mixes together two elements in a new combination. Their weight,
as a measure of their quantity, remains the same, but this quantita-
tive equality in no wise explains or describes the new odor, the new
color, or other new properties possessed by the compound. These
qualitative aspects, however, are certainly part of reality, although
they cannot be described by the law of substance nor comprehended
in a system which uses this law as the solution of all its problems.
The "law of substance" is for Haeckel but a necessary conse-
quence of mechanical causation. In fact the two for him are iden-
tical.11 Yet the relationship does not appear to be as simple as he
would have us believe. The law of substance, he tells us, is a
consequence of mechanical causation, yet his proof of the latter
rests largely on his supposed proof of the universality of the former.
Now the law of mechanical causation, involving the equivalence of
past and present, might lead naturally, though perhaps not inevitably,
to the "law of substance." On the other hand the "law of substance"
does not necessarily involve mechanical causation. It does indeed
preclude spontaneity, but it would be as compatible with teleology
as with mechanism, since it says nothing concerning the origin of
changes in matter or energy. The amount of energy and matter in
the universe might remain constant if their changes were due to a
desire for a future state as well as if they were due to a past stim-
ulus. Thus even the universality of the "law of substance" would
not prove the universality of mechanism. The latter theory must
stand on its own feet and be accepted or rejected on its own merits.
IV.
Haeckel declares that mechanical causation explains all phe-
nomena. To quote his own words: "The great abstract law of
mechanical causality, of which our cosmological law — the law of
substance — is but another and a concrete expression, now rules the
" Cf. The Riddle of the Universe, pp. 215, 366.
3<D2 THE MONIST.
entire universe as it does the mind of man; it is the steady im-
movable pole-star whose clear light falls on our path through the
dark labyrinth of the countless separate phenomena."12 "The monism
of the cosmos which we establish thereon proclaims the absolute
dominion of 'the great eternal iron laws' throughout the universe.
It thus shatters at the same time the three central dogmas of the
dualistic philosophy — the personality of God, the immortality of the
soul, and the freedom of the will."13
Before accepting Haeckel's conclusion concerning freedom the
adequacy of mechanism itself must be examined. Of the many
objections which might be made, and which have been made, to
universal mechanism, we shall confine ourselves to the following:
(1) the universality of mechanism cannot be proved; (2) the uni-
versality of mechanical causation would not, as Haeckel would have
us believe, necessarily preclude purpose and rational or ethical free-
dom; (3) mechanism by itself fails to give a satisfactory account
of experience as we actually know it.
We contend that mechanism cannot be proved. Experience
cannot show that mechanical causation is universal and necessary,
and reason does not disclose any logical necessity for insisting that
every aspect of reality shall be explained by reference to the past.
On the contrary, the concept of mechanical causation is full of
difficulties which force the mind beyond it.14 The universality of
mechanical causation is indeed a methodological postulate of sci-
ence, but not necessarily a universal principle of reality. Haeckel
makes many dogmatic assertions to the effect that mechanism is
universal, and that even the will is absolutely bound by causal law.
Thus the will is, he declares, the necessary outcome of heredity
and environment. Yet obviously he cannot prove that such is the
case. He cannot prove that A did a certain act because A had a
certain heredity and a certain environment, and that A could not
have done anything else. Indeed, in the case of human activities
so many complex conditions occur that it is practically impossible
to isolate any set of conditions in such a way as to establish a
uniform series of cause and effect. Hence the establishment of
causal connection (quite apart from the question of its universality
and necessity) is in such cases a task for the future, rather than
12 The Riddle of the Universe, p. 366.
13 Ibid., p. 381.
14 For a careful analysis of causation cf. Taylor, Metaphysics, pp. 158ff ;
Ward, Realm of Ends, pp. 273ff ; Bergson, Time and Free Will, pp. 199-221.
CRITICISMS AND DISCUSSIONS. 303
an accomplished fact. The possibility therefore remains that the
mental life may resist such causal treatment. An indication in this
direction is found in the comparative lack of success of psychology
in the use of scientific methods found fruitful in other fields.
In the second place, even though mechanism were proved to be
universal, this would by no means preclude the possibility of pur-
pose, of value, and of rational or ethical freedom. Haeckel's abso-
lute denial of all distinctions of value is evident in the following
quotations : "As our mother earth is a mere speck in the sunbeam in
the illimitable universe, so man himself is but a tiny grain of proto-
plasm in the perishable framework of organic nature."15 "Our own
'human nature,' which exalted itself into an image of God in its
anthropistic illusion, sinks to the level of a placental mammal, which
has no more value for the universe at large than the ant, the fly of a
summer's day, the microscopic infusorium, or the smallest bacillus.
Humanity is but a transitory phase of the evolution of an eternal
substance, a particular phenomenal form of matter and energy, the
true proportion of which we soon perceive when we set it on the
background of infinite space and eternal time."16 From Haeckel's
point of view, indeed, neither man nor the bacillus can have any
value for the "universe at large," since there is, in his opinion, no
purpose whatever in the universe. All is but the result of blind
forces, and even the progress of evolution is of no value to the
universe.17 Now this denial of value is explained by the fact that
Haeckel always associates teleology with a separate immaterial
principle. He regards it as an interruption of mechanical cau-
sation, and so feels that it is incompatible with monism. Whether
or not an absolute monism of any sort is compatible with dis-
tinctions of value and with freedom, at least it is plain that the
denial of teleology does not necessarily follow from the establish-
ment of mechanical causation. Mechanism, as many teleologists
tell us, may be the instrument of purpose. Far from being an-
tagonistic to teleology it alone makes teleology possible. Without
it purpose would be impotent. For example — to take an analogy
from human life — man can utilize natural processes for the carry-
ing out of his purposes only in so far as he can rely upon their
mechanical uniformity. Even a machine is an embodiment of pur-
pose. It \yorks in a mechanical way, but its construction can be
15 The Riddle of the Universe, p. 14.
18 The Riddle of the Universe, p. 244.
17 Cf Ibid., Chap. XIII.
3O4 THE MONIST.
explained only in terms of purpose. From this point of view,
teleology is not an external principle opposed to mechanism, but
rather is immanent in all natural processes, and includes and tran-
scends their merely mechanical aspects. The processes of the uni-
verse are describable in terms of mechanical causation, but these
series of mechanical changes are what they are by reference to their
value for the whole.
Again we object to mechanism taken as a sole explanation of
the universe on the ground that it fails to take into account many
facts of experience. Although supposed to be the direct outcome
of an acceptance of evolution, mechanism has been unable to give
a satisfactory explanation of evolution itself. Furthermore mech-
anism cannot explain the existence of values, purposes, and ideals,
and many other aspects of reality.
Bergson, perhaps better than any one else, has succeeded in
proving the first point. In his careful examination of theories of
evolution, he shows how mechanism is forced to take refuge in
a miracle to account either for the successive production and preser-
vation of millions of minute variations in the same direction, or for
the complementary changes of the various parts of an organ neces-
sary for the preservation and improvement of its functioning.
Moreover this same miracle must be repeated innumerable times as
the same change has taken place in many different lines of evolu-
tion.18 Furthermore, in every explanation of evolution, terms such
as adaptation and struggle for existence occur, but these are not
mechanistic terms, since they imply purpose, ends, value. The mech-
anist holds that all achievements of evolution are merely results
of external and internal forces, which are absolutely blind. Yet if
such is the case, why is the organism said to struggle for existence ?
Haeckel himself, indeed, often finds a place for the action of internal
forces and declares that the movement of molecules is due to an
inner will. "Even the atom is not without a rudimentary form of
sensation and will, or, as it is better expressed, of feeling and in-
clination— that is, a universal 'soul' of the simplest character."19
The term "inclination" suits Haeckel's purpose by its vagueness, but
if it is at all comparable to will it implies a reaching for the future
which is not explicable as merely the result of a previous force.
To do justice to this "inclination" Haeckel would be forced beyond
his rigid determinism.
18 Cf. Bergson, Creative Evolution, pp. 62-76.
19 The Riddle of the Universe, p. 225.
CRITICISMS AND DISCUSSIONS. 305
The discussion of the inadequacy of mechanism as an account
of evolution has led directly to our second criticism : that mechanism
fails to do justice to the existence of values and purposes which are
present not alone in our inner experience but which find an outer
embodiment in the great achievements of civilization.20 Surely the
painting of a great picture, the writing of a drama, or the founding
of a college cannot be accounted for as the result of purely natural
forces. Now the mechanist, of course, does not attempt to deny
the presence and power of ideals in human life. His contention is
simply that these ideals themselves are the result of purely mechan-
ical forces. Will, Haeckel says, is absolutely determined.21 Thus,
according to Haeckel, the psychologist can trace the behavior of
the self to causes in preceding conditions much as the physicist
traces causal connections between the motions of stones. Haeckel,
however, overlooks the fact that at this point we happen to be in a
peculiarly favored position. We can see the action from within as
well as from without, and as we do so we discover a process of
determination differing profoundly from the mode of determination
described by the scientist. In our own case our action cannot be
understood apart from our ideal. This ideal, although due to pre-
ceding conditions of one sort or another, does not act upon us as
an external compelling force, but influences us through the appeal
it makes to our own interests. It is an ideal for us because we
ourselves select it, and not because it is forced upon us by any ex-
ternal force. But this process of the selection of an ideal, or of
evaluation, is distinct from any process found in the purely physical
world and is not describable in mechanistic terms.22
Haeckel himself grows eloquent over the ideals of the good,
the true, and the beautiful, and urges us to put these ideals before
any false ideals promulgated by superstition. Such exhortations,
however, have apparently little place in an absolutely mechanistic
scheme where each self is absolutely determined by his heredity
and environment. Haeckel becomes indignant over what he regards
as superstitions, yet he should recall that all superstitions as well
as the despised dualistic philosophy are, on his scheme, natural
products and therefore as necessary as his own monistic utterances.
20 Cf. Bosanquet, The Value and Destiny of the Individual, pp. 109ff.
21 Cf. The Riddle of the Universe, p. 131 ; History of Creation, Vol. I, p.
237.
22 For a clear description of the distinction between personal and mechan-
ical determination, cf. Ward, Realm of Ends, pp. 179ff.
306 THE MONIST.
Furthermore the ideals of the good, the true, and the beautiful
must be, for Haeckel, purely human ideals, since no values exist
for the universe. But if man himself has as little value as Haeckel
gives him, it is strange that he should regard human ideals as worthy
of reverence and worship.
A final word must be added to our criticism of mechanism.
The theory of mechanism itself is not, as Haeckel must believe, a
purely natural product. It is due to the organizing activity of man's
intelligence and could not exist without it. Haeckel regards this
unifying and critical faculty of man as due to the "concatenation
of presentations."23 Yet the mere concatenation of presentations
could never of itself lead to the criticism and combination necessary
to bind together these various sensations under the law of causa-
tion. This unifying of experience demands, as Eucken has so clearly
shown, that man be able to separate himself from the chain of na-
ture in order to combine and order the presentations that come to
him. Hence the formulation of the theory of mechanism is a fact
which mechanism itself fails to explain, and the very existence of
the theory is evidence of its own inadequacy as a final explanation of
all facts in the universe.
Our examination of Haeckel's philosophy has shown the lack
of cogency of his denial of freedom. While this in itself furnishes
no evidence for the reality of freedom, it at least frees us from many
objections that are commonly raised against it. It indicates that the
problem cannot be disposed of in so summary a manner by science,
and thus affords ground for those who in the twentieth century,
in spite of Haeckel's dictum, maintain the possibility of freedom.
GERTRUDE CARMAN BUSSEY.
GOUCHER COLLEGE, BALTIMORE, MD.
DETERMINISM OF FREE WILL.
WITH REFERENCE TO THE PRECEDING ARTICLE.
There is a strange confusion about mechanicalism and freedom
of will which seems to have been constructed by our theological
school of educators on the basis of a misinterpretation of philo-
sophical thought, and errors thus derived are still perpetuated.
The idea of the will is perhaps the fundamental conception of
23 Cf. The Riddle of the Universe, pp. 121f.
CRITICISMS AND DISCUSSIONS. 307
ethics, and an important item for moral purposes is the freedom of
an acting person. But "free will" is nothing mysterious nor in-
credible ; it is that condition of a will which is not hindered by com-
pulsion. He is free who acts on his own account, according to his
own character, and is not interfered with by external circumstances
which would make it impossible for him to act as he wishes. A man
under compulsion is not responsible for his action; for his act is
the act of some one else, or is due to the circumstances which force
him against his own will. The external circumstances may be ever
so indirect and may be reducible to fear. A man threatened by the
consequences of the results of his act is no freer than a man who
is directly forced into acting contrary to his will by facing the re-
volver of a highway robber. If an act is committed because the
acting person wishes the act and also willingly accepts all of its
consequences, it is and ought to be considered an act of free will,
and there is scarcely any thinker who would not admit this definition
of free will.
Is there any one who denies that the act of a free will, as here
defined, is as much determined as any other event in this world in
which we live? If the free act of a man is really the result of de-
liberation and if it is performed according to the nature of the
actor's character, the result of this decision will be as necessary as
the act of an unfree man who acts under compulsion according to
motives of fear or any external force. Determinism is a general
feature of the world which expresses the truth that the law of
causation remains unbroken. According to the law of causation,
everything is determined, even the act of a free man.
Yet there are, or rather have been, some theologians who believe
in free will, not as free will necessarily must be, viz., an unhampered
will, but as a carte blanche or tabula rasa, a cause that is not caused,
or as a determinant which on its part is undetermined, which is
free in the sense that it is unformed, or a factor that is somehow
an exception to causation and not the product of the efficacy of
causation. They think that a man is not responsible if his actions
are determined or determinable and can be predicted, just as in
moving pictures only such consequences will happen as are on the
films, and the man who knows the film would naturally and neces-
sarily be able to tell what is going to happen in the next moment.
What an undetermined will is or would be, has never as yet been
clearly described; it is only declared to be an exception to the law
308 THE MONIST.
of causality, and being undetermined seems to be as much a mere
chance product as the haphazard cast of dies, in which case of
course the actor could no longer be regarded as responsible for a
deed not determined by himself.
The truth is that if we were omniscient we could predict the
history of the world from step to step just as the theatrical man-
ager of the movies knows the next act if he knows the film that is
to project it on the screen. If I know all the characters of the
acting persons, I will be able to predict the outcome of their activ-
ity under definite conditions, and there can be no quibbling about it.
We must not identify necessity and compulsion. Everything
is determined ; and all acts are determined with necessity, even the
free acts of a free man. Further it would be wrong to say that
man is compelled to act according to factors which are none of his
making, if he necessarily acts according to his will.
It is true that there are factors which have preceded him;
among them there are factors such as have determined his char-
acter. He has been determined and his will has been given him.
In this sense it is claimed that he is as unfree as any slave who is
not his own master. But is that not a wrong conclusion that here
too identifies necessity with compulsion? It is necessary that a
man should act according to his character if he is not under com-
pulsion. The acts of a free man are necessary because his will
necessarily and naturally follows the impulses of his own character.
To say that we are slaves because we follow necessarily our own
instincts is simply an illogical distortion of facts. The truth is that
in doing what we will we obey the behest of those factors which
shaped our will. However, granting that our will is not of our own
making, we will be obliged to confess that we are the continuation
of those factors which make us ; or in other words, our ancestors
whose will we incorporate are ourselves in a former generation.
Thus we ought to recognize openly and unhesitatingly that the
whole development of the world is not a piecing together of inde-
pendent individuals, but that we are mere fragments of a continuous
whole, we are pieces of a prolonged history of one and the same
aspiration which may be modified, improved, or even on the other
hand weakened and debased. Former generations have made us
of the present age, and future generations will be as much the
product of the present generation as we are of the past. Thus
if we speak of having been made by prior factors we must recognize
CRITICISMS AND DISCUSSIONS. 309
that the factors that made us are our own existence, as we existed
in former days, — yet the truth remains that a free will is definitely
determined. A free will which acts in an unhampered way is as
much determined as any will which suffers violence or acts under
compulsion.
Miss Bussey has taken up Haeckel and criticizes him for de-
nying freedom of will where he stands up for determinism. I do
not think that Professor Haeckel will take up the cudgel and defend
himself. On the other hand I grant that Professor Haeckel is an
enthusiastic defender of the monistic world-conception for which
he demands a strict and universal application of the mechanistic
theory to all events of existence. I will not deny that Professor
Haeckel sometimes accepts views which I myself would not endorse.
For instance he identifies God with matter and energy while I would
look upon God in contrast to matter and energy, as a religious
formulation of the world order which is the ultimate raison d'etre
of natural law throughout the sphere of existence, including also
the natural law that governs human society and is the basis of the
rules of conduct. But this is a point which could easily be recon-
structed or altered, for Professor Haeckel himself would scarcely
object to it.
In order to understand Haeckel one ought to interpret his
writings in the spirit in which he has written them, and ought not
imply mistakes which are rather incidental points, such as Miss
Bussey criticizes.
Miss Bussey in criticizing Professor Haeckel should consider
that he rejects the theory of free will because he understands by free
will the theological conception of an undetermined will, viz., that
kind of a free will that does not exist, because it is a self-contradic-
tory notion, an impossible and foolish conception of a misguided
brain. If he rejects it he does so only in the sense in which theo-
logians have misrepresented freedom of will as being exempt from
the law of causation. And in doing so he is certainly right in the
face of Miss Bussey or any one who believes in a freedom of that
kind, proclaiming that it is independent of causation.
There is no need of entering into the details of Miss Bussey 's
discussion. Any of our readers who knows Haeckel will be able
to form his own judgment. Only a few points shall be mentioned
here.
The universe has certainly to be explained from the highest
3IO THE MONIST.
product its development achieves and not from its lowest beginnings.
It is man that gives us the key to the appearance of the moner,
while the moner will not be able to tell what its evolution will bring
out in the end. On the other hand we have not solved the problem
unless we trace the development of a rational being step by step
in a mechanistic fashion of cause and effect. To deny it would
mean to abolish science in spots. I prefer to keep my trust in
science, for science to me is God's revelation. The most important
step for instance is the development of reason, and it has been ex-
plained in a mechanistic sense by Ludwig Noire when he shows how
the origin of language has produced reason and not the reverse ; or,
to express his principle in a popular way, "We think because we
speak" and not "we speak because we think." The mechan-
ical mechanism of speech came first, and it was the mechanism of
logic and grammar which has enabled us to think.
It is not a fault of Haeckel's if he holds the view that man
explains the nature and significance of the moner. It proves that
he is not onesided. His claim is but the natural consequence of a
consideration of evolution.
The law of the conservation of matter and energy is an a priori
law, which in its general' meaning is similar to mathematical postu-
lates. It is a demand of science and need not be proved in detail.
It is a pre-supposition just as much as is the law of causation
which the scientist assumes when he investigates natural phenom-
ena. That there is a purpose in the universe is a proposition which
would involve a belief that the universe as a whole is to be under-
stood as an individual personal being after the fashion of a man.
It would involve an anthropomorphic conception of God, and I
doubt whether even among our theologians there are now many
bold enough to take such a position. This, however, does not ex-
clude that the universe in its processes follows a definite direction,
a claim which is proved by the facts of evolution and is probably
not denied by either a theistic or atheistic interpretation of the
word.
Why the formulation of the theory of mechanicalism should
be a fact which mechanicalism itself fails to explain is unintelligible,
and why its own existence should be evidence of its own inadequacy
is hard to understand, unless the notion of mechanicalism be nar-
rowed to a limited field which does not include the entire construe-
CRITICISMS AND DISCUSSIONS. 3! I
tion of mechanicalism and its internal interrelations, such as for
instance the interrelations of logical rules and conditions.
We may be able to uphold the theory of free will but we shall
certainly not be able to deny the principle of determinism, and this
is a blessing for the ethicist who preaches morality and claims that
the freedom of will is essential for it, because if free will were
indeed an exception to the law of causation and the will were unde-
termined and not changeable by education but remained a tabula
rasa in spite of all attempts to change and improve it, or make it
definite in the right direction, what would be the use of wasting our
energies in promoting the welfare of mankind and eliminating evil
influences? Let us be glad that determinism is true, for otherwise
there would be no science, and principles of conduct would be a
meaningless play of a misguided and erring imagination.
Haeckel apparently commits a very grave mistake. His opin-
ions are "the result of the general presuppositions and prejudices of
the age." He and many others "believe whatever fits in with their
view of life and dismiss without a hearing anything which conflicts
with it." Miss Bussey claims that "in this age of science the scien-
tist has become the arbiter of all questions, and his view is com-
monly accepted as authoritative." In other words, we expect that
science shall solve our problems, and we are prejudiced enough to
bow down before science and accept its verdict. Haeckel for in-
stance is so prejudiced that he believes in the universality of natural
laws, and, says Miss Bussey, "It is a philosophic commonplace that
laws resting on experience can be universalized only by means of
the supposition of the uniformity of nature." It is a pity that
Haeckel follows this fallacy and accepts the uniformity of nature,
but the worst is that I too plead guilty. I believe not only in his
"supposition of the uniformity of nature," but also in science with
all that it implies, especially determinism which demands the de-
terminedness of everything, even the determinedness of an unham-
pered and, in this sense, free will. I can not help it. I am in the
same predicament as Professor Haeckel. May God have mercy on
our souls! EDITOR.
THE BELIEF IN GOD AND IMMORTALITY.
Professor James H. Leuba, professor psychology and peda-
gogy in Bryn Mawr College, has undertaken to write a book on
The Belief in God and Immortality. It is not a proof or disproof
312 THE MONIST.
of the doctrines essential in all positive religious creeds but a study
of psychological statistics as to frequency and distribution of be-
liefs in a personal God and a personal immortality, and he finds
that upon the whole in each group investigated as to their religious
beliefs, the more distinguished fraction includes by far the smaller
numbers of believers.
Professor Leuba's work is divided into three parts. The first
part enters into a discussion of the characteristics of a belief in a
continuation after death. He begins with the savage's idea of soul
and ghost, setting forth in his second chapter the origin of the
ghost idea, the appearance of ghosts in dreams and visions. He
distinguishes from the belief in soul-ghosts the belief in immor-
tality which he regards as late in the development of mankind.
The fourth chapter is devoted to "The Origin of the Modern Con-
ception of Immortality," beginning with a "translation to a land of
immortality." The fifth chapter enters into metaphysics, the deduc-
tions of which however are regarded as insufficient.
In later days more scientific methods have been used by relying
on physical and psychical manifestations and on the historical facts
on which the resurrection of Christ is taught.
In Part II the belief in the personality of God and immortal-
ity is made an object of statistical study, first (Chapter VII) among
the students of American colleges. In this it has become necessary
to make a distinction between the personal and impersonal con-
ceptions of God. The eighth chapter is devoted to an investigation
of the belief in immortality, including a comparison of the changes
taking place during college years. Here follows a detailed investi-
gation (introduced first by the causes of the failure to answer and
the interpretation of the questionnaire) of the beliefs held by
the scientists, the historians, the sociologists, the psychologists, and
the philosophers, concluding with a comparison of the signed and
unsigned answers. He comments on the results of his investigation
thus:
"The essential problem facing organized Christianity is con-
stituted by the wide-spread rejection of its two fundamental dog-
mas— a rejection apparently destined to extend parallel with the
diffusion of knowledge and the moral qualities that make for emi-
nence in scholarly pursuits."
The third part which might be considered as independent of
the first two is devoted to the question of the utility of the belief
CRITICISMS AND DISCUSSIONS. 313
in personal immortality and a personal God. Professor Leuba
asks the question, "Is humanity better off with than without that
belief (in a personal God and a personal immortality) ? He answers:
"The utility of the belief in immortality to civilized nations is much
more limited than is commonly supposed .... we may even be brought
to conclude that its disappearance from among the most civilized
nations would be, on the whole, a gain."
It is noteworthy that his results show that the desire for im-
mortality and the usefulness of the belief is rather disappearing
with an increase of intelligence. There is an increasing tendency
to disclaim any desire for immortality. This is in strong contrast
to the supposition formerly quite common that even disbelievers
yearn for immortality, but among the answers received to a ques-
tionnaire Professor Leuba finds even a relatively considerable num-
ber who abhor the idea of an endless continuation and he quotes a
number of instances. For instance a woman thirty years of age
declares that she has always felt death to be better than all else,
anticipating it as the best thing life has to offer; and concluding
with the sentence that death itself is a consummation devoutly to
be wished.
Another letter is quoted as stating, "I feel a great dread of the
possibility of having to live forever, or even again," and Professor
Leuba quotes from Swinburne's poem "The Garden of Proserpina"
the poet's hope of annihilation, where he says:
"Then star nor sun shall waken,
Nor any change, of light ;
Nor sound of waters shaken,
Nor any sound or sight;
Nor wintry leaves nor vernal,
Nor days nor things diurnal;
Only the sleep eternal
In an eternal night."
John Addington Symonds echoes the same ideas in prose. He
says:
"Until that immortality of the individual is irrefragably dem-
onstrated, the sweet, the immeasurably precious hope of ending
with this life, the ache and languor of existence, remains open to
burdened human personalities."
The greater stimulus for a desire for immortality comes in
cases of the death of beloved persons, and the most impressive
instance of this kind is quoted by Professor Leuba in the case of
314 THE MONIST.
a widow writing to her friend, the famous Professor Schleier-
macher. Quoting from Schleiermacher's Leben as quoted by James
Martineau in A Study of Religion, Vol. II, page 337:
"O Schleier, in the midst of my sorrow there are yet blessed
moments when I vividly feel what a love ours was, and that surely
this love is eternal, and it is impossible that God can destroy it;
for God himself is love. I bear this life while nature will; for I
have still work to do for the children, his and mine ; but O God !
with what longings, what foreshadowings of unutterable blessedness,
do I gaze across into that world where he lives! What joy for me
to die!
"Schleier, shall I not find him again? O my God! I implore
you, Schleier, by all that is dear to God and sacred, give me, if
you can, the certain assurance of finding and knowing him again.
Tell me your inmost faith on this, dear Schleier; Oh! if it fails, I
am undone. It is for this that I live, for this that I submissively and
quietly endure : this is the one only outlook that sheds a light on my
dark life, — to find him again, to live for him again. O God! he
cannot be destroyed !"
In commenting that the psychological state might have been
quite different in Schleiermacher's friend if she had remarried.
Professor Leuba says : "In that occurrence her former yearnings
for another life might have been replaced by dread of the time
when she would be face to face with two husbands."
Perhaps the most dignified expression of an impersonal im-
mortality has been expressed by George Eliot in her "Choir In-
visible," but the main and classical instance is the orthodox Bud-
dhist faith, and Professor Leuba quotes at length the text from
Buddhist scriptures as translated by Henry Clarke Warren, where
Buddha insists on not being born again and that the present life
is his final entry into Nirvana. It reads thus:
"And being, O priests, myself subject to birth, I perceived the
wretchedness of what is subject to birth, and craving the incompar-
able security of a Nirvana free from birth, I attained the incom-
parable security of a Nirvana free from birth; myself subject to
old age, .... disease, .... death, .... sorrow, .... corruption, I per-
ceived the wretchedness of what is subject to corruption, and, crav-
ing the incomparable security of a Nirvana free from corruption, I
attained the incomparable security of a Nirvana free from corrup-
tion. And the knowledge and the insight sprang up within me, 'My
CRITICISMS AND DISCUSSIONS. 31$
deliverance is unshakable; this is my last existence; no more shall
I be born again.' And it occurred to me, O priests, as follows :
" 'This doctrine to which I have attained is profound, recondite,
and difficult of comprehension, good, excellent, and not reached by
mere reasoning, subtile, and intelligible only to the wise. Mankind,
on the other hand, is captivated, entranced, held spell-bound by its
lusts ; and forasmuch as mankind is captivated, entranced, held
spell-bound by its lusts, it is hard for them .... to understand how
all the constituents of being may be made to subside, all the sub-
strata of being be relinquished, and desire be made to vanish, and
absence of passion, cessation, and Nirvana be attained.' "
It is peculiar that among scientists there was one who clung
with great insistence to the belief in immortality, and this is no less
an authority than the great biologist, Henri Pasteur, and he kept his
religious faith and science in two different departments of his
mind. He says:
"My philosophy is of the heart and not of the mind, and I
give myself up, for instance, to those feelings about eternity which
come naturally at the bedside of a cherished child drawing its last
breath.
"There are two men in each one of us: the scientist, he who
starts with a clear field and desires to rise to the knowledge of
Nature through observation, experimentation, and reasoning; and
the man of sentiment, the man of belief, the man who mourns his
dead children and who cannot, alas, prove that he will see them
again, but who believes that he will, and lives in that hope;....
the man who feels that the force that is within him cannot die."
Professor Leuba adds the following comment on Pasteur:
"I may remark incidentally upon the off-hand manner in which
Pasteur divides life into two spheres, that of science and that of
feeling, and apparently finds no use for logic and reason in the
latter. This is a shocking example of a dangerous practice which,
when carried to its logical consequence, would permit one to believe
whatever he pleases. When I attempt to understand this attitude
in a distinguished man of science, I can only conjecture that he
treated religion as something primarily intended to comfort anyway,
anyhow."
Professor Leuba's book does not decide the question of the
acceptability or unacceptability of the belief itself, but is simply a
statistical investigation and for that reason possesses virtue for
316 THE MONIST.
theists as well as unbelievers in helping to find out the psychological
state of things as it happens to be in our present generation, and
from that standpoint the book will retain its virtue whatever be the
position of the reader.
SIR OLIVER LODGE ON LIFE AFTER DEATH.
Sir Oliver has always been a believer in mediumistic experience
and in the spirit existence of man in the other world, and in spite
of his knowledge of physics he has taken a broad stand by coming
out squarely and unreservedly in showing his faith. Details of such
an expression might be amusing if it were not actually sad to see
a man of his significance stooping to views which otherwise prevail
only in the circles of half -educated people. His son Raymond died
at the front in Flanders on September 14, 1915, and the bereaved
father has published a book1 containing a summary of his own
philosophical views and a record of communications received from
Raymond since his death.
From this we learn that Raymond woke up in the other world
and got accustomed to his new surroundings. There are seven
spheres all above the earth and turning around with the earth, but
there is no consecutive night and day. It is always daylight except
when one desires darkness ; then night spreads according to one's
wishes. Raymond resides in a house which appears to be made
of brick, and spirit houses form streets in which the spirits walk
and move. People who have lost arms or legs develop new ones
as if by a kind of natural recuperation, so he tells his parents that
he has replaced a tooth, and comrades of his who had lost arms or
other limbs are restored to their original natural shape, but this res-
toration is not quite simple and there is a kind of spirit-doctors
who help with their restoration. There is a special difficulty in
restoring the spiritual body if the material body has been destroyed
before its regeneration in the spirit world, so Raymond gives a
definite warning that dead bodies should not be cremated before
father has published a book containing a summary of his own
they have been restored in the spirit plane of life.
The seven spheres which are built around the earthly plane
seem to revolve with it at different rates of speed, so that the first
sphere is not revolving at the same rate as the second, third, fourth,
fifth, sixth and seventh spheres. Greater circumference makes the
CRITICISMS AND DISCUSSIONS. 317
revolution more rapid and this increase of rotation has an actual
effect on the atmospheric conditions prevailing in different spheres.
When asked for details about the nature of the other world Ray-
mond said :
"What I am worrying about is how it is all made and of what it
is composed. I have not found out yet, but I have a theory. It is
not an original idea of mine. I was helped to it by words dropped
here and there. People who think everything is created by thought
are wrong. I imagined for a little while that one's thoughts over
here formed the buildings and flowers and trees and solid ground ;
but there is something more than that.
"There is something always rising from the earth — something
chemical in form. As it rises to ours it goes through various
changes and solidifies here. I feel sure it is something given off
from the earth that makes the solid trees, flowers, etc
"All the decay that goes on on the earth is not lost. It doesn't
just form manure or dust. Certain vegetable and decayed tissue
does form manure for a time, but it gives off an essence or a gas
which ascends and which becomes what you call a 'smell.' Every-
thing dead has a smell, if you notice ; and I know now that the smell
is of actual use, because it is from that smell that we are able to
produce duplicates of whatever form it had before it became a smell.
Even old wood has a smell different from new wood ; you may have
to have a keen nose to detect it on the earth plane.
"Old rags, cloth decaying and going rotten, all have smells.
Different kinds of cloth give off different smells. You can under-
stand how all this interests me. Apparently, so far as I can gather,
the rotting wool appears to be used for making things like tweeds
on our side. But I know that I am jumping; I'm guessing at it.
My suit, I expect, was made from decayed worsted on your side.
"Some people here won't grasp this even yet — about the material
cause of all these things. They go talking about spiritual robes
made of light, built by thoughts on the earth plane. I don't believe
it. They go about thinking that it is a thought robe they're wearing,
resulting from the spiritual life they led ; and when we try to tell
them it is manufactured out of materials they don't believe it. They
say, 'No, no; it's a robe of light and brightness which I manufac-
tured by thought.' So we just leave it. But I don't say that they
don't get robes quicker when they have led spiritual lives down
318 THE MONIST.
there; I think they do, and that's what makes them think that they
made the robes by their lives.
"You know flowers how they decay. We have got flowers
here ; your decayed flowers flower again with us — beautiful flowers."
They have not only spirit doctors but also manufacturers and
can provide you with materials if you so desire. Raymond himself
does not smoke, but a friend of his, a great smoker on the earth
plane, demanded cigars and he got them, but only about five; and
the things given him looked like cigars, but after smoking about
five cigars he no longer cared for more. He changed his habit and
got accustomed to a more spiritual mode of life.
Colors have their significance, and different colors have dif-
ferent effects upon the character of the spirits.
"There's plenty of flowers growing here, you will be glad to
hear. But we don't cut them here. They don't die and grow again ;
they seem to renew themselves. Just like people, they are there all
the time renewing their spirit bodies. The higher the sphere he
went to, the lighter the bodies seemed to be — he means the fairer,
lighter in color. He's got an idea that the reason why people have
drawn angels with long fair hair and very fair complexions is that
they have been inspired by somebody from very high spheres."
The information Professor Lodge publishes was received from
the medium Mrs. Leonard through her "control" known as "Fedo."
Incidentally we find a personal remark put in brackets and in
italics of which Sir Oliver is apparently the authority. It reads:
"A good deal of this struck me as nonsense, as if Peda has picked
it up from some sitter."
Mediums have said much nonsense in print as well as in private
seances, and the spirits of dead people have distinguished themselves
by silly utterances ; but the recent story of Raymond's communica-
tions rather excels all prior tales of mediumistic lore in the silliness
of its revelations. But the saddest part of it consists in the fact
that a great scientist, no less a one than Sir Oliver Lodge, has pub-
lished the book and so stands sponsor for it.
Sir Oliver Lodge is a scientist who has done much creditable
work and has written a number of books which exhibit keen thought
and a good grasp of his subject, his specialty being physics. The
books he has written are as folows:
Elementary Mechanics; Modern Views of Electricity; Pioneers
of Science; Signalling Without Wires; Lightning Conductors and
CRITICISMS AND DISCUSSIONS. 319
Lightning Guards; School Teaching and School Reform; Mathe-
matics for Parents and Teachers; Life and Matter; Electrons;
Modern Views of Matter; The Substance of Faith; Man and the
Universe; The Ether of Space; The Survival of Man; Parent and
Child; Reason and Belief; and Modern Problems.
CURRENT PERIODICALS.
In Vol. XIV (1915) of the fifth series of the Atti of the Royal
Academy of the Lincei at Rome is a publication in full of the treatise
De corporibus regularibus of Pietro Franceschi or Delia Francesca
which was found in 1912 in the Vatican Library by G. Mancini. To
this is prefixed a learned dissertation by Mancini to show that this
treatise was pilfered by Luca Pacioli in his work on mensuration, the
Divina proportione ; and a report by Gino Loria on Mancini's memoir.
* * *
The articles of greatest interest to philosophical mathematicians
in recent numbers of Vol. XVII (1916) of the Transactions of the
American Mathematical Society are as follows. In the number for
April, Robert L. Moore gives three systems of axioms for plane
analysis situs — the non-metrical part of the theory of plane sets of
points, including the theory of plane curves ; Charles N. Haskins
writes on the measurable bounds and the distribution of functional
values of "summable" functions — which here means functions which
are integrable in the generalized sense of Lebesgue; and Dunham
Jackson proves in another way an important theorem of Haskins.
In the number for July, L. L. Silverman discusses the generalization
of the notion of the summability of a series to the limit of a function
of a continuous variable ; G. H. Hardy develops a new and powerful
method for the discussion of Weierstrass's continuous function
which is not differentiable, and allied questions; and William F.
Osgood, to show that a theorem of Weierstrass for analytic func-
tions of n complex variables is true for other "spaces" than that of
analysis, lays down a general definition of "infinite regions," which
includes the cases of projective geometry, the geometry of inversion,
the geometry of the space of analysis, and so on.
* * *
In the Bulletin of the American Mathematical Society for June,
1916, Dr. A. Bernstein reduces the number of postulates which
32O THE MONIST.
Huntington gave in 1904 for Boole's algebra of logic from ten to
eight, and that of postulated special elements from three ("zero",
the "whole," and the "negative") to one (the "negative"). An
interesting and valuable address delivered before the University of
Chicago by Prof. Edward B. Van Vleck on "Current Tendencies
of Mathematical Research" is printed in the October number.
The number of the Revue de metaphysique et de morale for
May, 1916, contains a long and important article by A. N. White-
head on the relationist theory of space. This theory is developed
for a great part by help of the symbols of the author and Russell's
work. The other articles in this number are by F. Colonna d'Istria
(religion according to Cabanis), Leon Brunschvicg (the relations
of the intellectual and the moral conscience), R. Hubert (the Car-
tesian theory of enumeration : on the fourth Rule of the Discours) ,
and Georges Guy-Grand (impartiality and neutrality). In the July
number of the Revue Lionel Dauriac writes on contingence and
category, and tries to decide whether Kant was right or wrong in
not separating the necessary and the a priori. Gaston Milhaud dis-
cusses the famous mystical crisis through which Descartes passed in
1619. Henri Dufumier maintains that the algebra of classes in logic
only takes a systematic form if we consider it as a generalization
of the mathematical theory of aggregates. F. Buisson explains
"the true meaning of the sacred union." Finally, there is a necrol-
ogy of Victor Delbos (1862-1916).
In the eighteenth volume (1916) of Prof. Gina Loria's quar-
terly Bollettino di bibliografia e storia delle scienze matematiche ,
the most interesting articles in the first two numbers (April and
June) seem to be: J. H. Graf's collection of the correspondence
between Ludwig Schlafli and some of his Italian mathematical con-
temporaries (pp. 21-35, 49-64) ; and G. Vivanti's review of the
late Julius Konig's Neue Grundlagen der Logik, Arithmetik und
Mengenlehre of 1914 (pp. 37-39).
VOL. XXVII JULY, 1917 NO. 3
THE MONIST
THE ELECTRONIC THEORY OF MATTER.1
•
"Wic Alles sich zum Ganzen webt !
Eins in dem Andern wirkt und lebt."
— Goethe.
THE subject of the considerations that follow is pro-
posed as the sixth under the division of physics in
the published program of this congress. Unquestionably
the proposal was most timely and fortunate, for no theme
of purely scientific content is more important or more cen-
tral on the stage of interest or more worthy of the atten-
tion of the assembled savants of two continents. Surely
it is eminently appropriate that the New World should
foster the New Knowledge, should master it, acclaim it,
proclaim it, and advance it.
The most obvious criticism upon any attempt to treat
this theme on the present occasion would seem to be that
the barrel is too large for the hoop. So far and wide
reaching is the new doctrine of matter, so interpenetrative
of so many remote and alien disciplines, that any half-way
adequate presentation of even the most near-lying con-
siderations would necessarily pass swiftly beyond the
largest bounds to be assigned this paper and easily ex-
pand into a stately volume.
1 This paper, read (in Spanish") at the First Pan-American Scientific
Congress (Santiago de Chile, Dec. 25, 1908 to January 5. 1909), has appeared
thus far only in the Trabajos del Cuarto Congreso Cicntifico (i° Pan- Ameri-
cano), Vol. V, pp. 1-22, Santiago de Chile, 1910.
322 THE MONIST.
\Vc must begin then with renunciation. The attempt
can not be made to detail but only to suggest some of the
proofs (which are manifold and decisive) of the actual
existence of the corpuscle, sub-atom, or electron, as the
uniform elementary constituent of the visible universe is
variously named. The isolated independent subsistence of
this corpuscle is the central revelation of the new knowl-
edge. Tt was first discovered many years ago, and pro-
claimed to the world as the fourth or radiant state of
matter by Sir William Crookes, after whom the vacuum
tubes in which the green phosphorescence accompanying
the passage of an electric current were and still are named.
That something called cathode rays emerged from the
cathode or negative pole and moved in right lines, was
proved by the shadow cast by an interposed mica cross.
The English declared these rays were particles, shot out
from the cathode (pole) against the inside walls of the
tube ; but the Germans held it was only ether waves stirred
up at the pole and propagated rectilinearly. That the Eng-
lish were right was shown conclusively by subjecting the
rays first to magnetic and then to electric attraction, where-
by it appeared that they behave in all ways precisely as
minute particles laden with negative electricity. Amazing
is the control which the experimenter exhibits over these
flying hosts of electric atoms; by deft manipulation of his
infinitely fine magnetic or electric fingers he may turn the
stream of corpuscles as he will and even bend it into a
spiral or into a circular hoop far more supplely than one
might bend the superfinest Damascus blade. But incon-
ceivably more delicate still is the touch of the mathe-
matical reason, whereby even the individual electron is
caught in its flight and forced to tell the secret of its speed.
For one may subject the flying particles simultaneously
to opposite electric and magnetic influences by immersing
them in two coexistent and mutually annulling fields of
THE ELECTRONIC THEORY OF MATTER. 323
force, so that they fly undisturbed straight from the nega-
tive to the positive pole. These two self-destroying forces
are H ev and c X, whence v = X/H, whereby v the veloc-
ity of the corpuscle is known when we know H the magnetic
and X the electric force, both of which are readily meas-
ured. This velocity increases with the exhaustion of the
tube from eight thousand up to one hundred thousand kilo-
meters per second, which is many thousand times the mean
speed even of hydrogen molecules at the highest tempera-
ture ever yet attained.
But far more wonderful and incomparably more im-
portant than this determination of a variable velocity is
the determination of a constant, the most fundamental yet
discovered in nature. Science itself may be defined as the
eternal search for invariants amid the eternal flux of var-
iants, and this astounding constant of which I am about to
speak reminds us indeed of Plato's unwavering axis of the
universe turning forever in the lap of Necessity. In the
equation Hcv = X?, the symbol c denotes the negative
electric charge borne by the individual corpuscle. If we take
away the magnetic force, leaving only the electric, this latter
will bend the path of the flying corpuscle as gravity bends
the path of the level-aimed cannon ball into a parabola.
Now Galilei has taught us the formula for the amount (s}
of the bend or the fall in the time t ; it is s = V>at2 where a
is the acceleration in question. Here the acceleration is
the force X^ divided by the mass in of the corpuscle; the
time is the tube length / divided by the velocity 7'; and the
distance s is the descent of the green spot at the end of the
tube;
hence ^ +* V^(c/m} . (lz/vz\ whence c/m *= (2i^//2X),
where all on the right side is known. Hereby is determined
this ratio of the electric charge to the mass of the flying
corpuscle, and this ratio is found to be everywhere the
324 THE MONIST.
same (unless indeed the velocity of the corpuscle, of which
it is in strictness a function, approaches that of light).
The value of this remarkable constant (for all ordinary
velocities) is in the accepted C. G. S. system no less than
17,000,000 ( i .7Xio7). Why is it so large ? Is it because
the charge e is so great, or because the mass m is so small?
This question can be answered and has been answered by
the exquisitely beautiful experiments of the two Wilsons
(C. T. R. and H. A.) on the formation of clouds by con-
densation of vapor around nuclei. Not only does the water
collect around particles of dust but also around any par-
ticles charged with electricity: nay more, it refuses to
collect except around nuclei until the vapor reaches eight-
fold saturation. Now it has been found possible to free a
cylinder of air from dust, and supersaturate it with vapor,
and then to form in it suddenly a dense cloud by electri-
fying its particles with radiations from radium or still
better by charging its individual molecules with electrons
shot out from a metal plate played on by ultra-violet light.
By attracting electrically these drops coagulated around
these molecules one may suspend them in the air of the
cylinder like balloons or make them fall as slowly as one
will, so that their velocity of fall may be measured; and
Stokes has deduced the formula for this velocity, v -
2/g.ga2/\i where g is the known acceleration of gravity
and \i the known coefficient of viscosity ; hence a, the radius,
and thence the size of the drop is found; and hence by
measuring the amount of watery vapor deposited one finds
the number of the drops and so can count the number of
electrified molecules, that is the number of corpuscles, since
each molecule has but one negative electron. Plainly, if
by one chance in a trillion two corpuscles should light on
one molecule, their mutual repulsion would dislodge them
instantly.
By electrometric methods one may find the total charge
THE ELECTRONIC THEORY OF MATTER. 325
of electricity on the total water, the sum of the drops, and
dividing this by the number of drops or corpuscles one finds
the charge c on each, and then on dividing this by the con-
stant ratio there results the mass m of each corpuscle.
These numbers turn out to be appalling in their minuteness.
The charge e equals 3io/io12 of an electrostatic unit, or
one one-hundred-trillionth (icr-20) of an electromagnetic
unit, and is the long well-known approximate value of the
charge borne by an atom of hydrogen in the electrolysis of
dilute solutions. The mass in of the corpuscle proves to be
six hundred quintillionths of a gramme (6 X IO—28), a
degree of parvitude far beyond the utmost stretch of the
imagination. The same may indeed be said of the atom
of hydrogen, but the mass of this atom is 1700 times2 the
mass of the corpuscle. Hitherto this hydrogen atom has
been conceived as standing on the remotest confines of
matter, but the new knowledge shows us a still lower world
of corpuscles, nearly 2000 times smaller.
At this point it may be proper to enter a caution. It
is almost universal to speak of this corpuscle as of invari-
able mass bearing an invariable charge of negative elec-
tricity, and the calculations and experiments do indeed
yield results uniform within the limits of error. But we
must remember that these experiments and calculations
have always treated and apparently must always treat not
the individual corpuscle but millions on millions of cor-
puscles and atoms. The results then were only averages
of countless numbers of individuals, and the constancy of
such an average implies nothing at all as to the constancy
or inconstancy of the individual, just as the comparative
steadiness of the rates of birth, death, marriage, homicide
and the like, even in a population of a few millions, implies
- Later determinations raise this number to 1830 or even 1872. M. Perrm's
experiments on "visible molecules" indicate that the mass of a hydrogen atom
is 1.63 quadrillionths (1.63/1024) of a gram. Hence the mass of an electron
would be 0.8/1027 gram.
326 THE MONIST.
nothing whatever as to the rate in any particular family.
For all we know the range of individuality among atoms
and corpuscles may be quite as great as among suns or
planets or men, and this we must say even in face of the
famous dictum of Maxwell, that atoms of any one sub-
stance have all the marks of manufactured articles, being
all exactly alike.
Returning from this digression we must now ask what
is the mass and what is the charge of electricity of the
corpuscle? It is precisely here that the new knowledge
calls for the profoundest transformation of our concep-
tions, for it derives the phenomenon of mass in the cor-
puscle solely from the motion of the flying charge of nega-
tive electricity. We all know that work is needed to start
a body in motion, as a car even on a perfectly smooth track.
For any particular body having a particular velocity v,
the amount of work necessary is perfectly definite, namely,
V^ multiplied by a constant, M, called the mass of the
body. We say the kinetic energy imparted is HMf2. This
supposes the motion is in a vacuum, which is never the
case; in practice the motion is always in some fluid, as
water or air. Then we all know that more work is needed,
according to velocity. One fans oneself gently with ease,
but violently only with great effort. In fact, the fluid is
also set in motion as well as the body, and this calls for
energy or work. How much fluid is dragged or pressed
along with the body will depend on the body's size, shape
and speed and on the density of the fluid. Some of the sim-
plest cases have been studied. Sir George Gabriel Stokes
has found that in case of a sphere the work done on the fluid
is %M' V2, where M' is the mass of a volume of the fluid
half as large as the sphere (shown by Green, 1833), so that
the total energy imparted is %(M + M')V2-. Now all
motions take place in the all-pervading ether. It follows
then that, if the ether itself has mass, when put into move-
THE ELECTRONIC THEORY OF MATTER. 327
ment it must absorb energy or require work, and bence
that some perhaps infinitesimal part of the mass of a
moving body even in a vacuum must be due to the swirl
set up in the ether. In the case of the moving corpuscle
the analogy is not absolutely perfect, but exact enough to
make intelligible the statement that if a conducting sphere
of radius r, having a charge e and mass ;//, be set moving
with velocity v, the energy developed in the ethereal mag-
netic field has been proved to be Vzk(e/r} .z>2, so that the
total work done is V»[m -f- %k(c/r)]v2, and the ordinary
mass m is thus increased by %k(e2/r), which stands for the
inertia overcome in the ether. (This k is a factor due to
the crowding together of the lines of force into a plane
through the sphere center, and perpendicular to the mo-
tion, and increases rapidly as the velocity becomes great.)
Since e is extremely small, this increase is wholly imper-
ceptible in case of aft single bodies subject to our senses
or experiment. But for the corpuscle, when r becomes
inconceivably small, this so-called electric mass assumes
important proportions, yea, it accounts for absolutely all
the mass of the corpuscle, which must have this electric
mass and need have no other at all. For Kaufmann has
measured the value of e/m (or m/e) for the various cor-
puscles emitted with various velocities by radium ; and J. J.
Thomson has calculated k for these velocities. It turns
out that the calculated relative increase (due to rising
velocity) in the electrical mass is constantly equal to the
observed relative increase in the whole mass, whence one
must conclude that the electrical mass is the whole mass,
for if there were any ordinary non-electrical mass, however
small, it would certainly not thus increase apparently with
the increasing velocity. The mass of the corpuscle is thus
not located, at least in any appreciable degree, in the cor-
puscle itself, but only in the universal ether around it.
Imagine a sphere surface perfectly rigid but absolutely
328 THE MONIST.
void, empty even of ether itself, a mere round hole in
universal ether. If set in motion this hollow sphere would
gather mass as it gathered velocity, but the mass would
not be inside, it would be wholly outside, inwrought in the
universal eddy set up in the infinite ether. In this sense
the mass of the moving hollow sphere would be coextensive
with the whole space filled by ether, and in this sense we
may say the same of the mass of a flying corpuscle: it
reaches throughout the world. We may imagine it as a
mere needle-point from which Faraday tubes of force radi-
ate to the utmost stars. But since the ether bound by the
tubes varies as the squared density of the tubes, and hence
varies inversely as the fourth power of the distance from the
sphere center, it follows that the corpuscle mass is after all
highly concentrated round the corpuscle core. For an easy
reckoning shows that the corpuscle radius r is only about
five millionths of the molecule radius, which is commonly
taken as the hundred millionth of a centimeter ; that is, of
course, in order of magnitude. Hence from the surface of
a molecule to the surface of a corpuscle at its center the
mass intensity would increase more than a trillionfold.
It follows that one can no longer affirm with perfect
rigor the principle of the conservation of mass, for the
masses vary constantly with the velocities of the corpuscles.
But to our gross senses even when armed with the most
delicate instruments these variations might forever remain
imperceptible. However, Heydweiler claims to have actu-
ally detected a difference in the joint weights of water and
copper sulphate before solution and after, and Wallace
holds that the mass of water is changed by freezing —
highly interesting results, that await confirmation. But it
must not be supposed that the notion of mass itself is
hereby eliminated or e^en reduced to greater simplicity.
For all these results assume to start with the assumption
that the ether itself has mass. Calling then to one's help
THE ELECTRONIC THEORY OF MATTER. 329
•
the Faraday image of tubes of force and still more the
hydrodynamics of vortex rings, one may deduce from
ether-mass the mass of all material bodies; but mass itself
adheres along with time and space inexpugnably in our
reasonings.
Corpuscles therefore are; atoms also are; how then
shall we think them related? As the corpuscle mass is
only Vnoo of the hydrogen atom mass, shall we think this
smallest atom as compounded of 1700 corpuscles? There
are many reasons against such a doctrine, reasons that lead
one to think the number of corpuscles in the atom as always
small. But shall we think the atom as in any case com-
posed of corpuscles? There seems to be no escape from
such a conception, which lies directly across the path of
thought. For many experiments that cannot be mentioned
here show that corpuscles are all-pervasive. Metals heated
pour them forth, as do all other substances hot, and some
shoot them out even when cold, as rubidium, and at fearful
speed, as all radio-active substances; yea, if we had in-
struments fine enough we might detect them in every sub-
stance, and everywhere maintaining the constant ratio e/m
inviolate. Moreover, that the corpuscle is closely related to
the atom is clearly seen in the fact already mentioned that
the corpuscle's and the hydrogen-atom's charges of elec-
tricity are the same.
Before trying to construct imaginatively the atom out of
corpuscles we must recall that there are rays (as Goldstein's
Canals trahlen) of positive as well as of negative electricity,
that are deflected by a magnet oppositely to the negative
cathode rays. For them the ratio e/m is not constant
and never exceeds 10,000, which is also its value for hydro-
gen ions in electrolysis of dilute solutions. It is natural to
figure thus the positive corpuscle as a sphere of positive
electrification, about the size of an atom. Of course such
a sphere implies an equal and balancing amount of negative
33O THE MONIST.
•
electricity, and this we imagine distributed throughout the
sphere as equal corpuscles or units of negative electricity.
Since the atom is permanent, this distribution of negative
electrons in the positive sphere must form a system in stable
equilibrium, and the question arises, what arrangement of
the electrons would constitute such a stable system? The
problem is mathematico-mechanical, and its general solu-
tion lies beyond the range of our present powers of analysis ;
but if we propose the problem not for space but for the
plane, we may solve it and get a system of answers quanti-
tatively different but formally analogous to those that must
be rendered for threefold space.
At this point theory and experiment have joined hands
in a most friendly fashion. As early as 1881 the present
Cavendish professor of physics at Cambridge, stimulated
by the brilliant experiments of Crookes, in a long neglected
but now classical paper in the Philosophical Transactions,
discussed the motion of a charged sphere in an electric
field, thereby laying the foundations of the doctrine of elec-
tric mass. Twenty-three years later ( 1904) he advanced
to the discussion of the equilibrium of a system of negative
electric charges abandoned to their mutual attractions in
a positively electrified shell.* He showed that the configura-
tion of planar equilibrium is (in general) a regular poly-
gon concentric with the sphere, but for six particles the equi-
librium is unstable, one particle will break ranks and rush
to the center, while the other five form a regular pentagon.
Similarly if there be seven, eight or nine; if there be ten,
three will form an inner equilateral triangle, and so on up
to seventeen, when one of the inner ring will again break
ranks and fly to the center, leaving an inner ring of five,
and an outer ring of eleven. (Four corpuscles cannot be in
planar equilibrium at rest, but only when the four are in
rapid rotation. At rest they are at the corners of a regular
* For an additional note see page 480.
THE ELECTRONIC THEORY OF MATTER. 33!
tetrahedron whose edge equals the radius of the sphere.
Six corpuscles are balanced at six corners of a regular
octahedron.) When the number reaches thirty-two, the
three-ring system becomes unstable, again a particle
seeks the center, leaving an inner ring of five, a middle
ring of eleven, an outer ring of fifteen. Looking at it
another way one may ask how many must be put inside a
ring of n to make it stable. The answers are: for 5, o; for
6, 7, 8, each I ; for 9, 2; for 12, 8; for 13, 10; for 15, 15;
for 20, 39 ; for 30, 101 ; for 40, 232. You see how rapidly (as
the cube of n) the inside corpuscles multiply as the outer
ring increases in number. The structure must be compact,
densely peopled toward the center, not hollow like a shell.
The whole scheme of numbers has been worked out by
J. J, Thomson (Philosophical Magazine, 1904) mathe-
matically, and a beautiful experiment first made for an-
other purpose by the American Mayer, afterwards under
other forms by Wood and Monckmann, confirms his re-
sults. On a water surface any number of small equally
magnetized needles are made to float by being thrust
through cork discs, only like poles being above the water.
These repel each other like the negative electric corpuscles.
The sphere of positive electrification is represented by a
magnet hung above the water, the opposite pole pointing
downward. This holds the magnets in groups in stable
equilibrium, and the arrangements of the magnets actually
observed agree excellently with the arrangement required
and predicted by mathematical analysis. Provisionally then,
we may proceed on the working hypothesis that the atom is
a system of corpuscles composed of a number of concentric
sub-systems all held in stable equilibrium by an enclosing
sphere of positive electrification. This conception may be
somewhat vague and may hereafter require modification,
but it is far clearer and more precise than was possible
a few years ago and constitutes a notable advance in phys-
332 THE MONIST.
ical theory. We have spoken of the configuration as stable,
but this stability must not be understood too strictly nor as
always equally rigid. Since we may have an outer ring of
five, six, seven or eight with only one at the center, it is
plain that if there were seven in equilibrium, neither an
increase nor a decrease of just one would disturb the system
much; but if there were sixteen the arrangement would
be one of two rings, eleven in the outer, five in the inner ;
take away one and the arrangement persists with only ten
outside; but add one and the two-ring system is no longer
stable, a particle goes to the center, the rings remain un-
changed, a three-ring system is formed. A better though
more complex example is afforded by the group of eleven,
ranging from 58 to 68 corpuscles (Thomson) ; all these
have five rings, thus, in order:
19 20 20 20 20 20 20 20 20 20 21
16 16 16 16 17 17 17 17 17 17 17
13 13 13 13 J3 13 13 H 14 J5 15
8 8 8 9 9 10 10 10 id 10 10
_2_j2_3_3^3^3jl^_5_j$_5
58 59 60 61 62 63 64 65 66 67 68
Here we see that if a corpuscle be injected into the 58-
system it produces the least possible disturbance, place is
made for it in the outmost ring and the others remain as
they were, a 59-system results. But if a corpuscle be added
to this it can find no place in any ring but the central; it
must find its own way to the center ring or else dislodge
a corpuscle from the outer ring, which corpuscle will then
dislodge another from the next and so on till a corpuscle is
finally dislodged into the innermost ring from the one next
to it. Still another corpuscle may be injected and another
and another, profoundly altering the original arrange-
ment but preserving the outer ring unbroken.
THE ELECTRONIC THEORY OF MATTER. 333
So it goes on, the center becoming denser and the outer
ring more stable till the total number 67 is reached, having
the arrangement 20, 17, 15, 10, 5. Here the central mass,
though it may still make room for another corpuscle, is
very steady and stubborn, so that now when another cor-
puscle is injected, the outer ring yields, absorbs it and now
has 21. Accordingly this 67-system is like that of 58,
most stable, changing least from its original form. The
group of arrangements from 59 to 67 corpuscles forms a
series all having twenty in the outer row, the stability of
each system increasing up to the last, after which a new
group begins with similar properties but only eight mem-
bers. Now these corpuscles are units of negative elec-
tricity. As the number of these inside the atom increases,
the outer ring remaining the same, the stability, measured
by Q, the work necessary to disperse all the units infinitely
apart, increases; the more inside the more firmly the out-
side ring is held. Hence the 59-system will be least stable,
a unit would easily fly off leaving only the preceding 58-
system. If wre suppose this 59-system neutral, on losing
this negative unit it becomes comparatively at least electro-
positive; in fact the most strongly electro-positive of this
series of arrangements. The following members must lose
more and more negative units in order to become electro-
positive as this 59-system. The 67-system is charged with
the utmost negative excess and so is most electro-negative
or least electro-positive. The outer ring of 20 will in fact
admit no more negative units inside, but on still further
addition a new outer ring of 21 is formed and a new
series begins with a highly electro-positive system of 68
and again runs down to an electro-negative system of 77,
in each of which the outer ring is 21. Herewith then we
attain a new notion of valency. For the 59-system has
only just sufficient corpuscles inside to maintain its outer
ring of 20; the fifty-ninth in the system, the twentieth in
334 THE MONIST.
the ring, might easily break away leaving only 19 outside
and the atom positive from the loss of the negative unit.
But it could not remain positive for it would draw to
it another corpuscle and so restore its ring of twenty,
and this process might be repeated. But as many as 8
negative corpuscles might be injected into the ring of 20,
raising the total number to 67; hence we may say this
59-system corresponds to an atom of valency 8 for a nega-
tive unit charge and of valency o for a positive charge,
since it could not assume permanently the positive unit
charge. Consider next the 67-system. It is impossible to
drive a single negative unit within the 2O-ring ; if one
collide with the atom it stops in the outer ring making 21,
but this ring is very unstable and would easily lose this
electric unit ; hence we may say this system has no negative
valency or a negative valency equal to o. However, this
same 67-system might lose one, two or three. . . .or even
eight atoms, reducing its negative, t. e., raising its positive,
charge, though with harder and harder work ; it could not
lose more without changing its outer ring and passing into
another series. Hence we may say that its positive val-
ency is 8, just as its negative valency is o. It is needless
to dilate upon the intermediate members. Similar con-
siderations show that we may arrange them thus:
No. of cor-
puscles . . 59 60 6 1 62 63 64 65 66 67 68
\ -ho +i + 2 +3 +4 --3 —2 —i — o
Valenc . _g _
Electro-positive Electro-negative
Such series are actually found among known chemical ele-
ments. Such are helium, lithium, barium, boron, carbon,
nitrogen, oxygen, fluorin, neon, and neon, sodium, mag-
nesium, aluminum, silicon, phosphorus, sulphur, chlorin,
argon. Of course it is not meant for a moment that any
THE ELECTRONIC THEORY OF MATTER. 335
such planar arrangement is actually present in chemical
atoms ; their corpuscles must be arranged in tridimensional
space ; but it can hardly be doubted that relations analogous
to the foregoing, only more complicated, must characterize
spatial as well as planar arrangements. If we call the
tendency of a system to shed a corpuscle corpuscular pres-
sure (outward}, then it appears that this pressure changes
abruptly at the end of each series ; thus at 58 the pressure
is very low, at 59 very high ; at 67 low ; at 68 high : it falls
through the electro-positives down to and through the
electro-negatives. We might then define positive valency
of an electro-positive (or negative) as the greatest number
of corpuscules it can lose without abrupt fall in corpuscular
pressure; the negative valency of the electro-negative is
the number it can gain without sudden rise of corpuscular
pressure. Upon these definitions and conceptions has been
erected a most plausible theory of chemical combination,
into which we cannot enter. But one other aspect may not
be passed by in silence. While no one affirms that the
planar forms of equilibrium are the actual forms assumed
by corpuscles in atoms, it seems hardly possible that they
are not similar thereto, similar enough to allow a most
important conclusion. These forms are arranged in series,
and the members of these series bear striking resemblances.
There are rings outside of rings, and rings outside of these,
and so on. Thus around a central one there is a ring of 6,
giving 7; and around this a ring of n, yielding 18; and
around this a ring of 15 making 33 ; and around this a ring
of 1 8, making 51 in all ; and around this a ring of 21, which
makes 72; and around this still another of 24, or a total
of 96.
It seems impossible that atoms consisting of these
or any such systems of corpuscles should not have many
likenesses in property. We are reminded of a determi-
nant of a definite form, whose degree is raised by border-
336 THE MONIST.
ing it successively by parallel lines above and below,
on the right and left. The general properties of the de-
terminants remain the same. If then all possible forms
of equilibrium should actually be realized as chemical ele-
ments, of necessity these elements would fall into series
and in fact a twofold series which might be expressed by
a vertical and horizontal alignment, the elements in the
vertical rows being alike in their centraj rings but differing
in the number of rings; those in a horizontal row being
alike in the number of rings and in their outer ring but
different in their inner rings. Herewith then we are lifted
up to what is commonly regarded as the apex of chemical
theory, the periodic law of Mendelyeev, which thus ap-
pears not as an empirical observation, however great
or important, but as an inexorable necessity of the me-
chanical laws of configuration and equilibrium. It is
most remarkable also that herewith the law is explained
not only in its rigor where it is rigorous, but also- in its
laxity where it is lax. For there is no necessity that all
the possible forms of equilibrium should be actualized;
there might very well be gaps, even considerable gaps, in
both the vertical and horizontal series. In that case some
gap, say in a vertical row, might have next to it some
actual form in a near-lying parallel line, which would thus
present not an exact but only an approximate repetition
of property.
Thus the electronic theory of matter yields not only a
vivid idea of the necessary existence of a double system
of valences among atoms, and of the probabilities and
nature of chemical combinations, but also yields deduc-
tively in a highly acceptable form the confessedly highest
and most significant induction yet reached in chemical
theory. This conception of the atom not as an infinitesimal
grain, strong in solid singleness, as Lucretius fancied it,
nor yet as a vortex-filament in an incompressible friction-
THE ELECTRONIC THEORY OF MATTER. 337
less ether, so sleek and slippery as to wriggle out from
under the edge of the keenest knife and sharpest scissors,
as Helmholtz and Kelvin conceived it, but as a highly
organized community with members held together in unity
in stable equilibrium, not at rest but in a system of com-
plicated movements of inconceivable velocity, whose very
swiftness itself contributes indispensably to the stability
of the configuration (see p. 480), — this conception not only
imparts new grandeur to physics but aligns it on the one
hand with astronomy, on the other with biology and even
anthropology. For we are all familiar in general terms
with the planetary system and also with Bode's law, a
special case, it would seem, of some principle of balancing,
such as reaches from the atom to the constellation, from
the star dust of a nebula to the most complex organization
of human society. Indeed the principle of natural selection
would seem to be hardly less operative in the world of
brute matter than in the world of life.
More specifically, however, the electronic theory casts
a broad beam of light on some long outstanding enigmas
of astronomy. The motion of comets, presenting in the
vast sweep of their tails an apparent repulsion from the
sun, finds in this theory a long desiderated explanation.
At this point science has to thank a large number of widely
separated conceptions and personalities. It was the British
Maxwell who as early as 1873 confirmed the suspicion of
the German Euler (1746), that ethereal undulations must
produce a longitudinal pressure along the ray of heat, and
three years later the Italian Bartoli reached a similar, more
general conclusion by a wholly different path. The mathe-
matical prophecy declared this pressure to be E(i -(- r}/v,
where E is the energy and v the velocity of light, and r
the coefficient of reflection of the receiving medium. But
this phenomenon is so extremely subtile as long to have
eluded the keenest observation, though the unerring finger
338 THE MONIST.
of mathematics was pointed at it steadily for twenty-eight
years.
At length (1901) the Russian Lebedev succeeded in
detecting and even in measuring it. Two years later the
Americans Nichols and Hull not only repeated Lebedev's
experiment with far higher precision, but showed decisively
that the observed value of the repulsion agreed within the
limits of error with that foretold by the English clairvoyant
Maxwell. Of course this repulsive push is inconceivably
minute, and on even a very small sphere it would be im-
perceptible in comparison with the extremely feeble attrac-
tion of gravity. However, the latter decreases as the mass
or as the cubed radius, while the repulsion decreases as the
surface or as the squared radius, of a spherical particle.
No matter then how much greater the attraction than the
repulsion on any given sphere, as the radius decreases the
repulsion must finally gain the upper hand, the particle
sufficiently minute must be repelled by the light away from
the sun. A particle of earth o.ooooi of an inch in diameter
would hang balanced between the push and the pull; if
larger it would fall, if smaller it would rise and fly away
before the thrust of the light. Now we need not indeed
have recourse to electric corpuscles to find particles much
below this critical magnitude, and the phenomenon of com-
etary tails blown backward from the sun with inconceiv-
able velocity, as by the breath of a god, is readily intelli-
gible as resulting from the now demonstrated pressure of
light.
Into the details of this matter it is impossible to enter
here ; suffice it that the illustrious Swedish physicist, Svante
Arrhenius, has subjected the whole subject to rigorous
calculation which has been in the main verified, at certain
points amended, by Schwarzschild, so that the enigma of
cometary motion may now be regarded as solved. It is
found in fact that the maximum direct pressure at the sur-
THE ELECTRONIC THEORY OF MATTER. 339
face of the sun is 2% mg. per square centimeter. While
gravitation sets an upper limit to the diameter of the re-
pelled particle (0.0015 mm.) the diffraction of light sets
a lower, namely, about o.i the wave length of the light in
question. Only particles whose diameters lie between these
limits can be repelled, all others are attracted. The great-
est theoretic repulsion is 19 times the attraction of gravi-
tation on a particle of water; but the heterogeneity of
solar light reduces this by nearly half, leaving as maximum
only the tenfold of gravitation on a water sphere of
0.00016 mm. diameter. Since molecules fall far below this
size, it appears that Maxwell's law does not apply to gases.
But here again the corpuscle vindicates its great import-
ance in cosmogonic theory. For the gases near the sun
must be at least partially ionized, since its light is rich in
ultraviolet rays and these provoke the radiation of ions.
But these ionized gases, as proved by the Wilsons, are
readily beclouded by the vapor condensing around the ions
or corpuscles. The drops thus formed must be repelled by
the light pressure, or if too heavy must fall toward the
sun and bear away with them the negative electricity, leav-
ing the gas positively laden. Hence the great part played
by electricity in cometary phenomena.
While the second and third of Bredichin's groups of
comets are easily understood as composed of hydrocarbon
particles, the first group shows repulsion nineteen times as
great as gravity, and the comets of Rooerdam and Swift
(1893 II and 1892 I) even 37 and 40.5 times as great.
Such repulsion would require a specific gravity only one-
tenth that of water, but it may well be that hydrocarbon
spherelets subjected to expulsion of hydrogen under intense
solar heat may be turned into sponge-like carbon pellets
of the levity required.
The combined conceptions of the forward thrust of
light and the universal radiation of corpuscles give us a
34O THE MONIST.
widely imaginative vision of the heavens above us. The
corona of the sun with its greenish pearly light is no longer
a mystery. We think of it as composed of particles near
the critical sizes and hence held suspended in the sky, like
the coffin of Mohammed, between the pull of gravitation
and the push of the light, two forces obeying the same law
of inverse squares. Its tenuity transcends all conceiving,
it being five million times thinner than the head of the
comet, and its whole mass if of coal would be burned upon
earth in a week. Nevertheless it is something, and it
assures us that an endless drizzle of still finer dust is
steadily poured out by the sun into surrounding space. In
perhaps six thousand billion years the sun might dissipate
itself entirely. Unless time be finite it would follow that
at this rate all the suns in the universe would have dis-
solved unending ages ago and vanished like the baseless
fabric of a dream. Meantime, however, there has been
integration proceeding pari passu with disintegration. Since
the earth in flight round the sun sweeps up about twenty
thousand tons of meteoritic matter yearly, it is easy to
reckon that the sun must catch some three hundred milliard
tons in the storm that drives forever about him. These
meteorites are in fact the building stones of the suns. Of
what are they themselves built? According to Norden-
skjold, they are woven together of metallic atom on atom,
the universal floating dust of dissolved or dissolving worlds.
An awful, a tremendous cycle!
As the sun thus inspires and expires matter like some
stupendous lung, similarly it respires electricity. The cloud
particles repelled by light-pressure bear away negative
electric nuclei and leave a positive charge behind of nearly
250 milliard coulombs, enough to dissolve 24 tons of water.
As the solar dust aggregates into meteorites it dispels its
negative electrons and these are drawn in a ceaseless flood
toward the sun by its positive charge. The sun is thus at
THE ELECTRONIC THEORY OF MATTER. 34!
once a source and a sink of electrons streaming into and
from it, to and from the uttermost walls of the world.
We look aloft into the azure heavens and think to be-
hold a sky unflecked by the minutest cloud. But reason
perceives even there an eternal whirlwind of cosmic dust
swirling around planets and stars and darkling suns. Well
for humanity that the veil of this absorbent mist is flung
abroad over the whole heavens! Else would the dome of
the sky glow like a furnace, and the countenance of crea-
tion be withered and blasted. But not only do these nebu-
las and frozen stars shield the earth from planetary death ;
they are the very guardians of the universe itself (accor-
ding to Arrhenius) against that heat-death (Warmetod)
with whose frightful specter Clausius has so long appalled
the stoutest hearts of science and philosophy. For in his
second law of thermodynamics the German declared that
the entropy of the universe tends to a maximum while the
energy remains constant. Now this entropy (of a body)
is its total heat divided by its absolute temperature. It
measures the amount of heat turned into a body in
the process of exchange and so rendered unavailable for
outward effect. The one universally observed case is that
bodies at unequal temperatures tend to exchange heat
till they attain the same temperature, when all effective
interaction ceases. Applying this observation to the uni-
verse, Clausius declared that it was tending to such
equilibrium, which would suspend all effective activity,
and this universal uniform condition he named heat-death,
and taught that it was inevitable, merely a matter of
time — a fearful piece of reasoning, which it seemed
equally impossible to accept or to refute. But the facts
were against Clausius, for least of all men would he
deny the past was infinite, hence the universe had had time
to run down infinitely often, nevertheless it is manifestly
not run down but running on still. Since it has not met
342 THE MONIST.
the Wdrmctod in the infinite past, neither need it do so
in the infinite future. Maxwell imagined an intelligent
demon sorting out the atoms in a uniformly heated gas,
by opening the windows of an invisible partition for the
passage of each fast-moving particle and shutting them
against the slow-moving one, thus sifting them into two
apartments, the one hot and therefore able to work on,
the other cold. We know, however, o£ no such demons,
though it would be rash to deny them all existence. But
on the other hand molecules moving swift enough (more
than ii kms. per second) must tear themselves away from
the earth's attraction, leaving behind their slower fellows,
thus sorting out the two classes after the fashion of Max-
well's demons. In this way perhaps the moon has grad-
ually lost her atmosphere. So too may the nebulas lose
fast-flying molecules that wander into their outer parts
and retain only the slower ones and so remain or even
become cool. Meantime the fast flyers would be caught in
the widespread atmosphere of some condensing star and so
raise still higher its rising temperature. Whether or no
this be the exact fashion in which the universal activity
is maintained, we may be sure that it is maintained in
some fashion, and the dreadful presage of universal heat-
death that has so long oppressed the scientific conscious-
ness may now be dismissed as the nightmare of a fevered
dream.
Still other riddles of the heavens have yielded to the
divination of the corpuscular theory. Since we must now
think of the sun as sprinkling all space with an incessant
shower of dust, clearly the earth too must be thus sprinkled.
The atmosphere can no longer be thought as imperceptible
beyond 100 kms., but must certainly reach an average
height of 400 kms. Were it not for the electric charges
with which the particles are laden, the amount of sun-dust
that could reach the earth could hardly exceed two hundred
THE ELECTRONIC THEORY OF MATTER. 343
tons yearly, one one-hundredth only of what actually
reaches us in meteors and shooting stars. On the contrary,
Nordenskjold reckons the cosmic dust positively charged
that reaches the earth at ten million tons yearly, and
Chamberlain advocates a planitesimal theory that con-
siders the planets as mainly built up of collected meteors.
Be that as it may, the significance of the negatively laden
particles in the higher regions of our atmosphere is beyond
question. Of the luminous effects of wide-scattered dust
in the air, the appalling eruptions of Krakatoa (1883) and
Mount Pelee (1902), which reddened the sky for months,
have furnished examples. On a far grander and more
benignant scale the sun powders the upper air into un-
earthly radiance. Inhabitants of the polar regions have
the nightly vision of the Aurora Borealis in two forms
long confounded but now clearly distinguished: the one
a nearly steady phosphorescent gleam swelling in arch on
arch toward the apex of the sky; the other consisting of
beams, of fountains of light spouting their torrents of
splendor zenithward from the fluttering draperies of flame
that fringe the northern horizon. The explanations of
these two classes are of course not quite the same, and in
their detail one must distinguish between the maximal and
minimal years of sunspots ; but in general one may say that
the negatively laden particles shot out in perpetual tempest
from the sun must beat upon the atmospheric envelop of
the earth, which is speeding on as through a driving rain.
It is the equatorial regions that are full exposed to this
tempest. Great however as is the velocity of the particles,
they do not in general pierce through this envelop, but
are caught as it were in the net of the lines of the earth's
magnetic force. Round these they spin in descending con-
verging spirals toward the poles. On impinging upon
strata of air as dense as in a vacuum tube they act as in
such tubes and exhaust their energy in the fitful gleams
344 THE MONIST.
peculiar to cathode rays. In case however of a maximal
year of sunspots the velocity of the projected particles is
enormously increased and they drive on nearly in straight
lines in spite of the suction of the lines of force both in the
sun's and even in the earth's magnetic field. Hence the
Aurora may in these years descend even toward tropical
regions. It is at once perceived that the relation between
the two phenomena of sunspots and polar light must be
intimate and so it is, but into this there is no time to enter.
Still another astronomical puzzle, not however of polar
but of tropical observation, seems now in fair way toward
solution. The inhabitants of even higher latitudes may be-
hold morning and evening at the equinoxes a cone of light
rising from the horizon and called zodiacal from its con-
nection with the zodiac. It is certainly a luminous cloud
of particles glittering in the sunlight. Some have thought
it to be the last remnant of nebular dust still circling the
sun like a ring. But according to Arrhenius both the
zodiacal light itself and its still more perplexing Gegen-
schein are due to the same corpuscular storm that pours
steadily out from the sun and down upon it from all sur-
rounding space. Surely enough has now been said to show
that the corpuscular theory has the widest-reaching astro-
nomic and cosmogonic bearings.
May it not also have biologic significance? The prob-
lem of the origin of species on our planet has gone through
a variety of phases. Linnaeus held that the Infinite Ens
had in the beginning created just so many species, which
remained unchanged down to our own day. This rigid
conception was shaken by Lamarck and others a century
ago, but again restored to acceptance by the great author-
ity of Cuvier. Finally it went down forever before the
researches of Darwin and the school of evolution. More
recently De Vries has detected species in the very act of
transvolution, not however through the gradual accumu-
THE ELECTRONIC THEORY OF MATTER. 345
lation of infinitesimal variations, but by finite leaps or mu-
tations establishing a new "species in a single generation.
Meantime the deeper problem of the origin of life on our
planet has not been advanced toward solution, unless the
successive recognition of one proposed solution after an-
other as unsatisfactory may be said to be advancing.
Thinkers of the highest rank still cling to the notion of
"spontaneous generation," under the compulsion of such
reasoning as this : there was a time, no matter how remote,
when there was no life on the earth; now there is life;
therefore sometime between now and then life began to
be. Others however spy out another possibility, namely,
that life was imported from some other planet. The illus-
trious Kelvin insisted that the maxim omne vivum e vivo
was as sure as the law of gravitation and hence was driven
to maintain ( 1871 ) the hypothesis that life had been borne
to our planet on some meteorite, some disrupted fragment
of another world. But the difficulties that embarrass the
development of such a notion seem quite unconquerable.
More acceptable is the modern idea of panspermia, hinted
by Richter as early as 1865. In a word, this doctrine holds
that the germs of life are scattered as spores throughout
all the deepest abysses of space, that they are driven by
light-pressure along the sunbeams from planet to planet,
from system to system, on journeys that may last for thou-
sands or ten thousands of years. The intense cold of the
interstellar spaces need not chill them to death since they
are unaffected by a bath of liquid hydrogen ( — 252° C).
They need not dry out, they need not suffer any destructive
chemical change, as oxidation, on their solitary flight from
world to world, since evaporation and chemical processes
are suspended in that Lethean flood. They need not be
too large for the fingers of the light to push before it, since
they have been discovered having diameters between 0.0002
and 0.0003 mrn., and are doubtless often much smaller,
346 THE MONIST.
magnitudes well within the grasp of the sunbeam. But how
could they be lifted up into the higher regions of the plan-
etary atmosphere, and there surrendered to the propulsion
of the waves of ether? Currents of air might lift them
a hundred kilometers from the earth, but could never re-
lease them from the atmospheric envelope. Here again
one must invoke the omnipotence of the electron. The
negative-laden sundust that kindles the Aurora fires in
the upper air must also beat upon these spores and charge
them with electricity. So charged they must powerfully
repel each other in every direction and some would be
launched outward into the depths of ether and there seized
and sped onward by the impulsion of the light. Is the
electric field strong enough for this action? Assuredly.
An electric field of two hundred volts per meter would
suffice. Such are familiar near the earth's surface, and far
intenser ones must prevail in the regions of polar light.
Undoubtedly it is a most perilous voyage upon which
such a germ of life sets out from the system say of a
Centauri to that of our own sun. The immense majority
of these ether-farers would almost certainly be lost and
perish; but here and there some one would arrive after
nine thousand years safe at the utmost borders of our solar
world and there by chance light upon some grain of sun-
dust in the reflection of the zodiacal light and be borne
along therewith sunward within the planetary ring and
even into the atmosphere of some planet, as our own, which
would seize upon it with gravitation and slowly drag it
to the earth. Perilous too would be the landing on the
shores of Time, yet some lucky sailor would succeed and
so would establish a form of life upon this planet. Surely
a most tremendous conception, striking wonder and awe
through the hardest heart! Amazing too it is to reflect
that this prodigious idea, so carefully wrought out and
bulwarked at every point by the adamantine pillars of
THE ELECTRONIC THEORY OF MATTER. 347
mathematical calculation, should have been anticipated in
its grandest proportions by the brooding fancy of the pre-
Christian Gnostics! For in the Naassene scriptures pre-
served to us by the good bishop Hippolytus in his Refutatio
omnium haeresium we find creation described allegorically
in the parable of the Sower as the sowing down of seeds
from the unportrayable Godhead. Moreover in the deep-
thoughted Gnostic Basilides we find repeatedly the same
idea, and the new knowledge has actually adopted his fa-
vorite technical term "Panspermia" as its own, to express
this most recent of astronomico-biological ideas. Life then,
at least in its germs, is everywhere pulsing and throbbing
throughout the universe and when the finger of time points
to the accepted moment the myriad forms of life leap into
being out from the teeming womb of ether. We may also
see that these forms are probably nearly the same, at least
closely allied even at opposite poles of the Milky Way, and
man may feel his blood kinship with the tenants of the re-
motest world. It is impossible then to repress the sugges-
tion that the actual forms of life, being everywhere what
they are, have in themselves some deep-lying hitherto un-
suspected reason for being thus and not otherwise, some
reason profound as the properties of numbers or the logical
necessities of the geometry of Euclid.
May one not even venture to hint that the biological
import of the corpuscular theory is not yet exhausted?
That in the almost endless divisibility of the atoms into
sub-atoms or electrons there inheres some undiscovered
connection with the pangenesis required in theories of
heredity ? That the mysteries of Mendelism and Mutation
may yet be illuminated by flashes of electric light as in a
polar sky at midnight? But even were I able, there is no
time to pursue this thought further.
After all, however, what do we know of electricity?
•Lord Kelvin declared that in his age he understood it no
348 THE MONIST.
better than in his youth. The colossal theory we have been
considering assumes all the fundamental and hitherto in-
explicable laws of electric action. In particular, besides
the ether and its wonderful properties, it assumes the law
of inverse squares as the mode of all interaction of elec-
trons. Herein of course it is perfectly justified, but the
mind cannot lay the importunate query, Why does this
interaction vary precisely as the inverse square of the dis-
tance? Moreover, since all this action takes place in the
uniform universal ether whose regions are distinguishable
only by the motions that affect them, the mind is led irre-
sistibly to the conjecture that the problem is ultimately
hydrodynamical, that all these electric phenomena must
some way be thinkable as movements in an all-pervading
medium. Hence then the great significance of the vortex
theory developed by Thomson (Kelvin) from the central
property of vortex filaments discovered by Helmholtz.
Kelvin and his disciples thought to recognize the indis-
soluble atom in the indissoluble vortex-ring and imagined
that an exhaustive doctrine of knots would yield a list of
atoms or elements. The new knowledge shows indeed
that the atom is not indissoluble, nay, is in some cases
actually dissolving, but the vortex-ring or filament is not
thereby deprived of its scientific importance. It may yet
be that the sub-atom or electron is essentially a vortex in
ether, and the theoretic properties of such vortices may be
the observed properties of electrons. At this point then
the question arises : What is the relation between the equa-
tions of electricity and the equations of hydrodynamics?
Or between the electromagnetic and the hydrodynamic
fields? Or, finally, between the interactions of electric
units and of hydrodynamic elements? It seems hard then
to exaggerate the moment attaching to such researches as
those of the two Bjerknes (father and son) upon fields of
force, researches both experimental and mathematical.
THE ELECTRONIC THEORY OF MATTER. 349
They have proved that the relation in question is certainly
a close one, that nearly all the elementary actions assumed
in electromagnetic theory may be surprisingly simulated
by actual experiments on pulses in a liquid medium. True,
there presents itself a queer paradox: the hydrodynamic
field appears not as the direct but as the inverted image
of the electro-magnetic field, attraction and repulsion in
the one answering to repulsion and attraction in the other.
But in spite of this perversion, the results remain highly
interesting and point the way along which research must
follow till something more satisfying shall be suggested
or discovered.
Meantime it is no more important to see clearly the
wide range and immense perspective of the new knowledge
than it is to recognize unequivocally its necessary limita-
tions. Even when we suppose the hydrodynamic analogy
perfect, even were it possible to state all the facts of the
material and ethereal worlds, in a word, of the world of
space and sense, in terms of motion rotational or irrota-
tional in a universal ether of whatever properties, even
though the vision of the Laplacian Intelligence should thus
be actualized on a scale far grander than Laplace himself
ever dreamed of, it would still remain true that the real
problem of the world was just as far as ever from solution.
For let it be understood once for all that this problem is
not even to be stated finally in terms of mass and motion,
the sole concepts available in the corpuscular or any other
physical theory. Mass and motion are not ultimates in
human thinking, no physical concepts can be ultimates.
The supreme all-comprehending fact of the world is Mind,
Soul, Spirit, and the ultimates of all thinking, of all reality,
must be psychical. The physical world is an idea, a sensible
form, which the mind constructs at every instant by the
inherent law of its own activity. It is a splendid, a glorious
construct, a real construct, well worthy the everlasting
350 THE MONIST.
study and admiration of its own creator. Contemplating
this amazing creation the Spirit beholds its own image as
in a mirror, and it may even explore the depths of its own
being by interpreting backward its own image ; just as the
mathematician may translate the analytic (algebraic) prop-
erties of his equation into geometric properties of the cor-
responding locus and again may interpret the geometric
properties of this locus into corresponding algebraic prop-
erties of the original equation. But the equation is not the
locus, nor ever can be, nor would it cease to be, nor change
its properties, if the geometric construction were quite im-
possible. Speaking then in allegory one may declare that
the psychical world is a sublime equation of infinite degree ;
the physical world is its majestic geometric locus, its con-
struct in terms of time and space, mass and motion. Be-
tween the two there subsists or may subsist some one-to-one
relation. Let us study the grand image with unreserved
admiration and with unflagging zeal. But let us never for-
get that after all it is only an image, a stupendous parable.
Let us never forget the great word of Goethe: Alles Ver-
gangliche ist nur ein Gleichniss. Otherwise the brightest
achievements of physical and physiological research may
prove to be only traps for our unwary feet. Otherwise we
shall surely fall into the pit of materialism; we shall mis-
take the significant for the significate, we shall see in the
whole universe only an interplay of corpuscles, and shall
talk with Cabanis of the brain secreting thought as the
liver secretes bile. Such an issue would be deplorable be-
yond all words, it would indeed be a bankruptcy of science
absolutely hopeless. As secondaries electrons are invalu-
able; as primaries they are absolutely worthless. The
favorite maxim of Sir William Hamilton abides in full
validity :
"On earth is nothing great but man;
In man is nothing great but mind."
THE ELECTRONIC THEORY OF MATTER. 35!
But if even the sublimest flights of physical speculation
vouchsafe us no glimpse beyond the veil, how shall we
ever lift it? What gaze of reason shall ever penetrate its
folds? Who knows indeed that in the nature of the case
it can be lifted ? For my part I should be content to think
it a veil of Isis. Is there not a kind of inspiration in the
thought of revelation after revelation, forever and ever,
world without end?
''Higher than your arrows fly,
Deeper than your plummets fall,
Is the deepest, the Most High,
Is the All in All."
Chase without catch would indeed be disheartening, but
chase with nothing more to catch would certainly be empty
and uninviting. If the chase is to be eternal and yet in-
spiring, the quarry must be infinite, the supreme truth must
be forever approachable but never attainable! In the an-
cient myth it is said that Egeus concealed from Theseus the
secret of his birth, burying the evidence beneath a stone
by the seaside. The Father of heaven and earth has per-
haps secreted somewhere the proofs of the origin and na-
ture of all things ; but upon the shore of what ocean, O man
of science, has He rolled the stone that hides them ?3
WILLIAM BENJAMIN SMITH.
NEW ORLEANS, LOUISIANA.
3 Adapted from Maurice Guerin's Lc Ccntaure.
PURPOSE AS SYSTEMATIC UNITY.
i.
THE present investigation is undertaken for the sake
of the light which it may throw on the problem of
value. Assuming that value is a function of what may
broadly be termed "interest," it becomes imperative to get
at the fundamental or generic character of this phenom-
enon. What is that attitude or act or process which is
characteristic of living things, which is unmistakably pres-
ent in the motor-affective consciousness of man, and which
shades away through instinct to the doubtful borderland of
tropism ? Both the vocabulary and the grammatical struc-
ture of language provide for the teleological categories.
"Purpose," "means and end," "in order to," "for the sake
of," "with a view to" — these and many other kindred forms
of speech are evidently applicable to the same context.
There is something in our world to which they serve to call
attention. What is it?
I propose to view the matter objectively rather than
introspectively. What we wish to discover is the nature
of the thing, and not the nature of the consciousness of the
thing. It is fair, I think, to apply the analogy of mechan-
ism. One would not think of approaching this latter ques-
tion by an examination of the consciousness of mechanism.
Similarly, purpose is supposed to be a kind of happening
or chain of events differing in its determination from that
of mechanism. It may appear that consciousness is inci-
PURPOSE AS SYSTEMATIC UNITY. 353
dental to the purposive kind of determination. But in that
case we should begin with the process as a whole and work
in. We should not shut ourselves up in advance to the
view that purpose takes place only in the introspective
stream of consciousness. We cannot, in other words, de-
termine the role of consciousness in purpose unless we
first take that view of the matter in which both conscious-
ness and its physical context are taken into account. Be-
havior or conduct, broadly surveyed in all the dimensions
that experience affords, can alone give us the proper per-
spective. We want if possible to discover what it is to be
interested, not what it is merely to feel interested. What
is implied in being favorably or unfavorably disposed to
anything? It may be that it all comes to nothing more than
a peculiar quality or arrangement among the data of intro-
spection, and in that case a structural psychology of feel-
ing, will, desire or ideation will tell the whole story. But
such a conclusion would be equivalent to an abandonment
of the widespread notion that purposiveness is a kind of
determination of events differing from their mechanical
determination. The really important claim made in behalf
of purpose is the claim that things happen because of pur-
pose. Are acts performed on account of ends ? Is it proper
to explain what takes place in human or animal life, or in
the course of nature at large, by the categories of teleol-
ogy? The most exhaustive introspective analysis of the
motor-affective consciousness would leave this question
unanswered, and to confine ourselves to the data which
such analysis affords would be to prejudge it unfavorably.
It is, of course, permissible to suppose that even though
a case should be made for purpose in a physical or cosmic
sense, value should be limited to subjective purpose. But
it is evident that the question would then be merely one of
terms. If there are objective purposes as well as sub-
jective, it would be necessary to employ some term to
354 THE MONIST.
designate the objects of both attitudes. It would be im-
portant to construe subjective purpose as a species of this
broader genus, which would be accomplished best by taking
it as a kind of valuing. Furthermore objective or dynamic
purpose, if there be such, would be far too important in its
bearing on the special value-sciences to warrant our dis-
regarding it in a general theory of value.
II. NEGATIVE MEANINGS.
In looking for a clue to the meaning of dynamic or ob-
jective interest, we must free our minds so far as possible
from the purely negative associations which the teleological
terms have acquired. The case for teleology is prejudiced
by a suggestion of anti-scientific bias, or of unscientific
laxity. This is due no doubt mainly to the religious or
popular auspices under which it has been advanced. The
teleological hypothesis is often invoked to satisfy aspira-
tions, to flatter human nature or to conceal ignorance. In
the present controversy over vitalism the proofs of pur-
posiveness seem to consist mainly in the indictment of
mechanism. Purpose is not recommended on account of
its own success, but on account of the failure of something
else. When so invoked it means little more than that un-
known factor, x, which is needed to complete the explana-
tion of such phenomena as growth or organic equilibrium.
It is not surprising that vitalists should be regarded as
impatient scientists who cannot wait for a rigorous experi-
mental solution, but must needs invent an agency ad hoc;
or at best as irresponsible critics who remind plodding
science of its outstanding difficulties without assisting in
the serious work of overcoming them.
The party of teleology according to this view is a sort
of opposition party to the real scientists, who are sobered
by being in power. It is the function of this opposition
party to challenge and censure, rather than to legislate
PURPOSE AS SYSTEMATIC UNITY. 355
and administer. It can afford to be careless or premature
because it is not in office. For itself it has no policy, but
confines itself to impeaching those policies of mechanism,
determinism, naturalism and experimentalism which au-
thoritative science is patiently executing. There is doubt-
less a certain merit in this two-party system in science.
But certainly for our constructive purposes there can be no
virtue in a conception of purpose as merely not something
else. If the conception is to be of any use to us it must
have a positive explanatory value of its own.
Nor is it at all necessary to suppose that purpose is the
contradictory alternative to some other hypothesis such as
mechanism. Purpose must doubtless be different from
mechanism if it is not to lose its identity altogether ; but
that it should be incompatible with mechanism, or observed
only in its breach, does not follow. Through being thus
regarded from the outset as an antithesis to an established
and universally credited theory, teleology needlessly makes
enemies for itself. There is certainly no reason to suppose
in advance that teleology is less compatible with mechanism
than statics with dynamics, or the atomic theory with the
electro-magnetic theory of light. There is certainly an
empirical presumption to the contrary. A man who goes
to his journey's end in order to keep an engagement, does
not appear to violate the law of gravitation in so doing.
Let us therefore endeavor to get the positive sense of the
teleological type of explanation, and let us say of its com-
patibility or incompatibility with the mechanical type of
explanation, that that is as it may be.
III. PROVISIONAL DEFINITION.
We start with the popular supposition that there is a
peculiar and specific mode of explanation, which may cer-
tainly be employed in the case of the rational conduct of
man, which may probably be applied to lower forms of life,
356 THE MONIST.
which may for speculative reasons be extended to the cos-
mos as a whole, and for which the name is "purpose."
i. Let us first consider a case of human conduct. An
off-hand provisional view of this alleged mode of explana-
tion is afforded by Socrates's famous allusion to Anaxag-
oras in Plato's "Phsedo." Socrates, it will be remembered,
distinguishes two ways of explaining his being in prison.
On the one hand it is to be explained by reference to his
bones and muscles. But this, he thinks, would be an in-
appropriate explanation ; not untrue to be sure, since bones
and muscles do supply the necessary "conditions," — but
not the sort of explanation that touches the real cause of
a mind's acting. The second and preferred explanation
is in terms of Socrates's purpose of "enduring any punish-
ment which the law inflicts." A mind, in other words, acts
for the best, according to its lights. To explain its action,
therefore, it is necessary to discover what it deems best,
and then to construe the particular act as an instance of
that best. In the present case it is supposed that Socrates
is actuated by the general principle of submission to the
law, and that he has judged his remaining in prison to be
what under the existing circumstances that principle im-
plies.
Let us analyze the situation more carefully, lest we
omit any essential factor. In the first place, there must be
a general type of action, such as submission to law, of
which a particular act, such as remaining in prison, may
be regarded as an instance. In the second place, there must
be an agent possessed of a stable disposition or tendency
to perform acts of a certain class, under varying circum-
stances. The particular performances will differ according
to circumstances, but they must be consistent in some re-
spect. Then, thirdly, there must be some determinate re-
lation between the rule or type of action and the agent's
disposition. But what is this determinate relation? The
PURPOSE AS SYSTEMATIC UNITY. 357
simplest alternative is to suppose that the rule of action is
identical with the constant or consistent feature of the dis-
position. Thus we might suppose that Socrates tended
under varying circumstances to submit to the law. But
this will not do. For if it should happen that his remaining
in prison were as a matter of fact not what the law re-
quired ; if it should happen that there had been some error
in transmitting the commands of the authorities, or if it
should turn out upon reflection that Socrates's escape
rather than his passively yielding to tyrannical oppression
was more in keeping with his constitutional rights, that
would not disprove his purpose to submit to law. What
is necessary is that Socrates should mean to submit to
law, or that he should think his act to be a case of sub-
mitting to the law. The link between the rule and the
disposition is an act of interpretation or judgment. In
other words, one is said to be governed by a purpose M,
when M is some generalized form of action, and when one
is disposed consistently to perform what one believes
(whether correctly or mistaken) to be a case of M.
2. This, then, appears to be what is meant by purpose
when purposiveness is imputed to the rational or reflective
procedure of man. Let us now turn to what common sense
would regard as a more doubtful case of purpose, the case
of animal beliavior. The differentia of animal behavior
which was first remarked, was the power of self-motion.
Whereas an inanimate object merely submitted to motion
imparted to it from without by impact, a living thing
seemed to be an original source of motion. Associated
with this phenomenon was the relatively unpredictable
character of the action of living organisms. What they
did was so far due to internal and unobservable factors
that you could not rely on their yielding in any uniform
way to the operation of the external forces that you might
observe or apply. Living things had a way of moving of
358 THE MONIST.
themselves, without any apparent cause which might serve
to put you on your guard. Hence they were said to ex-
hibit "spontaneity."
It is still customary to characterize living things in this
way. Biologists describe the organism as "an active, self-
assertive, living creature — to some extent master of its
fate."1 But this spontaneity or self-motion no longer serves
to distinguish living from inanimate things, owing to the
development of the science of energy. We should now
speak of this apparent spontaneity as a release of stored
energy. The organism accumulates chemical energy by
the process of metabolism, and then discharges it when
subjected to some kind of stimulation from without. When
the discharge occurs it is out of all proportion to the stimu-
lation. Indeed in some cases there appears to be no ex-
ternal stimulus at all. In any case the internal factor is so
much more important than the external factor that the
latter affords no safe basis for prediction. As organisms
become more elaborate the discharge comes to depend more
and more upon the quantity and balance of its stored en-
ergies and less and less upon what is done to it from
without. But even so this phenomenon of release or dis-
charge does not differ in principle from what happens in
the case of combustion or in the case of the action of high
explosives. If the behavior of living things is spontaneous
in this sense, there is also "spontaneous combustion" in the
same sense. In the one case as in the other we now suppose
that the action would be predictable even with the utmost
quantitative precision if we knew the internal organization
of the acting body, as well as the character and intensity
of the stimulus. It is merely a question of the relative
preponderance of central over peripheral factors.
Hence we are at present inclined to look elsewhere for
the differentia of life, and to find it, not in the spontaneity
1 Thomson, Heredity, p. 172.
PURPOSE AS SYSTEMATIC UNITY. 359
of action, but in its direction toward something. The ex-
plosion, we say, is blind and aimless, — indifferent to con-
sequences; whereas life is circumspect and prophetic.
Forewarned is forearmed. This is what we mean When we
speak of living things as exhibiting intelligence. We do
not credit all living things with intelligence; but we have
no hesitation in imputing it to the higher forms of animal
life, and the phenomena of instinct and tropism have led
to our imputing at least a quasi-intelligence to the lower
animals and even to plants.
In so far as we impute intelligence to living things,
we feel the need of explaining their action in a peculiar
way. The explosion is satisfactorily accounted for as a
resultant of two physically existing factors, the internal
organization of stored energies and the external spark or
trigger. But in the case of intelligence it seems necessary
or at least appropriate to refer to the sequel, to that which
is merely in prospect at the moment when the action occurs.
Thus a dog moves rapidly away, or gets behind some inter-
vening obstacle, when his master takes down the whip.
In so far as this implies intelligence we think of it not in
terms merely of existing chemical energy and the light
impinging on the optic nerve. We take account also of
what is going to happen, namely the painful beating. We
say that that also explains why the animal is acting as he
does. Or we say that the animal is acting "in order to
avoid" the beating. But since the beating which is avoided
does not as a matter of fact occur, we are thus appealing
to a factor which is in some sense merely possible or hypo-
thetical. Over and above the animal's power of spontane-
ous motion, over and above the external action of the stim-
ulus, there is some additional factor which refers to this
mere possibility and which decisively determines the direc-
tion which the discharge takes. I do not mean to assert
that this third factor cannot be traced to the previous ex-
360 THE MONIST.
periences of the animal. Probably it can ; and this has led
comparative psychologists to associate intelligence with
docility, or the capacity to "learn by experience." But that
is not the point. However he may have come by it, the
animal is supposed at the moment of action to possess a
capacity for prospectively determined action. He acts not
because of what is or has been merely, but because of what
may be by virtue of his action, or what ivould be without
his action. He acts, we say, from fear of a painful whip-
ping, or from hope of immunity. There is no way of
describing either the fear or the hope, without admitting
it to be the fear or the hope of something, which something
is not upon the plane of past or present physical existence
as ordinarily conceived.
If, now, we put together the results of the analysis of
our two examples we shall have a provisional view of
interested or purposive action. In both cases there is an
organism with certain accumulated energies and certain
organized propensities. In both cases there is a specific
external situation which acts upon the organism and lib-
erates the energies and propensities. So far there is noth-
ing to distinguish these cases from such physico-chemical
analogies as I have cited. But in both cases there is a
third and differential factor which constitutes their pur-
posive aspect. The act is construed by the agent in terms
of something ulterior and non-existential. Socrates judges
his act to be of the general type of submission to law; to
the dog the whip is a sign of beating or pain-to-come, and
his flight is a response "as to" pain. In both cases the
agent views the situation whether by inference or asso-
ciation, in the light of some aspect or relation that tran-
scends given fact ; and his acting as he does is determined
by his viewing it as he does.
Jennings has termed this characteristic of behavior
"reaction to representative stimuli." "The sea urchin. . . .
ITRPOSE AS SYSTEMATIC UNITY. 361
responds to a sudden shadow falling upon it by pointing
its spines in the direction from which the shadow comes.
This action is defensive, serving to protect it from enemies
that in approaching may have cast the shadow. The re-
action is produced by the shadow, but it refers, in its bio-
logical value, to something behind the shadow.
"In all these cases the reaction to the change cannot
be considered due to any direct injurious or beneficial effect
of the actual change itself. The actual change merely
represents a possible change behind it, which is injurious
or beneficial. The organism reacts as if to something
else than the change actually occurring. The change has
the function of a sign. We may appropriately call stimuli
of this sort representative stimuli."2
The same general principle applies to the higher organ-
ism, Socrates. That which releases his action is a represen-
tation. His friends come to his prison and urge him to
escape. Their actions and words are a sign to him of
law-breaking and as such he resists them ; or his presence
in prison represents to him submission to law, and as repre-
senting that, he holds to it. Let us now refine this notion
of interest or purpose by comparing it with other notions
which approximate it, but in some respect fall short of it
or depart from it.
IV. THE PATHETIC FALLACY.
The most familiar error regarding purpose is the so-
called "pathetic fallacy." It will be worth our while to
inquire just wherein the fallacy lies. Suppose that in spite
of my most painstaking efforts to execute a powerful stroke,
the golf ball rolls ingloriously from the tee. I then turn
and rend, my new driver or call down maledictions upon it.
I am angry not with myself but with it. I feel resentment
- H. S. Jennings, Behavior of the Lower Organisms, p. 297.
362 THE MONIST.
toward it precisely as though it had meant to spite me.
I virtually attribute malice to it. Now this, as my less
heated partner may remind me, is unreasonable, because
the golf stick really didn't mean it or do it "on purpose."
It is true that in effect the stick thwarts me. The stick is
a cause of my displeasure. But the error consists in im-
puting that displeasure to it as a motive or ground for its
action. In other words, it is not sufficient for purposive
action that its effect should occasion displeasure; it is neces-
sary that this displeasure as a prospective contingency
should determine the act. Or take another example. Bask-
ing in its warmth, I praise the sun and feel gratefully dis-
posed to it. If I knew what the sun liked I would gladly re-
ciprocate. This is an innocent error, a kind of poetic license,
but error it is none the less. For I have responded to the
sun as though the pleasure which its rays were about to
give me had actuated the sun in shedding them; whereas
this effect upon my sensibilities is accidental and in no way
needed in order to account for the radiation of the sun's
light and heat.
But there is also a positive implication in this criticism.
My own action in each case is purposive. My addressing
the ball, or lying in the sun, is to be accounted for by
reference to the stroke or the bodily comfort that is to
come. My error lies not in employing such a mode of
explanation but in misapplying it. There is a human weak-
ness, doubtless one of the major motives in religion, which
prompts one to extend to all the agencies involved in any
event that purposive type of determination which really
holds only of one's own participation in it. In the case of
one's own agency the prospective sequel does account for
the act, but in the case of the other contributory agencies
this explanation is out of place ; or, some but not all antece-
dent agencies are determined by the sequel. Not to dis-
criminate is to commit the inverse of a common fallacy.
PURPOSE AS SYSTEMATIC UNITY. 363
It would not be inappropriate to term this characteristic
teleological error the fallacy of "ante hoc ergo propter hoc."
There is a further point which this error brings to light.
In so far as I like it the sun's warming my body is good.
The effect of the sun's action is therefore good; and it
might even be that the sun "tended" to warm my body and
so to do good. But that is evidently not sufficient to make
the sun's action purposive. Action resulting in, or tending
to, good is not ipso facto purposive action. It would be
purposive only provided that result were somehow ac-
countable for the action. In other words we are forced
to recognize the essentially dynamic character of purpose.
It is not the quality of the results, whether good, bad or
indifferent, that implies the purposiveness of its antece-
dents, but the function of that result as somehow partici-
pating in the determination of the process.
V. PURPOSE AND SYSTEMATIC UNITY.
i. Among the widely accepted notions of purpose or
interest which we shall find it profitable to examine, the
next is that which identifies purpose with systematic unity.
This notion is distinguished by the fact that it disregards
the time factor, or regards it as accidental. Purpose of this
sort may characterize the world sub specie eternitatis. It
may qualify a static whole, and appear in its mere structure
or arrangement, regardless of its origin or history. It fol-
lows that the purposiveness of any given reality may be
judged by internal evidence, even when it is supposed that
the reality in question was produced by conscious design.
A purposive object is believed, like Paley's watch, to ex-
hibit its "designedness" in its very form. This formal,
static purposiveness is identified with order, system, — the
interrelation of parts in a whole. Let us first consider
examples, beginning with an example in which the time
factor is clearlv eliminated.
364 THE MONIST.
An ellipse is more than a mere collection of individual
points; it is a curve having a distinctive character as a
whole, which may be expressed by the equation x -\- y = c.
Every individual point in the curve is a value of the vari-
ables in this equation, and its position is determined ac-
cording to the law by the position of the other points. Al-
though the position of each point differs from that of every
other point, there is at the same time a certain identical
character among them all, namely the "x -\- y" character,
or the sum of the distances from two fixed points called
the "foci." To call this a unified whole means that there
is a definite whole-character in terms of which all of the
constituents can be described. This whole-character is
the law of the parts, prescribes their positions, or, as it is
sometimes expressed, "generates" them. In the case of a
broken line or a curve having no equation, there is no
whole-character except the merely collective aspect of the
several points. In that case the parts are prior to the
whole, and to speak of them as parts of the whole is there-
fore circular or redundant. But in the case of the ellipse
the whole is prior to the parts, or comes first in the order
of explanation. The parts, therefore, are said to be gov-
erned by something ulterior to them. The ellipse does not
exist except in so far as all the points are in their proper po-
sitions, and yet their being so disposed is determined by the
nature of the ellipse. The ellipse is then said to be the pur-
pose which regulates the several points. Each point is deter-
mined by what is necessary in order that there shall be an
ellipse.
Let us now turn to examples in which time figures as
one of the internal factors of a unified whole. The whole
is not in time, but time is in the whole. First let us take an
example of what is commonly regarded as mechanism.
Suppose a body to be moving in a straight line at a uniform
velocity, governed by the law of inertia. Although each
PURPOSE AS SYSTEMATIC UNITY. 365
successive position of the body is new, a certain ratio of its
distance-interval and its time-interval measured from any
previous position is always the same. Its kinematic his-
tory as a whole exhibits a definite character which pre-
scribes what its position must be at each particular mo-
ment. It may in its actual behavior be construed as a reali-
zation of the principle of uniform velocity. This principle
in itself is a universal or ideal entity. It does not exist
except in and through the successive positions of a moving
body which obeys it. And yet these positions are them-
selves somehow determined by it.
Let us take one more example, one that is less precise
but is drawn from the context of life. Modern civilization
may be said to possess a characteristic flavor, which dis-
tinguishes it as a form of life. It is conditioned by the
co-presence and cooperation of a thousand factors, such
as the present phase of geological evolution, temperate
climate, fertility of soil, racial blend, cultural tradition etc.
But these many factors compose something. There is a
unique and simple quality which somehow supervenes when
all these factors are aggregated, — a quality which is iden-
tical with none of them and yet somehow takes them all
up into itself. In terms of this one quality we can construe
all the various conditions as contributing this or that to it.
Through it they become, not so many miscellaneous par-
ticulars, but various aspects or phases of one thing. This
resultant quality, or Gcstaltsqualitat, is their purpose by
reference to which they are now seen to be for something.
They may now be understood not merely severally but col-
lectively. There is a reason why they should be together ;
or, over and above that determination which accounts for
each by itself, there is a determination which accounts for
each in its relation to the others. But this determination
springs somehow from a character which does not come
into existence until after they are all in place.
366 THE MONIST.
These examples serve to give plausibility to the notion
that is now before us. Let us analyze them more carefully.
It will be found, I believe, that the notion of unity which
they illustrate is divisible into two types, which I shall call
"ideal" and "existential" unity. The first is based on the
conception of a universal, A universal unifies its instances.
Furthermore it has this peculiar relation to any instance
of itself: it explains the instance, or serves as a description
of it, and in that sense appears to be prior to it ; but on the
other hand it exists, or is exemplified only through the in-
stance, and in that sense appears to be posterior to it. So
that a case of a universal seems to be something that is
only through itself. Interrelation is an example of ideal
unity. When a number of terms possess a mutual relation
exclusively, that is, when they are related among them-
selves as none of them is related to any term without, they
compose a whole. Or they may all sustain a common rela-
tion to a term outside the group. Or they may be instances
of the same set of universals where the universals are them-
selves interrelated.
The second, or existential, type of unity consists of the
convergence or fusion of many existences into one. The
unity lies not in any universal or set of universals under
which many particulars may be subsumed, but in an ul-
terior particular. Whereas unity of the first type is intelli-
gible or apprehended by reason, unity of this second type
is sensible or is a matter of empirical fact. The several
particulars work together to produce a singular result, or
blend into an individuality that is directly felt. Let us in-
quire, then, whether either of these conceptions of unity,
that of universality or that of individuality, will serve as a
definition of purpose.
2. It is to be noted at the outset that purpose would be
an all-pervasive feature of the world we live in. Instead
of its being the exception it would be the rule. Instead of
PURPOSE AS SYSTEMATIC UNITY. 367
its being a residual aspect of the world, complementary to
that aspect with which the physical sciences have to do, it
would coincide with that orderly and law-abiding aspect
of nature of which physical science has been the principal
exponent. Instead of its being the antithesis to mechanism,
mechanism would itself supply the most perfect instances
of it. This will doubtless serve to recommend it in the
judgment of those who have a predilection for teleological
monism. But such philosophers cannot escape the price of
their easy speculative victory. For in so far as a conception
is universal it is relatively colorless. To characterize the
world as purposive in this general formal sense is to say
nothing more than every scientist or materialist asserts. It
does not differ from saying that it is determined and in-
telligible in terms of laws. Democritus and Spinoza would
then be as good teleologists as Plato or Leibniz. And quite
apart from its philosophical barrenness such a view would
be wholly inept for the purpose of a theory of value. It
would wholly disregard the peculiar or differential feature
of those phenomena which biology, economics, ethics and
esthetics study, and would be of no service whatever in
distinguishing and coordinating these sciences. Although
this pragmatic objection might be thought to justify our
dismissing it, it will be instructive to discover if possible
just wherein this view fails to agree with our provisional
conception.
3. Unity may be thought to constitute purpose, or to
imply a purposive origin. In other words the purpose in
question may be thought of as internal to the system, or as
external. When intelligible or ideal unity is thought of as
itself constituting purposiveness it is evident that the com-
mon view from which the teleological terms get their initial
meaning, is virtually abandoned. Consider first the simple
relation of a universal to its instance. A certain given
curve is, let us say, an ellipse. The universal ellipse gives
368 THE MONIST.
the curve its character, or serves as a description of it;
while on the other hand the curve gives existence or em-
bodiment to the general nature ellipse. There is no para-
dox here provided we distinguish the sort of status which
a universal enjoys from the status of existence. There is
a peculiar relation between a universal and its instance
whereby the first qualifies the second and the second real-
izes the first. Now it means nothing to say that the curve
exists in order to realize the ellipse. It simply does realize
the ellipse. The ideal nature of the ellipse explains what
the curve is; but it does not explain the fact that the curve
exists. Compare the case of Socrates cited above. The
purposiveness of Socrates'-s act lay not in the fact that it
was an instance of submission to law, but in the fact that its
being such in some sense accounted for its occurrence. We
express this by saying that Socrates performed the act be-
cause he deemed it such. In other words, the particular
case of being submissive to law which in fact ensued was a
condition of its own occurrence, through being referred to
as a hypothetical possibility by the mind of Socrates. To
construe the curve similarly it would be necessary to impute
to the curve as determining its existence some reference
to the possibility of its being an ellipse ; which would imply
a complexity of determination for which there is here no
justification.
In the case of existential unity or individuality, it is
admitted that a variety does possess a unitary aspect, but it
cannot be said that any term of the manifold exists for the
sake of that unity. The peculiar flavor which supervenes
upon an assemblage of historical conditions is not neces-
sarily accountable for any of them. It is not necessary to
suppose that the conditions were in any sense determined
by their composing a unity. This would be the case only
provided among the determining factors of each condition
there were one which referred to the composite sequel;
PURPOSE AS SYSTEMATIC UNITY. 369
which might, of coursej be the case, but could not be argued
merely from the fact of the supervening unity.
The situation is not altered if we suppose any degree
or any combination of these types of systematic unity. If
nature throughout observes the law of gravitation, or that
of the conservation of energy, so that every bodily event is
an instance of the same set of interrelated universals — if
it be possible to describe everything in nature by one form-
ula, this would not in the least imply that nature exists for
the sake of realizing the formula. If the world as a whole
should possess a simple flavor or quality to which every
existence and every event contributed an indispensable
condition, this would not in the least imply that such a
cosmic quale determined its conditions. In short mere unity
as such, whether it be a conceptual unity or a perceptual
unity, does not constitute purpose. This does not prove
that purpose does not involve unity, but only that its dif-
ferentia must lie in something else.
4. But it may still be supposed that unity argues an ex-
ternal agency of a purposive sort, that unity is a product
of purpose. In the first place, it is to be observed that unity
furnishes an almost irresistible opportunity for the pathetic
fallacy. There is a strong human interest in unity, an in-
tellectual and practical interest in ideal unity, and an es-
thetic interest in existential unity. When nature is found
to obey relatively simple laws, and so to be predictable and
workable, the mind rejoices and praises God. When sky
and sea and land compose a pleasing landscape, or when a
thousand different conditions conspire to bring about the
existence of fuel or food, one feels instinctively grateful.
And so strong is the instinct that it creates its own object.
But we may dismiss this impulse as an amiable weakness.
We have already seen that the fact that a state of things
is an object of interest, is no proof that that state of things
is owing to interest.
37O THE MONIST.
A second argument for the purposive origin of unity
is the argument from analogy, the argument that Paley
employed in the case of the watch. A thing of the type
which man makes on purpose is presumably made on pur-
pose, if not by man then by God. There is a curious para-
dox connected with this argument. Man is peculiarly ad-
dicted to making machines, or things which work uni-
formly and automatically. That being the case those parts
of nature which argue a purposive creation ought to be
those parts which are most mechanical, such as the periodic
motions of the stars, or the conservation of energy. A
living organism differs from the typical human artefact
just in so far as it is spontaneous and unpredictable; and
ought therefore to be the last thing to be attributed to a cre-
ative will. As a matter of fact, however, the mechanical parts
of nature are the originals of which human artefacts are
adaptations and imitations. Machines are made after the
analogy of nature, and t^heir machine-like character is due
to what they borrow from its independent and self-sufficient
forces. Invention does, it is true, correlate these forces in
new ways ; but there is nothing in the principle of correla-
tion that is new. One could not look for a prettier correla-
tion of forces than that between the centrifugal and centri-
petal forces of a planet moving in an elliptical orbit. The
fact is that man can contrive for his own ends physical
systems which resemble those which he finds in nature.
The remarkable or unaccountable thing is not that system-
atic unity should appear in the absence of purpose, but that
purpose should have anything to do with it at all. The orig-
inal mechanisms of nature are relatively intelligible, and
human artefacts relatively doubtful and obscure. Purposive
origination is not to be invoked as a helpful hypothesis to
account for a mystery; it is itself the mystery which the
mechanical laws of nature will presumably help to solve.
PURPOSE AS SYSTEMATIC UNITY. 371
If the argument from analogy is to be employed at all,
there is more justification for arguing from the case of
nature to that of human conduct than for arguing in the
reverse direction. If the hypothesis of purpose is needed
at all, it is needed to explain not the existence of systematic
unity in the world, but the peculiar case of human conduct
or animal behavior.
Nor is the case for the argument from analogy
strengthened if the emphasis is put on the aspect of utility.
A systematic unity which serves human needs does not
require an explanation which refers to these needs. The
periodic motions of the earth evidently provide the heat
and light and intervals of rest without which human life
would be impossible. Their utility exceeds that of any
man-made agency. But to suppose that they have come
about for the sake of this, is simply to lapse into that
pathetic fallacy which we have already dismissed.
5. There is one further argument from unity which
deserves consideration, the argument, namely which em-
ploys the notion of probability. It is argued that in pro-
portion as a coincidence is remarkable it must have been
designed. Thus, for example, Professor Henderson has
shown that the physico-chemical constitution of the natural
world is uniquely favorable to life. It constitutes a maxi-
mum of fitness.
"The fitness of the environment results from character-
istics which constitute a series of maxima — unique or
nearly unique properties of water, carbonic acid, the com-
pounds of carbon, hydrogen, and oxygen and the ocean —
so numerous, so varied, so nearly complete among all
things which are concerned in the problem that together
they form certainly the greatest possible fitness. No other
environment consisting of primary constituents made up
of other known elements, or lacking water and carbonic
372 THE MONIST.
acid, could possess a like number of fit characteristics or
such highly fit characteristics, or in any manner such great
fitness to promote complexity, durability, and active metab-
olism in the organic mechanism which we call life."3
The author then goes on to argue that "there is not one
chance in millions of millions" that all these properties
should simultaneously occur, and that they should be thus
uniquely favorable to life, unless we assume some general
law that determines them so to be.
Now, in the first place, this appears to be a misuse of
the principle of probability. It is not proper to infer a law
from a single simultaneity, but only from a succession of
simultaneities. If the first throw of a pair of dice happens
to be a double-six, that does not prove that the dice are
loaded, in spite of the fact that the chances were thirty-six
to one against that particular combination. There would
be ground for suspecting a partiality for double-sixes only
provided in the long run this combination turned up more
frequently than once in thirty-six times. The general or
original physico-chemical composition of the cosmos is
like a single throw of dice ; the chances are heavily against
it, but this proves nothing as to any determining principle
over and above chance. It would be possible to make such
an inference only provided it were possible to gather in the
cosmic elements and throw them again. It makes no dif-
ference whatever how heavy the odds are against any par-
ticular combination, provided there is only one instance of
the combination; for it is entirely in keeping with a com-
bination's unusual or remarkable character that it should
occur first. In other words, the principle of chance has to
do with the frequency of a combination and not with its
place in the series. Where the range of alternatives is
large the first combination will always be highly improb-
3 I.. J. Henderson, The Fitness of the Environment, p. 272.
PURPOSE AS SYSTEMATIC UNITY. 373
able; but this fact follows from the principle of chance,
and cannot create a presumption against chance.4
The same reasoning holds of the "fitness" of the en-
vironment for life. Let us suppose life to be a constant.
It will then be comparable to a die having the same num-
ber on all of its faces. The environment, on the other
hand, has millions of faces only one of which matches the
first die. That the two should match in any single instance
is highly improbable; the chances are millions to one
against it. But if it should happen that there was only
one trial, its happening to be successful would prove noth-
ing as to there beiag anything more than chance at work.
Professor Henderson insists that the relation of fitness
between life and its environment is reciprocal; but he ap-
pears to ignore this essential fact, that it is the environment
which is given once and for all, while the die of life is
thrown again and again. It may be argued that life agrees
with its environment too often to permit one to suppose
that on the part of life it is a matter of chance. But nothing
of the sort can be inferred on the part of the cosmic en-
vironment because that lies unchanged upon the board.
The relation of matching where one term is cast once and
the other repeatedly is not a reciprocal relation. If the
matching is uniformly successful, it may prove that the
matcher is not trusting to chance, but it proves nothing as
to the matched.
Suppose that we vary the illustration. It is a remark-
able fact that a given individual likes the world just as
he finds it. The world agrees with his taste. In view of
the vast range of possibilities, the countless worlds that
would offend him, this is prodigiously improbable. But it
does not follow that the world is determined to please him.
4 Bosanquet makes this clear when he says : "We have very small ground
for being surprised at the actual occurrence of that alternative which had
fewest chances in its favor; and absolutely none for being surprised at the
occurrence of a marked or interesting alternative which has against it enormous
odds." {Logic, second edition, Vol. I, p. 342.)
374 THE MONIST.
That would follow only provided the world came up again
and again according to his taste. But, unfortunately for
the argument, the world does not come up again and again,
but only once. Suppose, on the other hand, that sentient
beings come up again and again always liking the given
world. This, then, would argue that the taste of sentient
creatures was somehow determined with reference to their
environment, and did not originate independently of it.
Even this would not prove purpose. Suppose all the
impressions on a given area of sand to correspond exactly
and uniquely to the feet of a certain child that is at play
in the neighborhood. This would presumably not be an
accident ; but would be accepted as evidence that one of the
terms of the fitness relation, namely the feet of the child,
was the cause of the other, namely the impressions on the
sand. It would be necessary, however, to distinguish this
case from the. relation between the same child's feet and
the shoes in his closet. There is fitness in both cases; and
in both cases the fitness is determined, not accidental. But
in the latter case alone would one say that the fitness was
due to purpose. One would not argue the purposiveness
from the bare relation of fitness, or from the non-accidental
character of the fitness, but from the peculiar way in which
the fitness was in this case determined. The shoes in the
closet are of a certain shape because of being judged or
expected to fit their owner. And this might still be the
case even though they should as a matter of fact fit very
poorly.
6. We conclude, then, that purpose in the provisional
sense adopted at the outset, cannot be said to consist in the
structural unity of any system taken as a whole; nor can
it be inferred from such a unity, as necessary to account
for its uniqueness, maximal character, aptness or any other
peculiarity. The same condition of unity might or might
not have been due to purpose. It is necessary in each case
PURPOSE AS SYSTEMATIC UNITY. 375
to observe the actual course of its coming into existence.
In other words, purpose is not to be defined in general
formal terms, any more than chemical reaction. It is not
the same thing as determinateness or law in general. If
there be such a thing, it consists in a particular sort of
agency that appears in some cases of determination and
not in others. We dissent, then, from the view that pur-
pose is exhibited in all cases of system and unity ; being ex-
hibited most unmistakably in those realms of nature that
science has already set in order, and more doubtfully,
therefore, in the phenomena of life.5 We agree with those
who find purpose to be a peculiarity attaching to some parts
of the existent world, most unmistakably to the behavior
of man; purpose in the inorganic world being a doubtful
extension of a conception derived from the datum of life.
RALPH BARTON PERRY.
HARVARD UNIVERSITY.
5 I understand that this latter is the view to which "objective" idealists
incline, as illustrated by the case of Bosanquet. Cf. his "Meaning of Teleol-
ogy," Proceedings of the British Academy, Vol. II : "The foundations of
teleology in the universe are far too deeply laid to be accounted for by, still
less restricted to, the intervention of finite consciousness. Everything goes to
show that such consciousness should not be regarded as the source of teleol-
ogy, but as itself a manifestation, falling within wider manifestations, of the
immanent individuality of the real. It is not teleological because, as a finite
subject of desire and volition, it is 'purposive.' It is what we call 'purposive'
because reality is individual and teleological, and manifests this character partly
in finite intelligence, partly in appearances of a far greater range and scope"
(pp. 8-9). This "individuality of the real" which manifests itself in the larger
cosmic and historical processes, where we cannot suppose it to be designed
or commanded by any finite mind, would appear to consist in systematic unity
of the sorts which we have defined.
THE ORIGIN OF TAOISM.*
THE western world is apt to regard Chinese reflection
as predominantly ethical. This is due largely to the
fact that the system of Confucius is taken as typical.1 But
this view is misleading and requires to be supplemented.
In reality the Chinese mind is fundamentally concerned for
the health of the inner man, and accordingly it is more
properly described as ethico-spiritual. This appears to the
careful reader in the teachings of Confucius himself, and
it is notably true of the mystical doctrine of Lao-tze and
his more immediate followers.
Taoism is well named after the central principle (Tao)
which pervades this system of thought. The original mean-
ing of the term was "way" (path), which in the realm of
* Partial publication (Part I, revised and abridged) of thesis entitled:
"The Thought of Lao-tze; its origin, content and development," presented to
Northwestern University in partial fulfilment of the requirements for the
attainment of the degree of Doctor of Philosophy. The whole will appear in
book form in the publications of the Open Court Publishing Company.
1 The common view is well seen in Grube when he says ("Die chinesische
Philosophic," in Kultur der Gegemvart, I, v, p. 66, 2d ed., Berlin 1913) that
"iiberhaupt das Chinesentum in Konfuzius seine vollendetste und ausgeprag-
teste Verkorperung gefunden hat.... Will man die chinesische Kultur mit
einem kurzen ScMagwort charakterisieren, so wird man sie als konfuzianisch
bezeichnen." This is very misleading. Confucianism came to be dominant
over Taoism in China partly because of the royal edict of Wu-Ti (139-85
B.C.), which exalted this thought at the expense of all other, and partly
because of the universal difficulty of popularizing mysticism or adapting it to
institutional life. But while Confucius has had more visible effect in China
the effect of Lao-tze has been more profound. "It is not Confucianism so
much as Taoism which has most profoundly influenced the Chinese mind."
This statement by Chang-Tai-Yen, a noted scholar and my former revered
teacher, I believe gives the real truth of the matter, and it should be carried
in mind always in studying Chinese thought.
THE ORIGIN OF TAOISM. 3/7
moral inquiry came to mean "norm of conduct" ; in time it
was narrowed to mean "the rational principle in man," and
then later it was extended to signify "reason in man and
reality." This transformation was brought to definite ac-
complishment by the real founder of the system, as I be-
lieve, Lao-tze (sixth century B. C.), who was concerned
to find a metaphysical basis for his ethico-spiritual convic-
tions and to that end hypostatized the principle of Tao.
Thus a convenient analogue in western thought is Reason
or Logos mystically conceived.2
Concerning the life of the founder we know very little
in detail, and of his work we have only the Tao-Teh-king
which tradition attributed to him.3 Both the historicity
of Lao-tze and the authenticity of his work have been ques-
tioned. But it is my belief that, in the existing state of
our data, these doubts have been disposed of definitively by
Carus.4 Certainly the proper procedure here is first to at-
- The term "Tao" of course long antedates the time of Lao-tze. As early
as the Shu-King its ambiguity is already evident, where it means "way" (wan-
tao, or "royal way," as the norm to which all should conform) and also
"rational part of man" (tao-sin, or rational heart, as distinguished from jhren-
sin, or human heart). Herein lay the germ for the development from the
moral to the definitely metaphysical. The transition was therefore from "way"
to "right way of life," to "life according to reason," to life in accordance with
the rational principle of all reality, including man." It was this last idea which
was elaborated by Lao-tze in a world-view.
3 The Tao-Teh-King is accessible to the English reader in the excellent
translation by Carus (Chicago, 1898; [rev. ed. 1913]) where (pp. 95, 96) the
brief account of Lao-tze's life, by Sze-Ma-Chien, may also be found in English
translation. This account gives his place of birth, family, official connection
(custodian of the royal archives and state historian) and relates an encounter
with Confucius. "He practised reason and virtue" we are told, and that his
teaching was directed to "self-concealment and namelessness." When he fore-
saw the decline of his state he left for the frontier, where the custom-house
officer urged him to write a book before leaving his country. "Thereupon,"
concludes the account, "he wrote a book of two parts consisting of five thou-
sand and odd words, in which he discussed the concepts of reason and virtue.
Then he departed. No one knows where he died." The term Tao-Teh-King
was not employed before the second century A. D., but the sayings which con-
stitute this work were uniformly referred to Lao-tze as their author. It had
been customary to name books after the writer.
4 See his admirably judicious article, "The Authenticity of the Tao-Teh-
King," in The Monist, Vol. XI, 1901, pp. 574-601. It is my belief that the
western reader of Chinese literature is in danger of hasty conclusions from the
difficulty of understanding the Chinese way of thinking. The Chinese mind
378 THE MONIST.
tempt to justify the tradition before rejecting it because of
difficulties in the Tao-Teh-King. The determining factor
in this connection must be a real understanding of that
work. If it can be viewed as a unitary whole produced by
a single mind, the tradition may be taken as confirmed
beyond question. In considering its content systematically
I will hope to show that this can be done. For the present
my concern is to indicate how the thought of Lao-tze can be
considered in the historical continuity of Chinese reflection,
after the manner of the western treatment of the history
of philosophy. To that end we must deal with it as a
product of preceding thought and immediate environment
and the genius of our author.
The rise of a new viewpoint in the development of
thought cannot be an entirely isolated affair, however novel
the addition may be. This may be safely assumed for the
progress of Chinese thought as it is for that of the west.
Hence one may properly expect that a system such as that
of Lao-tze's in the Tao-Teh-King could not have appeared
without a preceding development and that accordingly it
should be studied in its historical setting.
The earliest Chinese reflection centered in the conduct
of man and is embodied in the Hong-Fan, which dates back
to 2205-2198 B. C. and forms a part of the Shu-King (the
oldest book of China). Therein we find rules laid down for
does not move normally in the channels of discursive reasoning because it is
essentially intuitionistic. Insight rather than dialectic engages their attention.
Hence the westerner may too readily suspect forgery in what appears to be
nonsense (cf. La Couperie, Western Origin of Chinese Thought, p. 124, where
he shrewdly observes this). The cautious reader will bear in mind the con-
ciseness of diction in the Tao-Teh-King as standing for thought far deeper
than appears, and also that the circumstances of writing precluded fuller
elaboration, as well as the inevitable errors of copyists where the thought of
the text is obscure in itself. In particular it is important to pay due regard
to the purity of style and soberness of thought which signalize the Tao-Teh-
King in contrast with the later works of the school. A stream cannot rise
higher than its source, and a forgery would necessarily have revealed those
fantasies and vagaries which are so conspicuous in the writings of the later
Taoists. The unsympathetic reader is apt to be robbed of insight both for
seeing this obvious fact and also for getting the real meaning at the heart of
the perversions and aberrations.
THE ORIGIN OF TAOISM. 379
the ordering of one's inner life, the securing of proper bal-
ance between conflicting tendencies in one's nature, the
relation that subsists between man and the natural world-
order as well as that between man and his fellows. In it
we find too the conception of the king as the embodiment of
the eternal moral principles, the "royal way" (wan-tao)
which was conceived of as the objective criterion to which
men should conform their personal preferences. And in
it we find the notion of Tao also as "rational part of man,"
as above indicated. The idea of Tao therefore goes very
far back in Chinese thought.
In addition to the Shu-King there is also the Yih-King,
or "Book of Changes."5 Therein is outlined the first
Chinese cosmological scheme, as well as an ethical doctrine
based on this cosmology. It posits an original principle
called Tai-Chi, the "Great Origin," and two primary forces
called Yin and Yang. It was thought that the world was
formed through the action and reaction between these two
principles. A cosmos was regarded as possible only when
there was a perfect balance between these two basic ele-
ments, otherwise chaos would ensue. The attendant eth-
ical doctrine centered in the notion of moderation. As in
the objective order so in man an equilibrium of opposite
forces was the aim. Going to extremes was regarded as
disastrous, because contrary to the course of nature. The
cosmology and the ethics of the Yih-King were therefore
constituent elements in Chinese reflection long before they
appeared in the Tao-Teh-King.
In addition to these two sources there were probably
other documents which were later lost, as the quotations
in the Tao-Teh-King would indicate. Moreover, the ac-
counts of the lives of ascetics make plain that from early
times there had been men who lived in seclusion, insulated
6 The rudiments of this work were in existence prior to the date of the
Shu-King, but were not elaborated until about 1200 B. C.
380 THE MONIST.
from the currents of social and political life. With the
advent of the period of storm and stress, at Lao-tze's time,
this ascetic spirit became much intensified. It took deep
hold on the thoughtful and serious-minded men of that
age, some of whom betook themselves to rural pursuits
while others moved about apparently without profession,
eccentric and mysterious in behavior.
In the Tao-Teh-King the connection with the past is
evidenced by certain expressions8 which indicate clearly
a consciousness of debt to preceding thought. This has
long been recognized by Chinese scholars and has been
largely responsible for the impulse to find the origin of
Taoism in reflection antecedent to Lao-tze. Thus Hwang-
ti, the legendary emperor of the Chinese, has been regarded
as the founder of Taoism, though on very meagre evi-
dence.7 Again it has been suggested that Lao-tze was
simply the transmitter of wise sayings and proverbs out
of the past.8 Another account makes Lao-tze to have sat
under a master, Shan Yung, who was already advanced in
years.9 Still another view finds the origin of Taoism in the
Yih-King, whose cosmology and ethics bear so striking
a resemblance to those of the Tao-Teh-King.10 In short,
Chinese scholars have been amply aware of a continuity
between preceding reflection and that of Lao-tze, and the
connection is so obvious that there is danger of thereby
overlooking his originality.11
6 Such, for example, as "The Ancients say," "The Poet says," "The Sage
says" and the like.
7 Based on the fact that a passage of the Tao-Teh-King is quoted from a
book attributed to Hwang-ti no longer extant. The same passage is found
at the beginning of the work of Lieh-tze. The existence of such a book was
denied by Hwai-Nan-tze.
8 By Chu-Hsi (1130-1200 A.D.)
9 See Hwai-Nan-tze (ch. 10) Lao-tze "learned the lesson of tenderness by
watching the tongue." The allusion is to old age when the teeth have fallen
out.
10 See Yih-King, especially Books III, VI and XI, Engl. transl. by Legge
(Sacred Books of the East, Vol. XVI).
11 Cf. Carus, op. cit., p. 31, and Strauss, Lao-tze's Tao-Teh-King, pp. Ixiii ff.
THE ORIGIN OF TAOISM. 381
That Lao-tze had free and full access to the literature
of his day is sufficiently attested by the tradition which
made him custodian of the royal archives and state his-
torian. This included the classical literature which has
survived and probably much that has since been lost.12 It
is inconceivable that a contact of this kind should have
failed to influence the development of his thought. In
addition there were certain records of the hermits or re-
cluses who preceded him and to whose general circle he is
supposed to have belonged. The contempt for temporal
goods, the effort to create a world of their own beyond
that of ordinary values, the spirit of thoroughgoing re-
nunciation which characterized this group are essentially
the marks of the thought of Lao-tze. Such influence of
his predecessors and contemporaries in thought must there-
fore be assumed if we are to avoid the impossible idea that
the construction of Lao-tze was wholly de novo.13
Thus it is clear that Lao-tze enjoyed the intellectual
heritage of his age. But we must recall that this heritage
reveals no such systematic character as may be found in
the Tao-Teh-King. This work is so characterized by sim-
plicity and unity, it so bears the impress of a single indi-
vidual, that it suggests inevitably to the reader who has
entered into its spirit a seamless fabric woven from the
deeply experienced convictions of a distinct personality. One
must therefore assume some genius operative in revital-
izing and bringing en rapport with his age the inherited
12 The Shi-King, Yih-King and Lih-King would have been accessible to
Lao-tze in their ancient form and not as revised by Confucius.
13 The possibility of foreign influence in the shaping of Lao-tze's thought,
either direct or indirect, I do not consider here. Where the effort is made
(e. g., by Harlez, Douglas, La Couperie, Strauss, Remusat, in varying degrees)
the proof rests upon mere resemblance in mystical or mythological or re-
ligious conceptions. Such procedure is too open to the charge of precipitate
generalization on the basis of fancied resemblance and too hazardous in the
absence of supporting external evidence to win more than doubtful assent.
It may be true that such foreign influence did exist in fact. But the state of
historical knowledge is at present entirely inadequate to furnish satisfactory
conclusions. It seems therefore to me more desirable to seek to account for
Lao-tze by reference to indigenous conditions.
382 THE MONIST.
thought of the past. As Eucken well says, with the western
philosophy in mind, "It is not so much the past which
decides as to the present as the present which decides as
to the past, and that in accordance with this, our picture
of the past continually changes, depending upon the spirit-
ual nature of the present.14 So in the Chinese constructive
activity of the sixth century B. C, for which the historical
evidence is ample, the living present served to stimulate
and illuminate the obscure potentialities of the past. Cer-
tainly the writer of the Tao-Teh-King was possessed of a
genius for illuminating even the homeliest wisdom in the
literature and tradition at hand, and by new insight into
the significance of Tao he was enabled to unfold the possi-
bilities lying inherent in this supreme principle.15
But with all his genius Lao-tze was a part of his age,
and hence he must be considered in relation to the con-
ditions then prevailing. What has been so distinctly true
in the progress of western philosophic thought again must
be taken to maintain in its degree for the development of
Chinese thought. "Philosophy," says Windelband, "receives
both its problems and the materials for their solutions from
the ideas of the general consciousness of the time and from
the needs of society. The general conquests and the newly
emerging questions of the special sciences, the movements
of the religious consciousness, the intuitions of art, the
revolutions in social and political life — all these give phi-
losophy new impulses at irregular intervals, and condition
the directions of the interest which forces, now these, now
those, problems into the foreground, and crowds others
aside for the time being."1 Here we have the course indi-
14 Main Problems of Modern Philosophy, 1912, p. 319.
15 The unfolding of the past by synthesis of the various elements therein
is perfectly familiar to the student of western philosophy in its development.
It is so much a condition of progress in that thought that its history is replete
with illustrations. I believe the same may be safely assumed for the develop-
ment of Chinese thought, however more measured its progress is.
I" History of Philosophy, 1893, p. 13.
THE ORIGIN OF TAOISM. 383
cated which must be followed in our inquiry concerning
the origin of Taoism. The rise of this system of thought
must remain an obscure mystery unless we regard the en-
vironment of Lao-tze, in connection with his heritage and
his genius, and seek to understand the Tao-Teh-King at-
tributed to him in relation to the cultural milieu in which it
arose. To this we turn now.
The first form of government that Chinese history dis-
closes to us may be designated an elective monarchy, in
the sense that the successor to the throne was chosen by
the nobles and ministers. In, this way Yao (2357-2255
B. C.) and Shun (2255-2205 B. C.) came to hold the im-
perial scepter. A change came with Yu (2205-2197 B. C.)
who chose his own son to succeed him and so departed
from the established mode of procedure, and who laid the
basis for the feudal system by assigning portions of the
empire to members of the imperial family. The exact
course of the ensuing development it is impossible to fol-
low. But with the Cheo dynasty (1122-249 B. C) feudal-
ism had become established as a well-defined political in-
stitution. As elsewhere in political history it consisted in
dividing the empire into fiefs or estates to be distributed
among the various nobles for the purpose of consolidating
the empire.
This feudal system worked well at first, largely because
strong emperors held the scepter of state and the fief-
holders served as a bulwark to the throne. But as time
went on the emperors forgot the labors of their forefathers
and turned more and more away from the responsibilities
of government to the gratification of personal desires. As
a result of this there came about gradually a decline of the
central power. The various nobles and princes, who had
theretofore been kept within control, began to show signs
of recalcitrancy and to assert their own powers. This
process of encroaching upon the royal prerogatives in-
384 THE MONIST.
creased more and more until the emperor became a mere
figurehead, a negligible factor, and the real power passed
into the hands of the vassals. With this came a contest
among the various states for supremacy, and so the nation
was precipitated into a tumultuous maelstrom of strife.
The balance between the forces which make law and order
possible had become violently disturbed. Factional strife
and internecine feuds became the order of the day. There
ensued a reckless rush for self-aggrancjizement and an
unscrupulous disregard of rights, and brute power replaced
reason. To supplement the military force the resources
of craft and cunning were pressed into service and the
Machiavellian attitude became dominant.
Along with the political decline went hand in hand a
cultural deterioration. In place of the earlier devotion to
peaceful pursuits, with its cultivation of arts and literature,
there arose an exaggerated emphasis upon material values,
and the earlier simplicity was supplanted by sophistication
both in thought and in action. In this rule of unreason the
complex social organization, which the first few rulers had
succeeded in building up, had completely collapsed. At the
beginning of the dynasty, especially in the reign of Chen-
Wang (1115-1079 B. C), there had been worked out an
elaborate system of etiquette, which in point of complexity
has no parallel in history.17 But in these troublous times
this fell to pieces. Neither the weaklings on the throne
nor the contending vassals were inclined to maintain this
elaborate system. And where all forces were working for
disintegration naturally all phases of the social life were
affected. The established ethical standards also broke
down to be superseded by personal whim and caprice. No-
17 In its ramifications it extended to every phase of social and political
life. Regulations were prescribed even for such details as mode of dress,
eating, toilet, form of address, etc., etc. Its apparently immutable and fixed
character testifies to the genius for organization of its author, Cheo-King, and
also accounts for the fascination which it exercised over the mind of Con-
fucius later who felt impelled to refer to that period as the great age of culture.
THE ORIGIN OF TAOISM. 385
where could universal rules of conduct be found, as in the
ancient days. Unjust laws were enacted in place of the
old regulations, which had been so nicely calculated to pro-
mote orderly life. The life of the people was made mis-
erable by all sorts of oppressive measures, and their very
life-blood was drained that the craving of the rulers for
military glory and the excitement of the chase might be
satisfied. In short, a condition of affairs existed which
was strikingly similar to that which prevailed in France
prior to the Revolution. Wherever one looks he is con-
fronted with unreason and disorder resulting from the
chase after worldly gain and the abuse of power.
Such were the conditions prevailing in the world into
which both Lao-tze and Confucius were born. The in-
tensity of the crisis may be measured by the fact that
China's two greatest creative thinkers arose at this time,
after whom really significant thought in that country con-
tinued to develop. The system of each was adapted to
solve from its angle the problem set by the aggravated
situation. Confucius was conservative and sought to re-
construct in harmony with the past, while Lao-tze was
radical and could be satisfied with nothing short of com-
plete breach. Each may be conceived as crystallizing the
spirit and thought of the type which he represented. The
temperament of the one was essentially institutional and
accordingly gave itself to reconstructing the social -fabric
as existing, as is abundantly clear out of all his writings.
The temperament of the other was wholly impatient with
all temporal expedients and would not stop short of per-
manent peace in some eternal principle; this he found by
reconstructing the ancient Tao as supreme principle of men
and reality, as also amply appears in his work, the Tao-
Teh-King.
The contrast between the two men was really antip-
386 THE MONIST.
odal18 and by reference to it the signficance of the genius
of our author stands out at its highest. Confucius was
characterized by moderation and sanity as the world of
common sense measures these qualities. In his efforts at
reform he confined himself wholly to the attainable, in
conformity with the sagacity of the plain man. His keen
sense for concrete reality forbade him to step forth with
anything like a Utopian program. He clung to the solid
ground, with never a desire to soar in the empyrean realms.
He was no doctrinaire, no mere theorist in any sense, but
a practical reformer. To mend the situation as he saw it
he set about to abolish the feudal system, as the source of
disintegration, and to reestablish the monarchy with its
stabilizing force of imperial power. To counteract the
forces that were making against law and order he set out
to revive the doctrines of the ancient sages, the system of
Cheo-li, whose exact and rigid orderliness very naturally
fascinated his type of mind. Hence his supreme emphasis
on ritual and his belief that the golden age lay in the past.
But the spirit of Lao-tze was radically different and
permitted no such direction as that of Confucius in his solu-
tion of the problem. His genius impelled him to make a
clean sweep and led him to a very different reconstruction.
He felt that the world had gone so far astray that it could
not be reformed by mere revival of ancient traditions or
by any other patching-up process. He demanded some
radical procedure, a complete reversal of the existing order.
He felt deeply the insecurity, nay, the utter collapse of
the foundations of life in his age, and he sought a basis so
18 This contrast is revealed in beautiful simplicity in the report by Sze-
Ma-Chien concerning the interview between the two men (Carus, Tao-Teh-
King, pp. 95, 96). The difference between 'these men is vividly portrayed by
Grube, who writes : "Auf der einen Seite ein Mann, der mit beiden Fiissen
auf dem Boden der Wirklichkeit steht und....nur nach dem Erreichbaren
strebt. Auf der anderen Seite das Wolkenkuckucksheim eines einsamen, welt-
fremden Denkers. Dort zielbewusstes Streben nach staatlicher Reform auf
sittlicher Grundlage, hier asketische Weltflucht und mystisches Versenken ins
ewige Tao."
THE ORIGIN OF TAOISM. 387
secure that it might not be shaken. Like Plato, so much in
this his fellow-spirit of the Occident a century and a half
later, he regarded the present order as wholly bad and
not to be compromised with. And like Plato he turned
away from the immediate world of strife to the life of
reflection and contemplation, to find a world that was char-
acterized by the eternal as opposed to the temporal. But
more mystical than Plato he found his solution by way of
the inner life and communing with nature. In revolting
against the existing order he was driven to withdraw from
externals like the true . mystic that he was. And in so
withdrawing he found within his inner self the supreme
principle of his own and of all being. Thus he was enabled
to give new life and meaning to the doctrine of Tao, as a
simple and unitary principle of all reality.
To this abiding principle he called his wayward people
to return. In opposition to the spirit of self-assertion that
pervaded the age, he called for complete renunciation, for
the surrender of the petty ambitions of the ego which only
in this way could realize Tao. Instead of the feverish and
scattered haste so common in his day, he enjoined quiet
confidence in the fundamental reason of the universal order.
Against over-regulation and the multiplication of laws and
statutes he therefore went the full length of a doctrine of
laissez-faire. He would have none of the ceremonies and
rules of etiquette on which the conciliating Confucius later
laid such stress ; they were for him the most prolific source
of the great evil of hypocrisy, being merely external show.
All parading of virtue or even conscious well-doing was
for him an evil. He would eliminate all virtue except
that of acting according to Tao and all knowledge save that
of Tao. This was the sum and substance of his thought.
And the solution which he disclosed to his age as the way
of salvation was an unfolding of this.
But Lao-tze did not stand alone in this negative atti-
388 THE MONIST.
tude toward the existing order of things. He was a true
spokesman for those fellow spirits of his race and day who
had also turned unreservedly to the inner life for refuge
from the storm of the external world. Like all great lead-
ers of thought, our philosopher gave form and body to
the longings and aspirations in the minds of the many less
gifted. He is clearly the concentrated embodiment of the
quietistic and mystical spirit of the recluses already re-
ferred to. They were in need of a spokesman to make
clearly articulate what they felt and experienced, and this
was supplied by Lao-tze. As the genius of Confucius
enabled him to serve as a constructive guide for the type
he represented, so the genius of Lao-tze enabled him to
create for and direct the less numerous but relatively wide-
spread number of the opposite type.19
Such then was the place of Lao-tze in the origin of
Taoism. He was its real founder because it was his genius
that established it. What had grown up during long cen-
turies and undergone gradual transformation was brought
by him to articulate formulation under the impulse of an
environment which pressed to a mystical solution. His
fundamental doctrine was the long familiar Tao, but its
central position and multiple unfolding in man and in
reality required the labor of genius for establishment. Lao-
tze was that genius, and so Chinese history has recorded
10 In the Confucian Analects alone reference is made to fourteen such
recluses who ridiculed the effort to reform a decadent society. The fortuitous
character of these meetings and the fact that they are recorded by Confucius
and his disciples attest how widespread the movement was. Strauss (op. cit.,
pp. xliii ff ) has suggested the ingenious theory that there was already in ex-
istence a Taoist sect (Tao-Gemeinde), whose teachings were reduced to writ-
ing by Lao-tze. There is no basis in fact for this conjecture, and it overlooks
the real ability of Lao-tze. But this is undoubtedly a more correct direction
for interpretation than that which disregards the widespread nature of the
movement.
In this connection it is of great importance to bear in mind, contrary to
a too prevalent misconception, that even Confucius had to give up his efforts
at reform in despair in his later years, and that he was forced to content him-
self with the more quiet work of teaching and of editing books. The real sig-
nificance of his work lay in this preparation for posterity rather than in his
actual effect on his own age.
THE ORIGIN OF TAOISM. 389
him as one of its two great creative thinkers. Accordingly
his doctrine, as set down in the Tao-Teh-King, is found
to exhibit the unity and simplicity which signalize that
work. It is essentially the reaction to. a most difficult situa-
tion of a born mystic who was able to give full expression
to the mysticism of his people. And what has been said
of the mystic in general maintains for Lao-tze in an emi-
nent degree. "What the world, which truly knows nothing,
calls 'mysticism,' is the science of ultimates,. . . .the science
of self-evident reality, which cannot be 'reasoned about,'
because it is the object of pure reason or perception."20
Herein is contained the key to the true understanding of
Lao-tze's work.
KING SHU Liu.
NANKING UNIVERSITY.
20 Quoted from Patmore by Underbill (Mysticism, 4th ed., 1912, p. 29).
THE CONTRIBUTIONS OF PARACELSUS TO
MEDICAL SCIENCE AND PRACTICE.
THERE appears to be little doubt as to the real value
of many specific contributions of Paracelsus to med-
ical knowledge and practice, although competent author-
ities differ widely as to the extent and character of his
influence upon medical progress. It may be admitted that
his vigorous assaults upon the degenerate Galenism of
his day were effective in arousing an attitude of criticism
and questioning which assisted greatly the influence of
other workers whose labors were laying less sensationally
but more soundly the foundation stones of scientific medi-
cine.
Vesalius, often called the founder of the modern science
of anatomy, and Pare, the "father of surgery," were both
contemporaries of Paracelsus, though their great works
appeared only after the death of Paracelsus. The "Greater
Surgery" of Paracelsus had appeared nearly thirty years
before Pare's classical work and had passed through sev-
eral editions, and it is said that Pare acknowledged his in-
debtedness to Paracelsus in the preface to the first edition
of his work.1
Admitting that none of the medical treatises of Para-
celsus has the scientific value of the works o^his great
contemporaries, it should nevertheless not be forgotten
1 Cf. Stoddart, The Life of Paracelsus. London, 1911, p. 65.
THE CONTRIBUTIONS OF PARACELSUS.
that his work may have had an influence for progress in
his own time much greater than its present value in the
light of later knowledge. Dr. Sudhoff records some nine-
teen editions of the "Greater Surgery" by the close of the
sixteenth century, in German, French, Latin and Dutch
languages, and other works of his shared in somewhat less
degree in this popularity.
The disapproval and hostility of the universities and the
profession toward Paracelsus should not be permitted to
mislead us into underrating his influence, as it may be re-
called that both Vesalius and Pare also suffered from this
hostility. Vesalius was denounced by his former teacher
Sylvius as an insane heretic and his great work on anatomy
was denounced to the Inquisition. Though he was not
condemned by that body his professorship at Padua be-
came untenable, and he was forced to return to his native
city Brussels and is said to have become a hypochondriac
as the result of his persecutions.
Pare was more successful in maintaining his profes-
sional position through official support though the faculty
of the University of Paris protested his tenure of office.
The history of medical science and discovery has been
the subject of more thorough study than most of the nat-
ural sciences, and a number of competent critics of early
medical history have estimated the place of Paracelsus in
the development of various departments of that science.
From such sources may be best summarized the contribu-
tions of Paracelsus.
Thus with respect to surgery, Dr. Edmund Owen in
the Encyclopaedia Britannica (eleventh edition, article
"Surgery") says:
"The fourteenth and fifteenth centuries are almost en-
tirely without interest for surgical history. The dead level
of tradition is broken first by two men of originality and
genius, Paracelsus (1493-1541) and Pare, and by the re-
392 THE MONIST.
vival of anatomy at the hands of Andreas Vesalius (1514-
64) and Gabriel Fallopius (1523-1562), professors at
Padua. Apart from the mystical form in which much of
his teaching was cast Paracelsus has great merits as a
reformer of surgical practice. It is not, however, as an
innovator in operative surgery, but rather as a direct ob-
server of natural processes that Paracelsus is distinguished.
His description of hospital gangrene, for example, is per-
fectly true to nature ; his numerous observations on syphilis
are also sound and sensible; and he was the first to point
out the connection between cretinism of the offspring and
goitre of the parents."
So also Proksch,2 the historian of syphilitic diseases,
credits Paracelsus with the recognition of the inherited
character of this disease and states that there are indeed
but few and subordinate regulations in modern syphilis-
therapy which Paracelsus has not enunciated. Iwan Bloch
also attributes the first observation of the hereditary char-
acter of that disease to Paracelsus.3 That Paracelsus de-
voted so much attention to the consideration of these dis-
eases was evidently made a subject of contemptuous criti-
cism by his opponents as may be inferred from his replies
to them in the-Paragranum.*
"Why then do you clowns (Gugelfritzen} abuse my
writings, which you can in no way refute other than by
saying that I know nothing to write about but of luxus
and venere? Is that a trifling thing? or in your opinion
to be despised? Because I have understood that all open
wounds may be converted into the French disease (i. e.,
syphilis), which is the worst disease in the whole world,—
no worse has ever been known, — which spares nobody and
attacks the highest personages the most severely — shall I
2 Quoted by Baas, Geschichtliche Entunckelung des arztUchen Standcs,
p. 210.
3 Neuburger und Pagel. Handbuch der Geschichte der Medisin, III, 403.
4 Paracelsus, Opera, Strassburg Folio, 1616. I, 201-2.
THE CONTRIBUTIONS OF PARACELSUS. 393
therefore be despised ? Because I bring help to princes, lords
and peasants and relate the errors that I have found, and
because this has resulted in good and high reputation for
me, you would throw me .down into the mire and not spare
the sick. For it is they and not I whom you would cast
into the gutter."
Dr. Bauer" calls attention to the rational protest of
Paracelsus against the excessive blood-letting in vogue at
the time, his objections being based on the hypothesis that
the process disturbed the harmony of the system, and upon
the argument that the blood could not be purified by merely
lessening its quantity.
''For the healing art and for pharmacology in connec-
tion therewith," says Dr. E. Schaer in his monograph on
the history of pharmacology,6 reform is in the first instance
attached to the name of Theophrastus Paracelsus whose
much contested importance for the rebirth of medicine in
the period of the Reformation has been in recent times
finally established in a favorable direction by a master
work of critical investigation of sources .... But however
much overzealous adherents of the brilliant physician may
have misunderstood him and have gone at times beyond
the goal he established, nevertheless the historical con-
sideration of pharmacology will not hesitate to yield to
Paracelsus the merit of the effective repression of the me-
dieval polypharmacy often as meaningless as it was super-
stitious and to credit him with having effectively called
attention to the pharmacological value of many metallic
preparations and analogous chemical remedies."
Dr. Max Neuburger7 thus summarizes the claims of
Paracelsus to a place in the history of the useful advances
in medicine:
r> G esc hie lite der Aderlasse, 1870, p. 147.
6 Neuburger and Pagel, II, 565-6.
7 Neuburger and Pagel, II, 36ff.
394 THE MONIST.
"Under the banner of utilitarianism Paracelsus ren-
dered the practical art of healing so many services that in
this respect his preeminent historical importance cannot
he doubted. In bringing chemistry to a higher plane and
in making the new accessory branch useful to medicine, in
comprehending the value of dietetics, in teaching the use
of a great number of mineral substances (iron, lead, cop-
per, antimony, mercury), and on the other hand in teaching
the knowledge of their injurious actions; in paving the
way to the scientific investigation of mineral waters (de-
termination of the iron contents by nut galls), in essen-
tially improving pharmacy (with his disciples Oswald Croll
and Valerius Cordus) by the preparation of tinctures and
alcoholic extracts. . . .he has achieved really fundamental
merit for all time."
It was also no unimportant service that Paracelsus
rendered to medical science in attributing to natural rather
than to the mystical influence of devils or spirits such
nervous maladies as St. Vitus' dance. It is doubtful per-
haps if his influence in this direction was very immediate
upon contemporary thought, at least if we may judge from
the sad history of the trials, tortures and executions of
witches during a century after the activity of Paracelsus.
Doubtless also the fantastic character of the philosophy
of Paracelsus itself served to diminish the effect of his
sounder and saner thought.
A distinguished student of the history of science, An-
drew D. White, thus characterizes the services of Para-
celsus in this direction.8
"Yet in the beginning of the sixteenth century cases of
'possession' on a large scale began to be brought within the
scope of medical science, and the man who led in this evo-
lution of medical science was Paracelsus. He it was who
first bade modern Europe think for a moment upon the
8 History of Warfare of Science and Theology, II, 139.
THE CONTRIBUTIONS OF PARACELSUS. 3Q5
idea that these diseases are inflicted neither by saints nor
demons, and that the 'dancing possession' is simply a form
of disease of which the cure may be effected by proper
remedies and regimen. Paracelsus appears to have escaped
any serious interference; it took some time, perhaps, for
the theological leaders to understand that he had 'let a new
idea loose upon the planet/ but they soon understood it
and their course was simple. For about fifty years the new
idea was well kept under, but in 1563 another physician,
John Wier of Cleves, revived it at much risk to his position
and reputation."
An interesting thesis maintained by Paracelsus was the
doctrine that every disease must have its remedy. The
scholastic authorities had pronounced certain diseases as
incurable, and they were accordingly so considered by the
profession. Rejecting as he did the ancient authorities,
Paracelsus naturally enough rejected this dogma as neces-
sarily true. Manifestly also he believed that he himself
had with his new remedies effected cures of certain of these
diseases, though he makes no pretension to be able to cure
all diseases. The history of medical thought and discus-
sion shows that this thesis of Paracelsus was a frequent
subject of partizan debate during the century after Para-
celsus.
Paracelsus sustains his thesis, however, not by the
method of modern science — upon evidence of experiment
and observation — but by the philosophical or rather meta-
physical argument of its a priori reasonableness in the
divine purpose, and by his interpretation of the doctrines of
Christ.
"Know therefore that medicine is so to be trusted in
relation to health — that it is possible for it to heal every
natural disease, for whenever God has entertained anger
and not mercy, there is always provided for every disease
a medicine for its cure. For God does not desire us to die
396 THE MONIST.
but to live, and to live long, that in this life we may bear
sorrow and remorse for our sins so that we may repent of
them."8
"There is yet another great error which has strongly
influenced me to write this book, — namely, because they
say that diseases which I include in this book are incurable.
Behold, now, their great folly: How can a physician say
that a disease is incurable when death is not present ; those
only are incurable in which death is present. Thus they
assert of gout, of epilepsy. O you foolish heads, who has
authorized you to speak, because you know nothing and
can accomplish nothing? Why do you not consider the
saying of Christ, where he says that the sick have need of
a physician? Are those not sick whom you abandon" T
think so. If then they are sick as proven, then they need
the physician. If then they need the physician, why do
you say they cannot be helped? They need the physician
that they may be helped by him. Why then do you say
that they are not to be helped? You say it because you
are born from the labyrinth [of errors] of medicine, and
Ignorance is your mother. Every disease has its medicine.
For, it is God's will that he be manifested in marvelous
ways to the sick."1
This is obviously setting dogma against dogma, and
opposing to scholasticism the methods of scholasticism.
Yet that this dictum of Paracelsus was not without in-
fluence upon contemporary thought is evidenced by a pas-
sage in the writings of Robert Boyle in the century follow-
ing.11
"Though we cannot but disapprove the vainglorious
boasts of Paracelsus himself and some of his followers,
who for all that lived no longer than other men, yet I think
9 Paracelsus, Liber de religione perpetua. Sudhoff, Versuch eincr Kritik,
etc., II, 415.
10 Par., Op. I, 253. "Die erste Defension."
11 Boyle's Works, Birch's ed., I, 481.
THE CONTRIBUTIONS OF PARACELSUS. 397
mankind owes something to the chymists for having put
some men in hope of doing greater cures than have been
formerly aspired to or even thought possible and thereby
engage them to make trials and attempts in order thereto.
For not only before men were awakened and excited by
the many promises and some great cures of Arnaldus de
Villanova, Paracelsus, Rulandus, Severinus, and Helmont,
many physicians were wont to be too forward to pronounce
men troubled with such and such diseases as incurable and
rather detract from nature and art than confess that these
two could do what ordinary physick could not, but even
now, I fear, there are but too many who though they will
not openly affirm that such and such diseases are absolutely-
incurable, yet if a particular patient troubled with them is
presented, they will be very apt to undervalue (at least)
if not deride those who shall attempt to cure them."
His rational consideration and treatment of wounds
and open sores is worthy of note. Instead of the customary
treatment of closing up by sewing or plastering, or cov-
ering them with poultices and applications, he advocated
cleanliness, protection from dirt and "external enemies."
and regulation of diet, trusting to nature to effect the cure.
"Every wound heals itself if it is only kept clean."12
There is no doubt that Paracelsus enjoyed a consider-
able reputation as a skilful and successful practitioner, and
there is contemporary testimony, as well as his own state-
ments, to show that he was frequently sent for even from
long distances to treat wealthy and prominent patients
whose maladies had baffled the skill of the Galenic phy-
sicians.
It is of course true that popular reputations of phy-
sicians are not always the true measure of ability even in
our day. Nevertheless there seems little reason to doubt
in spite of the assertions of hostile critics of his time, that
12 Cf. Helfreich in Neuburger and Pagel, III, p. 15.
398 THE MONIST.
with his new remedies, his keen observation, and his un-
usually open mind, he was indeed able to afford relief or
to effect cures where the orthodox physicians trammeled
by their infallible dogmas were unsuccessful. That his
new methods sometimes did harm rather than good is quite
possible. That would naturally be the result of breaking
radically new paths. And an independent empiricism — a
practice founded upon experiment and personal observation
seems to have been his practicce and his teaching, "Expe-
rentia ist Sciential It seems probable that in his dealings
with the sick, his fantastic natural philosophy was rather
subordinated to a native common sense and practical logic.
As stated by Professor Neuburger (op. cit., II, 35), "We
see in Paracelsus. .. .the most prominent incorporation
of that enigmatic, intuitive, anticipative intelligence of the
people, which drawing upon the unfathomable sources of
a rather intuitive than consciously recognized experience,
not infrequently puts to shame the dialectically involved
reasoning of scholasticism."
Paracelsus has indeed clearly expressed his opinion
that theories should not be permitted to dominate the prac-
tice of the physician.
"For in experiments neither theories nor other argu-
ments are applicable, but they are to be considered as their
own expressions. Therefore we admonish every one who
reads these, not to oppose the methods of experiment but
according as its own power permits to follow it out without
prejudice. For every experiment is like a weapon which
must be used according to its peculiar power, as a spear to
thrust, a club to strike, — so also is it with experiments. . . .
To use experiments requires an experienced man who is
sure of his thrust and stroke that he may use and direct
it according to its fashion."1
That he endeavored to keep an open mind toward the
» Chir. Bucher, Fol. 1618, pp. 300-301.
THE CONTRIBUTIONS OF PARACELSUS. 399
symptoms of his patients, not too much governed by pre-
conceived dogmas, is also indicated in his defense against
certain attacks of his opponents in which they accuse him
of not at qnce recognizing symptoms and treatment:
"They complain of me that when I come to a patient,
I do not know instantly what the matter is with him, but
that I need time to find out. It is indeed true that they
pronounce judgment immediately; their folly is to blame
for that, for in the end their first judgment is false, and
from day to day as time passes they know less what the
trouble is and hence betake themselves to lying, while I
from day to day endeavor to arrive at the truth. For ob-
scure diseases cannot be at once recognized as colors are.
With colors we can see what is black, green, blue etc. If
however there were a curtain in front of them we could
not recognize them. . . . What the eyes can see can be
judged quickly, but what is hidden from the eyes — it is
vain to grasp as if it were visible. Take, for instance the
miner; be he as able, experienced and skilful as may be,
when he sees for the first time an ore, he cannot know what
it contains, what it will yield, nor how it is to be treated,
roasted, fused, ignited or burned. He must first run tests
and trials and see whither these lead. . . .Thus it is with
obscure and serious diseases, that so hasty judgments can-
not be made though the humoral physicians do this."14
Admitting the value of the positive contributions of
Paracelsus to medical knowledge and practice, the net
value of the reform campaign which he instituted is vari-
ously estimated by historians of medicine. For it must be
remembered that Paracelsus fought against dogmas in-
trenched in tradition, by dogmas of his own. To the fan-
tastic theories of the Greek-Arabian authorities he opposed
many equally fantastic theories. That by his assault upon
the absurdities and weaknesses of the Galenic medicine of
14 Of. foi, I, 262. (Die siebente Defension.)
4OO THE MONIST.
his time he paved the way for greater hospitality to new
and progressive ideas is unquestionable, but that by this
assault he also did much to discredit the valuable elements
as well as the corruptions of ancient medical achievements
is also true. It is very difficult to balance justly the pro-
gressive and the reactionary influences he exerted upon
the progress of medicine, and naturally, therefore, author-
ities differ upon this question. Thus Neuburger (op. cit.)
appreciates the value of the accomplishments of Paracelsus,
yet doubts that he is to be considered as a reformer of
medicine in the sense that was Vesalius or Pare, that is,
he laid no foundation stones of importance and the real
value of much of his thought required the later develop-
ments of modern scientific thought for its interpretation.
His aim was to found medicine upon physiological and
biological foundation but the method he chose was not the
right method, and his analogical reasons and fantastic phi-
losophy of macrocosm and microcosm were not convincing
and led nowhere. The disaffection and discontent with
conditions in medicine produced by his campaign, can,
thinks Neuburger, hardly be called a revolution. That
was to come later through the constructive work of more
scientific methods.
In a similar vein Haeser (op. cit.) remarks "Scarcely
ever has a physician seized the problem of his life with
purer enthusiasm, served it with truer heart, or with
greater earnestness kept in view the honor of his calling
than the reformer of Einsiedeln. But the aim of his scien-
tific endeavors was a mistaken one and no less mistaken
was the method by which he sought to attain it."
A recent writer, Professor Hugo Magnus,15 presents
a more critical point of view :
"We must then summarize our judgment to this effect,
that Paracelsus keenly felt the frightful corruption which
15 Hugo Magnus, Paracelsus der Ucberarzt. Breslau, 1906.
THE CONTRIBUTIONS OF PARACELSUS. 4OI
medicine and the investigation of nature suffered from the
hands of the Scholastics, but that he did not understand
how to penetrate to the causes of this condition of his
science. Instead of seeking in the scholastic system the
root of this medical degeneration, he believed that it must
.be found exclusively in the healing art of the ancients. And
thus he sought to shatter in blind hatred all that existed,
without being in position to be able to replace the old theory
he maligned by a new and better concept of nature and
medicine. So Paracelsus wore away in unclear struggling,
his bodily and mental energy, and lived indeed as a re-
former,— a medical superman, in his own imagination, in
his own valuation, but not in the recognition of his own
times, nor in the judgment of posterity."
"If therefore I can find no relationship between the
general methods of medicine to-day and the Theophrastic
concept of nature, nevertheless our supercolleague must
be considered in an essentially limited respect, to be sure,
as the pioneer in certain modern points of view. He was
the first to attempt the consideration of the phenomena of
organic life in a chemical sense, and I do not need to em-
phasize that he thereby paved the way to a very powerful
advance in our science. In this respect was Paracelsus a
reformer, here he has pointed new paths in the valuation
of pathologic phenomena as well as in therapy, even if here
also he has theorized enough and allowed his neo-Platon-
ism to play him many a trick."
By discarding and condemning all the ancient author-
ities, thinks Magnus, Paracelsus assailed not only the cor-
rupted Galenism of his time but did much to discredit the
positive achievements of the Greeks, and although the orig-
inal Greek authorities were not the then prevailing texts,
they were at least accessible in newly translated versions,
and the good in them might have been incorporated and
built upon by Paracelsus if he had possessed the scientific
4O2 THE MONIST.
point of view. To the extent of his influence in this direc-
tion Paracelsus was therefore an opponent rather than a
promoter of the progress of medical science. "Through
his irrational theories he gave impulse to all sorts of mis-
taken notions among his followers, so that the wildest
vagaries existed among the Paracelsists of the succeeding
century."
The above will serve to illustrate the trend of modern
critical judgment of Paracelsus as a reformer of medicine.
However estimates may vary as to the extent of the
influence of Paracelsus as a reformer of medicine, credit
must certainly be given him as a forceful agent in the
downfall of the scholastic medical science of his time. The
real reform in medical science, its establishment upon a
basis of modern scientific method, was not the work of his
century nor of the century to follow. Indeed it may not
be too much to say that that great reform was mainly the
work of the nineteenth century, and was made possible only
through the patient labors of many investigators in the
domains of physics, chemistry, anatomy, and biology.
If, however, we cannot claim for Paracelsus the un-
challenged place of the reformer of medicine, we may at
least recognize in him an earnest, powerful, and prophetic
voice crying in the wilderness.
J. M. STILLMAN.
LELAND STANFORD JUNIOR UNIVERSITY.
THE ORIGIN OF THE MUTATION THEORY.
AP the time when Darwin published his book on the
Origin of Species biological science was in a very
different condition from what it is now. Hardly ten years
had elapsed since Schleiden and Schwann discovered the
fundamental law that all living organisms are built up of
one or more ordinarily almost innumerable cells.
Mohl's contention that protoplasm is the essential and
in fact the only living part of the cell is almost contempo-
raneous with Darwin's book (1849 and 1851). The pres-
ence of a nucleus within the cells began to be recognized.
Hereditary problems were almost only discussed by breed-
ers.
The Textbook of Botany by Julius Sachs appeared in
1868; it was the first to introduce into botany really scien-
tific methods. When I was a student at the University of
Leiden (1866-1870) systematic and descriptive morpho-
logical studies prevailed. Microscopical study of tissues
was new and cytology had hardly reached us. Under these*
conditions a student interested in the causal relations of
the phenomena of life naturally turned his mind to physics
and chemistry. The prominent question of those days
was the validity of physical and chemical laws in the living
body. The idea dawned upon us that this question chiefly
related to the protoplasm but hardly needed a proof for the
cell walls and the tissues built up of them.
Once convinced that the phenomena of life are regu-
404 THE MONIST.
/
lated by the protoplasm we naturally looked for methods of
studying this relation. Many different ways presented
themselves, and among these four seemed to me the most
promising. They were the study of respiration, of galls,
of osmosis and of variability. I tried all of them and at
the end chose the last. Respiration was the source of
energy; it was a phenomenon common to animals and
plants, and one of the main links which connected both
kingdoms in our knowledge at that time. I devoted many
years to its study, chiefly in a comparative way, and chose
it for the subject of my inaugural address when I was
called to the chair of plant physiology in the University of
Amsterdam (1878).
But galls seemed to promise far more. They are built
up of the ordinary qualities of the plants combined in a
new way to fit the requirements of their insects, and
this combination is brought about under the influence of
some stimulus given off by the insect. To discover the
nature of these stimuli and the laws by which they so effec-
tively change the growth of the tissues, seemed to me a
scope worth the devotion of a whole life. I made a large
collection of galls, in search of the species which \vould
be the most appropriate to attack this line of research.
I concluded for those of the willows, belonging to the
genus Nematus. But at that period I met with Mr. M. W.
Beyerinck who was far beyond me in the study of the life
history of the galls, and so I left this pathway. I have,
however, read a course upon galls and their bearing on the
broad problems of biology about every third year from that
time on.
The study of osmosis and of the turgidity of the cells
led to the discovery of the semi-permeable membranes of
the protoplasm and their significance for growth and move-
ments as well as for the study of isotonic coefficients and
the determination of atomic weights, as, e. g., in the case
THE ORIGIN OF THE MUTATION THEORY. 405
of the sugar raffinose. But its promise of elucidating
hereditary questions diminished with every new discovery.
In 1880 I started a course on variability. I had been
interested in this question chiefly by making a herbarium
of monstrosities, and monstrosities were at that time almost
all we knew of variability. Moreover I had visited the
celebrated agriculturist W. A. Rimpau at Schlanstedt in
Saxony and stayed repeatedly for some weeks on his estate
in order to study his selection of cereals and sugarbeets.
This induced me to take up a thorough study of agricul-
tural and horticultural selection and I soon found that
Darwin's books were the best guides for this literature.
Especially from the pamphlets of Vilmorin, Verlot and
Carriere I took a large part of the facts for elaboration of
my lessons.
I read this course every second year from 1880 to 1900,
and each time introduced into it the principles and methods
which I found in the literature. This consisted partly in-
rare pamphlets which I succeeded in collecting only grad-
ually, partly in articles scattered in agricultural and horti-
cultural journals. In the meantime I increased my collec-
tion of monstrosities but soon perceived that collecting is
not the right way to gain an insight into them. Therefore
I preferred revisiting the same spots in nature for succes-
sive years and found the monstrosities regularly repeated.
This induced the idea of their being heritable phenomena,
a conception wholly new at that time, although the in-
heritance of the cockscomb or Celosia was, of course, known
to every horticulturist. Then I turned to cultivation, made
races of fasciated and twisted forms and studied the in-
heritance of pitchers and analogous deviations.
Parallel to these experimental studies I tried to pene-
trate into the theoretical side of the question, and this led
to the publication of my book on Intracellular Pangenesis
in 1889, of which the Open Court Publishing Company
406 THE MONIST.
published an English translation by Prof. C. Stuart Gager
in 1910. Freed from the hypothesis of the transportation
of germs through the tissues, Darwin's pangenesis coin-
cided with my own conception of the material basis of
protoplasmic life and of the hereditary qualities. This
study brought about the conviction that variability must
at least consist in two essentially different principles. One
of them is the origin of new qualities and their accumula-
tion through geological times, producing the continuous
development of higher forms from lower. This form is
what we now call mutability. The other is our present
fluctuating variability. It determines the degree in which
the single qualities will show in different individuals. I
proposed this difference between mutability and fluctu-
ating variability at the conclusion of my book, but said to
myself: It is all right to deduce the theoretical necessity
of this conclusion, but it would be of far higher importance
to prove the actual existence of these two types of variation.
I set at work at once, first in the field but soon in the
garden. I cultivated over a hundred wild species, and
some of them through many years. Fluctuating variabil-
ity was everywhere present. Then I chanced to meet with
Quetelet's Anthropometrie, which had appeared in 1870,
applied his methods to plants and saw that here the same
general laws prevail. Different forms of curves of varia-
tion were determined in the corn marigold (Chrysanthe-
mum segetum) and other plants (1894-1899), and it be-
came clear that they changed the properties only in the
directions of more or less development, but gave no indi-
cation whatever of an origin of new qualities. Fluctua-
tion and mutability must therefore be principally distinct.
Mutations must of course be rare, but some few of them
occurred in my garden in well-guarded breeds. They were
sudden, without visible preparation or transitions. The
peloric toadflax appeared in 1894, the double corn marigold
THE ORIGIN OF THE MUTATION THEORY. 407
in 1896; they sufficed to prove the reality of mutations
and gave an experimental basis for the appreciation and
the study of the sudden appearance of new varieties in
horticulture.
Besides them, one species proved to be rich in such
sudden changes. It was Lamarck's evening primrose, a
species originally wild in the eastern United States and
collected there by Michaux, but which has since disap-
peared in America. It has, however, won an extensive dis-
tribution in England, Holland, Belgium and France, pre-
ferring the sand dunes along the coast. I observed its muta-
tions for the first time in 1888 and since then it has never
ceased to produce them. The number of mutants amounts
to more than a dozen, some of them being progressive, as
for instance the giant type or Oenothera Lamarckiana
gigas, published in 1900, others retrogressive like the dwarfs
and a brittle race called O. rubrineri'is. Ordinarily they
are constant from seed, but some show a splitting and are
therefore considered to be half-mutants only, as O. lata
and allied forms. The changes are always sudden and
without transitions and occur so regularly in about i% of
the individuals that they constitute an unexpected but ex-
cellent material for experimental researches.
In my course on variability I laid especial stress on the
pedigrees of. definite systematic groups. The families of
the euphorbiaceous and the umbelliferous plants afforded
a very demonstrative material, and the hypothesis of the
descent of the Monocotyls from the Dicotyls through types
allied with the common buttercups, proposed at that time
by Delpino, proved to be very convincing and instructive.
Systematic atavisms, as shown in the leaf-bearing seedlings
of the leafless species of Acacia and analogous instances
were added to these discussions. They showed that evo-
lution in nature is partly progressive and partly retro-
gressive. Progression means differentiation and speciali-
408 THE MONIST.
zation, it governs the main lines of the pedigree of the
animal and vegetable kingdoms. But retrogression, con-
sisting in the loss of previously developed qualities, must
be responsible for a large part of the diversity of forms in
nature. And since it is easier to lose a thing than to acquire
a new quality, the cases of retrogression must be far more
numerous in nature than those of actual progression.
Therefore there must be two kinds of mutations and
even in our experimental cultures progressive ones must
be rare, and retrogressive ones comparatively more fre-
quent. This is exactly what we see in the mutations of the
evening primrose.
Alongside of these studies I tried hybridization. Opium
poppies afforded a useful material and led to the rediscov-
ery of Mendel's law. At that time this conception was be-
lieved in by nobody, it was rather considered as an ideal-
istic fiction. But the splitting of the poppies confirmed that
of Mendel's peas, and numerous garden varieties behaved
in the same way. I was fortunate enough to be the first
to publish this result (1900) and pointed out that it is
especially retrogressive variations which follow this law,
whereas progressive ones produce constant hybrids, at
least in many instances.
Paleontological studies strengthened the idea of the
origin of species by means of sudden variations instead of
a slow and gradual development. This side of the question
has since been taken up by Charles A. White and other
paleontologists. From my own studies I deduced the con-
tention, that life on this earth has not lasted long enough
for such a slow development as Darwin's theory of selection
supposed. Darwin calculated some thousands of millions
of years as required for his theory, but geologists and
physicists only allow about forty or at most a hundred
millions of years for the development of all animals and
plants. The hypothesis of sudden mutations delivers us
THE ORIGIN OF THE MUTATION THEORY. 409
from this difficulty. And so it does for many other objec-
tions which were still being used as weapons against the
whole principle of evolution in the form proposed by Dar-
win.
It has always been my conviction that the improvement
of industrial practice is the main aim of all science. Bio-
logical science has to be a basis for agriculture and horti-
culture. The discipline of heredity should be crowned by
the advance in our knowledge concerning the breeding of
animals and plants. With Dr. Wakker I studied the dis-
eases of the flower bulbs cultivated all around Haarlem
(1883-1885), and since then I regularly sent contributions
to the journal of our agricultural society. From 1892 to
1894 I was editor of the journal of the Dutch Horticultural
Society in order to have an easy access to horticultural
establishments in the Netherlands as well as abroad, and
collected all the evidence I could find concerning practical
plant-breeding. As a matter of fact this was very scanty
but it led me to a connection with the Director of the Swed-
ish agricultural station at Svalof, Dr. Hjalmar Nilsson,
whose celebrated method of plant improvement rested on
the same scientific basis as my own experiments.
My book on the mutation theory is the combination of
all these preliminary studies into a regular discussion of
the main principle. I had the great advantage of my
steadily repeated courses on heredity, which constituted,
if I may say so, a first unpublished edition, with all the
many faults inherent to first trials on a new field. The
book appeared in 190x3, and an English edition,1 prepared
by Prof. J. B. Farmer and A. D. Darbishire, was published
by the Open Court Publishing Company in 1909. It tries
to show that the origin of species is a natural phenomenon
and that it is possible to subject it to experimental study.
In nature the mutations have produced the whole evolution
1 The Mutation Theory. 2 vols.
4IO THE MONIST.
of all living beings; in the garden we can, of course, only
expect to see their very smallest steps. The identity of
retrogressive mutations in nature, in horticulture and agri-
culture and in the experimental garden seems now to be
beyond doubt. But progressive changes, which are the
most important, are at the same time the rarest, in nature
as well as in cultivation. In regard to these the theory
relies on its broad arguments and the question whether
the directly observed progressive mutations afford a mate-
rial for the interpretation of the ways of nature is still
under discussion.
The theory is based upon arguments taken from widely
different branches of nearly all natural sciences. It con-
duces of necessity to experimental research, but this, of
course, is still in its first infancy. It promises, however,
to become some day of important service to science at large
as well as to the practice of breeders.
HUGO DE VRIES.
LUNTEREN, HOLLAND.
THE MANUSCRIPTS OF LEIBNIZ ON HIS DIS-
COVERY OF THE DIFFERENTIAL CALCULUS.
PART II (CONTINUED).
§§ XI— XV.
Between the date of the manuscript last considered
and the one which follows there is a gap of seven months,
for which Gerhardt does not appear to have found any-
thing. This is very unfortunate ; for in this interval Leib-
niz has attained to the important conclusion that the true
general method of tangents is by means of differences.
We saw that in November 1675 he had started to investi-
gate more thoroughly the direct method of tangents; but
the method is that of the auxiliary curve, and there is no
indication whatever of the characteristic triangle. Does
this interval correspond with the time taken by Leibniz
for his final reading of Barrow from Lect. VI to Lect. X,
comparing all the geometrical theorems with his own nota-
tion? Or is it only a strange coincidence that Leibniz's
order is the same as that of Barrow, first the auxiliary
curve, and lastly the method of differences? One could
form a more definite opinion, if Leibniz had given a dia-
gram for the first problem he considers, the one in the next
following manuscript, which amounts to the differentiation
of an inverse sine. Such a diagram he must have had
beside him as he wrote; for I think the reader will find
that he wants one to follow the argument; with the idea
412 THE MONIST.
of verifying this argument, I have not endeavored to supply
the omission.
The consideration of the direct method of tangents is
apparently, however, only as a means and not as an end;
for Leibniz harks back to the inverse method, and to the
catalogue of quadrible curves, which he seems to say he
has in hand. It is not until November 1676 that he seems
to be coming into his own; and it is not until July 1677 that
he has a really definite statement of his rules. On the other
hand, in July 1676, he is consistently using the differential
factor with all his integrals, and before the end of that
year he has the differential of a product, whether obtained
as the inverse of his theorem fy dx = xy — $x dy, or by
the use of the substitution x + dx, y -f- dy, is not certain ;
but this substitution appears in the manuscript for No-
vember 1676. Finally, in July 1677, appears the general
idea of the substitution of other letters, in order to eliminate
the difficulty caused by the appearance of the variable
under a root sign or in the denominator of a fraction ; and
with this the whole thing is now fairly complete for all
algebraical functions. There is as yet no equally clear
method for the treatment of exponentials, logarithms, or
trigonometrical functions; for the latter he refers to a
geometrical diagram, strongly reminiscent of Barrow.
§xi.
26 June, 1676.
Nova methodus Tangentium.
(New Method of Tangents.)
I have many beautiful theorems with regard to the method of
tangents both direct as well as inverse. D^scartes's method of
tangents depends on finding two equal roots, and it cannot be em-
ployed, except in the case when all the undetermined quantities
occurring in the work are expressible in terms of one, for instance,
in terms of the abscissa.
But the true general method of tangents is by means of dif-
THE MANUSCRIPTS OF LEIBNIZ. 413
ferences. That is to say, the difference of the ordinates, whether
direct or converging, is required. It follows that quantities that
are not amenable to any other kind of calculus are amenable to
the calculus of tangents, so long as their differences are known.
Thus if we are given an equation in three unknowns, in which x
is an abscissa, 3; an ordinate, and z the arc of a circle of which x
is the sine of the complement, e. g., the equation bzy = cx* + fz-. To
find the next consecutive y, in place of x take x + f$, and in place of
, Pr Pr (51)
z take 2 -as. or, since dz= -7 — — , we may take z- —~=-
/* 9 • / 9 9 '
hence we have
Hence the difference between y and (y) is given by
= t>* dy ;
dy =?2cxT}~r*^x**2/a:r t ttf
Therefore a=~ — 2 r« — ~g — - = - = — 2 — -^2 .
From this the flexure or sinuosity of the curve can be found,
according as now 2cz\Jrz -x*, now 2fzr predominates ; for when
they are equal, the ordinate on that side on which it was previously
the greater then becomes the less. It is just the same, if several
other undetermined quantities, such as logarithms and other things
occur, no matter how they are affected, as for instance in the equa-
tion b2y = cx2 + fz"+xzl, where s is supposed to be an arc, and / a
logarithm, .v the sine of the complement of the arc, and y the num-
ber of the logarithm, b being the radius and unity, equal to r. Also
it is just the same, whenever an undetermined transcendental has
been derived from some dimension or quadrature that has not been
investigated.32
For the rest, many noteworthy and useful theorems now arise
from the foregoing by the inverse method of tangents. Thus gen-
eral equations, or equations of any indefinite degree may be formed,
at first indeed in two unknowns, x and y, only. But if in this way
the matter does not work out satisfactorily, it will easily do so when
51 In this and the following line I have corrected two obvious misprints;
they are evidently not the fault of Leibniz, for the lines that follow from them
are correct.
82 There is some doubt here as to whether Leibniz could have given an
example ; but it must be remembered that these are practically only notes,
mostly for future consideration.
414 THE MONIST.
the tables which I am investigating are finished ; then it will be
possible to take one or more other letters, and to take the difference
as an arbitrary known formula, and when this is done it is certain
that finally in any case a formula will be found such as is re-
quired, and in this way also a curve which will satisfy the conditions
given ; but in truth the description of the curve will need diagrams
for these symbols, representing the sums of the arbitrarily chosen
differences.
Now once a curve is found having the tangent property that
we want, it will be more easy afterwards to find simpler construc-
tions for it. We have this also as a convenient means enabling
us to use many quantities that are transcendent, yet depending the
one on the other, such for example as are all those that depend
on the quadrature of the circle or the hyperbola. From these
investigations it will also appear whether or no other quadratures
can be reduced to the quadrature of the circle or the hyperbola.
Lastly, since the finding of maxima and minima is useful for the
inscription and circumscription of polygons, hence also, by employ-
ing these transcendent magnitudes, convergent series can be found,
and in the same way their terminations ; or of any quantities formed
in the same way. However in that case it may not be so easy to
argue about impossibility ; at least indeed by the same method.
Only I do not see how we can find whether from the quadrature of
the circle, say, any sum can be found, when no quantity depending
on the dimensions of the circle enters into the calculation.
§XII.
July, 1676.
Methodus tangentium inversa.
[Inverse method of tangents.]
In the third volume of the correspondence of Descartes, I see
that he believed that Fermat's method of Maxima and Minima is
not universal; for he thinks (page 362, letter 63) that it will not
serve to find the tangent to a curve, of which the property is that
the lines drawn from any point on it to four given points are to-
gether equal to a given straight line.
[Thus far in Latin; Leibniz then proceeds in French.]
Mons. des Cartes (letter 73, part 3, p. 409) to Mons. de Beaune.
"I do not believe that it is in general possible to find the con-
verse to my rule of tangents, nor of that which Mons. Fermat uses,
THE MANUSCRIPTS OF LEIBNIZ. 415
although in many cases the application of his is more easy than
mine ; but one may deduce from it a posteriori theorems that apply
to all curved lines that are expressed by an equation, in which one
of the quantities, x or y, has no more than two dimensions, even
if the other had a thousand. There is indeed another method that
is more general and a priori, namely, by the intersection of two
tangents, which should always intersect between the two points at
which they touch the curve, as near one another as you can im-
agine ; for in considering what the curve ought to be, in order that
this intersection may occur between the two points, and not on this
side or on that, the construction for it may be found. But there
are so many different ways, and I have practised them so little, that
I should not know how to give a fair account of them."
Mons. des Cartes speaks with a little too much presumption
about posterity ; he says (page 449, letter 77) that his rule for re-
solving in general all problems on solids has been without compari-
son the most difficult to find of all things which have been discovered
in geometry up to the present, and one which will possibly remain
so after centuries, "unless I take upon myself the trouble of finding
others" (as if several centuries would not be capable of producing
a man able to do something that would be of greater moment).
(Page 459.) The question of the four spheres is one that is
easy to investigate for a man who knows the calculus. It is due
to Descartes, but as it is given in the book, it appears to be very
prolix.
The problem on the inverse method of tangents, which Mons.
des Cartes says he has solved (Vol. 3, letter 79, p. 460)
[Leibniz then continues in Latin.]
EAD is an angle of 45 degrees. ABO is a curve, BL a tan-
gent to it ; and BC, the ordinate, is to CL as N is to BJ. Then
c BC = *y
BJ =y — x
ny n y—x x
hence, ,__, _. _ ml__t
hence, £ = '-«. bu, < J±;
y t y dy
therefore fe^-JL. or dx y—x dx = dy «;
dy y—x
hence §dxy- $xdx=-n$dy.
416
THE MONIST.
Now, fdy = y, and $.vdx = x*/2, and fdxy is equal to the
area ACBA, and the curve is sought in which the area ACBA is
equal to (x*/2) +ny= (AC2/2) +«BC B3
Let this .r2/2, i. e., the triangle ACJ be cut off from the area,
then the remainder AJBA should be equal to the rectangle ny.
The line that de Beaune proposed to Descartes for investigation
reduces to this, that if BC is an asymptote to the curve, BA the
axis, A the vertex, AB, BC, fixed lines, for BAG is at right angles.
B T A
Let RX be an ordinate, XN a tangent, then RN is always to
be constant and equal to BC ; required the nature of the curve.
This is how I think it should be done.
Let PV be another ordinate, differing from the other one RX
by a straight line VS, found' by drawing XS parallel to RN ; then
53 Leibniz has a footnote to this manuscript : "I solved in one day two
problems on the inverse methods of tangents, one of which Descartes alone
solved, and the other even he owned that he was unable to do."
This problem is one of them, the first mentioned in the footnote given by
Leibniz. But it requires a stretch of imagination to consider Leibniz's result
as a solution. For he ends up with a geometrical construction, that is at
least as hard as the construction that can be made by the use of the original
data. There is of course the usual misprint that one is becoming accustomed
to; but there is also the unusual, for Leibniz, mistake of using his data in-
correctly. Starting with the hypothesis that BC : CL = N : BJ, he writes CL =
N.BC/BJ (correcting the omission of the factor N), instead of CL ==
BC.BJ/N.
The solution of the problem is y-\-n\og(y — x+n)—Q, as originally
stated, or .r = «log(tt — y-\-x), if we continue from Leibniz's erroneous re-
sult dx/dy = n/(y — x).
The point to be noted, however, is that Leibniz does not remark that "this
curve appertains to a logarithm."
THE MANUSCRIPTS OF LEIBNIZ. 417
the triangles SVX, RXN are similar, RN -t-c, a constant, RX = y,
SY = dy, and therefore
- - = — - ; hence cy — I y dx or c dy = y dx. "
Ct X t — C , J
If AQ or TR = r, and AC = /, while
AC / TR z az
If dx is constant, then dz is also constant. Hence
c dy= jy dz, or cy= -, \ y dz , and ry dy^jy1 dz, therefore
>'2 a (* —
~o —j \ y1 dz- Hence we have both the area of the figure and the
'2
c
moment to a certain extent (for something must be added on
account of the obliquity) ; also
cz dy=-,yz dz , and therefore c I z dy— -7 I yz dz.
Also £&. — - dz, and hence, c I -y- = -z. Now, unless I am
y f J y f
greatly mistaken, J ^ is in our power.55 The whole matter reduces
J y
to this, we must find the curve5" in which the ordinate is such that
54 Leibniz does not see that this result immediately gives him the equation
that he requires. Thus jr = cLogy, as he would have written it; the usual
omission of the arbitrary constant does not matter in this case, so long as BA
is taken as unity, which is possible with Leibniz's data.
55 Here he seems to recognize that he has the solution. The next sentence
is, however, very strange. As long ago as Nov. 1675 he has written fa-/y as
Logy, and recognized the connection between the integral and the quadrature
of the hyperbola ; and yet he says "unless I am mistaken, fdy/y is always in
our power." Now notice that in the date there is no day of the month given,
contrary to the usual custom with these manuscripts so far; can it be possible
that this date was afterward added from memory, and that the manuscript
should bear an earlier date? If not we must conclude that Leibniz has not
yet attained to a correct idea of the meaning of his integral sign, and is still
worried by the necessity (as it appears to him) of taking the y's in arithmet-
ical progression.
56 The passage in the original Latin is very ambiguous, and it may be that
it is not quite correctly given ; I think, however, that I have given the correct
idea of what Leibniz intended. One has to draw an auxiliary curve, in which
y = dy/dx, and then find its area ; in that case it should be "divided by the
differences of the abscissae" instead of "divided by the abscissae."
41 8 THE MONIST.
it is equal to the differences of the ordinates divided by the ab-
scissae, and then find the quadrature of that figure.
1 (57)
d^ay = ,
Figures of this kind, in which the ordinates are dy/y, dy/y2
dy/y3, are to be sought in the same way as I have obtained those
whose ordinates are y dy, y~dy, etc. Now w/a -- dy/y, and since dy
may be taken to be constant and equal to ft,56 therefore the curve,
in which •w/a = dy/y, will give wy = aj3, which would be a hyper-
bola.53 Hence the figure, in which dy/y = 2, is a hyperbola, no mat-
ter how you express y, and if y is expressed by </r we have dy = 2<f>,
, 2<t> 2 Cdy a fcC-l
and -TO- = T . Now, c I - = -f z , and therefore - • - = z,
•r • •* J y f a J y
which thus appertains to a logarithm.00
Thus we have solved all the problems on the inverse method
of tangents,61 which occur in Vol. 3 of the Correspondence of Des-
cartes, of which he solved one himself, as he says on page 460,
letter 79, Vol. 3 ; but the solution is not given ; the other he tried
to solve but could not, stating that it was an irregular line, which
in any case was not in human power, nay not within the power of
the angels unless the art of describing it is determined by some other
means.
§XIII.
This manuscript bears no date: however, it was prob-
ably written very shortly after his call on Hudde at Am-
sterdam, on his way home from England (the second visit)
57 An interpolated note, marking a sudden thought or guess ; for the next
sentence carries on the train of thought that has gone before. Query, some
interval of time, either short (such as for a meal) or long (continued the next
day), may have occurred here.
58 This cannot be referred back to the present problem, since Leibniz has
already assumed in it that dz and dx are constant. This may account for the
fact that he has hesitated to say that the integral represents a logarithm.
59 This working is intended to apply to the auxiliary curve mentioned
above, w standing for dx, and /3 for dy ; hence the curve is not a hyperbola ;
Leibniz seems to have been misled by the appearance of the equation suggest-
ing xy = constant.
60 Here apparently he leaves the muddle, in which he has entangled him-
self, and returns to his original equation ; he then remembers that he has found
before that the integral in question leads to a logarithm.
61 He has not solved either of them ; nor can it be said from this that
"Leibniz in 1676 sought and found the curve whose subtangent is constant."
Of all the work that Leibniz has done hitherto, there is none that is so incon-
clusive as this in comparison.
THE MANUSCRIPTS OF LEIBNIZ. 419
to Hanover. Leibniz stayed in Holland from October 1676
to December of that year; hence the date may be fairly
accurately assigned.
Hudde showed me that in the year 1662 he already had the
quadrature of the hyperbola, which I found was the very same as
Mercator also had discovered independently, and published. He
showed me a letter written to a certain van Duck, of Leyden I
think, on this subject. His method of tangents is more complete
than that of Sluse, in that he is able to use any arithmetical pro-
gression, as in a simple equation, whereas Sluse and others can
use only one. Hence constructions can be made simple, while terms
can be eliminated at will. This also can be made use of for elim-
inating any letter with greater facility, for numerous-equations of
all sort are thereby rendered fit for elimination.
x3 + px* + qx =0 x*+ xy + y1 + x + y+ *= Q
y'2 y, 2xdx + xdy + 2ydy+dx+dy = 0
y3 ydx
*
2yx* + yx y dy y~+2x+l
yzx
What I had observed with regard to triangular numbers for
three equal roots, and pyramidal numbers for four, was already
known to him, and indeed even more generally,
-10123456
-3-1 0 0 1 3 6 10 15
-4-1 0 0 0 1 4 10 20
Here it must be observed that the number of zeros increases, as
this is of the greatest service in separating roots.
He has also rules for multiplying equations, so that they are
not only determined for equal roots, but also for roots increasing
arithmetically, or geometrically, or according to any progression.
Hudde has a most elegant construction for describing two
curves, one outside and the other inside a circle, which are capable
of quadrature, and by means of these curves he finds the true area
of a circle so nearly, that with the help of the dodecagon, in
a number of six figures, there is an error of only three units, or
3/100000.
42O THE MONIST.
He has a method for finding the real roots of equations, having
some roots real and the rest impossible, by the help of another
equation having all its roots real, and as many in number as he
previously had of real and impossible together.
He had an example of a beautiful method of finding sums of
series by the continuous subtractions of geometrical progressions.
He subtracts geometrical progressions whose sums are also geo-
metrical progressions, and thus he can find the sums of the sums,
and so he obtains the sum of the series. This method is excellent for
a series whose numerators are arithmetical, and denominators geo-
metrical, such as,
1 2 3 ±
2 4 8 16
He has three series, like those of Wallis, for interpolations for the
circle. He says that there are no more by that method, I think.
Also he can very often write down the quadratures of irra-
tionals, as also their tangents, without eliminating irrationals, or
fractions, etc.
8 XIV.
November, 1676.
Calculus Tangentium differentialis.
[Differential calculus of tangents.]
dx=l, dx- = 2x, dx3 = 3;r2, etc.
U l';; Jl 2 ,1 _ 3
a — -- 9 , a o = -- n , « •» — o , etc.
x xl x x x6 x1
1
d V*= /- , etc.
\x
From these the following general rules may be derived for the
differences and sums of the simple powers:
_ s* — x'+i
dx' — e,x*-1 , and conversely I x' = -- .
J e+l
~T 2
Hence, d-^-dx'"1 will be — 2*~3 or — —.,
x3 Xs
and d Jx or dx* will be— \x~'A or —4/1.
\*
Let y = X*, then dy = 2* dx or ^ = 2 x .
dx
THE MANUSCRIPTS OF LEIBNIZ. 42!
This reasoning is general, and it does not depend on what the pro-
gression for the .r's may be.0" By the same method, the general
rule is established as:
d*
- = ^ x , am
dx
Suppose that we have any equation whatever, say,
ay2 + byx + cs2 + f2x + g-y + h3 = 0,
and suppose that we write y + dy for y, and x + dx for x, we have,
by omitting those things which should be omitted, another equation
ayz + byx + ex2 + f2x + g2y + h3 = 0 ~*
a2dyy + by dx + 2cxdx + f2d.v + g-dy
bxdy
> = 0
ady2 -f bdxdy + cdx2 = 0
This is the origin of the rule published by Sluse. It can be extended
indefinitely: Let there be any number of letters, and any formula
composed from them ; for example, let there be the formula made
up of three letters,
ay2 bx2 cs2 fyx gyx hxs ly mx nz. p = 0.
From this we get another equation
ay2 bx2 cz~ fyx simi- ly mx simi- p
2adyy 2bdxx 2cdzz fydx larly Idy mdx larly
fxdy
a dy2 bdx2 cdz2 fdxdy
It is plain from this that by the same method tangent planes
02 AT LAST ! The recognition of the fact that neither dx nor dy need
necessarily be constant, and the use of another letter to stand for the function
that is being differentiated, mark the beginning, the true beginning, of Leib-
niz's development of differentiation. Later in this manuscript we find him
using the third great idea, probably suggested by the second of those given
above, namely, the idea of substitution, by means of which he finally attains
to the differentiation of a quotient, and a root of a function.
It is very suggestive that this remarkable advance occurs after his second
visit to London, while he is staying in Holland. Did some one tell then of
the work of Newton, or of Barrow's method (which is geometrically an exact
equivalent of substitution), pointing out those things of which he had not
perceived the drift, or is it the result of his intercourse with Hudde? For
the date is that of his stay at The Hague. (For the answer to this query see
an article to follow, entitled "Leibniz in London." — ED.)
03 This is Barrow all over; even to the words omissis omittendis instead
of Barrow's -rcjcdis rcjicietidis. Lect. X, Ex. 1 on the differential triangle at
the end of the lecture.
422 THE MONIST.
to surfaces may be obtained, and in every case that it does not
matter whether or no the letters x, y, z have any known relation,
for this can be substituted afterward.
Further, the same method will serve admirably, even though
compound fractions or irrationals enter into the calculation, nor is
there any need that other equations of a higher degree should be
obtained for the purpose of getting rid of them ; for their differences
are far better found separately and then substituted ; hence the
ordinary method of tangents will not only proceed when the ordi-
nates are parallel, but it can also be applied to tangents and any-
thing else, aye, even to those things that are related to them, such
as proportions of ordinates to curves, or where the angle of the
ordinates changes according to some determined law. It will be
worth while especially to apply the method to irrationals and com-
pound fractions.64
d fa + t>z + cz* . Let a + bs + cz* = x ;
then dv/x = — ;r~7~ > and ~r = b + 2cz •
2\/x dz
therefore d V a + bz+ c£ = —
Taking any equation between two letters x and y for a curve,
and determining the equation of the tangent, either of the two let-
ters x or y can be eliminated, so that all that remains is the other
together with dx and dy ; and this will be worth while doing in all
cases to facilitate the calculation.
If three letters are given, say x, y and z, and the value of dz
is expressed in terms of x or y (or even of both), an equation for
the tangents will at length be obtained, in which again there will
be left only one or other of the letters x or y together with the
two,sd.r and dy ; sometimes z itself cannot be eliminated. Also
this can be deduced in all cases of an assumed value of dz, and in
the same way more additional letters can be taken. Thus, bringing
together every general calculus into one, we obtain the most general
of them all. Besides, the assumption of a large number of letters
may be employed to solve problems on the inverse method of tan-
gents, with the assistance of quadratures.
64 Here we have the idea of substitutions, which made the Leibnizian
calculus so superior to anything that had gone before. Note that he still has
the erroneous sign that he obtained for the differentiation of \ I x at the be-
ginning of this manuscript. Also that the ds is wrongly placed in the denom-
inator of the result.
THE MANUSCRIPTS OF LEIBNIZ. 423
Thus, if the following problem is set for solution : It is given
that the sum of the straight lines CB, BP or
we have
dx + dy=xdx
_ -v"'
J ~~" fy
Thus we have the curve in which the sum of CB + BP (multi-
plied by a constant r) is equal to the rectangle AB.BC.
There are two marginal notes by Leibniz that must be referred to, in this
manuscript. The first reads :
It is especially to be observed about my calculus of differences that, if
b. ydx -f- xdy -\- etc. = 0
then byx -\- { etc. = 0, and so on for the rest. It is to be seen what is to be
done about the A3. For the purpose of making these calculations better, the
equation ay2 -\- byx + ex2 -f- etc- can be changed into something else by means
of another relation of the curve, and if it turns out all right it may be compared
to another calculation of the differences, since it comes to the thing as by the
first. The two points to be noticed are that Leibniz now for the first time rec-
ognizes the need of considering the arbitrary constant of integration, though
he hardly grasps how it arises, and that even now he cannot refrain from
harking back to his obsession of the obtaining of several equations for com-
parison. This note is not made any the easier to understand by its being
starred by Gerhardt for reference to the differentiation of x-, whereas it ob-
viously (when you come later to the passage) refers to the differentiation of
the equation of the second degree.
The second note refers to the substitution of x + d x for x and y -\- dy for
y, and reads :
Either dx or dy can be expressed arbitrarily, a new equation being ob-
tained ; and either dx or dy being taken away, x, or y, say, can be otherwise
expressed in terms of the quantities. It is not true, I think, that this is so, for
then a catalogue of all curves capable of quadrature would result, by sup-
posing one or other of them to be constant.
The point to be noticed in this rather ambiguous statement is that Leibniz
is still thinking of his catalogue, and is not himself convinced of the com-
pleteness of his method for all purposes.
§XV.
There is an interval of nearly seven months between
the date of the manuscript last considered and the one that
now follows. This interval has been full of work; for we
now find a clear exposition of the rules for the differentia-
424 THE MONIST.
tion of a sum, difference, product, quotient, etc., though
these are without proof, or indication of the manner in
which they have been obtained. There is also no rule
given for a logarithm, an exponential, or a trigonometrical
ratio. Leibniz may have known them, but even then it
would not be surprising to find them left out; for Leibniz's
great idea was the use of his method to facilitate calcula-
tion. We must conclude therefore that these rules are a
development of the method of substitution outlined in the
preceding manuscript.
This essay has several peculiar characteristics of its
own, which distinguish it from those that have gone before.
It is written throughout in French; it is to some extent
historical and critical, having the appearance of being
prepared for publication, or possibly as a letter; this is
corroborated by the fact that there is an original draft and
a more fully detailed revision. Could it be that this is the
original of Leibniz's communication of this method to New-
ton and others? If so, Leibniz is very careful not to give
much away. The figures are strongly reminiscent of Bar-
row, but the context does not deal with subtangents, which
are such a feature in all Barrow's work.
The start from the work of Sluse is peculiar ; it seems
to suggest that Leibniz is pointing out that his method is
a fuller development of that of the former. Leibniz has
already hazarded two different guesses at the origin of
the rules given by Sluse; the second, namely, by substitu-
tion of x -f- dx for x, etc., being the more probable. Is
Leibniz trying to draw a red herring across the trail, the
real trail that leads to Barrow's a and e?
1 1 July 1677.
Methode generate pour mener les touchantes des Lignes Courbes
sans calcnl, et sans reduction des quantites irrationelles et
rompues.
Til i: MANUSCRIPTS OF LEIBNIZ.
425
[General method for drawing tangents to curves without cal-
culation,and without reducing irrational or fractional quan-
tities.]
Slusius has published his method of finding tangents to curves
without calculation, in which the equation is purged of irrational
or fractional quantities.
For example, a curve DC being given, in which the equation
expresses the relation between BC and AS, which we will call y,
and AB or SC, which we will call x; let this be
a + bx + cy + dxy + ex- + fy- + gx-y + hxy2 + kx3 + ly* + etc. = d
One has only to write
0 -i- t>£ + c v + dxv + 2?*£ + 2fyv + gx'*y + hy1^ f 3&*2£ + 3fy''2t>
dy£ 2sx)'£ 2hxyv
(65;
nxy
ry
that is to say, if the equation is changed to a proportion,
$ _ c + dx + 2fy -H KX* + 2hxy + 3()'2 + 2ntxly + etc.
v b + dy + 2fx + 2gxy + hy* + 3^'2 + etc.
f TR
and, supposing that - expresses the ratio -
v BC =x
or
CS=j>
SV '
then TB or SV can be obtained, if BC and SC are supposed to
be given. When the given magnitudes, b, c, d, e, etc., with their
proper signs, make the value of £/v a negative magnitude, the tan-
gent will not be CT which goes toward A, the start of the abscissa
AB, but C(T) which goes away from it. That is all that has been
05 This line represents the "etc." of the original equation, and is set down
for the purpose of getting the derived terms ; the complete derived equation
therefore consists of the two lines above and the two below. Note the omis-
sion of the negative sign, when changing from the equation to the proportion.
426 THE MONIST.
published up to the present time, easy to understand by any one that
is versed in these matters. But when there are irrational or frac-
tional magnitudes, which contain either x or y or both, this method
cannot be used, except after a reduction of the given equation to
another that is freed from these magnitudes. But at times this
increases to a terrible degree the calculation and obliges us to rise
to very high dimensions, and leads us to equations for which the
process of depression is often very difficult. I have no doubt that
the gentlemen06 I have just named know the remedy that it is neces-
sary to apply, but as it is not as yet in common use, and is I believe
known to but a few, also because it gives the finishing touch to the
problem that Descartes said was the most difficult to solve of all geo-
metrical problems, because of its general utility, I have thought it
a good thing to publish it.
Suppose we have any formula or magnitude or equation such
as was given above,
a <r b.v + cy + d.vy + ex* + fy2 + etc. ;
for brevity let us call it o>; that which arises from it when it is
treated in the manner given above, namely,
b£ + cv + d.vv + dy£ + etc. ;
will be called </».>; and in the same way, if the formula is A or /*,
then the result above will be d\ or dp, and similarly for everything
else. Now let the formula or equation or magnitude w be equal to
A//*, then I say that dw will be equal to M ~ . This will be
sufficient to deal with fractions.
dw
z
Again, let w be equal to j/ <*> , then d<» — z.\-l/a> ; and this
V
will be sufficient for the proper treatment of irrationals.
Algorithm of the new analysis for maxima and minima, and
for tangents.
Let AB = .r, and BC = y, and let TVC be the tangent to the
curve AC ; then the ratio TB or SC=* will be called — .
EC = y SV ay
86 Leibniz, at the beginning, first wrote, "Hudde, Sluse, and others" ; but
later he struck out all but Sluse. (Gerhardt.)
THE MANUSCRIPTS OF LEIBNIZ.
427
Let there be two or more other curves, AF, AH, and suppose
u
that BF = z; and BH = w, and that the straight line FL is the tangent
T F* itr
to the curve AF, and MH to the curve AH ; also - - , and
FB dv
— ; then I say that d\, or dwv, will be equal to vdw + wdv \
BH aw
and if v~w = x, and y = vw = x2, then by substituting x for v and
for w, we shall have dvw = 2xdx.
(This will also hold good if the angle ABC is either acute or
obtuse ; also if it is infinitely obtuse, that is to say, if TAG is a
straight line.)
[Of this rough draft there is the following revision, and this
obviously comes within the same period. (Gerhardt.)]
Fermat was the first to find a method which could be made
general for finding the straight lines that touch analytical curves.
Descartes accomplished it in another way, but the calculation that
he prescribes is a little prolix. Hudde has found a remarkable
abridgment by multiplying the terms of the progression by those
of the arithmetical progression. He has only published it for equa-
tions in one unknown ; although he has obtained it for those in two
unknowns. Then the thanks of the public are due to Sluse; and
after that, several have thought that this method was completely
worked out. But all these methods that have been published sup-
pose that the equation has been reduced and cleared of fractions
and irrationals ; I mean of those in which the variables occur. I
however have found means of obviating these useless reductions,
which make the calculation increase to a terrible degree, and oblige
us to rise to very high dimensions, in which case we have to look
428
THE MONIST.
for a corresponding depression with much trouble ; instead of all
this, everything is accomplished at the first attack.
This method has more advantage over all the others that have
been published, than that of Sluse has over the rest, because it is
one thing to give a simple abridgment of the calculation, and quite
another thing to get rid of reductions and depressions. With respect
to the publication of it, on account of the great extension of the
matter which Descartes himself has stated to be the most useful
part of Geometry, and of which he has expressed the hope that there
is more to follow — in order to explain myself shortly and clearly,
I must introduce some fresh characters, and give to them a neiv
Algorithm, that is to say, altogether special rules, for their addition,
subtraction, multiplication, division, powers, roots, and also for
equations.
Explanation of the characters.
Suppose that there are several curves, as CD, FE, HJ, con-
nected with one and the same axis AB by ordinates drawn through
one and the same point B, to wit, BC, BE, BH. The tangents CT,
FL, HM to these curves cut the axis in the points T, L, M ; the
\ \
X-J n\
point A in the axis is fixed, and the point B changes with the
ordinates. Let AB = .r, BC = y, BF = w, BH = t'; also let the ratio
of TB to BC be called that of dx to dy, and the ratio of LB to BE
that of d.\- to div, and the ratio of MB to BH that of dx to dv.
Then if, for example, y is equal to vw, we should say dvzv instead
of dy, and so on for all other cases. Let a be a constant straight
line ; then, if y is equal to a, that is, if CD is a straight line parallel
to AB, dy or da will be equal to 0, or equal to zero. If the magni-
tude dxfdiv comes out negative, then FL, instead of being drawn
THE MANUSCRIPTS OF LEIBNIZ. 429
toward A, above B, will be drawn In the contrary direction, be-
low B.
Addition and Subtraction. Let y = v±w(±)a, then dy will be
equal to dv±dw(±}Q.
Multi-plication. Let y be equal to atw, then dy or dawv or a dvw
will be equal to avdzv + aivdv.
Division. Let y be equal to JL then dy or d JL
aw aa>
1 . v w dv — v dw
or —d — will be equal to - — 5 •
a w aw
The rules for Powers and Roots are really the same thing.
Powers. If v = w*, (where r is supposed to be a certain number),
then dy will be equal to z, w=~l, div.
dw
Roots or extractions. If y = */», then dz= z *~ / .
1/tP
Equations expressed in rational integral terms.
a H- fo> + c v 4- toy + ez/2 + /y2 + gv 2y + hvy2 + kv3 + ly3
4 ntv-y2 + nv*y 4- pvy3 + qv* + ry* = 0,
supposing that a, b, c, t, e, etc. are magnitudes that are known and
determined ; then we should have
0 = bdv + cdy + tvdy 4- 2cvdv + 2fydy + gv2dy + h\2dv
tydv +2gvydy +2hvydy
4- Zly-dy + 2mv2ydy 4- mPdy + py*dv + 4qv*dv + 4r\:'dy
4- 2mvy*dv + Znv-ydv 4- 3py-rdy
This rule can be proved and continued without limit by the pre-
ceding rules ; for, if
a 4 bv 4 cy 4 tvy 4 evz 4 fyz 4- gv*y + etc. = 0,
then da 4 dbv + dcy + tdvy 4 edv2 + /rfv2 4 ^rf?'2 v + etc. will also be equal
to 0. Now da = 0, dbv = bdv, dcy - cdy, di'y = vdy + ydv\ also dv2 =
2vdv, since dv* is equal to s,vs~l,dv, that is to say (by substituting
2 for z] 2vdv\ and dv3y = v2dy 4- 2vydv, for, supposing that 7^ = ^,
then dv"y will be rfwv, and fairy =ydw+i#dy, and rfw or dv- = 2vdv,
hence in the value of dwy, substituting for w and dw the values found
•
43O THE MONIST.
for them, we shall have dv2y = v"dy + 2vydv, as obtained above.
This can go on without limit. If in the given equation a + bv+cy
+ etc. =0, the magnitude v were equal to x, that is to say if the line
JH were a straight line which when produced passed through the
point A, making an angle of 45 degrees with the axis, then the
resulting equation, transformed into a proportion, would give the
rule for the method of tangents, as published by Sluse ; and, in
consequence, this is nothing but a particular case or corollary of
the general method.
Equations complicated in any manner with fractions and irra-
tionals. These could be treated in the same way without any calcu-
lation, by supposing that the denominator of the fraction or the
magnitude of which it is necessary to take the root is equal to a
magnitude or letter, which is to be treated according to the pre-
ceding rules.07
Also, when there are magnitudes which have to be multiplied
by one another, there is no need to make this multiplication in
reality, which saves still more labor. One example will be suffi-
cient.
[No example is given, however; but the following seems to
have been added later, according to Gerhardt.j
Lastly this method holds good when the curves are not purely
analytical, and even when their nature is not expressed by such
ordinates, and in addition it gives a marvelous facility for making
geometrical constructions. The true reason for an abridgment so
admirable, and one that enables us to avoid reductions of fractions
and irrationals, is that one can always make certain, by means of
the preceding rules, that the letters dy, dv, dw, and the like, shall
not occur in the denominator of the fraction, or under the root-
sign.
§XVI.
The next manuscript appears to be a more detailed
revision of the one last considered. It bears no date; but
it is safe to say that it belongs to a considerably later period
than that of July 1677. For in this are given, by means
of the infinitely small quantities dx and dy, proofs of the
67 The complete statement of the method of substitutions.
THE MANUSCRIPTS OF LEIBNIZ. 43!
fundamental rules for the first time; the figure notation
is changed from the clumsy C, (C), ((C)) to the neat
iC, 2C, 3C; the notation for proportion is now a: b:: c: d ;
and there are several other changes that readers will notice
as they go along. The ideas of Leibniz are now approach-
ing crystallization, as is evidenced by the fact that fy dx is
clearly stated for the first time to be the sum of rectangles
made from y and dx. It is rather astonishing, however, in
this connection to find J> + y — v = $x + fy — fv,
which can have no significance according to the above
definition; and also to find the whole thing explained by
arithmetical series, in which however it is to be observed
that dx is not taken to be constant. But for this one might
almost place this later than the publication of the method
in the Acta Eruditonim in 1684; in this essay Leibniz gave
a full account of his rules without proofs, and is evidently
trying to get away from the idea of the infinitely small, an
effort which culminates in the next, and last, manuscript
of this set.
If then we guess the date to be about 1680, probably
we shall not be very far out.
A remarkable feature of this manuscript is the omission
of really necessary figures, without which the text is very
hard to follow. Of course this manuscript was written
for publication, and the suggestion may be made that the
diagrams were drawn separately, just as in books of that
time they were printed separately on folding plates; but
then, why has he given three diagrams? The only other
suggestion that can be made as far as I can see is that he
was referring to texts, in which the diagrams were already
drawn, by Gregory St. Vincent, Cavalieri, James Gregory
(one of whose theorems he quotes), Barrow ( who strangely
enough also quotes the very same theorem), Wallis, and
others. For he mentions many of these authors, but there
432 THE MONIST.
is never a word about Barrow. I consider that he was
looking up their theorems to show how much superior his
method was to any of theirs.
It is to be observed that not even in this manuscript is
there any mention of logarithms, exponentials, or trigono-
metrical ratios. We shall see later that Leibniz is reduced
to obtaining the integral of (a2 -f- xzy/2 by reference to a
figure and its quadrature; that is to say, he is apparently
unable to perform the integration analytically. It there-
fore follows that, if he got a great deal from Barrow, he
was unable to understand the Lect. XII, App. I of the
Lectiones Geometricae.
The final conclusion that I personally have come to,
after completing this examination of the manuscripts of
Leibniz, as far as they are given by Gerhardt is this:
As far as the actual invention of the calculus as he
understood the term is concerned, Leibniz received no help
from Newton or Barrow ; but for the ideas which underlay
it, he obtained from Barrow a very great deal more than he
acknowledged, and a very great deal less than he would
like to have got, or in fact would have got if only he
had been more fond of the geometry that he disliked. For,
although the Leibnizian calculus was at the time of this
essay far superior .to that of Barrow on the question of
useful application, it was far inferior in the matter of
completeness.
(No date.)
Elementa calculi novi pro differentiis et siimmis, tangentibus et
quadratures, maximis et minimi s, dimensionibus linearum,
super ficierum, solidormn, alilsque communem calculwn trans-
cendentibus.
[The elements of the new calculus for differences and sums, tan-
gents and quadratures, maxima and minima, dimensions of
lines, surfaces, and solids, and for other things that transcend
other means of calculation.]
THE MANUSCRIPTS OF LEIBNIZ.
433
Let CC be a line, of which the axis is AB, and let BC be ordi-
nates perpendicular to this axis, these being called y, and let AB
be the abscissae cut off along the axis, these being called x.
Then CD, the differences of the abscissae, will be called dx;
such are tC XD, 2C,D, 3C3D, etc. Also the straight lines jD2C,
2D3C, 3D4C, the differences of the ordinates, will be called dy.
If now these dx and dy are taken to be infinitely small, or the
two points on the curve are understood to be at a distance apart
that is less than any given length, i. e., if iD2C, 2D3C, etc. are con-
sidered as the momentaneous increments68 of the line BC, increas-
ing continuously as it descends along AB, then it is plain that the
straight line joining these two points, 2C XC say, (which is an element
of the curve or a side of the infinite-angled polygon that stands
for the curve), when produced to meet the axis in /T, will be the
tangent to the curve, and iTjB (the interval between the ordinate
and the tangent, taken along the axis) will be to the ordinate jB jC as
,CjD is to xD-jC; or, if jTjB or L.T.,B, etc. are in general called /,
then t:y : : dx : dy. Thus to find the differences of series is to find
tangents.
For example, it is required to find the tangent to the hyperbola.
a a
Here, since y= — , supposing that in the diagram, x stands for
X
AB the abscissa along an asymptote, and a for the side of the
power, or of the area of the rectangle AB.BC; then
aa
dy — — — ax.
xx
68 Leibniz has evidently seen Newton's work at the time of this composi-
tion ; also the use of the word "descends" in the next line again suggests
Barrow, while the figure is exactly like the top half of the diagram given by
Barrow for Lect. XI, 10, which is the theorem of Gregory that is quoted by
Leibniz also. For this figure, see the note to that passage.
434
THE MONIST.
as will be soon seen when we set forth the method of this calculus ;
hence dx : dy or t : y : : - xx :aa : : - x : — : : -x\y; therefore t = -y,
that is, in the hyperbola BT will be equal to AB, but on account of
the sign -x, BT must be taken not toward A but in the opposite
direction.
Moreover, differences are the opposite to sums; thus 4B4C is
the sum of all the differences such as 3D 4C, 2D 3C, etc. as far as A,
even if they are infinite in number. This fact I represent thus,
fdy = y. Also I represent the area of a figure by the sum of all
the rectangles contained by the ordinates and the differences of the
abscissae, i. e., by the sum tB jD + 2B 2D + 3B 3D + etc. For the nar-
row triangles jC ^ 2C, 2C 2D 3C, etc., since they are infinitely small
compared with the said rectangles, may be omitted without risk ;
and thus I represent in my calculus the area of the figure by fy dx,
or the sum of the rectangles contained by each y and the dx that
corresponds to it ; here, if the dx's are taken equal to one another,
the method of Cavalieri is obtained.
But we, now mounting to greater heights, obtain the area of
a figure by finding the figure of its summatrix or quadratrix ; and
of this indeed the ordinates are to the ordinates of the given
figure in the ratio of sums to differences ; for instance, let the curve
of the figure required to be squared be EE, and let the ordinates
to it, EB, which we will call e, be proportional to the differences
of the ordinates BC, or to dy; that is let aB jE : 2B 2E : : jD 2C : 2D 3C,
and so on; or again, let AjBijBjC, 1C1D:1D2C, etc., or dx:dy
be in the ratio of a constant or never-varying straight line a to tB JL
or e; then we have
d.v :dy :: a:e, or e dx = a dy ;
•'• §e dx = fady.
But c dx is the same as e multiplied by its corresponding dx,
such as the rectangle 3B 4E, which is formed from 3B 3E and 3B 4B ;
hence, fed* is the sum of all such rectangles, 3B 4E + 2B !E + 3B 2E
+ etc., and this sum is the figure A 4B 4EA, if it is supposed that the
THE MANUSCRIPTS OF LEIBNIZ.
435
d.r's, or the intervals between the ordinates e, or BC, are infinitely
small. Again, ady is the rectangle contained by a and dy, such as
is contained by 3D4C and the constant length a, and the sum of
^x
"v f
<t^
.1
\ .a
}\
"\ B
1 \£
•c,\*
\,c
these rectangles, namely fady, or 3D4C.a + 2D3C.a + 1D2C.
is the same as gD^ + aDgC + ^oC + etc. into a, that is, the same
as 4B4C.a; therefore we have fady = afdy = ay. Therefore §edx
= ay, that is, the area A4B4EA will be equal to the rectangle con-
tained by 4B 4C and the constant line a, and generally ABEA is
equal to the rectangle contained by BC and a.09
Thus, for quadratures it is only necessary, being given the line
EE, to find the summatrix line CC, and this indeed can always be
found by calculus, whether such a line is treated in ordinary geom-
etry or whether it is transcendent and cannot be expressed by alge-
braical calculation ; of this matter in another place.
Now the triangle for the line I call the characteristic of the
line, because by its most powerful aid there can be found theorems
about the line which are seen to be admirable, such as its length,
the surface and solid produced by its rotation, and its center of
gravity ; for jC 2C is equal to -\/d.r.d.v+ dy.dy. From this we have
69 Leibniz does not give a diagram, but it is not difficult to construct his
figure from the enunciation that he gives for it. The whole of this paragraph
should be compared with the following extract from Barrow (Lect. XI, 19),
piece by piece.
"Again, let AMB be a curve of which the axis is AD and let BD be
perpendicular to AD; also let KZL be another line such that, when any point
M is taken in the curve AB, and through it are drawn MT a tangent to the
curve AB, and MFZ parallel to DB, cutting KZ in Z and AD in F, and R is
a line of given length, TF: FM = R : FZ. Then
the space ADLK is equal to the rectangle con-
tained by R and DB.
For, if DH = R and the rectangle BDHI
is completed, and MN is taken to be an indefi-
nitely small arc of the curve AB, and MEX,
NOS are drawn parallel to AD; then we have
NO : MO = TF : FM = R : FZ ;
NO.FZ = MO.R and FG.FZ-ES.EX.
Hence, since the sum of such rectangles as
FG.FZ differs only in the least degree from
the space ADLK, and the rectangles ES.EX form the rectangle DHIB, the
theorem is quite obvious.
T A
fi
Z.
F
L
U---- R .
\
^
N
0
0 H
S x
N.
E. x
M
V
B |
436
THE MONIST.
at once a method for finding the length of a curve by means of
XX
some quadrature ; e. g., in the case of the parabola, if y=-^~ , then we
have d\'= , and hence iC zC=-
a a
the ordinate of the hyperbola Vaa + •*'•*" is to the constant line a;
I r
that is, - I dx^aa + xx , a straight line equal to the arc of a
a J
parabola, depends on the quadrature of the hyperbola, as has already
been found by others ; and thus we can derive by the calculus all
the most beautiful results discovered by Huygens, Wallis, van
Huraet, and Neil.70
I said above that t : y :: dx\dy; hence we have t dy = y Ax, and
therefore §t dy= §ydx. This equation, enunciated geometrically,
gives an elegant theorem due to Gregory.71 namely that, if BAF is a
right angle, and AF = BG, and FG is parallel to AB and equal to
BT, that is, 1F1G = 1B1T, then ftdy, or the sum of the rectangles
contained by t (e.g., 4F4G or 4B4T) and dy (3F4F or 3D4C) is
equal to the rectangles 4F3G + 3F2G + 2F jG + etc., or the area of the
70 All the things given are to be found in Barrow, but his name is not even
mentioned.
71 This is the strangest coincidence of all ! For, Barrow also quotes this
very same theorem of Gregory, and no other theorem ; also it occurs in this
very same Lect. XI that has been referred to already ! Leibniz does not give
a diagram; nor from his enunciation could I complete the figure required, until
I had referred to the figure given by Barrow !! ! The two diagrams are given
below for comparison, Barrow's figure being the one referred to in the note
above. Query, is Leibniz's figure taken from Gregory's original, which I have
not been able to see, or is it the Leibnizian variation of Barrow's?
THE MANUSCRIPTS OF LEIBNIZ. 437
figure A 4F 4GA is equal to fy dx, that is, to the figure A 4B 4CA ;
or generally, the figure AFGA is equal to the figure ABCA.
Again, other things, which are immediately evident on inspec-
tion, from a figure, are readily deduced by the calculus ; for instance,
in the case of the trilinear figure ABCA, the figure ABCA together
with its complementary figure AFCA is equal to the rectangle
ABCF, for the calculus readily shows that fydx+fxdy = xy.
If it is required to find the volume of the solid formed by
rotation round an axis, it is only necessary to find Cy- dx ; for the
solid formed by a rotation round the base, $x-dy ; for the moment
about the vertex, fyxdx; and these things serve to find the center
of gravity of a figure, and also give the frusta of Gregory St.
Vincent, and all that Pascal, Wallis, De Laloubere, and others have
found out about these matters.
For, if it is required to find the centers of lines, or the surfaces
generated by their rotation, e. g., the surface generated by the rota-
tion of the line AC about AB, it is only necessary to find
J y V dx. dx + dy. dy
or the sum of every PC applied to the axis at the point B that
corresponds to it, (thus »P 2C will be applied perpendicular to the
axis AB at 2B), producing in this way a figure of which the above
represents the area. Thus the whole thing will immediately reduce
to the quadrature of some plane figure, if, instead of y and dy, their
values, obtained from the nature of the ordinates and the tangents
to the curve, are substituted. Thus, in the case of the parabola,
if y is equal to ~\/2ax, then dy= - - (as will be seen directly) ;
hence we get
aa r I r I
dxdx + — dxdx or J dx^Ayy + aa or j dx^l 2ax + aa ,
which depends on the quadrature of the parabola (for every
-\/2ax + aa or PC can be applied to a parabola, if it is supposed that
AC is the parabola, and AB its axis, provided in that case the
figure is changed and the curve turns its concavity toward the
axis) ;72 and this may be obtained by ordinary geometry, and there-
72 The Latin here is rather ambiguous ; query, a misprint. But I think I
have correctly rendered the argument. It is to be noted that the parabola
was at this period always thought of in the form we should now denote by
the equation y = xz, and the figure referred to by Leibniz is that which Wallis
calls the complement of the semiparabola.
438 THE MONIST.
fore also a circle will be found equal to the surface of the parabolic
conoid ; but this is not the place to deduce it at full length.
Now these, which may seem to be great matters, are only the
very simplest results to be obtained by this calculus ; for many
much more important consequences follow from it, nor does there
occur any simple problem in geometry, either pure or applied to
mechanics, that can altogether evade its power. Now we will ex-
pound the elements of the calculus itself.
The fundamental principle of the calculus.
Differences and sums are the inverses of one another, that is
to say, the sum of the differences of a series is a term of the series,
and the difference of the sums of a series is a term of the series ;
and I enunciate the former thus, §dx-x, and the latter thus,
d^x-x.
Thus, let the differences of a series, the series itself, and the
sums of the series, be, let us say,
Diffs. 1 2 3 4 5 dx
Series 0 1 3 6 10 15 x
Sums 0 1 4 10 20 25 . . $x
Then the terms of the series are the sums of the differences, or
x=(dx; thus, 3=1 + 2, 6=1+2 + 3, etc.; on the other hand, the
differences of the sums of the series are terms of the series, or
d^x-x\ thus, 3 is the difference between 1 and 4, 6 between
4 and 10.
Also da = Q, if it is given that a is a constant quantity, since
a-a = 0.
Addition and Subtraction.
The difference or sum of a series, of which the general term
is made up of the general terms of other series by addition or sub-
traction, is made up in exactly the same manner from the differ-
ences or sums of these series ; or
x + y - v = §dx + dy- dv, §x + y-v = §x + fy- fv.
This is evident at sight, if you take any three series, set out their
sums and their differences, and take them together correspondingly
as above.
THE MANUSCRIPTS OF LEIBNIZ. 439
Simple Multiplication.
Here dxy = xdx + ydy, or xy=fxdx+fydy.
This is what we said above about figures taken together with their
complements being equal to the circumscribed rectangle. It is
demonstrated by the calculus as follows:
dxy is the same thing as the difference between two successive
xy's; let one of these be .vy, and the other x + dx into.y + dy; then
we have
dxy = x + dx . y + dy- xy = xdy + y dx + dx dy ;
the omission of the quantity dx dy, which is infinitely small in com-
parison with the rest, for it is supposed that dx and dy are infinitely
small (because the lines are understood to be continuously increas-
ing or decreasing by very small increments throughout the series
of terms), will leave xdy + ydx; the signs vary according as y and x
increase together, or one increases as the other decreases ; this
point must be noted.
Simple Division.
y x dy— y dx
Here we have d - — - .
x xx
.y y + dy y x dy —y dx , . , , , . ,
For, d - = - . _ i = _ _ which becomes (if we
x x+dx x xx + x dx
write xx for xx + xdx, since xdx can be omitted as being infinitely
small in comparison with xx} equal to — - — - — — ; also, if y = aa.
xx
then dy = 0, and the result becomes , which is the value we
xx
used a little while before in the case of the tangent to the hyper-
bola.
From this any one can deduce by the calculus the rules for
Compound Multiplication and Division; thus,
dxvy = xy dv + xv dy + yv dx,
, y _ xv dy —yv dz — yz dv
a— — ;
vz vv.zz
as can be proved from what has gone before ; for we have
dv = x dy—y dx .
X XX
hence, putting zv for x, and sdv + vdz for dx or dzv in the above,
we obtain what was stated.
44° THE MONIST.
Powers follow: dxz = 2xdx, dx* = Zx*dx, and so on. For, putting
y = x, and v=x, we can write dxz for dxy, and this is (from above)
equal to xdy + ydx, or (if x = y, and consequently dx = dy) equal
to 2xdx. Similarly, for dxs we write dxyv, that is (from above)
xydv + xvdy + yvdx, or (putting x for y and v and d.r for cfy and
dv) equal to 3x*dx. Q. E. D. By the same method, in general,
dx' = e.x'— dx, as can easily be proved from what has been said.
.1 h dx
Hence also, a -— * = A+T •
For, if — = jv', then e= — h, and x'~l— ->— , , as is well known to
any one who understands the nature of the exponents in a geo-
metrical progression. The same thing will do for fractions. The
procedure is the same for irrationals or Roots. dt\/xh — dx''r,
(where by h:r I mean h/r, or h divided by r), or dxe (taking e
equal to h/r), or e.x~ dx, by what has been said above, or (by
substituting once more h : r for e, and h-r:r for e - \ ) — . x*-rr .dx;
and thus finally we get the value of d>\/xH.
Moreover, conversely, we have
f
r
f.t*Wxm~, f-e<tx=—=-l , f<i
J e + V J xe e-1.*-1 J
«jr*i±Z.
These are the elementary principles of the differential and
summatory calculus, by means of which highly complicated formu-
las can be dealt with, not only for a fraction or an irrational quan-
tity, or anything else ; but also an indefinite quantity, such as x or y,
or any other thing expressing generally the terms of any series,
may enter into it.
§ XVII.
The next manuscript bears no date; but this can be
easily assigned to a certain extent, from internal evidence.
It is for one thing later than the publication in the A eta
Eruditorum of Leibniz's first communication to the world
of his calculus in 1684. The manuscript is an answer, or
rather the first rough draft probably of such an answer,
to the animadversions of Bernhard Nieuwentijt against
the idea of the infinitesimal calculus. The latter stated
that (i) Leihniz could explain no more than Barrow or
THE MANUSCRIPTS OF LEIBNIZ. 44!
Newton how the infinitely small differences differed from
absolute zero; (ii) it was not clear how the differentials
of higher order were obtained frorh those of the first
order; (iii) the differential method cannot be applied to
exponential functions. Leibniz answers the first point skil-
fully, fails over the second through erroneous work, which
I think he afterward perceived; for he has a note that the
whole thing is to be carefully revised before publication.
It almost seems that he was not quite confident in his own
powers of completely answering these objections, for he
also notes that the rudeness of language in which the
answer is commenced must be mollified.
On the third point he is silent; in the later written
Historia, we have seen he is able to get, not over, but round
the difficulty of the exponential function; but the silence
here would seem to say that Leibniz could not manage ex-
ponentials as yet.
The success of the answer to the first point is due to
the underlying principle that the ratio dy : dx ultimately
becomes a rate; when this idea is muddled by an admixture
of the infinitesimal idea in the last paragraph the result
is almost disastrous. Leibniz, however, looked on his cal-
culus as a tried tool more than anything else.
When my infinitesimal calculus, which includes the calculus of
differences and sums, had appeared and spread, certain over-precise
veterans began to make trouble ; just as once long ago the Sceptics
opposed the Dogmatics, as is seen from the work of Empicurus
against the mathematicians (i. e., the dogmatics), and such as
Francisco Sanchez, the author of the book Quod nihil scitur, brought
against Clavius ; and his opponents to Cavalieri, and Thomas Hobbes
to all geometers, and just lately such objections as are made against
the quadrature of the parabola by Archimedes by that renowned
man, Dethlevus Cluver. When then our method of infinitesimals,
which had become known by the name of the calculus of differences,
began to be spread abroad by several examples of its use, both of
my own and also of the famous brothers Bernoulli, and more espe-
442 THE MONIST.
cially by the elegant writings of that illustrious Frenchman, the
Marquis d'Hopital, just lately a certain erudite mathematician,
writing under an assumed name in the scientific Journal de Trevoux,
appeared to find fault with this method. But to mention one of
them by name, even before this there arose against me in Holland
Bernard Nieuwentiit, one indeed really well equipped both in
learning and ability, but one who wished rather to become known
by revising our methods to some extent than by advancing them.
Since I introduced not only the first differences, but also the second,
third and other higher differences, inassignable or incomparable
with these first differences, he wished to appear satisfied with
the first only; not considering that the same difficulties existed
in the first as in the others that followed, nor that wherever they
might be overcome in the first, they also ceased to appear in the
rest. Not to mention how a very learned young man, Hermann
of Basel, showed that the second and higher differences were
avoided by the former in name only, and not in reality ; moreover,
in demonstrating theorems by the legitimate use of the first differ-
ences, by adhering to which he might have accomplished some
useful work on his own account, he fails to do so, being driven to
fall back on assumptions that are admitted by no one ; such as
that something different is obtained by multiplying 2 by m and by
multiplying m by 2 ; that the latter was impossible in any case in
which the former was possible; also that the square or cube of a
quantity is not a quantity or Zero.
In it, however, there is something that is worthy of all praise,
in that he desires that the differential calculus should be strength-
ened with demonstrations, so that it may satisfy the rigorists ; and
this work he would have procured from me already, and more
willingly, if, from the fault-finding everywhere interspersed, the
wish had not appeared foreign to the manner of those who desire
the truth rather than fame and a name.
It has been proposed to me several times to confirm the essen-
tials of our calculus by demonstrations, and here I have indicated
below its fundamental principles, with the intent that any one who
has the leisure may complete the work. Yet I have not seen up
to the present any one who would do it. For what the learned
Hermann has begun in his writings, published in my defence against
Nieuwentiit, is not yet complete.
For I have, beside the mathematical infinitesimal calculus, a
method also for use in Physics, of which an example was given in
THE MANUSCRIPTS OF LEIBNIZ.
443
the Nouvelles de la Republique des Lettres; and both of these I
include under the Law of Continuity ; and adhering to this, I have
shown that the rules of the renowned philosophers Descartes and
Malebranche were sufficient in themselves to attack all problems
on Motion.
I take for granted the following postulate:
In any supposed transition, ending in any terminus, it is per-
missible to institute a general reasoning, in which the final terminus
may also be included.
For example, if A and B are any two quantities, of which the
former is the greater and the latter is the less, and while B remains
the same, it is supposed that A is continually diminished, until A
becomes equal to B ; then it will be permissible to include under a
general reasoning the prior cases in which A was greater than B,
and also the ultimate case in which the difference vanishes and A
is equal to B. Similarly, if two bodies are in motion at the same
time, and it is assumed that while the motion of B remains the
same, the velocity of A is continually diminished until it vanishes
altogether, or the speed of A becomes zero ; it will be permissible
to include this case with the case of the motion of B under one
general reasoning. We do the same thing in geometry, when two
C
i
(O
straight lines are taken, produced in any manner, one VA being
given in position or remaining in the same site, the other BP passing
through a given point P, and varying in position while the point P
remains fixed ; at first indeed converging toward the line VA and
meeting it in the point C; then, as the angle of inclination VGA
is continually diminished, meeting VA in some more remote point
(C), until at length from BP, through the position (B)P, it comes
444 THE MONIST.
to fiP, in which the straight line no longer converges toward VA,
but is parallel to it, and C is an impossible or imaginary point.
With this supposition it is permissible to include under some one
general reasoning not only all the intermediate cases such as (B)P
but also the ultimate case (3P.
Hence also it comes to pass that we include as one case ellipses
and the parabola, just as if A is considered to be one focus of an
ellipse (of which V is the given vertex), and this focus remains
fixed, while the other focus is variable as we pass from ellipse to
ellipse, until at length (in the case when the line BP, by its inter-
section with the line VA, gives the variable focus) the focus C
becomes evanescent73 or impossible, in which case the ellipse passes
into a parabola. Hence it is permissible with our postulate that a
parabola should be considered with ellipses under a common rea-
soning. Just as it is common practice to make use of this method
in geometrical constructions, when they include under one general
construction many different cases, noting that in a certain case the
converging straight line passes into a parallel straight line, the
angle between it and another straight line vanishing.
Moreover, from this postulate arise certain expressions which
are generally used for the sake of convenience, but seem to con-
tain an absurdity, although it is one that causes no hindrance,
when its proper meaning is substituted. For instance, we speak of
an imaginary point of intersection as if it were a real point, in the
same manner as in algebra imaginary roots are considered as ac-
cepted numbers. Hence, preserving the analogy, we say that, when
the straight line BP ultimately becomes parallel to the straight line
VA, even then it converges toward it or makes an angle with it,
only that the angle is then infinitely small ; similarly, when a body
ultimately comes to rest, it is still said to have a velocity, but one
that is infinitely small ; and, when one straight line is equal to
another, it is said to be unequal to it, but that the difference is
infinitely small ; and that a parabola is the ultimate form of an
ellipse, in which the second focus is at an infinite distance from the
given focus nearest to the given vertex, or in which the ratio of
PA to AC, or the angle BCA, is infinitely small.
Of course it is really true that things which are absolutely
equal have a difference w^hich is absolutely nothing ; and that
straight lines which are parallel never meet, since the distance
73 The term is here used with the idea of "vanishing into the far distance."
THE MANUSCRIPTS OF LEIBNIZ. 44$
between them is everywhere the same exactly ; that a parabola is
not an ellipse at all, and so on. Yet, a state of transition may be
imagined, or one of evanescence, in which indeed there has not yet
arisen exact equality or rest or parallelism, but in which it is
passing into such a state, that the difference is less than any assign-
able quantity ; also that in this state there will still remain some
difference, some velocity, some angle, but in each case one that is
infinitely small ; and the distance of the point of intersection, or
the variable focus, from the fixed focus will be infinitely great,
and the parabola may be included under the heading of an ellipse
(and also in the some manner and by the same reasoning under the
heading of a hyperbola), seeing that those things that are found to
be true about a parabola of this kind are in no way different, for
any construction, from those which can be stated by treating the
parabola rigorously.
Truly it is very likely that Archimedes, and one who seems
so have surpassed him, Conon, found out their wonderfully elegant
theorems by the help of such ideas ; these theorems they completed
with reductio ad absurdum proofs, by which they at the same time
provided rigorous demonstrations and also concealed their methods.
Descartes very appropriately remarked in one of his writings that
Archimedes used as it were a kind of metaphysical reasoning
(Caramuel would call it metageometry), the method being scarcely
used by any of the ancients (except those who dealt with quad-
ratrices) ; in our time Cavalieri has revived the method of Archi-
medes, and afforded an opportunity for others to advance still
further. Indeed Descartes himself did so, since at one time he
imagined a circle to be a regular polygon with an infinite number
of sides, and used the same idea in treating the cycloid ; and Huy-
gens too, in his work on the pendulum, since he was accustomed
to confirm his theorems by rigorous demonstrations ; yet at other
times, in order to avoid too great prolixity, he made use of infini-
tesimals ; as also quite lately did the renowned La Hire.
For the present, whether such a state of instantaneous transi-
tion from inequality to equality, from motion to rest, from con-
vergence to parallelism, or anything of the sort, can be sustained
in a rigorous or metaphysical sense, or whether infinite extensions
successively greater and greater, or infinitely small ones successively
less and less, are legitimate considerations, is a matter that I own
to be possibly open to question ; but for him who would discuss
these matters, it is not necessary to fall back upon metaphysical
446 THE MONIST.
controversies, such as the composition of the continuum, or to
make geometrical matters depend thereon. Of course, there is no
doubt that a line may be considered to be unlimited in any manner,
and that, if it is unlimited on one side only, there can be added
to it something that is limited on both sides. But whether a straight
line of this kind is to be considered as one whole that can be re-
ferred to computation, or whether it can be allocated among quan-
tities which may be used in reckoning, is quite another question
that need not be discussed at this point.
It will be sufficient if, when we speak of infinitely great (or
more strictly unlimited), or of infinitely small quantities (i. e., the
very least of those within our knowledge), it is understood that
we mean quantities that are indefinitely great or indefinitely small,
i. e., as great as you please, or as small as you please, so that the
error that any one may assign may be less than a certain assigned
quantity. Also, since in general it will appear that, when any small
error is assigned, it can be shown that it should be less, it follows
that the error is absolutely nothing; an almost exactly similar kind
of argument is used in different places by Euclid, Theodosius and
others ; and this seemed to them to be a wonderful thing, although
it could not be denied that it was perfectly true that, from the
very thing that was assumed as an error, it could be inferred that
the error was non-existent. Thus, by infinitely great and infinitely
small, we understand something indefinitely great, or something
indefinitely small, so that each conducts itself as a sort of class,
and not merely as the last thing of a class. If any one wishes to
understand these as the ultimate things, or as truly infinite, it can
be done, and that too without falling back upon a controversy about
the reality of extensions, or of infinite continuums in general, or
of the infinitely small, ay, even though he think that such things
are utterly impossible; it will be sufficient simply to make use of
them as a tool that has advantages for the purpose of the calcula-
tion, just as the algebraists retain imaginary roots with great profit.
For they contain a handy means of reckoning, as can manifestly be
verified in every case in a rigorous manner by the method already
stated.
But it seems right to show this a little more clearly, in order
that it may be confirmed that the algorithm, as it is called, of our
differential calculus, set forth by me in the year 1684, is quite
reasonable. First of all, the sense in which the phrase "dy is the
THE MANUSCRIPTS OF LEIBNIZ. 447
element of r," is to be taken will best be understood by considering
a line AY referred to a straight line AX as axis.
Let the curve AY be a parabola, and let the tangent at the
vertex A be taken as the axis. If AX is called x, and AY, y, and
the latus-rectum is a, the equation to the parabola will be xx = ay,
and this holds good at every point. Now, let A J£ = x, and 1XlY = y
and from the point jY let fall a perpendicular XYD to some greater
ordinate 2X 2Y that follows, and let jX 2X, the difference between
A XX and A 2X, be called dx ; and similarly, let D 2Y, the difference
between tX XY and 2X ,Y, be called dy.
Then, since y = xx : a, by the same law, we have
y + dy = xx + 2x dx + dx dx, : a ;
and taking away the y from the one side and the xx:a from the
other, we have left
dy : dx = 2x + dx :a ;
and this is a general rule, expressing the ratio of the difference of
the ordinates to the difference of the abscissae, or, if the chord tY 2Y
is produced until it meets the axis in T, then the ratio of the ordinate
XX jY to T tX, the part of the axis intercepted between the point
of intersection and the ordinate, will be as 2x + dx to a. Now,
since by our postulate it is permissible to include under the one
general reasoning the case also in which the ordinate 2X oY is moved
up nearer and nearer to the fixed ordinate iX iY until it ultimately
coincides with it, it is evident that in this case rf.r becomes equal to
zero and should be neglected, and thus it is clear that, since in this
case T tY is the tangent, ,X ^Y is to T XX as 2x is to a.
Hence, it may be seen that there is no need in the whole of our
differential calculus to say that those things are equal which have
a difference that is infinitely small, but that those things can be
taken as equal that have not any difference at all. provided that
the calculation is supposed to be general, including both the cases
in which there is a difference and in which the difference is zero ;
448
THE MONIST.
and provided that the difference is not assumed to be zero until the
calculation is purged as far as is possible by legitimate omissions,
and reduced to ratios of non-evanescent quantities, and we finally
come to the point where we apply our result to the ultimate case.
Similarly, if x'A = aay, then we have
x3 -f 3xx dx + 3x dx Ax + dx d.v dx - aay + aa dy,
or cancelling from each side,
3xx dx + 3x dx dx -i dx dx dx - aa dy,
or 3xx + 2>x dx + dx dx, : aa - dy : dx = XX tY : T tX ;
hence, when the difference vanishes, we have
But if it is desired to retain dy and dx in the calculation, so that
they may represent non-evanescent quantities even in the ultimate
case, let any assignable straight line be taken as (dx), and let the
straight line which bears to (dx) the ratio of y or aX tY to aXT be
called (dy) ; in this way dy and dx will always be assignables
bearing to one another the ratio of D 2Y to D 1Y, which latter vanish
in the ultimate case.
[Leibniz here gives a correction for a passage in the Ada
Eruditorum, which is unintelligible without the context.]
On these suppositions, all the rules of our algorithm, as set
out in the A eta Eruditorum for October 1684, can be proved without
much trouble.
Let the curves YY, VV, ZZ be referred to the same axis AXX ;
and to the abscissae A tX (=*) and A 2X (=x + dx) let there cor-
respond the ordinates 1X1'Y(=y) and 2X 2Y (=y + dy), and also
the ordinates 1XlV(=v) and 2X2V (=v + dv), and the ordinates
THE MANUSCRIPTS OF LEIBNIZ. 449
lX1Z(=rr) and2X8Z(=s + <k). Let the chords /Y-jY, 1V2V, ^Z,
when produced meet the axis AXX in T, U, W. Take any straight
line you will as (d)x, and, while the point 4X remains fixed and
the point 2X approaches jX in any manner, let this remain constant,
and let (d}y be another line which bears to (d}x the ratio of 3; to
jXT, or of dy to dx; and similarly, let (d)v be to (d)x as v to jXU
or dv to rf.r; also let (d}z be to (d}x as r to tXW or dz to rfjr;
then (d)x, (d}y, (d}z, (d)w will always be ordinary or assignable
straight lines.
Nor for Addition and Subtraction we have the following:
If y-s = v, then (d)y- (d)s = (d)v.
This I prove thus: y + dy-s-ds=v + dv, (if we suppose that as v
increases, 2 and v also increase ; otherwise for decreasing quantities,
for 2 say, -ds should be taken instead of ds, as I mentioned once
before) ; hence, rejecting the equals, namely y-s from one side,
and v from the other, we have dy-dz-dv, and therefore also
dy - dz : dx = dv : dx. But dy : dx, dz : dx, dv : dx are respectively
equal to (d)y:(d).v, (d}z:(d}x, and (d}v:(d)x. Similarly, (d~)s
:(d)y and (d}v: (d)y are respectively equal to dz:dy and dv.dy.
Hence, (d)y-(d)s, :(d)x = (d)v :(d}x\ and thus (d}y-(d)z is
equal to (d}v, which was to be proved ; or we may write the result
as (d)v :(d)y=l- (d}z :(d}y.
This rule for addition and subtraction also comes out by the
use of our postulate of a common calculation, when tX coincides
with ,X, and ,YT, jYU, ,YW are the tangents to the curves YY,
VV, ZZ. Moreover, although we may be content with the assign-
able quantities (eOy, (d)v, (d}z, (d}x, etc., since in this way we
may perceive the whole fruit of our calculus, namely a construction
by means of assignable quantities, yet it is plain from what I have
said that, at least in our minds, the unassignables dx and dy may be
substituted for them by a method of supposition even in the case
when they are evanescent : for the ratio dy : dx can always be
reduced to the ratio (d)v '-(d)x, a ratio between quantities that
are assignable or undoubtedly real. Thus we have in the case of
tangents dv : dy = 1 -dz : dx, or dv=dy- dz.
Multiplication. Let ay = xv, then a(d)y = x(d)v + v (d)x.
Proof. ay + ady = x + dx, v + dv=xv + xdv + vdx
and, rejecting the equals ay and xy from the two sides,
450 THE MONIST.
a dy - xdv + v dx + dx dv,
or
a dy x dv
— £ = _— + v + dv ;
dx dx
and transferring the matter, as we may, to straight lines that never
become evanescent, we have
a(d)y , x(d}y
~TT\ r '/j\ + v + dv;
(d)x (d)x
so that, since it alone can become evanescent, dv is superfluous,
and in the case of the vanishing differences, as in that case dv = Q,
we have
a(d}y = x(d}v + v(d}x, as was stated,
or (d}y : (d}x = x + v, :a.
Also, since (d)y \(d}x always -dy.dx, it will be allowable to sup-
pose this is true in the case when dy, dx become evanescent, and to
say that dy :dx = x + v:a, or a dy = x dv + v dx.
Division. Let s: a -v.x, then (d}z: a-v(d}x-x(d}y, :xx.
Proof z f dz: a-v + dv, : ,x + dx;
or clearing of fractions, xz f xds + sdx + dsdx -av + adv ; taking away
the equals xs and av from the two sides, and dividing what is left
by dx, we have
adv- x dS) :dx = z + ds,
or a(d}v-x(d)z, :d.v = s + ds;
and thus, only ds, which can become evanescent, is superfluous.
Also, in the case of vanishing differences, when XX coincides with
2X, since in that case ds = Q, we have
a (d)v - x(d)z, : (d)x = s - av : x ;
whence, (as was stated) (d}s = ax(d}v-av(d}x,\ xx,
or (d)z: (d}x= (a:.r) (d)v: (d)x-av\xx.
Also, since (d}z:(d}x is always equal to dz\dx, on all other
occasions, it is allowable to suppose this to be so also when dz, dv,
dx are evanescent, and to put
dz : dx = ax dv - av dx, :xx
For Powers, let the equation be aa^x'^y" f then
(d)y *•*?•
(d)x n.yn-^'
THE MANUSCRIPTS OF LEIBNIZ. 451
and this I will prove in a manner a little more detailed than those
above, thus:
MXJL - € /_ i • € \C ~~~ A * _*» * • C\€ ~~ J .c w f — -\ r f t
a "-', -*' + ^ rf* + -r-^~ xf=?dxdx + - .. ' - x-^ dxdxdx
JL JL J i~ J i^c>
(and so on until the factor e-e or 0 is reached)
, W-.o-i . 7Z,W — 1 __2 , #,M — 1,W — 2 „_, .
= j y" + i^— dy + — j-y- y".2<tydy + - 1 2'3 -- >>V ###
(and so on until the factor n-n or 0 is reached) ;
take away from the one side a"-f xe , and from the other side vn,
these being equal to one another, and divide what is left by dx,
and lastly, instead of the ratio dy : dx, between the two quantities
that continually diminish, substitute the ratio that is equal to it,
(d)y:(d)x, a ratio between two quantities, of which one, (d}x,
always remains the same during the time that the differences are
diminishing, or while 2X is approaching the fixed point jX and
we have
€ f __ i € • € ™""~ JL * _ o . CtC ~~" ' -L > c ~~~* w . _ -i , _
r- x ' + -V^~ ^^ ^ + — — j;— rf^rf^ + etc.
\~ I jW J » «- )*J
. «.« — !,«— 2 «_ , ,
= f ^ (i)i~w~^ ?Si+ Tw~'^ W"i** * etc'
Now, since by the postulate there is included in this general rule
the case also in which the differences become equal to zero, that
is when the points 2X, 2Y coincide with the points jX, jY respec-
tively ; therefore, in that case, putting dx and dy equal to 0, we have
e , n n-i(dly
I* ~\y (d)x'
the remaining terms vanishing, or ((f)y : ('/)^ = e.x*~l : n y^L.
Moreover, as we have explained, the ratio (d)y:(d}x is the same
as the ratio of v, or the ordinate XX ,Y, to the subtangent iXT,
where it is supposed that Tj Y touches the curve in tY.
This proof holds good whether the powers are integral powers
or roots of which the exponents are fractions. Though we may
also get rid of fractional exponents by raising each side of the
equation to some power, so that e and n will then signify nothing
else but powers with rational exponents, and there will be no need
of a series proceeding to infinity. Moreover, at any rate, it will be
permissible, by means of the explanation given above, to return to
the unassignable quantities dy and dx, by making in the case of
evanescent differences, as in all other cases, the supposition that
the ratio of the evanescent quantities dy and dx is equal to the ratio
452
THE MONIST.
of (d)y and (rf).r, because this supposition can always be reduced
to an undoubtable truth.
Thus far the algorithm has been demonstrated for differences
of the first order: now I will proceed to show that the same method
will hold good for the differences of the differences. For this
purpose, take three ordinates, iXiY, 2X 2Y, 3X3Y, of which jX/Y
remains constant, but ,X 2Y and 3X 3Y continually approach jX.^Y
until finally they both coincide with it simultaneously ; which will
happen if the speed with which 3X approaches jX is to the speed
with which 2X approaches ,X is in the ratio of !X3X to tX,,X.
Also let two straight lines be assigned, (d).v always constant for
any position of 2X, and .2(d}x for any position of 3X ; also let (d)y
always be to (rf).r as D2Y is to jX 2X, or as v (i. e., jX^) is to
,XT; thus, while (d}x remains always the same, (d)y will be
altered as 2X approaches jX ; similarly, let 2(d)v be to *(d}x as
2D3Y to 2X 3X or as y+ dy (i. e., 2X 2Y) to 2X 2T ; thus while 2(</).r
remains constant, 2(rf)v will be altered as 3X approaches tX.
Also let (d)y be always taken in the varying line ,X 2Y, and
let 2X !<w be equal to (d) y, and similarly take 2(d}y in the line 3X 3Y,
and let 3X .,w be equal to 2(d}y. Thus, while 2X and 3X continually
approach to the straight line jX jY, 2X tw and 3X 2o> continually
approach it also, and finally coincide with it at the same time as
2X and 3X. Further, let the point in the ordinate jX 1Y, which jw
continually approaches and with which it at last coincides, be
marked, and let it be Q ; then jXft is the ultimate (d)y, which bears
to (rf).f the ratio of the ordinate jX jY to the subtangent jXT,
where it is supposed that T XX touches the curve in XY, because
then indeed XY and 2Y coincide. Now, since all this can be done,
THE MANUSCRIPTS OF LEIBNIZ. 453
no matter where ,Y may be taken on the curve, it is evident that
•a curve Qft will be produced in this way, which is the differentrix
of the curve YY ; just as, conversely, the curve YY is the summatrix
curve of QQ, as can be readily demonstrated.
By this method, the calculus may be demonstrated also for the
differences of the differences.
Let ,X ,Y, ,X 2Y, 3X:!Y be three ordinates, of which the values
are v, y + dy, y + dy + ddy, and let tX2X (dx) and 2X 3X (dx + ddx)
be any distances, and D.,Y(dy) and 2D3Y (dy+ddy) the differ-
ences. Now the difference between (d)y and 2(d)y, or between
,X« and ,X2n is 8,Q, and that between iXoX and 2X3X is ddx;
also let
(d)dx: (d)x = dx: .,(d).v, 74 and similarly let
(d)dv:(rf).v = an8:1X2X or ^O^XT.
Now, for the sake of example, let us take ay = xv. Then we
have ady = xdv + vdx + dxdv, as has been shown above; and simi-
larly,
ady + addy = (x + dx} (dv + ddv) + (v + dv) (dx + ddx) "
+ (dx+ddx) (dv + ddv)
= x dv + x ddv + dx dv + dx ddv + vdx + v ddx
+ dvdx + dv ddx + d.v dv + dx ddv
+ ddx dv + ddx ddv.
Taking away a dy from one side, and x dx + v dx + dx dv from the
other, there will be left in any case
ddy _ xddy v 2 dxdv 2^dv 2 dx ddx ddv
ddx a ddx a a ddx a a ddx a
In this it is evident that the ratio between ddy and ddx can be
expressed by the ratio of the straight line (d)dy to (d)x, the straight
line assumed above, which we have supposed to remain constant
as 2X and 3X approach ,X. Also, since (d)dx, (since it bears an
assignable ratio to (d)x, however nearly 2X approaches to XX, or
74 This makes (d)dx an inassignable. It may be a misprint due to a slip
of Leibniz, or of Gerhardt in transcription ; for there is no similarity between
it and the statement in the next line. I cannot however offer any feasible
suggestion for correction.
75 This is quite wrong. Leibniz has evidently substituted x -f- dx for x,
etc.; which is not legitimate unless SXSY is taken as y -f dy -f- d(y-\- dy),
and so on ; even then fresh difficulties would be introduced. As it stands, this
line should read
o dy + o ddy = x(dv -f- ddv) + v(dx -f ddx) -f (dx -f ddx) (dv + ddv).
On account of this error and that noted above, there is not much profit in
considering the remainder of this passage.
454 THE MONIST.
however much dx, the difference between the abscissae, is dimin-
ished), is not evanescent, even when, finally, d.v and ddx, dv and
ddv, are all supposed to be zero. In the .same way, the ratio of
ddv to ddx may be expressed by the ratio of an assignable straight
line (d)dv to the assumed constant (d)x; and even the ratio of
dvdx to a ddx may be so expressed; for, since dv: dx=(d}v:(d}x,
therefore dvdx:>dxdx=(d}v:(d} x. Henoe, i f a new straight
line, (dd)x, is assumed to be such that a ddx: dx dx=(dd}xs(d}x,
then the new straight line will be assignable, even though dx, ddx,
etc. become evanescent. Since therefore dvdx:dxdx=(d}v:(d)x
and dvdx:addx=(d)x:(A&)x, it follows that dv dx \ a ddx =(d)v :
(dd)x, an thus at length there is prouced an equation that is freed
as far as possible from those ratios that might become evenescent,
namely,
(d)dy _ x(<T)dy y 2 (d)y 2 dv 2dx (</) dy ddv
(d)dx ~ a (d)dx a (dd)x a a (d)dx a
Thus far all the straight lines have been considered to be assign-
able so long as jX and 2X do not coincide ; but in the case of coin-
cidence, dv and ddv are zero, and we have
(d}dy _x(d}dv v 2 (d)y 0 2 (d*)dv 0 0
(d~)dx a (d)dx a (dd)x a (d~)dx a a '
or, omitting terms equal to zero,
(d~)dy i x (d)dv v
a (d)dx a (dd~)x
Hence, if dx, ddx, dv, ddv, dy, ddy, are by a certain fiction imagined
to remain, even when they become evanescent, as if they were in-
finitely small quantities (and in this there is no danger, since the
whole matter can be always referred back to assignable quantities),
then we have in the case of coincidence of the point tX and 2X the
equation
ddv x ddy v 2 dx dy
ddx a ddx a a ddx
J. M. CHILD.
DERBY, ENGLAND.
LIBRA:
THE ETERNAL BALANCE OF GOOD AND ILL.
I.
FROM everlasting is the Universe,
And unto everlasting shall extend;
Without beginning is it; without end
Its morrows ever yesterdays rehearse;
Not first nor last but only midst it knows;
As never young, so never old it grows.
ii.
Yet is the secret of its permanence
Not rest but striving, not a dead repose,
No peace of mutually slaughtered foes,
Nor truce of wearied, but a strife intense,
Deathless, of powers that charge and countercharge
Ever, yet never may their bounds enlarge.
in.
Not progress is the secret of the sky,
And not decay the withering doom of earth ;
Though, out of star-mist, systems round to birth,
And a dead moon mirrors earth's destiny,
The star shall sink in darkness whence it came,
And earth's grim desert be reborn in flame.
IV.
Tt is the wave with endless rise and fall,
456 THE MONIST.
It is the tide with ceaseless ebb and flow,
The changing moons, the hours with gloom and glow,
That hold the mystery of each and all,—
The rhythmic secret, wherein man has part
Even from the first pulsation of his heart.
v.
The pendulum with its untiring swing
Not only metes out time, but it reveals,
Babbling, the word eternity conceals,
Though to men deaf with their own questioning;
The lilting ripple of the poet's song
Itself contains the clue he sought life-long.
vr.
Nothing can be unfolded but has first
Been folded in, and shall be so again ;
Nor yet can aught in equipoise remain,
But ever driveth toward the best or worst;
Nature keeps neither full nor empty cup,
And the half-filled she drains or fills it up.
VII.
Yet what had no besrinninq- always is
«Z> C5 f
And never can become; no inward change,
However wide its outward motions range,
Can touch its heart; despite man's fantasies,
The Universe exists, not merely seems
An everlasting see-saw of extremes.
vm.
These two extremes man knows as More and Less,
As Good and 111, lastly as Right and Wrong;
Feels them as Love or Hate his pulses throng;
THE ETERNAL BALANCE OF GOOD AND ILL. 457
Sees them with Beauty clothed or Ugliness,
And names them from their power to bless or ban
God and Devil, Ormuzd and Ahriman.
IX.
The righteous Paul lamented in his heart
The Good by Evil thwarted. So in thine
The False and True, the Cruel and Benign,
The Pure and Impure make thee what thou art
And what the All is: tiger, dove, and man,
Seraph and fiend, are fashioned on one plan.
x.
Even as the Universe, mid seeming change,
Really is locked in iron permanence,
So, everywhere, despite our cheated sense,
From one self-nature may it never range:
One is it, one in body and the soul,
And every part is parcel of the whole.
XI.
Behind all forces hides the primal Force,
The Unconditioned, which is bad and good
Impartially, and its divided mood
The single spirit of the Universe,
Of you and me and all men and the earth
And all the worlds Infinify wheels forth.
,*
XTT.
But mortal life displays not one but two,
Shows Good all-perfect warring against 111,
Which yet abides unconquerable still,
And in this duel sets for man a part,
And teaches he must choose the side of Good,
Or rank below the cleft, insensate wood.
458 THE MONIST.
XIII.
Had it been destined to be otherwise,
Long since it would have been so; nay, for we
Deal not with time but with eternity,
It would have been so always; had our skies
Been fated to o'erarch a perfect earth,
They would have overarched it from their birth.
XIV.
This is the revelation; this alone
Rained ever from the Milky Way adown,
Or flamed from Vega and the Northern Crown,
Even this that written on my heart I own.
Not ours to ask if unto me or you
The word be welcome, but if it be true.
xv.
What then must be the Universe, ideal?
Never and nowhere; but endurable,
A place where on the whole 'tis fairly well,
Where at least men can live; in short, the real.
Had it been more, there were no need to ask;
Had it been less, not ours had been the task.
XVI.
If this be true, as Life forbids to doubt,
Is low then one with high, is conscience vain?
Forever no! But, though I shall not gain
After short strife a glorious mustering out,
My privilege more glorious is to be
A soldier of the Right eternally.
XVII.
Yet what avails my battle for the Right,
THE ETERNAL BALANCE OF GOOD AND ILL. 459
You ask, if through eternity shall still
Be kept the balance between Good and 111?
Me much avails it, for 'tis mine to fight
On the Lord's side, being birthmarked with his seal;
My joy, my life is in that battle peal.
XVIII.
More can I ask? Shall some far eon see
The Evil quelled, the Good supreme prevail?
Not if our world have told us a true tale.
But can we hear and judge it rightfully?
Our torch is feeble; but at least its light
Reveals us friend and foeman in the fight.
XIX.
The rest is God's. Yet who would change that could
Doom so divine, which loftiest souls must bear,
Though archangelic? — in all worlds to share
The warfare of the soldiers of the Good,
Though marching under orders ever sealed,
And battling ever on a doubtful field!
HARRY LYMAN KOOPMAN.
PROVIDENCE, R. I.
CRITICISMS AND DISCUSSIONS.
LOGIC AND PSYCHOLOGY.
The nature and purpose of symbolic or mathematical logic,
which began to be developed by Leibniz and was continued quite
independently by Boole and others, is tolerably well known by now.
Logical reasoning is translated by it into what Leibniz called a "real
characteristic" which is very analogous to ordinary algebra, and
helps swiftness and accuracy of reasoning — even complicated rea-
soning— in much the same way as the signs in algebra do. This
tendency culminated in the very ingenious and useful "mathematical
logic" of Peano. Peano's system was far more complete than
Boole's, for the whole of a piece of reasoning which included
algebraic formulas and equations could be put into a symbolical
form in which ordinary words — which are not part of a "real char-
acteristic"— are not used. In this direction Peano's system met the
much earlier system devised by Frege. However, Frege's system
was not thought out so much with a view to rapidity of reasoning
and convenience of writing as with a view to emphasizing slight
and important logical distinctions in very similar concepts and
deductions and consequently a scrupulous accuracy in deductions.
It may thus be noticed, by the way, that the purpose of Frege's
symbolism was different from that of all previous symbolisms in
logic and mathematics, for Frege wished to lay stress upon the
differences in various analogous ideas and deductions rather than
upon their analogies. Broadly speaking, Russell and Whitehead's
work may be characterized by saying that it is formed under the
influence of a combination of the two tendencies represented by
Frege and Peano. The convenient symbolism of Peano is retained
wherever possible and the superior analysis and subtlety of Frege
is fully used. We ought to add also that nearly all of Frege's dis-
CRITICISMS AND DISCUSSIONS. 461
coveries were made independently by Russell himself, Frege's great
work having been neglected by philosophers and mathematicians.
There is one great point in which Russell's works differs much
from that of Frege: full use is made of the enormously important
researches of Georg Cantor on transfmite numbers. While putting
on a firm basis the treatment of infinite classes and numbers, Can-
tor's work led to the recognition of forms of a paradox absolutely
fundamental in logic. After many vain attempts by various mathe-
maticians and philosophers, this paradox has been satisfactorily
solved by the thorough remoulding of logic given in Whitehead and
Russell's Principia Mathematica.
From Peano's various Fonnitlaires to the work last mentioned
the subject-matter is principally the collection of truths which we
can reach by logical deduction from logical principles. This body
of truths is not a description of psychological methods of discovery
or psychological results, but is of course reached by psychical
processes, like most other discoveries in a purely intellectual domain.
It is then simply irrelevant to complain that there is no place in
the Fonnulaires or Principia for that "intuition" which brings about
mathematical discoveries. It would be just as much to the point
to complain that in what is excavated we do not discover the tools
used for excavating or the method of excavation. And yet this is
what the rather superficial and amusing discussions of Henri Poin-
care are mostly about. And these discussions are what Prof. J. B.
Shaw in the number of The Monist for July, 1916, refers to (p.
397) as Poincare's "successful attacks on logistic." We might
reasonably, it seems to me, have expected that Professor Shaw
should make some reference to the reply by Louis Couturat to
Poincare which was translated in The Monist for October, 1912,
and which is quite conclusive on so many points. Professor Shaw,
in his eloquent and somewhat inaccurate (both from the points of
view of history and logic) attack on mathematical logic, urges
what are, at bottom, the very same irrelevant arguments. I shall
try to point out some of these inaccuracies, both because they are
fairly common even now among mathematicians, and because it
is surely the duty of every one to contribute as far as he can to
the clarification of notions in America above all other countries ;
for it is from America that we expect an exceedingly large pro-
portion of the work of the intellect in future now that Europe has
deliberately handicapped herself.
462 THE MONIST.
Professor Shaw's slighting remark on the impotence and boast-
ing power of logistic (p. 411) is the result of a strange miscon-
ception. Logistic deals with logical entities and deductions which
are fundamental to mathematics, and it is unjust to try to make
people believe that logistic ever claimed to be the overlord of mathe-
matics. There seems, in fact, to be a note almost of personal dislike
for logistic in those mathematicians who attack it. And yet the
question is wholly concerned with logical facts, and is not to be
answered by rhetorical appeals to prejudice or sentiment. If logic
is more fundamental than mathematics, why should there be any
objection to the — successful as it happens — attempt to define mathe-
matical entities in terms of logical ones? If mathematics is more
fundamental than logic, the first thing to do is to draw up a scheme
showing that logical entities can be deduced from specifically math-
ematical ones. Until this is done, and certain objections to it are at
once obvious, it is quite unconvincing to disparage cultivators of
logistic. After all, logisticians are working at mathematics in much
the same way that other mathematicians are. They are concerned
with more fundamental problems and problems which do not so
easily appeal to the public, as, say, a proof of Fermat's great
theorem would, but they discover truths just as much as any other
mathematicians. They introduce conceptions to work with. We
may mention the idea of pro positional function actually mentioned
by Professor Shaw in terms of commendation (p. 411), which was
introduced implicitly by Boole and MacColl — both early mathemat-
ical logicians — and explicitly by Frege, Peano and Russell — all
logisticians. A small acquaintance with such a work as that of
Frege will give plenty of examples of other powerful new ideas
introduced. And then as to truths discovered by logisticians, we
may remind Professor Shaw that the solution of ''the paradoxes
of logic" is wholly due to them, while mathematicians who were
unacquainted with logistic hopelessly floundered in the search for a
solution. Twelve years ago I was one of these flounderers myself,
and my "solution" had been accepted as satisfactory by many
mathematicians.
The real fact is that these results of logistic do not strike
some mathematicians as nearly so important as some of the results
of the theory of functions, for instance. I think they forget that
it is only in virtue of all truths being really of equal "nobility"
that Jacobi was right in claiming that a theorem in the theory of
CRITICISMS AND DISCUSSIONS. 463
numbers was just as fine as a very striking result in mathematical
astronomy.
'Perhaps the greatest mistake made by Professor Shaw is the
extraordinary statement about the nature of truth near the top of
page 409. It is surely quite evident that truths themselves do not
develop. That twice two are four was just as true last year as it
will be next year, — even if no people at all are left alive on the
earth next year. Professor Shaw finds fault with something I
wrote because he thinks that I maintained that ideas are not created
by man. It is quite evident from what I said in the context that
I only held that truths are not created, though I certainly said in
a slipshod and inaccurate way that "we do not really create anything
in science." Really Professor Shaw shows afterward that he agrees
with me that truth itself is not created, and his remark that doubt-
less I thought that words and ideas waited in the mines of thought
for the lucky prospector does not appear to be either logical or a
good guess (see pp. 409-411). However, at the top of page 409
he remarks that the world of universals changes in time. I suppose
that he means that our ideas, say of an "integral" or "continuity"
have changed ; but I hardly think that he ought to have fallen into
the error of mistaking the thing itself for a result of our groping
after the thing. I take it also that he does not intend to say that
truth evolves, for that rests on a confusion between a proposition
and a propositional function, such as in thinking that such a func-
tion as "Dr. Wilson is President of the United States" is a propo-
sition and not a function of the time which becomes a proposition
when any instant is specified and is then constantly true or false
eternally. What is the case seems to me to be that in logic and mathe-
matics the world we are concerned with is a world of facts, not of
conceptions. Conceptions are formed by us for the purpose of
stating truths, and in the world of pure mathematics we only come
across facts and form and variables. In this I think that I shall
have the support of one at least among philosophers: I refer to
Dr. Cams, who has always maintained that mathematics is essen-
tially concerned with the ideas of form and "anyness."
We now come to the last inaccuracy in Professor Shaw's paper
that I shall deal with. This is the question about the logic of in-
finity. The inaccuracy of the statements on page 412 appears
clearly if we give a short statement of the facts in the treatment
of infinity by mathematicians and logicians. Georg Cantor, in a
464 THE MONIST.
series of works dating from 1871 to 1897, succeeded in founding
a new and immensely important theory of transfinite numbers.
The use of the lowest transfinite cardinal numbers did not audioes
not present any difficulty whatever to mathematicians or even logi-
cians ; but, as Burali-Forti, Russell, and others noticed in various
forms the whole series of transfinite numbers presents difficulties
which were later found to be fundamental logical difficulties of the
same nature as that of the Cretan who said that all Cretans were
liars. Such problems were discussed at length in Russell's Prin-
ciples of Mathematics of 1903 and in the years after the publication
of this book were satisfactorily solved by him and Whitehead.
These solutions may be found in the Principia of 1910, and in the
almost wholly symbolical form of the book last mentioned it is
naturally impossible, even if it were not superfluous, that the claims
made in the earlier work should be repeated. Thus it is unjust to
conclude (p. 404) that the Principia is an abandonment of the
claims of the Principles, brought about because of the difficulties
found in Cantor's work. One might just as well conclude that the
difficulties of a solution of the great difficulty of "Cantorism" had
made Russell give up joking, for there are many jokes in the Prin-
ciples and only one in the Principia. There is one more point. It
is only what we may call a "boundary problem" about Cantor's
numbers that gives rise to difficulty: the resolve that any object
about which we talk or reason must be defined in a finite number
of words (p. 413) does not succeed in putting out of court all
classes that have an infinite number of members. Infinite classes
of objects each of which can be finitely defined can be defined in
a finite number of words, or better symbols of a "real characteristic."
The class of prime numbers is such a class. If indeed we may use
the notion of any (which is represented by one word) or the
notion of a variable in general, we cannot avoid admitting definitions
of infinite classes by a definite number of words. If also we may
use a sign for a variable, there is no earthly difficulty in giving a
general rule for correspondence in a way that is denied by Professor
Shaw on page 413. The rule, for example, if n is an integer, given
in the formula n + p, where p is another integer, indicates precisely
another class of integers which is correlated to the whole class of
integers considered first.
There is a small logical error committed by Professor Shaw,
at least if he considers philosophy to be the same thing as meta-
CRITICISMS AND DISCUSSIONS. 465
physics, which may explain why he is so satisfied with himself for
ignoring philosophy. On page 409 he characterizes a certain as-
sumption as "philosophical" and explicitly divides "philosophy"
from "mathematics." On page 414 he agrees with Lord Kelvin
that "mathematics is the only true metaphysics." Thus he would
seem to hold that there is no such thing as philosophy at all ; this
would certainly explain why philosophical assumptions are so little
worth serious discussion. Such discussion would in fact be as
foolish a problem as to investigate the birthplace of Jack the Giant
Killer's hen. But if we try to take a somewhat broader view, and
are not satisfied with dividing our knowledge into arbitrary water-
tight compartments labeled "Philosophy," "Mathematics," and so
on, we see that there are certain logical questions which can be and
have been solved by symbolical methods which strongly remind us
of algebra, which are absolutely fundamental in mathematics, and
which when formulated in ordinary language sound so like what
professional philosophers have often talked about that many are
tempted to hurry them out of sight into the "philosophical" compart-
ment. These are some of the questions with which logistic deals.
Logistic never claimed to be able to run without the guidance of a
human intellect (see p. 411) any more than the sciences of mathe-
matics or logic or chemistry did. What it does claim to do is, like
ordinary mathematics, to save our minds the labor of performing
again each elementary reasoning which requires no talent but only
memory — often a prodigious memory when the reasoning is compli-
cated ; so that we can reserve all the talents we may possess for
overcoming those obstacles to a discovery of truth that have not been
hitherto overcome. Then again, unlike ordinary mathematics, logis-
tic seeks to point out differences in analogous ideas and reasonings
which play an even greater part then analogies when we come to
consider really subtle reasoning. Thus the analogy between impli-
cation between propositions, inclusion between classes, and inclusion
between relations breaks down in certain cases, and we see that
Russell in his later work forsakes the identical form of the symbols
expressing these relations. Peano, as we know, kept to the same
symbol on account of the very close analogy between the relations
spoken of.
If we are content to accept without examination the arbitrary
classification of people who were unacquainted with modern logic
into exclusive "mathematical" and "philosophical" compartments,
466 THE MONIST.
we must be prepared to think we see what we think are the rigid
foundations of mathematics being eaten into by philosophy, and if
we wish still to maintain that the foundations of mathematics are
rigid we shall have continually to give the new name of "philosophy"
to parts of what were hitherto considered to be mathematical. This
state of things was actually brought about by what Poincare called
"Cantorism" : truths which were hitherto considered solid and math-
ematical seemed to be thrown into doubt by the advance of philos-
ophy. Of course this was not really so: the logical questions at
the foundation of mathematics are capable of scientific investigation
just as much as the theory of numbers of the differential calculus,
and it is unnecessary and ridiculous to narrow the scope of our
investigations because we shall meet logical difficulties if we do not.
What would be thought of a tradesman who thought he could calm
the mind of his assistants by maintaining that the ravages of a bull,
although they seemed to be in his own china shop, were really in
a drapery department which had somehow extended into that part
of his shop where plates were sold? This is what those mathe-
maticians do who dismiss awkwardness to "philosophy" and think
that thereby they have kept mathematics pure and free from all
"metaphysical" discussions.
Miss Dorothy Wrinch has sent the following comments on
Professor Shaw's article in The Monist for July, 1916:
"The chief thing that I quarrel with in Professor Shaw's article
is his idea of one: selecting one pencil from a pile is really rather
different from considering the class whose members are the classes
'living kings of England,' 'fathers of A,' etc. Further, it would be
difficult to give a definition of one or ttvo which is not a statement
in which one or two appears: he does not attempt to say that the
other constituents of this 'statement' have not been defined, or that
the definiendum is not unique. These could be his only grounds for
attacking a definition, which is merely a statement in symbolic
form of cases in which the number one or the number two appears.
Also it seems a pity (line 9, p. 407) that he should fall into the
error that he deplores in mathematical logicians, viz., the error
of introducing the notion of truth (and truth value) when 'in no
place. . . . they are defined.'
"I suppose that it was 'in the intoxication of the moment' that
Professor Shaw called a prepositional function of two variables
a relation (p. 404, line 14), and let out of the bag the existence of
CRITICISMS AND DISCUSSIONS. 467
a difference — hitherto, apparently, kept dark by mathematicians —
between the properties of the roots of a quadratic equation and
the properties of quadratic functions of x.
"Professor Shaw makes some strange remarks on page 413.
If the collection of all integers does not exist (line 8) it seems
hardly necessary to refute the proposition that it is possible to cor-
relate the collection of all integers to some other infinite collections.
"It seems rather unsportsmanlike to rely upon people's short
memories and call Poincare's attacks on logistic successful. Might
it not be well to remind people of the conclusions to which M.
Couturat came at the end of his article in The Monist for October,
1912: 'Admitting the principles and primitive ideas of the logisticians,
M. Poincare has maintained that, setting out from these data, they
cannot build up mathematics without another postulate — an appeal
to intuition or a synthetic a priori judgment ; and he has thought
that he has discovered in their logical construction certain paral-
ogisms (beggings of the question or vicious circles). I believe that
I can conclude from the above discussion that not one of these
theses is proved, and that, in particular, the logisticians have not
committed any of the logical errors that are so lightly imputed
to them/"
PHILIP E. B. JOURDAIN.
FLEET, HANTS, ENGLAND.
THE CAL-DIF-FLUK SAGA.
EDITORIAL INTRODUCTION.
Mr. J. M. Child has given, in the following "Saga," an amusing
description of the results he has arrived at in his book on Barrow,
just published in the series of "Open Court Classics." The closing
lines represent the opinion he has formed from a consideration of
the manuscripts of Leibniz, an annotated translation of which has
been appearing in current numbers of The Monist, beginning with
October, 1916, and continued in the April number and the present
one.
The saga evidently refers to the question of the invention of
the infinitesimal calculus. Isa-Roba is Barrow, Isa-Tonu is Newton,
Zin-BH is Leibniz, while Cavalieri is mentioned under the name of
Ler-a-Cav. Gen-Tan-Agg stands for Barrow's Gen-eral method of
gents and of ^^-regates ; while Shun-Fluk and Cal-Dif ob-
468 THE MONIST.
viously refer to the methods of Newton and Leibniz. Batnac is
the ordinary abbreviation of the Latin for Cambridge, Cantab., with
its letters reversed ; and the allusion in the next line is to Barnwell
Pool, where it is stated that an undergraduate whose boat had
overturned was saved from drowning, but died soon afterward
from blood-poisoning! Terangel is a transformation of Angleterre,
i. e., England. Ris-Pah is Paris, where Leibniz lived at the time
of the invention of the calculus.
In the second stanza, the allusions to ''burning midnight oil,"
the quill pen, incandescent gas mantle, and the electric light are all
fairly obvious ; while the Swan may be taken to refer to a well-
known make of fountain pen. Stanza 5 refers to the publication of
a book. The archery in the first method of training alludes to the
ancient definitions of a tangent and a normal to a curve ; and the
sword-play recalls Euc. I, 10 and Euc. I, 1, while the allusions in
the second are easily referred to the method of indivisibles.
In Stanza 9. the dagger refers to the differential triangle, which
Barrow only included in the first edition of his work on the advice
of Newton ; the knobs on the hand-grip refer to Newton's "dot"
notation.
The two weapons of Zin-Bli are the signs invented by him for
differentiation and integration. Lastly, Li-Nu-Ber is John Ber-
noulli, who stated that Leibniz got the whole of his fundamental
ideas from Barrow, whereas Leibniz himself denied any indebtedness
to Barrow.
THE CAL-DIF-FLUK SAGA.1
1. Saga of sons of a Goddess, of Thought and Learning the fountain,
(Haply in that which I sing, a real historical meaning.
Wrapped in a fanciful garb, and oddly disguised as a saga.
Those who are skilled in lore, and erudite more than their fellows,
Knowing the facts of the case, if they diligently seek may discover.)
Dwelt She, She dwells upon Earth, and henceforth for ever and ever
Dwell so She will among mortals. 'Tis thus decreed by the All- Wise.
2. Oil from the Midnight Lamp the sacrifice burned on her altars,
Plumes from the wing of the Goose her now peculiar token ;
Not so at first was it thus, and not in the times that are coming
1 From a manuscript found in 1916 A. D., while searching an ancient
tumulus or "barrow," and made out from the original by J. M. Child.
CRITICISMS AND DISCUSSIONS. 469
Will it be Oil and Plume ; I see with the eye of the seer
Wondrous visions of Light, enwrapped in a Mantle resplendent,
Torn from the heart of a stone, the essential soul of the Sun-god
Prisoned for ages therein ; and globes of crystal translucent
Glowing with filaments bright kept hot by the Spirit of Lightning,
Swan of the Golden Beak instead of the Goose for her token.
3. Sent upon Earth to dwell with mortals by will of the All-Wise,
Children divine to bear to those who Her fancy might capture.
Ardent and long was the wooing, both strong and patient the lover,
Ere he received his reward, or ere She presented him offspring.
Else as a mark of Her love to him She had chosen to honor, —
Chosen for womanly whim, for some unaccountable reason
Honored above all else, who never had courted her favor —
Sent She on lighting wings the soul of Her heart, Inspiration.
4. Children of fathers of Earth, but endowed with the life of the
Mother,
Destined as Heroes to wage perpetual warfare on all things
Troubling the minds of men desiring to widen the limits
Set on the realm of We-Know, by the race of the children gigantic,
Issue of Never-Before out of We-Xever-Heard-of the-Method.
Children begotten from Her are known by the names of their fathers,
More by the deeds of the sons are the fathers so held up to honor ;
Accurate records are kept; thus long through the ages that follow,
Known by the deeds of the sons are the fathers so held in remem-
brance.
Rightly was this the Law, for responsible he for the training,
Fitting the son for the fight for freedom and fuller perception.
5. Till 'twas such time as was meet, the custom obtained that in
secret
(Jealous that others might see not fully developed the power
Promising greatness to come), this fatherly training continued
Day after day for an eon ; until with a flourish of trumpets,
Front of the eyes of all, tattooed with the symbols of Learning,
Clad in a mantle of calf -skin, bearing on back and on bosom
Plainly for all to observe, in resplendent gold letters, his title,
Son of the Goddess of Thought, was he set as a champion of
Knowledge.
47O THE MONIST.
6. The methods of training were two, at least only two were ac-
counted.
Oldest and best known of all was the method derived from the An-
cients,
Cumbrous, exhaustive and long ; horizontal and parallel bar work,
Drawing of cord of the bow, and the rings were considered essential ;
Accurate hand and eye were developed by shooting an arrow,
Grazing the cheek of a figure, or forth from it standing erected ;
Cleaving a bar into twain, so each part as to balance the other
( Nought but two measuring swings ere the cut was delivered allowed
him).
Such like in days of old had fitted the Heroes for battle.
Founded on this was the second, but strangely unlike it in practice ;
Suppleness rathe.r than strength was the object and creed of the
trainer.
Straight-edged still was the sword ; with it blocks were sliced into
shavings,
Shavings were sliced into threads, and threads were chopped into
pieces,
Parts of ineffable smallness, divisible reckoned no further.
Masonry part of the course, in which arches with bricks were
fashioned,
Leaving the corners undressed ; as the pupil advanced in his training,
Smaller and smaller the bricks, indivisible finally counted.
Specially fitted for Heroes, prepared for attack on the giant
Clans of A-Re-A and Vol-Yum, the brood of Cur-Va-Rum and
Mez-zur.
Failed jf the fatherly training, the Goddess in sorrowful anger
Took from the child his soul, the gift which at birth She had given,
Worthier father to bless, if ever another such won Her.
7. Once in the days now gone, there lived on the banks of the Batnac,
Renowned for its smells and its mud, where pollution enters at Well-
Barn
(Truly not then was this fame, nor yet at the time of this writing
Thus had it won a repute, 'tis a prophecy sure that I utter),
Land of Terangel within, a mortal yclept Isa-Roba.
Many and varied his loves, his fickleness surely a drawback ;
Truly a wonder it was that the Goddess e'er let him approach Her.
Bare She however a son, Isa-Roba undoubted the father,
Fair both in face and in form, a divine conception befitting ;
CRITICISMS AND DISCUSSIONS. 47!
Ne'er such a babe before was born with so splendid a future ;
Seemed that the soul of his Mother had enter'd the Child at his
birth-time :
Best that She had to give, best that She can give for all time,
Gave She this son of Her heart ; Gen-Tan- Agg Isa-Roba did name it.
8. Trained he the boy in a manner that savored of that of the An-
cients,
Discipline rigorous keeping, yet toned with a method that fore-time
Ler-A-Cav brought to perfection, a mingling of first and of second
Systems of training recounted ; however 'twas foredoomed to failure.
9. Hercules never so strong as the youth Gen-Tan-Agg, no, nor
Samson.
Armed with his two-handed weapon he met many giants in combat ;
Numerous clans he defeated, by slaying their general doughty.
Nevertheless were his muscles too stiffened by reason of rigor,
Due to the manner in which Isa-Roba conducted his training.
Love for the two-handed broadsword, with which Isa-Roba had
armed him,
Made him neglect the superior weapon that hung at his waist-belt,
Sharper by far than the sword-blade, a steel of superior temper ;
Seems Gen-Tan-Agg only used it preparing the shafts of his arrows ;
Nigh came to leave it at home as he set out upon his first journey,
Girding it on at the last, not perceiving in it that a weapon
Ready to hand he had got against which no armor of mortals
Could for a moment prevail ; for piercing the joints of the harness.
Off' ring no passage to sword-blade, it reached his opponent's main
vitals ;
Forced him to give up his treasure, the secret protected for ages.
10. Happened it thus that a Hero, high-blessed by the Goddess his
Mother,
Spoiled by the weapon mistaken his anxious sire recommended,
Fame and renown and great honor did miss for ever and all time,
Losing the chance that was offered, a name and a high reputation.
Lastly, by father discarded (who fickly returned to a first love),
Languished and nigh came to perish, unhonored, unsung and neg-
lected.
11. Some of the records of giants the youth Gen-Tan-Agg had de-
feated
472 THE MONIST.
Chanced Isa-Roba, however, had told to a friend Isa-Tonu ;
Agile by nature, the latter immediate saw that the dagger,
Superior far to the broadsword, was a weapon of magical value ;
('Twas Isa-Tonu's advising that just at the very last moment
Caused Isa-Roba to add to his offspring's armor the dagger). .
Pity, perhaps, for the youth, or a covetous eye for the poignard,
Caused Isa-Tonu to take neath his fostering care the young stripling,
Freeing his father from trouble, unhampered to follow his fancy.
Thus Isa-Roba the story departs from, unhonored for all time ;
Save and if only in future, this tomb may be opened by some one
Trying to find out the truth of the Hero's father and birthtime.
Under the fostering care of a trainer less hide-bound by nature,
Slowly at first, then apace, did the Hero recover his power.
Changed was his armor, the sword altogether replaced by the dagger,
Changed was the dagger in form, for a knob, sometimes two, on the
hand-grip
Gave it a far better balance. Obsessed by his special requirements,
Secretly long Isa-Tonu did bind Gen-Tan-Agg to his service.
Later ungratefully hiding the name of the Hero who served him,
Swearing that all had been done by his own bastard offspring, young
Shun-Fluk.
12. Thus once again was the Hero discarded and left for to languish,
Shun-Fluk attaining the fame that should his have been truly and
rightly.
Nemesis, son of old Equity, sternest of Gods and the justest,
Saw Isa-Tonu's deception, and straightway the Goddess of Learning
Sought He and told Her the story. In sorrowful anger the Goddess
Listened with eyes that flamed at the failure that followed Her off-
spring,
Due to his father's bad training, and then Isa-Tonu's enslavement ;
Listened and cursed the first, for the other a punishment thought out.
13. "Punishment dreadful and dire !" So she spake, the while Neme-
sis listened,
Listened and nodded and smiled, as approved He the plan She sug-
gested.
"Lives there a mortal in Ris-Pah, who long has courted my favor;
Often of late have I thought that at last I'd rewrard his devotion.
Lacks he but one little thing, only one thing to render him fitting
• CRITICISMS AND DISCUSSIONS. 473
Trainer of offspring of mine; but the lack mean I now to forgive
him.
Never again could I bear such a child as I bore Isa-Roba ;
Certain is that ; but immortal the soul that at birth-time I gave him,
Breath of my life, Inspiration, again, Gen-Tan-Agg expiring,
Can, if I will it, enlighten the child which I'll offer to Zin-Bli.
Thus is he called by mortals, an inventor of weapons and symbols.
One has he fashioned already, in shape like a chopper for fire-wood,
Straight in the shaft, with a hand-stop to stay it from slipping,
Circular edge to the axe-blade, to shaft is it fastened by bolt-head ;
Much like the symbol that mortals set fourth in the lower-case
system.
This shall he teach my offspring to use to more delicate purpose.
Binds he his sticks all together with cord made out of the sum-omn ;
Lurking however in thought is the germ of a better invention.
Rod with curl at each end, slightly bent, so that clipped round the
bundle,
Binding the whole into one, he is able to thus grasp it firmly.
Armed with each of these twain, shall his offspring forth stand as
a Hero."
Spake She, and Nemesis nodding to all His approval, it was so.
14. Cal-Dif named Zin-Bli the child, and he trained him these weap-
ons to master ;
Speed, at all rates, with the first he created new records completely,
Nor did he stay at that ; with the second, the brood of the giants,
Laid he them low in the dust, so that never again should they trouble.
All that the Goddess had said, so performed She; the credit of
Cal-Dif
Famed through the kingdoms of mortals, became a renown for the
father,
Ne'er to be equalled till Earth is devoid of reasoning mankind.
15. Swelled as to head by renown, though Zin-Bli well knew Inspi-
ration
(Could he forget this?) had wrought in a magical manner the marvel,
Yet could not bear it for others to know whence the source of his
wisdom ;
Denied he the source whence it came, Isa-Roba's offspring discarded.
Nemesis saw what he did, and he stirred up the folk of Terangel,
474 THE MONIST.
Shun-Fluk to accuse him of stealing and sending him forth as his
Dif-Cal.
None seemed to have guessed the truth, save a man by the name of
Li-Nu-Ber.
16. Ye who perchance may consider this saga in future far ages,
Know now the truth ye may ; that the soul of the Goddess of Learn-
ing
Entered at first Gen-Tan-Agg, but he languished for lack of good
training ;
Afterwards, renamed Shun-Fluk, he recovered some of his birth-
right ;
Dying, his soul was then given to an ordinary child of a mortal,
Rendering its face and its form like one of divine conception.
17. Accepted as such by all, till the day that this saga's discovered,
Haply e'en then, for foretell I that Cal-Dif
Unfortunately, the manuscript, which consists of another couple
of sheets that were outermost in the roll, here becomes indecipher-
able through being destroyed by damp; it would have been inter-
esting, and useful in the light of judging of the truth of the facts
given, to have verified how far the prophecies were fulfilled by
events since the time at which they were written down and the
manuscript hidden in this old burial-mound.
J. M. CHILD.
DERBY, ENGLAND.
NOTES ON DE MORGAN'S BUDGET OF PARADOXES.
In a work requiring the large amount of reading involved in
editing a book like the Budget of Paradoxes, and particularly in the
condensing of the results to the proper proportions for footnotes
to aid the reader, it was, of course, inevitable that a certain number
of inaccuracies would occur. It is also evident that many more
notes might profitably have been added to elucidate the meaning
of the text, or to correct the original where this would be warranted.
De Morgan was a careless writer and many of his errors are
mentioned in the footnotes ; but numerous others exist, some of
which are patent to any reader and others of which might profitably
CRITICISMS AND DISCUSSIONS. 475
have been set forth by the editor. It is also a serious question as
to whether the translation of common phrases is not more of a
hindrance than a help to even the casual reader, and whether the
space used by such translation might not have been more profitably
devoted to a further elucidation of the text. This is the feeling of
one or two critics.
Since the work was published, several friends have called
attention to a few misprints, a few generous critics have suggested
helpful changes, and one or two others have objected to certain of
the notes. It therefore seems proper to present a few emendata
and errata which may assist the reader of the work.
In the matter of emendata to De Morgan's text itself and of
suggestions as to further helpful notes I am indebted chiefly to
Prof. A. E. Taylor of St. Andrews, Scotland, who has gone over
the work with great care and has kindly given the Open Court
Publishing Company the benefit of his reading. The following
notes on De Morgan's text are due to him.
Vol. I, page 3. De Morgan should not have attributed to
Spinoza the anonymous Philosophia sanctae scripturae interpres. It
was probably the work of his friend and physician Lodowick Meyer.
Vol. I, page 41. De Morgan's version of the passage from the
commentary of Eutocius on the tract by Archimedes on the meas-
ure of the circle is not satisfactory. The Cerii of Porus should
be the Ceria (/ojpia, honey combs) of Sporus. He probably used
the Wallis edition of Eutocius and quoted only the first four words
of the passage (Archimedis Opera Omnia, III, p. 300, of the 1881
edition of Heiberg) : «s dKpi/^eorepovs apidfiovs dyayetv TWV W 'Ap^i-
fHj8ov^ clprjfifvwv, rov re. £" <f>r)fj.l KOI r<av i oa". The restoration adopted
by Heiberg makes the statement of Eutocius correct: "a more ac-
curate evaluation than that of Archimedes, i. e., than the fractions
^4 and l%\" According to Sporus, Philo of Gadara had found
closer limits. Archimedes had given 3% as the upper limit and
31(%i as the lower limit of TT, the £" and oa" representing merely the
fractional parts.
Vol. I, page 96. De Morgan's language seems to imply that
the Convocation of the University of Oxford is, or was, a body of
ecclesiastics of the Anglican Church, but it is not an ecclesiastical
body at all. It consists of all masters of arts who qualify by the
regular payment of their university dues. Professor Taylor suspects
that De Morgan may have confused the Convocation of Oxford
with the Convocation of the Clergy of the Province of Canterbury.
4/6 THE MONIST.
Vol. II, page 274. For De Morgan's translation of oviAov /ue'Aos,
read "a song of bale" (oAoov /xe'Aos).
Vol. II, page 277. De Morgan overlooks the true reason why
Pope scans Mathcsis as Mdthesis, namely, that like all writers of
his day he pronounced Greek names according to their accent, not
as we now do with an adjustment of the stress accent to the quantity
of the vowels.
Vol. II, page 322. De Morgan is incorrect in his statement as
to Bohme's division of Mercurius. Bohme divides it Mer-cu-ri-us,
not Merc-u-ri-us.
Vol.11, page 340. It would be interesting to know whether De
Morgan's complaint that Walter Scott did not know what "Napier's
bones" were is well founded.
Professor Taylor suggests various other interesting notes re-
lating to the text, and of course such a list could easily be extended.
In the extensive bibliography given in the notes it was inevitable
that certain slips of the pen should have occurred. In Vol. I, page
105, I followed Bierens de Haan in giving the spelling "Johannem
Pellum." ^My friend Herr Enestrom has a copy of the edition in
question and the spelling there given is "loannem Pellivm." He
also calls my attention to the proof given in the Bibliotheca Mathe-
matica recently that Mydorge was not the author of the Recreations
mathematiques as published in Boncompagni's Bullettino.
Among the slips of the pen which I have noticed since the work
appeared is the name of D'Alembert for that of De Lalande in
Vol. I, page 41 ; "condemned" for "contemned" on page 92 ; and,
in Vol. II, "blata" for "beata" on page 61.
Professor Taylor calls attention to the further slips of "fellow
of Cambridge" for "fellow of Trinity College, Cambridge" and of
"Derion" for "Denon" (Vol. I, page 76) ; "Viscount of Palmerston"
for "Viscount Palmerston" (page 290) ; "closed" for "classed" (in
the text, Vol. II, page 148) ; "tolo" for "toto" in the text (page
344) ; and ± for ± 1 in the text (page 368).
I am also indebted to Professor Taylor for several suggestions
of betterment of the translations, matters which should have been
attended to by me in the preparation of these particular notes even
though I entrusted this work to another. The following changes
are not to be attributed to him, although changes (sometimes more
extended) were suggested by him.
In Vol. I, page 3, for "what it was" read "that it was" ; page
40, for "its appointed path" read "the appointed path" ; for the free
CRITICISMS AND DISCUSSIONS. 477
translation in verse on pages 53-54, for "And lacking nothing but a
start, and lacking nothing but an end," read "The only one without
a start, the only one without an end" ; page 339, for "think himself
to die" read "feel himself dying."
In Vol. II, page 23, n. 4, for "He was wont to indulge in" read
"He has a habit of refreshing his reader by"; page 151, for "con-
demned soul" (literal) read "hack" (colloquial) ; page 154, change
the translation of the familiar legal phrase to bring out the pun
upon J. S., "Summum J. S. (for jus) summa injnria" (the height
of law — J. S. — the height of wrong) ; page 200, change "sleeping
power" to "sleep-producing power" ; page 228, translate 8io<> ei/xi rj
i7pas, as "of Zeus I am, or Hera," and ^ /uWa as "mass" ; page 260,
translate the quotation from Acts xix. 38, as "the courts are sitting" ;
page 262, for "according to which" read "relatively" ; page 283, for
the manifest error in the note on "ab ovo" read "from the egg,"
probably relating to the passage in Horace, "nee gemino bellutn
Trojanmn ordititr ab oro,'' or possibly to "ab oro usque ad mala" ;
page 365 for "slayst" (misprint for "slayest") read "keepst."
Professor Taylor also suggests that Hobbes lived only about
eleven years in France (Vol. I, page 105) ; that Burnet left England
to avoid being involved in the ruin of the Whigs (page 107) ; that
Street acted in accord with the law (page 124) ; and that there was
nothing strange in Laud's patronage of Palmer (page 145). The
details of these emendata and certain other suggestions of change
would trespass too much upon the space which the editor of The
Monist has kindly allowed me.
DAVID EUGENE SMITH.
TEACHERS COLLEGE, NEW YORK.
BOOK REVIEWS AND NOTES.
REFLECTIONS ON VIOLENCE. By Georges Sorel. Translated, with an introduc-
tion and bibliography, by T. E. Hulme. London, George Allen & Unwin,
1916. 7s. 6d. net.
SoreFs book is exceedingly difficult to discuss in a short review. Its sub-
stance is a very acute and disillusioned commentary upon nineteenth-century
socialism, and upon the politics of the French democracy for the last twenty-
five years. It contains also two elements which must not be confused, Sorel's
own political propaganda (if he would allow it to be so called) and his phi-
losophy of history formed under the influence of Renan and Bergson. And it
expresses that violent and bitter reaction against romanticism which is one
of the most interesting phenomena of our time. As an historical document,
Sorel's Reflections gives, more than any other book that I am acquainted with,
an insight into what Henri Gheon calls "our directions."
Doubtless many readers will be disposed to consider the book under its
first aspect only. But the study of Sorel's political observations requires an
accurate knowledge of government and parliamentary activities since the Drey-
fus trial, and does not in itself make the work of importance to the English
and American public. What Sorel wants is not a political, but a social form.
One must remember that his creed does not spring from the sight of wrongs
to be redressed, abuses to be cured, liberties to be seized. He hates the middle
classes, he hates middle-class democracy and middle-class socialism; but he
does not hates these things as a champion of the rights of the people, he hates
them as a middle-class intellectual hates. And the proletarian general strike
is merely the instrument with which he hopes to destroy these abominations,
not a weapon by which the lower classes are to obtain political or economic
advantages. His motive forces are ideas and feelings which never occur to
the mind of the proletariat, but which are highly characteristic of the present-
day intellectual. At the back of his 'mind is a scepticism which springs from
Renan, but which is much more terrible than Renan's. For with Renan and
Sainte-Beuve scepticism was still a satisfying point of view, almost an esthetic
pose. And for many of the artists of the eighties and nineties the pessimism
of decadence fulfilled their craving for an attitude. But the scepticism of the
present, the scepticism of Sorel, is a torturing vacuity which has developed
the craving for belief.
And thus Sorel, disgusted with modern civilization, hopes "that a new culture
might spring from the struggle of the revolutionary trades unions against the em-
BOOK REVIEWS AND NOTES. 479
ployers and the state." He sees that new political disturbances will not evoke this
culture. He is representative of the present generation, sick with its own knowl-
edge of history, with the dissolving outlines of liberal thought, with humanita-
rianism. He longs for a narrow, intolerant, creative society with sharp divisions.
He longs for the pessimistic, classical view. And this longing is healthy. But
to realize his desire he must betake himself to very devious ways. His Berg-
sonian "myth" (the proletarian strike) is not a Utopia but "an expression of a
determination to act." The historian knows that man is not rational, that
"lofty moral convictions" do not depend upon reasoning but upon a "state of
war in which men voluntarily participate and which finds expression in well-
defined myths." It is not surprising that Sorel has become a Royalist.
Mr. Hulme is also a contemporary. The footnotes to his introduction
should be read. i?
THE NEW INFINITE AND THE OLD THEOLOGY. By Cassius J. Keyser. Yale
University Press, New Haven, 1915. Price 75 cents.
In this essay Dr. Keyser shows many interesting ways in which some of
the most difficult problems of theology may be partly or wholly overcome by
mathematical means.
The relation between religion and science is discussed, the author showing
that while science belongs to the middle zone, or rational world, religion
belongs to the over-world or superrational. Then follows a brief discussion
of the relation of theology to religion, theology being primarily a science, in
a word "the science of idealization." From the purely theological standpoint,
"God is an hypothesis." In all definitions of God the notion of infinity is
foremost. Therefore the essay develops the mathematical concept of infini-
tudes and through many examples makes clear the denumerable type of in-
finite manifolds ; then far surpassing this in glory, the continuum type, and
points to types of even higher orders. "The infinite of theology is the limit
of the endless sequence of more and more embracing infinitudes presented
by science.''
The contradictions of theology are of two kinds, foreign and domestic.
Theology may rid herself of the foreign variety by casting out all illegitimate
postulates. In the world of infinitudes the part of a group may be just as
numerous as the whole group. So in the realm of theology, the seemingly
contradictory ideas of omniscience and freedom may be reconciled; for the
dignity of omniscience is as great as omniscience itself. The same line of
reasoning is applied to the doctrine of the Trinity. The essay closes with a
reference to the so-called domestic difficulties, and shows that a being may
have many contradictory aspects and yet viewed in a large way all these
aspects may be true; just as in comparing different systems of geometry built
on various foundations the mathematician finds contradictory facts, yet does
not doubt the truth of any of these facts.
Dr. Keyser's careful, earnest style of writing makes it a pleasure to read
his works, and any one who has the "mathematical spirit which is simply the
spirit of logical rectitude" will enjoy this unusual essay.
EMMA K. WHITON.
480 THE MONIST.
THE STUDY OF RELIGIONS. By Stanley A. Cook, M.A., Ex-Fellow and Lec-
turer in the Comparative Study of Religions and in Hebrew and Syriac,
Gonville and Cains College, Cambridge. London, A. and C. Black, 1914.
Price, 7s. 6d. net.
Mr. Cook is very long-winded, but in spite of dryness and abstractness of
style he has written a valuable book. Much thought has evidently gone into
it, and its defects are due to a difficult manner of exposition, not to poverty of
ideas. This is not an "Introduction" of the type of Jevons's book ; it gives no
data for the beginner, nor, as one is apt to expect from the title, does it deal
chiefly with primitive religion. It is rather the comments of a scholar — Mr.
Cook is a recognized authority in his field — on the aims and methods of his
study. He has a great deal to say, and much that is extremely good, on the
evolution of religion — as is indicated by several chapter headings : Survivals,
The Environment and Change, Development and Continuity. "The doctrine
of survivals," Mr. Cook says, "is entirely inadequate when it forgets that we
are human beings and do not accept beliefs merely because they happen to lie
within our reach. The doctrine of survivals, is, in fact, a very handy and
cheap explanation of some one else's beliefs and practices — hardly of our
own !" Survivals are not simply "left behind," they are subconsciously se-
lected. Mr. Cook warns very wisely against arguing from the part to the
whole, against constructing a hypothetical system into which every survival
must fit. He warns also against confusing the evolution of beliefs with the
evolution of environments, in judging apparent retrogressions. On the crit-
ical attitude, on the acceptance of data, Mr. Cook has some excellent observa-
tions, and on the historical versus the religious importance of critical revisions.
He holds that the present is a time of religious unrest, though like most of
us, he cannot point to any definite theology for the future. His conclusion is
as follows : "The unbiased student of religions can hardly escape the conviction
that the Supreme Power, whom we call God, while enabling man to work out,
within limits, his own career, desires the furtherance of those aims and ideals
which are for the advance of mankind." *?
Just as we are going to press we receive two additional notes from Dr.
W. B. Smith to be inserted in his article as indicated respectively on pages
330 and 337.
Page 330: "For which Rutherford's 'nucleus theory,' apparently required
by the facts in the scattering of 'alpha rays' (of helium atoms) in passing
through laminae, substitutes a positive electric core, extremely minute, for
gold only one trillionth of an inch in diameter, in volume one billionth of the
atom itself. It would seem that the negative electron is nearly six thousand
million times as large as the positive hydrogen core. For Thomson's later
views see Philos. Mag., 1913, p. 892."
Page 337: "Why do the members fall together to the center as their
energies are dissipated in electric radiation? Bohr (Philos. Mag., 1913, pp.
1, 476, 854) invokes Planck's 'Quantum'-hypothesis in solving this riddle.''
VOL. XXVII. OCTOBER, 1917 NO. 4
THE MONIST
WHAT IS A DOGMA?*
EDITORIAL INTRODUCTION.
The primary significance of a dogma is not its speculative con-
tent, but the speculative truth of dogma is expressed in terms of
action. Such is the proclamation of a Roman Catholic thinker
which has evoked a lively discussion, and although his work has
been placed on the Index, this has evidently been for other reasons
than any connected with the charge of heresy. For this thesis de-
fines the general concept of dogma in the expressions of the well-
known philosophy of action originated by Maurice Blondel and
published in his book L'action which appeared in 1893, and as far
as we know his book was not placed on the Index. "Perhaps,"
writes Father E. Bernard Allo, O.P., "the thesis sketched by Le Roy
is not so different, perhaps the divergencies are less in idea than
in expression, in the significat itself than in the modus significandi"
(Foi et systeme, Paris: Bloud et Cie., 180, 181), and this is con-
firmed by Le Roy himself in a footnote on page 70 of his Dogme et
critique. A. Houtin in his history of Catholic modernism mentions
the Rev. A. D. Sertillanges as expressing the same opinions in the
referendum on Le Roy's article on dogma as Father Allo, and so
far as we can ascertain, th:ir writings have not been placed on the
Index. Further, for a book to be placed on the Index does not mean
that it is condemned, but the authorities intend to say that for
some reason hie ct nnnc the book is not to be generally read.
This article of M. Edouard Le Roy entitled "Qu'est-ce qu'un
dogme?" has even been looked upon with favor in some quarters
by representative ecclesiastical authorities; and being of great im-
portance, not only for Roman Catholicism, but also for Protestant-
* Translated by Lydia G. Robinson from the sixth French edition of tht
author's book Dogme et critique.
482 THE MONIST.
ism, yea generally for all religion, we take pleasure in rendering it
accessible to English readers.
It first appeared in the French fortnightly journal La Quinsaine
of April 16, 1905, where it was accompanied by an editorial note
as follows: "Without expressing any decision on our own part
with regard to the opinions of M. Le Roy it seems to us both inter-
esting and useful to take a text from his work by which to invite
theologians to furnish the public with the elucidation he asks for.
Hence we address a special invitation to all the authorized special-
ists in Catholic theology, to the professors of our liberal universities
and of the larger seminaries, to religious orders, and to the priests."
The invitation was eagerly accepted, and seven later numbers
of La Quinzaine contained communications of varying importance
on the subject. But these formed only a small part of the discussion
raised by this striking article. Its publication was followed by a
vast array of controversial writings which continued with increasing
violence throughout an entire year. Twenty or more other journals
opened their pages to the subject ; not only such distinctly clerical
journals as Etudes, Revue thomiste, Revue du clerge frangais, La
Croix, etc., but also general philosophical reviews, La Pensee con-
temporaine, Revue de philosophic, and such liberal journals as La
Justice sociale, Le Peuple fran^ais, and La Verite frangaise. And
not only these religious and critical periodicals devoted their pages
to the subject but a well-organized opposition to the offending
article rushed into print through the daily press.
Still the question which the author put to the clergy in def-
erence to them as being officially charged with the instruction of
the people did not receive a satisfactory answer. Many heaped
M. Le Roy with malicious calumnies, and many honestly misunder-
stood him. Many too misjudged him because they knew of the
article only through garbled reports or hostile criticisms. He there-
fore considered it necessary to put the article in permanent form,
and so he published it in a book entitled Dogme et critique (in the
series Etudes de philosophic et de critique religieuse with Librairie
Bloud et Cie.) together with his published replies to the most im-
portant of his adversaries, a careful bibliography of the contro-
versy and a more detailed development of the most significant points
of his thesis in fourteen brief additional chapters.
* * *
Religion is a practical affair, and its main purpose is to serve
WHAT IS A DOGMA? 483
us as a guide through life. Religion as a sentiment is practically
universal and we may consider it to be innate. It is a panpathy or
all-feeling which produces in every individual a deep-felt longing
to be at one with the whole universe of which each is a part.1 As
every material particle is an embodiment of gravitation in propor-
tion to its weight, and is possessed of a well-apportioned pressure
somehow and bent some whither, so the souls of things existent
feel themselves parts of the great whole in which they live and move
and have their being.
This panpathy in its historical development under definite con-
ditions assumes a definite form, and so religion leads necessarily
and naturally to church life and church formation, with dogmas
and regulations of conduct.
The dogmas of the church are collected in what has been called
the symbolical books which accordingly contain the several con-
fessions of faith. They are called symbolic because they served as
symbols, or tokens of recognition to the members of the church.
The man who could recite the symbol was welcome in the congrega-
tion as a brother who cherished the same faith, having found the
same solutions of the world problem as the whole church and hav-
ing accepted the same formulation of it.
The dogma is a symbol, but it is more than a symbol ; it is an
appropriate symbol. It is a statement satisfactory to the whole
congregation and in so far as it is satisfactory to the whole con-
gregation it has become to them a truth.
Dogmas are truths. Being religious truths they are holy truths,
and since they are taken seriously, they have often become the
cause of much controversy and have led to quarrels and bloodshed,
to persecution and warfare, to the establishment of the inquisition
and denunciation of heretics. We now learn that the intellectual
feature of the dogma is derived from the main and essential feature,
its practical value. This is an enormous gain, for it introduces into
the nature of dogma a condemnation of all intolerance and estab-
lishes an unlimited freedom of interpretation without, however,
detracting a hair's breadth from the practical significance of the
dogma. Not one jot or one tittle shall pass from it, but a thinker
is allowed to construct its meaning as best he can, provided he
recognizes and holds on to its practical application.
God is our father; he is called upon in prayer as a personality
1 For a more complete definition of religion in its several phases see
Carus's Dawn of a New Era, pp. 96-97.
484 THE MONIST.
— not a human personality, but a divine personality. The inter-
pretation of personality is a problem by itself, but the significance
of the dogma "God is a person" means that we should adjust our
relation to God in such a way as to make it a personal relation,
and this practical application constitutes the primary and underived
significance of the dogma.
This view is not a loose way of treating the dogma ; for the
freedom of interpretation gives much liberty of speculation, but
not an unlimited license. It is restricted and allows the dogma to
stand and remain unalterable as the only possible, the only allow-
able, expression of a truth. Though the dogma is not absolute it
is definite, and any other formulation of it would be wrong and
must be rejected. Thus the view of dogma here represented by
M. Le Roy remains as uncompromising as ever and would not
allow any dillydallying for the benefit of speculative minds.
It will be sufficient to characterize the author's effort and the
misunderstandings created in the broad problem in his own words.
They will show first the sincerity of his undertaking and explain
the situation of his own mind, and secondly they will describe his
critics and their inability to grasp M. Le Roy's point of view. A
faithful Catholic's understanding of the nature of dogma is char-
acterized by the article itself and for a summary of this phase of
religious thought it is fully sufficient.
This is what our author says in speaking of himself (Dogme
et critique, pp. v-x) :
"On April 16, 1905, I published in the Quinsaine an article
entitled 'What is a Dogma?' in which, speaking as a philosopher
who desires to think his religion, I addressed various questions to
theologians and apologists.
"Why did I use the form of interrogation instead of a direct
exposition? In deference to those who have official charge of in-
struction. It seemed to me desirable that the reply should come
from them. In this way I hoped to manifest my intention to act
always in conformity with the hierarchical principle divinely estab-
lished in the church. Although I have scarcely been able to con-
gratulate myself on the reserve and courtesy I thus showed, since
some have been pleased to see in it only a caution lacking in cour-
age and candor, still I retain to-day the same way of looking at
things. But be assured this does not in the least mean that I
experience the slightest difficulty for my own part in reconciling
WHAT IS A DOGMA? 485
faith and reason, nor that I hesitate or doubt the least bit in the
world with regard to my duty as a Catholic.
"My aim was, briefly, to expose certain facts which I had had
the opportunity to observe around me, and also to report an ex-
perience I had had in my relations with the unbelieving intellectual
world. It was for the theologians, I thought, to declare them-
selves after discussing the plan which I submitted to them. As for
myself, I was only a witness testifying to what he had seen and
come in touch with, a Christian soul relating some of the steps it
had taken.
"This attitude has been misunderstood. It has been regarded
as craftiness or malice, as a challenge or an irony. Some one spoke
with reference to it of a question 'irreverently and even imper-
tinently stated.'2 Was not 'importunately' meant instead, without
daring to say it, or admitting it? For, I beg to inquire, how may
one set about being more deferential than I have been? Unless the
only deference that is acceptable and sufficient is the deference of
an indifferent or heedless silence. Is it true that the question asked
was indiscreet? Certain papers hastened to make the claim, and
the Si&cle for instance was much diverted at the idea of Catholics
not being able to agree on defining a dogma. These are certainly
not my own sentiments. In asking an explanation I never intended
to be, nor do I think I was, a trouble maker, disturbing slumber
or ruffling tranquillity. But words like those I have mentioned
tend to justify this ill-natured hypothesis, and therefore it is they
which in the final analysis I find lacking in courtesy.
"For my part, on re-reading what I have written and feeling
ready to write again, I declare with M. Fonsegrive:3 'Have we been
wrong in saying these things out loud and, being Catholics, in
having enough confidence in our religion, in the power of truth, to
dare speak frankly, clearly, even vigorously? Would we have
shown more regard for our beliefs if we had spoken timidly and
feebly as one speaks at the bedside of the dying?' One must indeed
stand up for oneself. We are neither dissembling Protestants nor
disreputable rationalists. We are only searching always for the
greatest religion, without concessions or haggling. We do not wish
in the least to be either rebels or even eccentric persons. But our
faith is firm enough for us not to fear to look the facts in the face
and to speak out clearly what they show us ; and we attach enough
2 La vtrite franqaise, Dec. 20, 1905. 8 Quinsainf, Jan. 1, 1906, p. 30.
486 THE MONIST.
value to the divine word to wish to think with all the strength of
our soul, assured in advance that there we will find life and light
without other limitations than our own. Moreover we feel that
we are enough protected by the living supremacy of the church to
preserve the most complete internal peace throughout our most
venturesome inquiries. We are, in fine, sure enough of our obedi-
ence to legitimate authority to have no fear in running the com-
mendable risks which the experience of life always entails. But
the obedience we intend to render is not a simple obedience of
formulas and motions, it is a profound obedience which lays hold
of our whole being, heart, will and intelligence — in short, an obedi-
ence of reasonable men and free agents, not of slaves or mutes.
"Nevertheless, as soon as the article 'What is a Dogma?' ap-
peared a vast array of controversial writings began which continued
with increasing violence during one whole year. Not only did the
reviews take part, as was their natural business, but the daily papers
as well. For after having reproached me for opening a discussion
on such a subject before a public which though educated was not
professionally qualified, they had nothing more urgent to do than to
force the discussion before the eyes of a crowd which this time
had neither proficiency nor culture. The organization of the ex-
posure was perfect and the matter was abundantly exploited by
those who make orthodoxy a monopoly or a standard and who are
always to be found upon the heels of any one who takes the liberty
of thinking for himself.
"To polemics conducted in this way I shall make no reply.
Their authors, in spite of the pretensions they parade, are repre-
sentative of nothing in the church, and as, on the other hand, they
do not discuss but condemn and anathematize, substituting injury,
slander or denunciations for arguments, they are representative of
nothing from the intellectual point of view. What separates us
from them is a question of morality much more than a question of
critique.
"Fortunately other questioners have made their voices heard,
loyal and disinterested questioners of broad minds and upright
hearts, striving to understand and seeking nothing but the kingdom
of God, the welfare of souls, the light of truth. The present volume
is dedicated to them, to them and all those, whether known or un-
known, who are like them. Is there any need of justifying oneself
otherwise than by the words of Fenelon, which he might have taken
WHAT IS A DOGMA? 487
for a motto: 'Every Christian, far from entering controversies,
ought instead to explain his position more and more to try to satisfy
those who have had trouble with the first explanation.' If this
motive is not sufficient I may add that I cannot remain indifferent
in the face of the opinions that have been attributed to me. Too
many people have become acquainted with my article only through
incomplete analyses, through prejudiced reports or through refu-
tations which may well confuse them ; it is important that I should
publish an authentic text with comments made necessary by the
publicity the controversy has attained.
"For the rest, I still retain the same attitude I had at the be-
ginning. I wish to put a question, nothing more. The accompany-
ing comments and reflections are only to elucidate the meaning and
the scope ; to show also that it has not in the least been adequately
answered ; finally to furnish a definite theme for discussion and
investigation. Who would dare to find occasion in this to accuse
me of heresy?
"And now I have finished my task on this point ; I have said
what I had to say. The question has been asked, and nothing could
prevent it from being asked. Henceforth the ideas will make their
way of themselves and nothing will stop them. Let the future
answer. Perhaps we shall soon see what has often happened be-
fore, that what was once regarded as bold and disgraceful will end
by being universally accepted as a very simple and commonplace
matter."
According to Le Roy the intellectual feature of the dogma is
not denied nor abrogated. On the contrary it remains in force and
takes about the same place in religion as the laws of nature in nat-
ural science which formulate uniformities of facts but are not the
actual phenomena as experienced. They both have their positive
significance. It seems to we that in this way this conception of the
dogma is helpful to educated people.
It is not necessary to make the interpretation of religion be-
come a product of the Aristotelian philosophy. It would change
theology into an ancilla of medieval thinking and deprive it of the
liberty to adopt the scientific spirit.
While Le Roy's theory resembles pragmatism, one cannot
characterize it as purely pragmatic, and we should consider that the
papal decree, Lamcntabilc sane exitu of July 3, 1907, condemns the
THE MONIST.
views of those who claim that dogma is exclusively a regula prae-
ceptiva actionis, and that it is not a regula fidei. Nor is Le Roy an
agnostic. He positively affirms that we can know God in relation
to ourselves, and also that we can know him as he is in se. The
essence of the dogma according to him is not exhausted in its
moral significance, but includes also the cnunciatio speculative
The distinction between the actionists and analogists is more
one of words than of actual meaning, for both agree in presenting
the truth concerning God in terms of intellectual conception and in
terms of action, and thus both sides insist on a real cognition of
God, each in his own terms. The whole controversy turns on this
question, "Is practical truth contained in the speculative, or the
spculative in the practical?" while we might say they are both two
phases of the same. p. c.
THIS title, "What is a Dogma?" is only a simple ques-
tion and by no means does it promise an answer. It
is a question from the philosopher to the theologian calling
for an answer from the theologian to the philosopher.
It would indeed be vain to pretend to give here a com-
plete and definite answer to this complex question. Such
problems cannot be solved in a few pages. Therefore the
reader must not look for a settled doctrine in the short
article which is to follow, nor even for categorical theses
on any point. If he sometimes find that I speak in too
affirmative a tone let him be kind enough to admit that I
do so only for the sake of greater clearness in my questions.
In fact I wish to confine myself to simple suggestions
which I present merely as rough drafts of solutions offered
for the criticism of those who have authority to judge of
the subject. And moreover I can justify this attitude of
mine by an imperative reason, namely that I am not a theo-
logian and do not like to decide matters in which I am not
proficient.
Perhaps some one will ask, why then do I take the
WHAT IS A DOGMA? 489
trouble to treat a subject of which I admit I have no par-
ticular knowledge ? Here is my reason. In our day every
layman is called upon to fulfil the duty of apostleship in the
incredulous world in which he lives. He alone can serve
efficiently as the vehicle and intermediary of the Christian
message to those who would not trust the priests. There-
fore it is inevitable that some problems of apologetics
should be laid before him, problems whose solution is an
absolute necessity for him if he does not wish to fail in the
task which the force of circumstances has laid upon him
without possibility of escape, if he wishes to be always
ready, following the counsel of the Apostle, to satisfy those
who ask him the reason for his faith. It is only natural
therefore that I desire to be informed; and if I formulate
my question publicly it is because I am not the only one in
this situation, and because there is a general interest that
the answer shall also be a public one.
Besides I have another motive for acting as I am. If
I freely acknowledge my incompetence in a matter which is
properly theological, yet on the other hand I consider that
I am well situated to appreciate correctly the state of mind
in contemporary philosophers that is opposed to the under-
standing of Christian truth. And it is to this that I bear
witness in saying frankly, even brutally (if I must in order
to be fully understood), what I know, what I have ob-
served, what perhaps are not always sufficiently compre-
hended, namely the exact reasons why unbelieving philos-
ophers of to-day repulse the truth that is brought to them,
and the legitimate causes (agreeing in this with the Chris-
tian philosophers themselves) why they are not satisfied
with the explanations that are furnished them.
My ambition goes no farther than to point out certain
opinions, perhaps to suggest certain reflections, especially
to particularize the statement of certain problems. If the
present work bring a useful contribution to the studies of
4QO THE MONIST.
religious philosophy, if it furnish documents and materials
which others can turn to account, I shall have attained my
end. It is not a question of upholding a system nor of
aligning arguments for or against this or that school, but
only of elucidating certain fundamental ideas whose con-
sideration is imposed upon every system and upon every
school. An effort toward light in the bosom of Catholic
truth, faithfully accepted in its completeness and rigor —
this is what I submit to the decision of those who have been
charged with the duty of defining and interpreting it.
What I desire above all, I repeat, is to make better
known the state of mind of those contemporaries who think,
the nature of the questions they ask themselves, the ob-
stacles that hinder them and the difficulties that perplex
them. It cannot be denied that the classical replies no
longer satisfy them; there is no use in disputing over so
obvious a fact. The experience of cultivated non-Christian
circles (I might even say a personal experience) has dem-
onstrated to me that the proofs brought forward as tradi-
tional have no effect on intellects accustomed to the dis-
cipline of contemporary science and philosophy. Now why
this new impotence of old methods which have sufficed so
long? The reason appears to me to be, at least in great
measure, that the old apologetics assumes the greater part
of the problems to be solved in advance which the moderns,
on the other hand, judge to be essential and primordial.
The real difficulty for the moderns comes in altogether
before the arguments begin by which the theologians flatter
themselves they can convince them ; it lies in the postulates
taken for granted and in the very manner in which the in-
vestigation is approached.
It will be well to see how the questions ought to be put
to-day ; this should be the first result to be obtained. It is
the chief result, for without it we would never arrive at
anything serious. Thus is imposed the preliminary task
WHAT is A DOGMA? 491
of coming in contact with the minds whom one wishes to
address and whom one claims to understand. It is neces-
sary that the various chapters of the apologetic should be
taken up successively from this point of view in order to be
brought to general attention; and in examining here the
idea of dogma1 I only give a first example of the kind of
work that I think ought to be generally undertaken.
Let no one think such a task profitless or superfluous.
On the contrary, nothing is of greater urgency to-day nor
of more pressing necessity. It is strange and lamentable
how little we on the Catholic side know or how greatly we
fail to appreciate the state of mind of the opponents to
whom we try to speak.2 Nor are we listened to or under-
stood. What we say has no response and carries no weight.
We exert ourselves in silence and in a void without even
giving rise to any criticism or refutation. In short we
only reach those who do not need to be reached — I mean
those who are convinced beforehand or whose difficulties
are not of a theoretical kind. We must not deceive our-
selves. Catholic thought at the present day is without
notable influence on the various intellectual movements
which are developing around us. It sometimes follows
them at a distance and after having resisted them for a
long time ; but nowhere does it appear capable of directing
them, much less of promoting them. There is nothing
more sad than to confess so many efforts expended without
result on the one hand, and on the other hand so many
sincere questions asked which remain unanswered.
Doubtless one might say, and indeed some have said,
that there is no need of taking into account modern de-
mands because they proceed from a perverted and mis-
guided judgment. Wretched subterfuge! What contem-
poraneous thought is asking for beyond what it receives
1 1 will say once for all that by "dogma" I mean especially the "dogmatic
proposition," the "dogmatic formula," not at all the reality which underlies it.
2 I would say the same, moreover, of our opponents with respect to us.
492 THE MONIST.
is perfectly legitimate, and there is no justification in pre-
tending to refuse to grant it. Men of to-day are within
their rights in not consenting to be held down to the point
of view of the thirteenth century. It would indeed be
strange if any one should ask for a proof to support a
truth of this kind.3 After all, is it not the very mission and
the raison d'etre of apologetics to address itself to the
disordered, if such there be? It must take people as they
are and not require of them that they first come of their
own accord where it may prefer. Once again, it would be
strange if one had no right to make a cure except with
certain remedies.
Hence there may be some interest and some profit in
the testimony of those whose situation has put them in a
position to know the modern mind, its needs and its re-
quirements. These may try to tell how they have come
to think what they believe, how they have succeeded in
practically overcoming, and of their own accord, the diffi-
culties that they have met like the others. I do not say
that we must accept the conclusions of their experiences
uncritically; but after all, these experiences offer the ad-
vantage of furnishing living documents, not dead opinions,
and that is something. I here make no further claim.
One more word before I begin. Perhaps the reader
will be surprised to find so long a preamble introducing so
short an article. The reason is first of all that I wished
to write a sort of general preface for other similar articles
intended to follow this one, and also because I wished in
this way to forestall any possible misunderstanding. What-
ever opinion may be held on the ideas which I shall put
forth, it must not happen that any one will try to answer
me by charging me with heresy. I affirm nothing in this
work except facts easily verifiable by everybody. As to
3 The object of faith always remains the same but not the manner of
thinking it or of complying with it.
WHAT IS A DOGMA? 493
the rest, that is to say the sketches of theories, whatever
the form of the language which I have adopted in order to
make myself clear, I give them expressly as simple inter-
rogations addressed to whomsoever they may concern. In
a word, I do nothing but state some problems ; it is for the
apologists and theologians to solve them.
* * *
We no longer live in the day of partial heresies. For-
merly a purely logical and dialectic argumentation might
suffice because certain common principles \vere always ad-
mitted on both sides. But the case is no longer the same
to-day, when these principles go by default, when the fun-
damental difficulty is to establish a point of departure upon
which both sides may agree. To-day denial does not attack
one dogma any more than another. It consists above all
in a preliminary and total demurrer. The question is not
whether a proposition is a dogma or not; it is the very
idea of dogma which is repugnant, which gives offense.
Why is that?
When we examine the ordinary motives of this repug-
nance we find four principal ones which I shall briefly
enumerate, endeavoring to present them in all their force :
i. A dogma is a statement presented as being neither
proved nor provable.4 Those who declare it to be true
declare at the same time that it is impossible ever to arrive
at the point of grasping the intimate reasons of its truth.
Now modern thought, faithful to the precept of Leibniz,
endeavors more and more to demonstrate the old so-called
axioms. At least it wishes to justify them with Kant by
a critical analysis which shows them to be necessary con-
ditions of consciousness implied a priori in every act of
reason. It is distrustful of those evidences, pretending to
be direct, which were so numerous in former times. Often
enough it discovers in them simple postulates adopted for
4 I mean here to speak of intrinsic proof.
494 THE MONIST.
an end of practical utility more or less unconsciously per-
ceived.6 In short everywhere and always it calls for long
and detailed discussions before believing itself authorized
to draw conclusions. And it is not just any more or less
roundabout proofs that it thus demands, but direct specific
proofs. It does not like too general arguments which look
upon vast assemblages as a whole and proceed by whole-
sale demonstrations, because it has had experience too
many times with the illusions, mistakes and oversights
which they ordinarily conceal. Nor does it like any better
external, extrinsic arguments which end in proofs of a
negative character, in reductiones ad absurdwn founded
on judgments of contradiction or impossibility, because it
has also had experience6 too many times with their im-
prudent and hazardous character to declare either impos-
sible or contradictory a thing which may appear so to us
only from habit. Therefore it seems that in order to re-
main faithful to the tendencies which have assured its suc-
cess in all domains modern thought can do no less than
condemn absolutely the very idea of a strictly dogmatic
proposition. In what system acceptable to reason could
such a proposition find room without violence? Is not the
first principle of scientific method incontestably, according
to Descartes, that it must hold as true only what clearly
appears to be true ? What justification would there be for
making an exception of just those propositions which pass
as the most important, the most profound and the simplest
of all ? When affirmations are of the greatest consequence
and refer to the most difficult and recondite subjects it is
certainly not fitting to show oneself less attentive to the
exactness of "the rules which constitute our protection
against error. On the contrary it is just then that it would
5 Compare the Philosophic nouvclle edition of Bergson's works.
6 Especially in the sciences.
WHAT IS A DOGMA? 495
be legitimate to be even more exacting, more scrupulous,
more particular than usual.
2. It will doubtless be said that dogmatic propositions
are never affirmed without proof. In fact an indirect dem-
onstration has been attempted over and over again. One
certain apologetic which is regarded as purely traditional7
claims to prove that these propositions are true, although
it realizes that it is incapable of bringing fully to light the
how and the why of their truth. There is some analogy,
it seems, between such a proceeding and that of the mathe-
matician who limits himself at times to the theorems of
simple existence, or that of the physicist who often accepts
facts of which he cannot give any theoretical explanation,
or yet again of the historian who always receives knowl-
edge only by the path of testimony. Thus would end the
first objection.
Yes, here we would have a very simple solution, but
there is one misfortune, namely that the analogy pointed
out proves upon reflection to be absolutely inaccurate. The
difficulty we wish to avoid reappears in toto when we try to
justify postulates on which the alleged indirect demonstra-
tion rests. When a mathematician is satisfied with estab-
lishing a theorem of simple existence,, I mean a theorem
affirming the existence of a solution inaccessible in itself,
he reasons no less rigorously than in other branches of his
science. Now here we have nothing like that. It would
be necessary to prove directly that God exists, that he has
spoken, that he has said this and that, that we possess his
authentic teachings to-day. This amounts to the same
thing as saying that the problem of God, the problem of
revelation, of the inspiration of the Bible and of the author-
ity of the church, must be solved by a direct analysis. Now
these are questions of the same kind as the strictly dogmatic
7 This method of extrinsic demonstration is regarded, as traditional. Here
is a historical point on which much might be said, but such a discussion is
foreign to my subject.
496 THE MONIST.
questions, questions with reference to which it is indeed im-
possible to produce arguments comparable to those of the
mathematician. Likewise when a physicist accepts a fact
to which he can give no theoretical explanation this fact
corresponds, at least for him, to certain definite experi-
ences, to certain manipulations that can be practically car-
ried out, in short to a group of motions of which he has
direct knowledge. What similarity is there here? And
finally even the historian does not consent to receive truth
by testimony except because he is dealing with phenomena
of the same kind as those of which he has a direct view by
some other means. He still regards his science as always
conjectural and uncertain so long as it treats of somewhat
profound causes or of events that are more or less remote.
How much more ought one draw the same conclusion in
the case of dogmas which reflect only facts that are mys-
terious, strange and disconcerting, and to which no anal-
ogy in our human experience corresponds! It has been
well done. The alleged indirect proof has inevitably for
its basis an appeal to the transcendence of pure authority.
It claims8 to introduce the truth into us fundamentally
from the outside in the fashion of a ready-made "thing"
which might enter into us forcibly. Thus any dogma what-
ever seems like a subservience, like a limit to the rights of
thought, like a menace of intellectual tyranny, like a shackle
and a restriction imposed from without upon the liberty of
investigation — all of which is radically opposed to the very
life of the spirit, to its need of autonomy and sincerity, to
its generative and fundamental principle which is the prin-
ciple of immanence.
Let us insist a little upon this last point, for the prin-
ciple of immanence has not always been rightly understood.
Too often it has been made out a monster, whereas nothing
8 Or at least appears to claim, which is the form under which it is too
often presented.
WHAT IS A DOGMA? 497
is more simple nor on the whole more clear. We may say
that to have gained a clear consciousness of it is the essen-
tial result of modern philosophy. Who refuses to admit it
is from that time forth no longer counted among the num-
ber of philosophers, who does not succeed in understanding
it indicates thereby that he has not the philosophic sense.
And this is what constitutes the principle of immanence.
Reality is not made of separate pieces put in juxtaposi-
tion, but everything is within everything else; in the
smallest detail of nature or of science analysis recognizes
all of science and all of nature. Each of our states and
of our actions comprises our entire soul and the total-
ity of its powers. Thought, in a word, is wholly in-
cluded in each of its moments or degrees. In short, there
is never for us a purely external fact like some sort of
raw material. Such a fact indeed would remain abso-
lutely unassimilable, unthinkable; it would be a nonentity
to us, for where could we take hold of it? Experience it-
self is not in the least an acquisition of "things" which
previously were entirely unknown to us. No, it is much
more a transition from the implicit to the explicit, a pro-
found movement revealing to us the latent requirements
and actual abundance in the system of knowledge already
explained, an effort of organic development, putting to use
its reserves or arousing needs which increase our activity.
Thus no truth ever enters into us except as it is postulated
by that which precedes it as a more or less necessary com-
plement; just as an article of food to become valuable as
nourishment presupposes in the one who receives it certain
preliminary dispositions and preparations, for instance, the
appeal of hunger and the ability to digest. In the same way
the statement of a scientific fact presents this character,
no fact having meaning nor, consequently, existing for us
except by a theory in which it is born.
On these various points a critical examination of the
498 THE MONIST.
sciences has recently come to confirm the reflection of the
philosophers. It is obvious that I could not enter here into
detail,9 but the little that I have said will doubtless suffice
to give a glimpse at least of how that which has been called
extrinsicism10 is opposed in spirit, attitude and method to
modern thought.
3. In spite of what we have just said let us admit, how-
ever, the instruction of dogmas by simple affirmation of a
doctrinal authority which is accepted almost without criti-
cism. Nevertheless, in order to be acceptable these dogmas
would need to be perfectly intelligible in their statements,
leaving no room for any ambiguity of interpretation or any
possibility of error with regard to their real meaning. Now
this is not the case. In the first place their formulas often
belong to the language of a particular philosophical sys-
tem which is not always easily understood, which does
not always escape the danger of equivocation or even of
contradiction. There is no doubt, for instance, that the
doctrine of the Word in origin and context is closely con-
nected with Alexandrian neo-Platonism ; that the theory
of substance and form in the sacraments and that of the
relations between substance and accidents in the dogma of
the real presence, are really closely connected with Aristo-
telian and scholastic conceptions. Now these diverse phi-
losophies are sometimes doubtful as to their basis and ob-
scure as to their expression. In any event they have long
been antiquated, fallen into disuse among philosophers and
scholars. Would it therefore be necessary, in order to be
Christians, to commence by being converted to these philos-
ophies ? This would be a difficult undertaking, before which
9 See the Bulletin de la societe franqaise de philosophie, meeting of Febru-
ary 25, 1904.
10 Blondel uses the term extrincesisme together with historicisme to denote
two kinds of apologetics which he condemns. See his article on "Histoire et
dogme" in La Quinzaine of Jan. 15, Feb. 1 and Feb. 15 of 1904.
WHAT IS A DOGMA? 499
many believers themselves would feel strangely embar-
rassed. And moreover even this would not suffice, for the
confusion of many languages resulting from heterogeneous
philosophies constitutes still another difficulty no less
troublesome than the first.
But this is not all. Aside from this, dogmatic formulas
contain metaphors borrowed from every-day matters, for
instance when they speak of the Divine Fatherhood or Son-
ship. It is impossible to give an exact intellectual inter-
pretation of these metaphors, and consequently to deter-
mine their precise theoretical value. They are images
which cannot be converted into concepts. It would require
anthropomorphism to take them literally, and at the same
time it would be difficult to give them any deep significance.
One cannot even handle them without reserve, nor follow
them to a conclusion without arriving too quickly at ridicu-
lous consequences and absurdities. Hence arises a great
uncertainty that continues to increase the confusion of im-
aginative symbols with the abstract formulas of which we
were just speaking.
After all, the first difficulty with regard to dogmas
which many people find to-day consists in the fact that they
do not succeed in discovering a thinkable meaning in them.
These statements tell them nothing, or rather seem to them
to be indissolubly connected with a state of mind which
they no longer possess and to which they think they are no
longer able to return without degenerating. Moreover
many believers are virtually of the same opinion, and pre-
fer to refrain from all reflection, foreseeing certain ob-
stacles that they would meet in thinking what they believe
under the forms laid before them. A contemporaneous
philosopher has said: "What would most embarrass the
greater number of believers would be if, before asking
them for a proof of what they believe, one were simply to
5OO THE MONIST.
call upon them to define exactly what it is they affirm and
what they deny."1
4. Finally, let us pass over these difficulties. Even after
they are disposed of there still remains a last objection
which seems very grave, namely that in any event dogmas
form a group incommensurable with the whole of positive
knowledge. Neither by their content nor by their logical
nature do they belong to the same system of knowledge as
other propositions. They therefore could not be arranged
with others in a way to form a coherent system, so that if
one accepts them the result is an inevitable breach of unity
in the mind, a disastrous necessity of playing a double part.
Being unalterable they appear foreign to progress, which
is the very essence of thought. Being transcendent they
exist without relation to effective intellectual life. They
bring no increase of light to any of the problems which
occupy science and philosophy. Thus the least reproach
that one can cast upon them is that they seem to be without
profit, to be useless and barren — a very grave reproach in
a period when it becomes more and more perceptible that
the value of a truth is measured above all by the services
that it renders, by the new results that it suggests, by the
consequences which it brings forth, in short by the vivi-
fying influence it exerts on the entire body of knowledge.
Such, briefly summed up, are the principal reasons why
the idea of any dogma whatever is repugnant to modern
thought. I have endeavored to present them in all their
force, taking the same point of view in setting them forth
as those who regard them as conclusive and speaking, so
to say, not in my own name but in theirs. It remains now
to investigate some conclusions and some lessons which we
ought to be able to derive from them.
11 Belot, Bibliotheque du congrcs international de philosophic de 7900. Paris :
Armand Colin.
WHAT IS A DOGMA? $01
These reasons, it must be recognized, are perfectly
valid. I do not see any legitimate way of refuting the
preceding line of argument.12 The principles which it in-
vokes seem to me no more contestable than the deductions
which it draws from them. In fact I do not see that it has
ever been answered except by worthless subtleties or rhe-
torical artifices.13 But eloquence is not a proof, neither is
diplomacy. Hence our only real resource is to prove that
the idea of dogma which is condemned and rejected by
modern thought, is not the Catholic idea of dogma.
Perhaps it will be found that in speaking in this way
I depart from the role in which I have promised to confine
myself, that this time decidedly I am stating theses and
not asking questions. This would be a mistake. There is
no doubt that I am affirming something here, but what?
Nothing but facts. It is a fact that the unbelievers of
to-day are halted in the face of dogmas by the foregoing
objections. It is also a fact that whoever (even among
believers) has truly comprehended the spirit and the meth-
ods of contemporary science and philosophy, cannot but
give his assent to these objections. Now please note : those
very people who submit most completely and most cordially
to the authority established over them could not be affected
by it. No authority indeed could bring it about or prevent
that I find an argument valid or weak, nor especially that
this or that notion has or has not any meaning for me.
I not only say that no authority has any right in the world
to do so, but that it is absolutely impossible; for after all
it is I who do the thinking and not the authority that thinks
for me. No argument could prevail against this fact. I
can neither force myself to feel satisfaction nor prevent
myself from feeling it at the evidence on one side or an-
12 I say refuting, but it could be cut short by destroying the postulate which
is its root.
13 It would be interesting to enter into a detailed discussion of these an-
swers, but there is no room for it here.
5O2 THE MONIST.
other. To be sure I admit that authority imposes upon me
this or that belief with the result that it makes me follow
this or that line of conduct, but how could it compel me by
virtue of such a proof to believe what I do not regard as
convincing ? And how would I be able to obey it if it com-
manded me to understand this or that declaration which I
did not understand at all? As well might it require me
to cease thinking. No reason can be founded on faith.
Here we have an identity pure and simple. There is no
such thing as revealed logic.
Hence I come back to what I said a while ago, and,
speaking as a philosopher, I declare myself incapable of
thinking differently from our adversaries on the above-
mentioned points.
Moreover in making this declaration I consider that I
am doing nothing but stating a problem. The state of
mind which I have described exists, it is triumphant to-day ;
even those who belie've the most firmly share it. These are
the facts which it is impossible not to take into account and
which constitute, I repeat, the statement of a question to
be solved. Let us see exactly what this question is.
I shall henceforth regard it as granted that the objec-
tions summed up above cannot be evaded so long as the
idea of dogma which they contain is preserved. Does this
mean that we must conclude definitely that there is an
absolute incompatibility between the idea of dogma and
the essential conditions of reasonable thought? That in
order to think as a Christian it is necessary to cease think-
ing altogether? I certainly do not believe so. But to
avoid the objections in the case and to obtain the desired
harmony I ask myself if it is not the very manner in which
the idea of dogma is presented that is the real cause of the
contention, and if consequently we have not reason to
change this manner.1*
14 I beg the reader to give heed to the limits within which this question is
WHAT IS A DOGMA? 503
Now when we examine the conception of dogma which
the four objections above enumerated assume and imply,
we are surprised to find that it is common to the greater
number of Catholics and their opponents. It is a distinctly
intellectual conception. It regards the practical and moral
meaning of the dogma as secondary and derived and places
in the first rank its intellectual meaning, believing that this
constitutes the dogma whereas the other is merely a con-
sequence of it. In a word, it makes of a dogma something
like the statement of a theorem — an intangible statement
of an undemonstrable theorem, but a statement having
nevertheless a speculative and theoretical character and
relating above all to pure knowledge. This is the common
postulate that one discovers by analysis at the foundation
of both of the two opposed doctrines, the one that accepts
and the one that rejects the idea of dogma. Here I believe
is the crux of the difficulty. From this unexpressed postu-
late and from the conception which flows from it originate,
in my opinion, both the abuses to which the idea of dogma
can give rise and the conscientious objections that it raises.
Indeed it is inevitable that one would finally draw the con-
clusion that all dogma was illegitimate, for he would at
the same time define it as a theoretical statement while
nevertheless attributing to it characteristics the very oppo-
site of those which make statements correct. It is very
curious that the apologists are not more often informed of
a fact of such great importance as that their conception
of dogma would destroy in advance the theses that they
wish to establish. On the other hand, the same intellec-
tualist idea of dogma leads to two very regrettable and un-
fortunately very frequent exaggerations; one consists of
confusing dogmas properly so called with certain opinions
comprised. It does not discuss in any way the modification of the content
of dogma, nor even its traditional religious interpretation, but only the deter-
mination of the modality of the dogmatic judgment and of the qualification it
possesses.
504 THE MONIST.
and certain theological systems, that is to say, with intel-
lectual accessory representations; the other, in failing to
see that a dogma could never possess any scientific signifi-
cance and that there are no more dogmas concerning for
instance biological evolution than there are concerning the
movements of planets or the compressibility of gas.
From a thorough study of these various points we reach
the conviction that the problem of dogma is usually badly
stated;16 and perhaps we will see at the same time how it
ought to be stated in order to render possible a satisfactory
solution.
* * *
From this point I enter at once into the domain in which
I must keep myself in an interrogatory attitude. This is
my definite intention although to insure clearness I may
keep the didactic tone. What follows must be taken as a
simple exposition of what I ordinarily reply to those who
ask me what I think of the idea of dogma. Am I wrong
to speak in this way ? I am quite ready to acknowledge it if
any one will show me that it is not the right way in the
eyes of the church.
First of all I say that a dogma cannot be compared to
a theorem, of which we only know the statement without
its proof and whose proof can only be guaranteed by the
assertion of a teacher. Nevertheless I know that this is
the most common conception. We like to think of God in
the act of revelation as a very wise professor whose word
we must believe when he communicates to his audience
results whose proof that audience is not capable of under-
standing. But this appears to me to be hardly satisfactory.
We say that God has spoken. What does the word "speak"
mean in this case? Most certainly it is a metaphor. What
is the reality which it conceals? Herein lies the whole
difficulty.
15 At least in books in current use and in elementary education.
WHAT IS A DOGMA? 505
Without recurring to the general considerations I have
already developed let us take some examples that will serve
to specify what we have hitherto looked upon only in large
outlines.
"God is a person." Here we have a dogma. Let us
try to see in it a statement having above all an intellectual
meaning and a speculative interest, a proposition belong-
ing first of all to the order of theoretical knowledge. I
' pass over the difficulties aroused by the word "God," but
let us consider the word "person." How must we under-
stand it?
If we grant that the use of this word bids us conceive
the divine personality in the form shown to us by psycho-
logical experience on the model of what common sense
designates by the same name, as a human personality,
idealized and carried on to perfection, we have here a
complete anthropomorphism, and Catholics would certainly
agree with their opponents in rejecting such a conception.
Moreover to carry such a thought to its extreme limits is
a very delicate thing, very likely to induce error or at least
mere verbiage, incapable in any event of producing any-
thing more than very vague metaphors and perhaps even
eventually contradictory results.
Shall we limit ourselves to saying that the divine per-
sonality is essentially incomparable and transcendent ? Very
well, but if so it is very badly named, and in a way which
seems made expressly to induce delusion. For if we de-
clare that the divine personality does not resemble in any
respect that with which we are acquainted, what right
have we to call it "personality"? Logically it should be
designated by a word which would belong only to God,
which could not be employed in any other instance. This
word would therefore be intrinsically undefinable. Let us
imagine any assemblage whatever of syllables deprived of
all possible significance. Let A be this assemblage. Then
506 THE MONIST.
by our hypothesis "God is a person" does not have any
other meaning than "God is A." Is this an idea?
The dilemma is unsolvable for any one who is seeking
an intellectualist interpretation of the dogma "God is a
person." Either he will define the word "personality," and
then he is fatally sure to fall into anthropomorphism; or
he will not define it, and then he will fall none the less
fatally into agnosticism. Here we have a circle.
The same remarks hold with regard to the propositions
"God is conscious of himself ; God loves, wills, thinks, etc."
Let us take another example, the resurrection of Jesus.
If this dogma, whatever may be eventually its practical
consequences, has for its first aim to increase our knowl-
edge in guaranteeing to us the accuracy of a certain fact,
if it constitutes before all a statement of an intellectual
character, the question to which it first gives rise is this:
What precise meaning does it assume is to be attached to
the word "resurrection" ? Jesus, after having experienced
death, has once more become alive. What does this mean
from the theoretical point of view? Doubtless nothing
except that after three days Jesus reappeared in a state
identical with that in which he was before he was nailed
on the cross. Now the Gospel itself tells us exactly the
opposite. The resurrected Jesus was no longer subject to
ordinary physical or physiological laws; his "glorified"
body was no longer perceptible in the same conditions as
before, etc. What does this mean? The idea of life has
not the same content when applied to the period preceding
the crucifixion as to that which followed it. Now what
does the word represent with relation to this second period?
Nothing that can be expressed by concepts. It is simply a
metaphor which cannot be converted into specific ideas.
Here again, to be exact, it would be necessary to create a
new word, a word reserved for this single case, a word
WHAT IS A DOGMA ? 507
consequently to which it would not be possible to give any
regular definition.
Let us borrow a final example from the dogma of the
real presence. Here it is the term "presence" which must
be interpreted. What does it usually signify? A being- is
said to be present when he is perceptible, or when though
he himself cannot be grasped by perception he yet manifests
himself by perceptible effects. Now according to the dogma
itself neither of these two circumstances is realized in the
case in hand. The presence in question is a mysterious
presence, ineffable, unique, without analogy to anything
that one ordinarily understands by that name. Now I ask
what idea is there here for us? A thing that can neither
be analyzed nor even defined could not be called an "idea"
except by an abuse of the word. We wish a dogma to be a
statement of an intellectual order. What does it state?
It is impossible to say exactly. Does not this fact condemn
the hypothesis?
Finally the pretension of conceiving dogmas as state-
ments whose first function would be to communicate cer-
tain theoretical bits of knowledge would run against im-
possibilities on every hand. It seems to end inevitably in
reducing dogmas to pure nonsense. Perhaps it must for
this reason be resolutely abandoned. Let us therefore see
what different kind of significance remains possible and
legitimate.
* * *
First of all, if I do not deceive myself, a dogma has a
negative meaning. It excludes and condemns certain er-
rors instead of positively determining the truth.16
Let us once more take up our former examples. We
shall first consider the dogma "God is a person." I nowhere
see in it any definition of the divine personality. It teaches
16 We shall shortly see how dogmas are more and greater than this. But
at the start I shall place myself in a strictly intellectualist point of view.
508 THE MONIST.
me nothing about that personality. It does not reveal its
nature to me nor furnish me with any explicit idea. But
I see clearly that it tells me, "God is not impersonal" ; that
is to say, God is not simply a law, a formal category, an
ideal principle, an abstract entity, any more than he is a
universal substance, or some unknown cosmic force dif-
fused throughout the world. In short, the dogma "God is
a person" does not bring to me any new positive conception
nor does it any more guarantee to me the truth of any par-
ticular system among those which the history of philosophy
shows to have been successively proposed, but it warns me
that this or that form of pantheism is false and ought to
be rejected.
I would say the same with regard to the real presence.
The dogma does not tell me any theory about that presence,
it does not even teach me in what it consists ; but it tells me
very clearly that it must not be understood in such or such a
way as were formerly proposed, that for instance the con-
secrated host must not be regarded solely as a symbol or
a figure of Jesus.
The resurrection of Christ gives rise to the same re-
marks. This dogma does not teach me in any degree what
was the mechanism of this unique fact nor of what kind
the second life of Jesus was. In short it does not communi-
cate a conception to me. But on the contrary it excludes
certain conceptions that I might be tempted to make. Death
has not put an end to the activity of Jesus with reference
to the things of this world. He still mediates and lives
among us, and not at all merely as a thinker who has dis-
appeared and left behind a rich and living influence and
whose work has left results through the ages ; he is literally
our contemporary. In short, death has not been for him,
as it is for ordinary mortals, the definite cessation of prac-
tical activity. This is what the dogma of the resurrection
teaches us.
WHAT IS A DOGMA? 509
Shall I insist further? It does not seem advisable at
this time. The foregoing examples are sufficient to make
the principle of interpretation that I have in mind clearly
understood. Of course long expositions would be neces-
sary if we would enumerate in detail all the consequences
of this principle and all its possible applications, and an
enumerative study of the different dogmas would therefore
become indispensable. But this is not my real purpose. I
wish to confine myself simply to indicating an ideal. This
is why I do not undertake either to multiply examples or
even to develop any one of them completely.
Moreover the idea is not a new one. It belongs to the
most authentic tradition. Is it not indeed the classical
teaching of theologians and scholars that in supernatural
matters the surest method of investigation is the via nega-
tionist Permit me to recall in this connection a well-known
text of St. Thomas : "But the via remotionis is to be used
chiefly in considering divine substance. For divine sub-
stance by its immensity exceeds every form which our mind
can touch ; and so we cannot grasp it by knowing what it
is, but some sort of a notion of it we have by knowing what
it is not/'17
Nevertheless I ought to point out one objection which
might occur to the mind. We will easily grant that the
dogmatic formulation promulgated by the church in the
course of history has especially a negative character, at
least when looked upon from an intellectual point of view
as we are doing at this time. In fact, the church itself
declares that its mission is not in the least to produce new
revelations but only to maintain the depositum revelationis,
and the negative method here adopted is entirely suitable
for this mission. And yet, of what does this depositum
17 "Est autem via remotionis utendum praecipue in consideratione divinae
substantiae. Nam divina substantia omnem formam, quam intellectus noster
attingit, sua immensitate excedit; et sic ipsam apprehendere non possumus
cognoscendo quid est, sed aliqualem ejus habemus notitiam cognoscendo quod
non est." — Contra Gentiles, I, xiv.
5IO THE MONIST.
consist if not of a certain collection of original affirmations?
Take the primary expression of Christian faith, the Credo.
What could be more positive? Now here is the basis of
doctrine, that which characterizes and constitutes it. More-
over when we say "revelation" we certainly say affirmation
and not negation.
Certainly we do. I do not contradict it in the least,
but we must make a distinction. The creed of Nicaea and
Constantinople contains many traces of a negative dog-
matic elaboration: for instance, on the divinity of Christ
as against the Arian heresy; on the procession of the Holy
Ghost in opposition to the Macedonians, etc.18 Consequently
there is nothing on this head to contradict our conclusions.
It is only the grammatical form which is affirmative here ;
in reality we are treating of errors to be excluded rather
than theories to be formulated. But let us take the Apos-
tles' Creed. Here indeed we have nothing negative but
neither do we have anything properly intellectual and theo-
retical, nothing which belongs properly to the order of
speculative knowledge, nothing in short which resembles the
statement of theorems. It is a profession of faith, a declara-
tion of attitude. We shall soon examine dogmas from this
practical point of view (which I hasten to say is in my
eyes the principal point of view), yet we shall stop a mo-
ment at the intellectual point of view. The Apostolic Creed
in its original form affirms the existence of realities of
which it gives not even a rudimentary representative the-
ory, hence its only role with reference to abstract and
reflective knowledge is to state objects and therefore prob-
lems. Finally we see that the proposed objection is not
valid and we can maintain our thesis until further notice.
Thus in so far as they are statements of a theoretical
order dogmas have all a negative meaning. History proves
18 It would be easy to insist on the example of consubstantialem or of
Filioque.
WHAT IS A DOGMA? 51 1
this when it procures our assistance at the birth of one
after another of them in relation to the several heresies.19
The rise of all dogmas has always followed the same
course, has always presented the same phases: at the be-
ginning purely human speculations, some explanatory sys-
tems very similar to other philosophical systems, in short,
attempts at theories relating to religious facts, to mysteri-
ous realities experienced by Christendom in its practical
faith; then only come the dogmas for the purpose of con-
demning certain of these attempts, of taxing certain of
these conceptions with error and of excluding certain of
these intellectual representations. Hence it follows that
dogmatic formulas often borrow expressions from different
philosophies without taking the trouble to fuse together
and unify these heterogeneous languages.
This offers no more disadvantages than does the use
of concepts derived from different origins, from the mo-
ment that dogmas do not tend to constitute by themselves
a rational theory, an intelligible system of positive affir-
mations, but confine themselves to opposing certain excep-
tions to certain hypotheses and conjectures of the human
mind. On the other hand it is natural that each dogma
should put itself in the point of view belonging to the doc-
trine that it lays under an interdict, in order to attack it
directly without danger of ambiguity. Hence it also fol-
lows that dogmatic formulas can enact laws on the incom-
parable and the transcendent and yet not fall into the con-
tradictions of anthropomorphism or of agnosticism. It is
man who with his opinions, his theories and systems, gives
to dogmas their intelligible substance;20 these are confined
to pronouncing a veto at times, to declaring at times that
19 Compare the usual formula of the decrees of council : "5"* quis dixerit. ..,
anathema sit."
20 From the theoretical point of view, understand. Dogmas are thought
in terms of the human systems which they oppose. [This view is recently en-
dorsed by Catholic theologians of such recognized authority as Cardinal Billot.
S.J.-Tr.]
512 THE MONIST.
"such an opinion, such a theory, such a system, is not al-
lowed," without ever pointing out why they should not be
accepted, nor by what they must be replaced. Thus nega-
tive dogmatic definitions do not limit knowledge nor put an
end to progress ; in short they only close up false paths.
From the strictly intellectual point of view it seeems
to me that dogmas have only the negative and prohibitive
sense of which I speak. If they formulated absolute truth
in adequate terms (to assume that such a fiction has a
meaning) they would be unintelligible to us. If they gave
only an imperfect truth, relative and mutable, they would
not be justified in obtruding themselves. The only radical
way to put an end to all the objections on principle against
dogma is to conceive of them, as we have already said, as
being undefinable in so far as they are speculative propo-
sitions, except with relation to previous doctrines upon
which they promulgate an unwarranted judgment. More-
over is it not the teaching of theologians, including the
most intellectualist, that in a dogmatic statement the rea-
sons which can be incorporated in the text are not in them-
selves objects of faith imposed upon belief?
There is one important consequence resulting from the
foregoing, namely, that the true method of studying dog-
mas (from the intellectual point of view, understand) is
the historical method. The science known as positive the-
ology, or rather the history of dogma, seeks to perform
this task. The method has an effective apologetic value
much greater than purely dialectic dissertations. Because
in any event it is impossible to comprehend dogmatic state-
ments, there is the greater reason for justifying them if one
would commence by plunging them once more into their
natural historical environment without which their authen-
tic meaning becomes more and more vague and finally ends
by vanishing entirely.
Nevertheless dogmas do not have merely a negative
WHAT IS A DOGMA? 513
meaning, and even the negative meaning that they offer
when regarded from a certain direction does not constitute
their essential and primary significance. This is true be-
cause they are not merely propositions of a theoretical
character, because they must not be examined solely from
the intellectual point of view, from the point of view of
knowledge. This is what we shall now elucidate further.
Here more than ever I insist that the intention and
tendency of the pages to follow must not be misunderstood.
I repeat that the affirmative tone is used only as a means
for clearness. At bottom the question is always the same
as I specified at the beginning. Here, if I may say so, is
the form in which experience has shown me that the notion
of dogma is most easily assimilable to the minds of to-day:
A dogma has above all a practical meaning. It states
before all a prescription of a practical kind. It is more
than all the formula of a rule of practical conduct. This
is its principal value, this its positive significance. This
does not mean, however, that it must be without relation to
thought, for ( i ) there are also certain duties concerned
with the act of thought; (2) it is virtually affirmed by the
dogma itself that under one form or another reality con-
tains wherewith to justify the prescribed conduct as rea-
sonable and wholesome.
I take pleasure in quoting in this connection the follow-
ing passage from R. P. Laberthonniere:21 "Dogmas are
not simply enigmatical and obscure formulas which God
has promulgated in the name of his omnipotence to mortify
the pride of our spirits. They have a moral and practical
meaning; they have a vital meaning more or less accessible
to us according to the degree of spirituality we possess."
After all, when converts, in spite of good intentions,
21 Essais de philosophic religieuse, p. 272. Paris : Lethielleux.
514 THE MONIST.
themselves create part of the theoretical difficulties under
discussion do we not answer them daily: "Never mind all
that, it is not important. Do not believe that God requires
so many formalities. Come to him fairly, frankly, simply,
according to the wise words of Bossuet. Religion is not
so much an intellectual adherence to a system of speculative
propositions as it is a living participation in mysterious
realities." Why not then make theory agree with practice?
Let us keep the same examples. They represent well
enough the different types of dogmas. "God is a person"
means, "Conduct yourself in your relations to God as in
your relations with a human person." Likewise "Jesus
has risen" means, "Be in relation to him as you would
have been before his death, as you are with a contem-
porary." In the same way again the dogma of the real
presence means that one must have the same attitude
toward the consecrated host as one would have toward
Jesus had he become visible, and so on. It would be easy
to multiply these examples, and also to develop each of
them farther.22
That dogmas can and ought to be interpreted in this
way there is no doubt, and the fact will not be contested
by any one. In fact, it cannot be repeated too often that
Christianity is not a system of speculative philosophy but
a source and regimen of life, a discipline of moral and
religious action, in short the sum total of practical means
to obtain salvation. What then is surprising in the fact
that its dogmas primarily concern conduct rather than pure
reflective knowledge?23
I do not think it is necessary to insist farther upon this
22 I do not claim in the least that the foregoing comments exhaust the
meaning of the dogmas mentioned : they will suffice to point out a line of
inquiry.
28 This is why assent to dogmas is always a free act and not the inevitable
result of a compelling line of argument.
WHAT IS A DOGMA? 51$
point, but I wish to indicate in a few brief words the most
important consequences of the principle here laid down.
First of all it is clear that the general objections
summed up at the beginning of this article do not affect
this conception of dogma to the same extent and in the
same degree as they do the usual intellectualist conception,
for that provokes the conflict and renders the difficulty
insurmountable, whereas on the other hand we may now
catch a glimpse of a possible solution. As there is no ques-
tion of obtaining a theoretical statement in conditions rad-
ically opposed to those prescribed by scientific method, we
no longer find ourselves face to face with a logical stum-
bling-block but only with a problem referring to relations
between thought and action — a difficult problem certainly,
but not unapproachable and one which at any rate does not
appear absurd after it is stated.
Of course there are always important questions to be
solved. It is necessary to supply the dogma in some way
with a demonstration and justification, and this is by no
means a perfectly easy matter. Nevertheless one of the
greatest obstacles has been smoothed away. Practical
truths are established differently from speculative truths.
Recourse to authority which is entirely inadmissible in the
realm of pure thought seems a priori less shocking in the
domain of action, because if authority has legitimate rights
anywhere it certainly has in the domain of practical affairs.
The Council of the Vatican tells us: "If any shall say
that no true mysteries properly so-called are contained in
divine revelation, but that all the dogmas of faith can be
comprehended and demonstrated through reason duly per-
fected by natural principles, let him be anathema."24 Now
if faith in dogmas were first of all knowledge, an ad-
herence to some statements of an intellectual kind, one
24 "Si quis dixerit in revelatione divina nulla vera et proprie dicta mys-
teria contineri, sed universa fidei dogmata posse per rationem rite excultam a
naturalibus principiis intelligi et demonstrari, anathema sit."
5l6 THE MONIST.
could not comprehend either that assent to unsolvable mys-
teries could ever be legitimate or even simply possible, or
in what it might consist, or what sort of utility or value it
might have for us, or how it might constitute a virtue. On
the other hand all this can be understood if faith in dogmas
is a practical submission to commandments which have to
do with action. Nothing is more normal than activity plac-
ing mysteries before intelligence.25
The Council of the Vatican tells us further: "If any
shall say that assent to the Christian faith is not free. . . .
let him be anathema."2 This text is generally explained
by recognizing that the reasons for believing, the motives
of credibility, are not of insuperable force, a mathematical
evidence, and that in consequence a decisive act of the will
or of the heart is always necessary to conclude the investi-
gation definitely. Is this not virtually admitting that one
cannot see in belief in dogmas an act which should first of
all be intellectual without making it thereby inferior to the
ordinary acts of thought? How would such an act — an
act performed under conditions contrary to the nature of
thought — be even legitimate or merely possible? But on
the other hand it is easy to believe that the practical ac-
ceptance of commandments relating to action depends on
our free will and gains in perfection by not being able to
manifest itself by necessary consequence. Let us insist a
little upon this point, for it is of highest importance in the
problem of the relations between reason and faith.
From the beginning apologetics is confronted with a
grave difficulty which perhaps cannot always be satisfac-
torily disposed of. On the one hand it is clearly understood
that an act of faith is a free act and that its object, as well
as its supreme motive, is supernatural. But on the other
25 Submission to dogmas then from one point of view is for the believer
what submission to facts is for the scholar.
28 "Si quis dixerit assensum fidei Christianae non essc liberum , ana-
thema sit."
WHAT IS A DOGMA? 517
hand an act of reason ought to precede and prepare the act
of faith, for it is reason alone by which the obligation and
necessity of overreaching reason can be recognized. And
an act of reason must also constantly accompany the act of
faith, for it is necessary that the human mind shall have
some sort of hold upon the dogma if it wishes to ac-
cept it. St. Thomas said well: "Those things which are
under the faith .... no one would believe unless he sees
they ought to be believed."27
Now how shall we reconcile these two opposite require-
ments in a system of intellectualist interpretation? Either
we would maintain (as there are some who do) that the
apologetic proofs are absolutely positive and exact; and
then what would become of the liberty of the act of faith ?
Or in order to safeguard that liberty we would call them
insufficient and only more or less probable; and then our
faith would lack any basis, for after all an insufficient
proof is not an acceptable proof, especially in so important
and difficult a matter. An intellectualist attitude becomes
disarmed in the face of this dilemma since liberty does
not belong to the domain of pure intelligence and has no
place or part in the proceedings of discursive reason. But
with the other attitude the dilemma can be solved because
this time the dialectic in the case is action and life not
simply argument, and liberty revives with life and action.
Likewise we have here the objection relating to the
intelligibility of dogmatic formulas. Although these for-
mulas are hopelessly obscure, even inconceivable, when we
want them to furnish positive determinations of truth from
a speculative and theoretical point of view, they neverthe-
less show themselves capable of clearness if we are careful
not to- ask of them anything but instruction as to practical
conduct. What difficulty, for instance, do we find in under-
standing the dogmas of the divine personality, of the real
27 "Ea quae subsunt fidei. . . . nemo crederet nisi videret ea esse credenda .*'
5l8 THE MONIST.
presence, or the resurrection in the practical system of
interpretation just outlined? Although these dogmas are
mysteries for the intelligence that demands explanatory
theories they are nevertheless susceptible of perfectly clear
statement as to what they prescribe for our actions. Hence
the language of common sense has its place as well as the
use of anthropomorphic symbols and the employment of
analogies or metaphors, and neither the one nor the other
gives rise to unsolvable complications since this time it is
a question only of propositions relating to man and his
attitudes.
We also see now what the relation is between dogmas
and efficient life. We predict for them a possibility of ex-
perimental study and of gradual research which has here-
tofore escaped us. Finally we understand how they can be
common to all, accessible to all, in spite of the inequality
between intellects, whereas to conceive them in the intel-
lectualist way one would be inevitably led to make a dis-
tinction of an intellectual aristocracy. I have not room
here to develop these different considerations as much as
I should like, but I imagine that a simple indication after all
may be sufficient for the time being, and that the reader
can carry the process on for himself without any difficulty.
Nevertheless it seems necessary to me to prevent a possible
objection in order to avoid all misapprehension.
I have spoken of practice. This word must be rightly
understood. I take it in the widest acceptation of the
term. Action and life are here synonymous. Hence the
word does not in the least mean a blind step, without rela-
tion to thought or consciousness. In fact there is an act
of thought which accompanies all our actions, a life of
thought which mingles throughout our life ; in other words,
to know is a function of life, a practical act in its way.
This function, this act, is also called experience, a name
which indicates at the same time that we are not at all
WHAT IS A DOGMA?
dealing with actions performed without any sort of light
but that the light in question is not that of simple argumen-
tative reason.
I have also spoken of the activity which places mys-
teries before intelligence, and by way of elucidation I have
cited the example of scientific facts. To comprehend what
I mean by this, one must not forget that a scientific fact is
not a thing to be submitted to passively. If there is any
semblance of a purely external fact, of a mystery totally
opaque, of a violent commandment from without, it is so
with respect to argumentative understanding. But the
thought-action of which I was just now speaking avoids
this appearance. It infinitely exceeds the purely intellec-
tual thought. I have not heard anything to affirm other-
* 9R
wise.28
Hence there is a necessary relation between dogmas
and thought. It is at the same time both a right and a
duty not to be content with a blind belief in dogmas but to
strive also in proportion to one's strength to think them.
The system of separation, of tight partitions, of the twofold
accountability of conscience, is not desirable nor, to speak
truthfully, possible. It is contrary to the demands of that
faith which wishes to hold every man ; it is contrary to the
requirements of philosophy which desires a spiritual unity;
and finally it is contrary to the requirements of morality
which cannot approve an action that is systematically un-
considered.
But thought when applied to dogmas should not mis-
understand their primarily practical meaning. The path
to be followed is the test of practical experience and not
an intellectual dialectic. The inspiring principle is per-
fectly expressed in the sacred word, qui facit veritatem
venit ad lucem.
28 The reader who desires to pursue this point further may refer to several
articles I have published since 1889 in the Revue de metaphysique et de morale,
and in the Bulletin de la societe fran^aise de philosophic.
52O • THE MONIST.
Thus translated into terms of action the traditional
methods of analogy and eminence assume a very clear sig-
nificance. Under the guise of metaphors and images they
affirm that supernatural, reality contains the wherewith to
make obligatory by law that our attitude and our conduct
with regard to it should have such or such a character.
The images and metaphors — which are hopelessly vague
and fallacious when one tries to see in them any approxi-
mation whatever of impossible concepts — become on the
other hand wonderfully illuminating and suggestive after
one looks to find in them only a" language of action trans-
lating truth by its practical echo within ourselves.
It remains finally to specify the relations of dogmas,
understood in the way we have described them, to theo-
retical and speculative thought, to pure knowledge. In
what respect do they govern our intellectual life ? How does
their intangible and transcendent character leave the full
liberty of research intact as wfell as the undeniable right of
the mind to repulse every conception which tries to impose
itself from without? We shall easily see%
The Catholic is obliged to assent to the dogmas without
reservation. But what is thereby imposed upon him is not
in the least a theory, an intellectual representation. Such
a constraint indeed would inevitably lead to undesirable
consequences : ( I ) The dogmas would in that case . be
reduced to purely verbal formulas, to simple words whose
repetition would constitute a sort of unintelligible com-
mand; (2) Moreover these dogmas could not be Common
to all times nor to all intelligences.29
No, dogmas are not at all like that. As we have seen,
their meaning is above all practical and moral. The Cath-
olic, obliged to accept them, is not restrained by them ex-
cept as regards rules of conduct, not as regards any par-
29 In the two words "esotericism" and "Pharisaism" would be the inevi-
table double rock upon which they would split.
WHAT IS A DOGMA? 521
ticular conceptions. Nor is he condemned to accept them
as simple literal formulas. On the contrary, they offer
him a very positive content, explicitly intelligible and com-
prehensible. I will add that this content, having to do
solely with the practical, is not relative to the variable
degree of intelligence and knowledge; it remains exactly
the same for the scholar and the ignorant man, for the
exalted and the lowly, for the ages of advanced civilization
and for the races still in barbarism. In short it is inde-
pendent of the successive states through which human
thought passes in its effort toward knowledge, and thus
there is only one faith for everybody.
This being granted, the Catholic after having accepted
the dogmas retains full liberty to make for himself what-
ever theory/ or whatever intellectual representation he
wishes of the corresponding objects — the divine personal-
ity, the re'a) presence, or the resurrection, for instance. It
remains wtfti him to grant his preference to the theory
which best agrees with his own views, to the intellectual
representation which lie deems the best. His position in
this respect rs the same as that toward any scientific or
philosophical speculation, and he is free to adopt the same
attitude in both cases. Only one thing is imposed upon
him, only one obligation is incumbent upon him ; his theory
must justify the practical rules expressed by the dogma,
his intellectual representation must take into account the
practical edicts prescribed by the dogma. Thus in a word
it appears almost like the statement of a fact with regard
to which it is possible to construct many different theories
but which every theory must take into account, like the
expression of a truth many of whose intellectual represen-
tations are legitimate but of which no explanatory system
can well be independent.80
30 It is at this point that we must distinguish between intellectual formula
and the underlying reality in the dogma.
522 THE MONIST.
From this naturally follows the step that we have rec-
ognized as usual with religious thought in its effort at
elaboration. Let us take any dogma whatever, Divine Per-
sonality, the real presence or the resurrection of Jesus. By
itself and in itself it has only a practical meaning. But
there is a mysterious reality corresponding to it and there-
fore it presents to the intelligence a theoretical problem.
The human intellect at once takes possession of this prob-
lem; and obeying simply and solely the laws of its own
nature it imagines the explanations, the answers, the
systems codified in the precepts of scientific method and
the principles of reason.31 As long as the theory con-
structed in this way respects the practical significance of
the dogma it is given carte blanche. Hence to pass judg-
ment on the theories remains the task of pure human specu-
lation, and any authority exterior to the thought itself has
neither the right nor the power to interfere.82 But once
let a theory arise which makes an attack on dogma in its
own domain by altering its practical significance, and the
dogma would immediately array itself against it and con-
demn it, thus becoming a negative intellectual statement
superimposed upon the rule of conduct which at first it
was, purely and simply.
Hence one sees positively how the two meanings of a
dogma, the practical meaning and the negative meaning,
are reunited, the latter being subordinated to the former.
Moreover we see how dogmas are immutable and yet how
there is an evolution of dogmas. What remains constant
in the dogma is the orientation that it gives to our practical
activity, the direction in which it inflects our conduct. But
31 In this respect the Middle Ages had an independence and a boldness
which we have forgotten.
32 Religious authority which has souls in its charge can indicate certain
theories as dangerous, as long as they run the risk of being wrongly under-
stood and thus of reacting injuriously upon conduct. Hence arise censures
of an inferior note to those of heresy. But these condemnations are not prop-
erly dogmatic.
WHAT IS A DOGMA? 523
the explanatory theories, the intellectual representations,
change constantly in the course of the ages according to
individuals and epochs, freed from all the fluctuations and
all the aspects of relativity -manifested by the history of
the human mind. The Christians of the first centuries did
not profess the same opinions on the nature and personality
of Jesus as we, and they did not have the same problems.
The ignorant man to-day does not have at all the same
ideas on these lofty and difficult subjects as the philosopher
does, nor the same mental preoccupations. But whether
ignorant men or philosophers, men of the first or the twen-
tieth century, every Catholic has always had and always
will have the same practical attitude with regard to Jesus.
It is time to conclude and I will do so in as few and
brief words as possible.
Two main results seem to me to have been attained by
the foregoing discussion:
1. The intellectualist conception which is current to-day
renders the greater number of objections raised by the
idea of dogma unsolvable.
2. On the other hand, a doctrine of primacy of action
permits a solution of the problem without abandoning
either the rights of thought or the requirements of dogma.
If these conclusions were admitted, the apologetics of
our days would be under the irresistible necessity of modi-
fying many of its arguments and methods.
Now, can these conclusions be admitted without loss to
faith ? It is for the theologians to tell us, and in case their
response is negative to teach us how they expect otherwise
to prepare to surmount the obstacles which perplex us.
EDOUARD LE ROY.
LEIBNIZ IN LONDON.1
KIBNIZ paid two visits to London from Paris, where
he was staying from March, 1672, to October, 1676:
the first visit, which was in connection with the embassy
from the Elector of Mainz, was from January n to the
beginning of March, 1673; the second was made on his
way home to Germany, when he stopped in London for
about a week in October, 1676.
Leibniz had a habit of writing out all the important
scientific points in the correspondence that he kept up with
noted people, so that he might thus impress them the more
deeply upon his memory. I have discovered among his
manuscripts three folio sheets on which he has written
down the things worth noting in connection with these
two visits to London.2 The sheets which relate to his second
visit have been known to me for some time; but the other
ones, referring to the first visit, I came across only during
my last stay in Hanover in the summer vacation of the year
1890.
In what follows, 1 have only paid attention to the con-
tents of these sheets which refer to mathematics.3
1 Translated by J. M. Child, from an article by Dr. Gerhardt in the Sitzungs-
berichtc der Koniglich Preussischen Akademie der Wissenschaften zu, Berlin,
1891, pp. 157-165. The notes are by the translator.
2 These highly important documents ought to be photographed and pub-
lished in facsimile.
3 It seems a pity that Gerhardt has not given the contents of the section
labeled "Mechanica," unless indeed this is all non-mathematical; there may
LEIBNIZ IN LONDON. 525
The sheet relating to Leibniz's first visit to London, of
which I have added a partial transcript under the heading
I, is divided on both pages into sections [the word used in
the original is Felder — columns, but it will be seen that,
according to the transcript given later, the sections are
horizontal and not vertical], in which Leibniz has en-
tered all that he considered to be worth noting. While
the sections labeled "Chymica," "Mechanica," "Mag-
netica," "Botanica," "Anatomica," "Medica," and "Mis-
cellanea" are filled up with an extraordinary number of
memoranda, the first sections, which are allotted to mathe-
matical subjects, are very poorly filled. That labeled
"Geometrica" contains a note that is especially worth re-
marking : "Tangents to figures of all kinds. Development
of geometrical figures by the motion of a point in a moving
straight line."4 In all probability it may be supposed that
this refers to the lectures of Barrow, delivered on his
method of tangents at the University of Cambridge down
to the year 1669. As is well known, the method of Bar-
row is only applicable to such curves as can be expressed
by rational functions.5 Newton's name was mentioned in
be in it some intimation that would lead to a clue as to the origin of Leibniz's
use of the word moment, meaning thereby, not Newton's use of the word, but
the idea now familiar to us in the determination of the center of gravity of
an area, expressed by the equation
x = "ZaxfZa,
where a is the element of the area distant x from the axis, x the distance of
the center of gravity from that axis, and "Sax is the sum of the 'first moments
of the elements' or 'the first moment of the whole area.' See note 16, later.
4 "Tangentcs omnium figurarum. Figurarum geometricarum explicatio
per motum puncli in moto lati."
5 In a footnote, Gerhardt asserts that "Barrow's Lectiones Geometricae
appeared in 1672." This is incorrect ; for they were published, combined with
the second edition of the Lectiones Optics, in 1670 ; nor can Gerhardt be referring
to the second edition, for that appeared in 1674 and then as a separate volume.
Also, I have, in the little book on The Geometrical Lectures of Isaac Barrow,
published by the Open Court Publishing Co., given reasons for supposing that
these lectures were never delivered as Lucasian Lectures, though they may
have formed the subject-matter for college lectures at Gresham and Trinity.
Again, it is not true, although "well known," that "the method of Barrow was
only applicable to such curves as can be expressed by rational functions" ;
this remark is even only partially true about the differential triangle method ;
for, as I have shown in the above-mentioned book, Barrow had a complete
calculus, which included, among other things, the important idea of substitu-
526 THE MONIST.
the "Optica." Leibniz has the remark: "They told me
about a certain phenomenon that Barrow confessed he
was unable to solve. Newton's difficulty has so far not
been solved, Father Pardies having given it up."6 Ob-
viously this remark applies to Newton's experiment on the
refraction of light by a prism and to the decomposition of
white sunlight, and especially to the fact that a circular
solar image becomes after refraction a long spectrum.
Father Pardies of Clermont had published in opposition
to Newton his "Two Letters containing Animadversions
upon I. Newton's Theory of Light," in the Philosophical
Transactions of 1672, together with a letter from Newton.
It cannot be said for certain that Leibniz, during his
first stay in London, met with any of the great English
mathematicians ; Wallis lived at Oxford, while Barrow and
Newton resided at Cambridge.7 Indeed, it is made a matter
of plaint by Brewster, the biographer of Newton, that the
Royal Society of London at that time numbered few men
of distinguished talents who were in a position to perceive
the truth of the optical discoveries of Newton. In the
letter which Leibniz addressed to Oldenburg, the Secretary
of the Royal Society, during his visit to London, he men-
tion, which is all that is necessary to complete the "a-and-e" method and make
it applicable to surds and fractions, and probably was thus applied by Barrow
in working out his constructions ; but the whole thing was geometrical, which
apparently hid the inner meaning until recently.
To my mind, the mention of but "tangents and local motion" points out
that, on Leibniz's first reading of Barrow, he only perused at all carefully the
first five lectures, which are relatively unimportant; or rather it confirms an
opinion I had already expressed to Mr. P. E. B. Jourdain.
6 "Locuti sunt mihi de phaenomeno quodam quod Barrovius fatetur se sol-
vere non posse. Newtoni difficultas soluta hactenus non est, P. Pardies manus
dante."
7 It seems however that Leibniz attended the meetings of the Royal So-
ciety ; at any rate once, when he exhibited the model of his calculating machine.
It would be interesting if the roll of members present on all occasions during
this period could be obtained, as doubtless they were kept. For such men as
Ward were members at the time and attended the meetings, and Ward was,
if not in the same class as the three whose names are given, an excellent math-
ematician ; and, Leibniz, being somewhat of a notable, on account of his con-
nection with the Embassy from Mainz, would surely be introduced to all emi-
nent members present.
LEIBNIZ IN LONDON. 527
tioned that he had met by accident the mathematician Pell
at the house of Boyle, the chemist. The conversation fell
upon those number-series which in elementary mathematics
were called the higher arithmetical series and whose sums
and terms were found by the help of differences. Leibniz
showed that he had gone deeply into the study of such
series and had partly found out some new methods for
calculating the terms.8 Leibniz's letter to Oldenburg was
dated Feb. 3, 1673 (1672 O. S.).9
From the preceding it appears that what Leibniz learned
with reference to mathematics from his first visit to Lon-
don was quite unimportant.10 The chief aim of his stay in
London was to be elected as a Fellow of the Royal Society ;
and this came to pass, owing in part to an exhibition of a
model of his calculating machine, and in part to the friendly
offices of Oldenburg.
After his return to Paris at the beginning of March,
1673, 'Leibniz was able to find more leisure to follow up
his studies without hindrance; the political mission which
was the cause of his being sent to Paris, was now at an
end.
It may be regarded as certain that, before his first visit
to London, Leibniz made the personal acquaintance of the
men with whom he corresponded before he came to Paris,
and especially Antoine Arnauld and de Carcavy. The
8 The account given by Leibniz himself in the Historic (see The Monist
for October, 1916) reads thus: "He" [for Leibniz wrote in the third person,
under the guise of "a friend who knew all about the matter"] "also came
across Pell accidentally, and described to him certain of his own observations
on numbers, and the latter stated that they were not new, but it had been
recently made known by Nikolaus Mercator. . . . This made Leibniz get the
work of Nikolaus Mercator." As a matter of fact the suggested plagiarism,
or what Leibniz took for such a suggestion, was from Mouton and not from
Mercator. This is an instance of the lack of memory from which Leibniz
suffered ; such lack as caused him to make notes of all important points.
9 See Note 32, on the introduction of the Gregorian calendar.
10 I cannot see what reason Gerhardt has for this statement, considering
the contents of Barrow's book, which we know that Leibniz had purchased ;
that is, unless we assume either that Leibniz, as I have suggested, did not at
that time read the whole of Barrow, or failed to grasp what Barrow had given
owing to his (Leibniz's) incomplete knowledge of geometry.
528 THE MONIST.
latter belonged to the circle in which Pascal moved
Whether at that time Leibniz had made the acquaintance
of Huygens is not quite so certain ; at any rate he did not
come into close relations with him until after his return
from London. Huygens presented him with a copy of his
great work, Horologium Oscillatorium, which had just
(1673) been published. The recognition that his mathe-
matical knowledge at that time was insufficient to enable
him to understand the contents of this book, combined with
a reawakening of his former love for mathematics, had the
effect of making Leibniz devote himself with the greatest
fervor to the study of mathematcial subjects. Cavalieri's
method of indivisible magnitudes, the writings of Gregory
St. Vincent, the letters of Pascal (which were especially
recommended to him by Huygens), were used by him as
guides in his studies. As the first-fruits of these studies,
he obtained the theorem that, when the square on the
diameter of a circle was taken as unity, the area of the
circle was expressed by the infinite series
1-i + i- i+ ad inf. ™
O O I
He obtained it thus: Instead of dividing the circle, as in
the method of Cavalieri, into trapezia by means of parallels,
he divided it into triangles by lines radiating from a point ;
the areas of these triangles being proportional to certain
lines. With these lines as perpendicular ordinates a curve
could be constructed that was divided by these ordinates
into trapezia, each of \vhich is double the corresponding
triangle. In this way Leibniz obtained a curvilinear fig-
ure12 whose area was double that of the circle, but which
was expressed by a rational function, x = y2/(i -f- j2),18
11 Leibniz's own date for the discovery of this result, usually alluded to
by him as the "Arithmetical Tetragonism," is 1674; '"But in the year 1674 (so
much it is possible to state definitely) he came upon the well-known Arith-
metical Tetragonism; " (See Historia, in The Monist, Oct., 1916.
12 See the first critical note, page 536.
13 See the first critical note, page 536.
LEIBNIZ IN LONDON. $2$
of its coordinates; and, using a method that was similar
to that employed by Mercator for the equilateral hyperbola,
this area could be found (Ouadratrix).14
For the rest of Leibniz's treatment, see the hitherto
unpublished manuscript, given under II in the appendix
that follows.
As was often the case in the first scientific studies of
Leibniz, intimations of the great problems that occupied
his attention his whole life through are found here in his
first efforts in the domain of higher mathematics. First
is it to be remarked that Leibniz abandoned the division
of curvilinear figures into trapezia, as was done by Cava-
lieri, and instead divided them into triangles; from this
he was led to the "characteristic triangle,"15 which formed
the foundation in the application of the differential calculus.
Further, Leibniz constructed, instead of the proposed curve,
another of which the area could be found (the "quadratrix"
as he called it) ; this method of procedure frequently oc-
curs in the later works of Leibniz on the integral calculus.
Closely connected also with this is the solution of the in-
verse method of 'tangents, that is, given the tangent, to
find the curve.
In these first efforts of Leibniz in the domain of higher
mathematics is clearly to be seen the influence of his study
of the writings of Pascal.18 The French mathematicians
Roberval and Pascal did not consider that Cavalieri's
14 Observe that Leibniz (or Gerhardt) employs this word in a different
sense from that of Barrow, with whom it means the special curve whose equa-
tion is v = (r — ;r)tan"Mr/2r, a curve that is particularly connected with the
circle.
15 This contradicts both Gerhardt and Leibniz himself, who said that he
got it from a consideration of a figure used by Pascal in finding the content
of the sphere. See also the critical note referred to in 12, 13 above.
16 I hope to consider this influence in a later number of The Monist, in
connection with an essay by Gerhardt on this very point ; when I shall en-
deavor to substantiate an opinion I have formed with regard to the earlier
manuscripts of Leibniz, which were discovered by Gerhardt, and of which
translations appear in The Monist (April, 1917). I suggest that these do
not represent so much the record of his original investigations as notes made
while using the works of his predecessors as text-books.
530 THE MONIST.
method was consistent with the rigorous requirements of
mathematics;17 they reverted to the study of the Greek
mathematicians, and especially to the writings of Archi-
medes, combining with their method the developments
which Kepler, in particular, had brought about by the in-
troduction of infinitely small magnitudes into geometry.
Moreover, in connection with Pascal, it is to be observed
that he generalized into a "barycentric calculus" the proce-
dure used by Archimedes for the quadrature of the parab-
ola by means of the equilibrium of the lever.18 This
"calculus" enabled him to solve problems on the cycloid
which his contemporaries had vainly attempted.19 It was
not unknown to Leibniz that, since the time of Pappus of
Alexandria, quadratures and cubatures had been calcu-
lated by the aid of the center of gravity (Guldin's rule,
"Centrobaryca") ; certainly he was now led, by the works
of Pascal, again to notice the methods for the determina-
tion of the center of gravity, and was also induced to at-
tempt to extend and perfect them. The manuscript of
17 I fail to see how this statement can be completely reconciled with the
following well-known quotation from the "Lettre de A. Dettonville a Carcavy"
(1658):
"J'ay voulu faire cet advertissement pour monstrer que tout ce qui est
demonstre par les veritables regies des indivisibles se demonstrera aussi a la
rigueur et a la maniere des anciens; et qu'ainsi I'une de ces Methodes ne differe
de I'autre qu'en la maniere de parler; ce qui ne peut blesser les personnes
raissonnables quand on les a une fois avertyes de ce qu'on entend par la" (Vol.
VIII, p. 352).
Pascal also says on p. 350: ".... la doctrine des indivisibles, laquelle ne
peut estre rejettee par ceux qui pretendent avoir rang entre les Geometres."
That is, the method of indivisibles does not differ from the method of
exhaustions, except in the way the argument is put; and that the former must
be accepted by any mathematician with pretensions to rank among geometers.
The page reference is to the edition of Pascal's Works in 14 volumes, in
the series, Les Grands Ecrivains de la France (pub. Hachette et Cie., Paris,
1914).
18 Pascal calls it "la balance." It is worth noting in this connection that
Pascal uses the word "force" and not "moment" for the product of one of his
weights and its lever-arm ; so that we must look elsewhere for the clue to the
use of the word "moment" in this sense by Leibniz.
19 Several of the problems proposed were solved by Huygens, de Sluze,
and Wren; but by special methods, which did not satisfy Pascal, who called
for a general method. Later (1670) Barrow gives the rectification of the arc,
as a special case of a general theorem (Lect XII, App. 3, Ex. 2, see my Bar-
row, p. 177).
LEIBNIZ IN LONDON. 531
Leibniz which is dated October 25, October 26, October 29,
November I, 1675, and which contains the investigation
on the center of gravity, is headed, "Analysis Tetragonis-
tica ex Centrobarycis."2
It is worth remarking that in this Leibniz continues
the method by which he had found the series for the area
of the circle. Incidentally these studies were the first occa-
sion for the introduction of the symbol for a sum, i. e., the
integral sign (October 29, 1675); from this as the an-
tithesis, the sign for the difference, i. e., the symbol for
differentiation, resulted.21 The equation in which Leibniz
first introduced the sign of integration was, in the notation
of that time:
omn. /LU n /
n omn. omn. -
2 a
that is,
(omn./)2 / .
v ' = omn. omn. - ;
2 a
for which Leibniz writes
•>>>
2
that is, when / = ofy,
After his return to Paris in March, 1673, Leibniz was
in constant communication with Oldenburg, the Secretary
of the Royal Society; the subjects being almost entirely
mathematical. In this way he obtained his knowledge
of the work of the English mathematicians. Oldenburg's
mentor on all mathematical questions was John Collins,
who possessed a very wide acquaintance among English
20 A translation is given in The Monist, April, 1917.
21 See the second critical note, page 543.
532 THE MONIST.
mathematicians; and it was through him that what they
had done was communicated. In this respect special men-
tion is to be made of the letter from Oldenburg to Leibniz,
dated July 26, 1676, in which Collins informed him of a
collection of letters from English mathematicians that he
had in his possession. Collins mentions in it particularly
that script of Newton, of December 10, 1672, in which the
latter makes a communication about his method for tan-
gents to curves, which are given by an explicit algebraical
equation; he remarks that the method is only a corollary
to a general procedure for solving other problems, such
as those relating to rectification, determination of centers
of gravity and so on.22 Collins stated in addition that, be-
sides what this letter showed, nothing further was known
at that time about Newton's method. It was on account of
these communications, and probably also on account of a
letter from Newton to Oldenburg, of which Oldenburg sent
a copy to Leibniz at Paris, that Leibniz was moved to make
his return journey to Germany in October, 1676, by way of
London. Leibniz stayed there about a week; he made the
acquaintance of Collins, who willingly let him have access
to /his collection of treatises and letters.23 What Leibniz
found in them that he thought worth noting he set down
22 Leibniz, in the Ada Eruditorum for the year 1700, says, "I can affirm
that, when in 1684 I published the elements of my Calculus, I did not know
any thing more of Mr. Newton's inventions in this kind, than what he formerly
signified to me by his letters, viz., that he could find tangents without taking
away surds;. ..." As Newton says in the article in Phil. Trans., Vol. XXIX,
No. 342, Anno 1714 (usually called the "Recensio") this "is very extraordinary,
and wants an explanation."
23 This is feasible, but there is another alternative given by Dr. H. Sloman
(The Claim of Leibniz to the Invention of the Differential Calculus, English
edition, pub. Macmillan, 1860), which strikes me as even more probable. Slo-
man's points are as follows: (1) It is highly probable that Leibniz's week in
London was the last week of that month. (2) Oldenburg had then in his
possession two letters from Newton for Leibniz, dated Oct. 24 and 26 ; these
he showed to Leibniz. (3) As Newton himself mentions, these were blotted
and hastily written ; and thus Leibniz asks, on this account, that Oldenburg
should let him see the tract of Newton to which they refer ; which tract Leibniz
knew was in the possession of Oldenburg, that is, a copy of it. For the details
of the argument, occupying ten quarto pages, see the above-mentioned book
by Sloman, pp. 97-106.
LEIBNIZ IN LONDON. 533
on two folios ; the one has the heading, "Excerpta ex trac-
tatu Newtoni de Analysi per aequationes numero termi-
norum infinitas." This is the paper which Newton sent in
June, 1699, to Barrow, from whom Collins received it on
July 30, 1699. Collins made a copy of it, and sent the
original back; and the original was printed in the year
1711. The other sheet has the heading, "Excerpta ex Com-
mercio Epistolico inter Collinium at Gregorium." A partial
transcript of both these sheets follows under the head-
ing III.
With regard to the extracts from Newton's paper, it
is to be remarked that Leibniz was interested in the treat-
ment of algebraical expressions of powers and in the turn-
ing of irrational expressions into the form of series by
means of division and root-extraction. He noted indeed
many examples in their entirety. How to get to quadra-
tures was known to him; he merely indicated the process
by the sign of a sum, i. e., by the symbol of integration.
On the other hand, the part on the numerical solution of
adfected equations was new to him, and this he copied out
well-nigh \vord for word; this is the well-known New-
tonian method of solution of equations by approximations.
Leibniz passes over as well known to him the remark, made
by Newton at the close of the quadratures, that the prob-
lems of rectification, determination of the content of solids,
determination of the centers of gravity, can be solved in
the same way, and also the general indication of the process
to be followed in such cases. Then follows the solution of
inverse problems, for instance, to find from the area the
base, that is the axis of the curve. This Leibniz copied
out word for word. In the same way Leibniz has extracted
the conclusion of Newton's paper, "Demonstratio resolu-
tionis aequationum affectarum." At the end of his manu-
script Leibniz adds: "I extracted this from the letter of
534 THE MONIST.
Newton, August 20, 1672, addressed to Newton."84 Prob-
ably this means that from the letters referring to Newton,
Leibniz picked out the letter dated August 20, 1672, ad-
dressed to Newton.25 So far as the script can be de-
ciphered,26 its contents were a graphic representation of
Newton's method of solution of equations by approxima-
tions by means of Gunter's scale. Gunter's line had been
noted by Leibniz on his first visit to London.
Of quite special interest to Leibniz were the letters of
mathematicians which Collins had collected; on a second
folio he made excerpts from letters from James Gregory.
In two letters from Gregory (1670) was Isaac Barrow
extolled as the greatest, not only among living writers,
but also among all those that had written before him
(Barrow). Further Leibniz found among these letters the
letter mentioned above of Newton to Collins of December
10, 1672 ;27 he extracted what Newton had mentioned with
regard to his method of finding the expression for the tan-
gent to a curve. Leibniz added at the end of this extract,
"This method differs from that of Hudde as well as from
that of Sluse, in that irrationals need not be eliminated."28
24 The Latin, "Excerpsi ex Epist. Neutoni 20 Aug. 1672 ad Neuton," as
given by Gerhardt, seems somewhat unintelligible ; especially the word Neuton.
What Collins had (or what Oldenburg, as suggested by Sloman, had) was a
copy of a manuscript that Newton had sent to Barrow. Gerhardt says, "so
far as the script can be deciphered" ; perhaps the word Neuton is an error
of transcription, or maybe an error on the part of Liebniz, due to the juxta-
position of the Neutoni which comes just before. In any case, note 25 applies.
25 I do not think Gerhardt's translation of the word excerpsi is correct.
26 Gerhardt does not state whether the extract is badly written (this would
show that it had been done in a very great hurry, for Sloman says that Leibniz,
in his matter for publication, wrote a beautiful hand), or whether spoilt by
age; in the latter case, as old-time inks contained salts of iron, the manuscript
might be restored by photography, by means of a special plate, that I under-
stand is sometimes used for detecting forgeries in deeds and notes.
27 The letter was sent to Barrow to be sent on to Collins, probably with
the object of being communicated through the latter to others; Collins seems
to have been the regular channel of communication at this period, in a similar
way to Mersenne.
28 So we find in a manuscript, dated July 11, 1677, first of all an allusion
to Sluse's method of tangents, "in which the equation is purged of irrational
or fractional quantities" ; then the remark, "I have no doubt that the gentlemen
LEIBNIZ IN LONDON. 535
From these extracts it follows that the contents of New-
ton's letter were unknown to him at that time (Oct., 1676)."
Regarding the verbal communications that Leibniz
had from Collins during the second stay in London, Collins
wrote to Newton from London on March 5, 1677 (1676
O. S.), that the representation of the roots of an equation
by a series was discussed between them.
It is clear that Leibniz during his second stay in London
had made himself more familiar with the results obtained
by English mathematicians than he was before. The ques-
tion now arises: What specially occupied his attention?
What had particular influence upon his studies ? It is seen
that what Leibniz found in Collins's collection relating to
algebraical analysis was new to him and excited his in-
terest; also the verbal exchange of ideas between himself
and Collins was upon the same subjects.
On the other hand, as regards the infinitesimal calculus,
Leibniz obtained nothing during his second visit to Lon-
don; he had made a progress, by the introduction of his
algorithm into the higher analysis, beyond anything that
came to his knowledge in London.30 Also these algebraical
results, at least for the next period, left behind no lasting
impression; for among Leibniz's papers is to be found an
extensive treatise, written on board the ship that carried
him from London to Holland, wherein he considered the
I have just mentioned know the remedy that is necessary to apply"; then fol-
lows the rule for a quotient, and the remark that this will be sufficient for
fractions ; lastly the rule for powers, with the remark that this will be sufficient
for irrationals. Later, he says, "This method has more advantage over all
others that have been published than that of Slusius over all the rest, because
it is one thing to give a simple abridgment of the calculation, and quite another
thing to get rid of reductions and depressions."
Thus, after the sight of Newton's paper, his whole business has been to
improve the method of Sluse.
29 I read it quite otherwise ; he has had information of some kind, whether
from Oldenburg direct or from Tschirnhaus, while in" Paris, and visits London
with the express intent of seeing the original papers.
30 See the third critical note, page 546.
536 THE MONIST.
fundamental principles of motion, in the form of a dia-
logue.31
It was in the letter to Oldenburg written from Amster-
dam on November 18/28, 1676™ which Collins spoke of
in the letter to Newton mentioned above, that Leibniz first
refers to the subject of the problem of tangents, and re-
marked that the method of Slusius was not yet very per-
fect.33
KARL IM MANUEL GERHARDT.
CRITICAL NOTES ON GERHARDT'S ESSAY.
BY THE TRANSLATOR.
NOTE I. The origin of Leibniz's "transmutation of figures"
(Referred to in footnotes 12, 13, IS.)
In the manuscript, which follows under heading II, Leibniz
appears to attach very considerable importance to the method of
transmutation of figures, and to claim that he had originated it.
This claim is not incontestible ; indeed I am almost inclined to think
it is a deliberate plagiarism to start with ; but Leibniz has perceived
31 Could this possibly have had its rise in an effort on the part of Leibniz
to understand fluxions, or rather the idea of fluxions as he had found it in
Newton's paper?
32 In 1582, Gregory XIII had directed 10 days to be suppressed from
the calendar, then in accordance with the Julian system of intercalation, in
order to allow the error which had crept into the time of the vernal equinox,
by which Easter-day was settled, to be put right. The Gregorian calendar
was introduced into all Catholic countries the same year, in Scotland in 1600,
in the protestant states of Germany in 1700, but not in England until 1752. At
the same time the commencement of the legal year in England was altered
from May 25 to January 1 ; thus we frequently find two years given for dates
between January 1 and May 25 ; while there are two days of the month given
for all months of the year. For instance, February 1673 in the new Gregorian
calendar would be only February 1672 in the Julian, distinguished by the letters
O. S. (Old Style) ; and this date was written February 1672A- Similarly the
date November "/2», 1676, was the 28th of November in the New Style, and
the 18th in the Old Style, the number of the year being the same, since the day
did not lie between the 1st of January and the 25th of May.
83 "Methodus Tangentium a Slusio publicata nondum rei fastigium tenet."
These are Leibniz's words ; Gerhardt omits to translate the word publicata,
which probably refers to the publication in the Phil. Trans, of 1672, by Slusius,
of the rules of his method, illustrated by examples. Sluse had probably im-
proved upon this before 1676, but there is no evidence on this point. It would
seem as if the subsequent work by Leibniz, culminating in the manuscript of
July 11, 1677, was largely an attempt to perfect the rule of Sluse as a rule,
and that Leibniz, if ever, did not appreciate the idea fundamental in the cal-
culus, namely that of rates, until very much later.
LEIBNIZ IN LONDON. 537
in it something which the original author did not. Can it by any
chance be the case that, in conformity with several other instances
of Leibniz's bad memory for details, he is confusing author and
subject, when he speaks of "the great light that suddenly dawned
on him, which the author had missed," the reference being to Pascal
and the discovery of the differential triangle? Can it be that the
true connection is that in considering the original work of the author
of such transmutations of figures, he perceived the method for the
arithmetical quadrature ? For here he really has found a thing that
the author missed though it was almost staring him in the face,
his discovery being due to a habit that Leibniz had of writing down
everything that he could get out of any particular figure or bit of
work that he had in hand, whether it was relevant or irrelevant.
Wallis and Pascal had both hinted at the method, i. e., had used
it in special cases, namely for proving the equivalence of the parab-
ola and the spiral ; and Leibniz was familiar with both these authors.
Again, James Gregory had, in the words of Barrow (Led. Geom.,
Lect. XII, App. 3, foreword to Prob. IX), "set on foot a beautiful
investigation about involute and evolute figures," i. e., polar and
rectangular figures equal in area to one another. Of course, Leib-
niz may not have seen this work of Gregory until later ; probably
not, although in one of his manuscripts he gives a theorem of
Gregory ; this however does not count for much, for the very same
theorem is given by Barrow (see my Barrow, p. 130) and we know
that Leibniz had a Barrow in his possession. This book, judging
by his words, "as in Barrow, when his Lectures appeared, in which
I found the greater part of my theorems anticipated," Leibniz wishes
to make his friends believe was the 1674 edition, and not the edition
of 1670, which he bought on his first visit to London. Why did
Leibniz wish to conceal this fact ? I. assert that the reason for doing
so was the fear that seemed always to overshadow him, the fear of
being accused of plagiarism, whether such was a true or a false
charge. I am firmly convinced that Leibniz got his transmutation
of figures from Barrow ; to this conclusion I have only just come, it
never having entered my head to look for it at the time that I wrote
my articles for The Monist of October, 1916, April and July, 1917.
Before I bring forward my arguments, it is right to state as a
preliminary that, just as in calculus nowadays we usually draw a
curve with its convexity downward, and draw the tangent to meet
the horizontal axis beneath the curve, so Barrow drew his curves
with the concavity downward in many cases, mostly, I think, in
order to fit the diagrams conveniently on the old-fashioned folding
plates of diagrams, that in those days were added in batches at
the end of a book (see a specimen I have given at the end of my
Barrow) ; in other cases, he draws his figure on the left-hand side
of the axis. Whichever figure he draws, he always did one thing,
namely, he drew any supplementary figure he had need of on the
other side of his axis or base. Leibniz almost invariably drew
538
THE MONIST.
his curve on the right-hand side of a vertical axis, and supplemen-
tary figures on the same side. Hence, in the extract from Barrow
given below, I am to be excused for failing to notice before what
is more than a mere similarity.
In the following extract from Barrow (Lect. XI, Prop. 24),
I have added Barrow's proof, which I thought unnecessary to give
in my book; the figures given are Barrow's own on the left, which
has been "up-ended" on the right ; the latter is to be compared with
the several figures by Leibniz.
Barrow's Lectiones Geometricae, Lect. XI, Prob. 24.
If DOK is any curve, D a given point on it, and DK any
chord ; also if DZI is a curve such that when any point M is taken
in the curve DOK, DM is joined, DS is drawn perpendicular to
DM, MS is the tangent to the curve, DP is taken along DK equal
to DM, and PZ is drawn perpendicular to DK, so that PZ is equal
to DS ; in this case the space DZI is equal to twice the space DKOD.
Fig. 1.
Fig. 2.
For let KP be considered to be indefinitely small, and let DT
be perpendicular to DK and KT the tangent to the curve DOK.
Then, drawing the arc MP, we have as before,
KP : PM = KD : DT = KD : KI, and hence KP . KI = PM . KD.
Take another small part PQ and, with center D, draw an arc QN
through Q cutting the chord DM in R ; then as before,
MR:RN = MD:DS, PQ:RN = MD:PZ, PQ.PZ = RN.MD;
and so on one after the other. Therefore, it is evident that the
sum of all the rectangles KP.KI, PQ.PZ, etc., is equal to the ag-
gregate of all the spaces PM . KD, RN . MD, etc. ; that is, the space
DKI = 2 times the space DKOD.
The words I have italicized refer to Prop. 22, in which he uses
LEIBNIZ IN LONDON.
539
a similar though rather more complicated figure to reduce a polar
area to a rectangle of which one side is a given straight line, and
explains that the reasoning depends on the fact that the line DK is
divided into infinitely small parts. Compare the words I have ital-
icized with the description of Leibniz's method : "the areas of these
triangles being proportional to lines.
Further, Barrow proceeds in Prop. 25 to prove the equivalence
of the spaces formed (i) by applying each MS to the base and (ii)
by applying each chord to the arc, previously rectified. And he winds
up with the words: "Should any one explore and investigate this
mine, he will find very many things of this kind. Let him do so
who must, or if it pleases him."
This all suggests that Leibniz did explore this mine, that he
did not invent the method of transmutation of figures for himself,
Fig. 3.
Fig. 4.
that he did find very many things of this kind, and that Barrow had
missed the arithmetical quadrature construction ; this Leibniz ob-
tained through his regular practice of working every mine right
out, to keep up Barrow's simile. Further comment is needless, I
think, after a comparison of Barrow's figure (the up-ended version)
with the figures of Leibniz given above.
Fig. 3 occurs in a manuscript November 21, 1675, which ac-
cording to Leibniz is at least a year after he had discovered the
arithn::tical quadrature; and yet it has a heading, "A new kind of
Trigonometry of indivisibles, etc." In this figure it is to be noticed
that he has the perpendicular to the chord BC, agreeing with Bar-
row's DS and DT, but has not the tangent at the vertex that was
necessary for the demonstration of the arithmetical quadrature. In
the working in connection, he considers the similarity of all the tri-
54O THE MONIST.
angles possible, and notes as one point that "the sum of all the tri-
angles or the area of the figure is equal to the products of the AB's
into the CE's, which is Barrow's proof of Prop. 24 above.
Fig. 4 is the figure given in the Historia (see Monist for Oct.,
1916), in connection with the explanation of how he found the area
of the circle. Notice the difference between this figure and the
one given in the manuscript that follows under the heading II, also
that the description there given of the way in which he was led to
it is much more natural. This is probably the true version, for the
use of the notation B, (B), ((B)), points out that it was written
at a comparatively early period, before Leibniz had adopted the pre-
fix notation, 1B, 2B, 3B. In the account in the Historia, to which
Fig. 4 applies, Leibniz says, "he once happened to have occasion to
break up an area into triangles formed by a number of straight lines
meeting at a point, and he perceived that something new could
be readily obtained from it." I suggest that the occasion was most
probably while he was digging in Barrow's mine ! This is the reason
why he has in the Historia given the figure more according to his
usual practice, and different from the figure in the earlier manu-
script, which is too much like a copy of Barrow's (query, where did
Barrow get it from?). With regard to the figure and proof in the
manuscript which follows, we find that the reasoning there given is
unsound, unless Gerhardt has given us a slightly erroneous diagram ;
for Leibniz apparently does not perceive that the ordinates BA,
which are equal to the corresponding CE, must pass through the
respective points D, before he can say that one figure is double the
other. Hence I conclude that at the date of this manuscript, the
demonstration was imperfect and that he had no proof until he dug
in Barrow's mine ; in support of which conclusion I will quote from
the Recensio, mentioned in footnote 22. "This quadrature, com-
posed in the common manner, he began to communicate at Paris in
the year 1675. The next year he was polishing the demonstration
of it, to send it to Mr. Oldenburg, in recompense for Mr. Newton's
Method, as he wrote to him May 12, 1676; and accordingly in his
letter of August 27, 1676, he sent it, composed and polished in the
common manner." This polishing, I take it, consisted in making
the slight but important alterations in the demonstration and figure,
from those given in the manuscript II that follows, to those given in
the Historia.
What had he then got in July 1674, when he wrote to Olden-
burg saying that he had got a wonderful Theorem, which gave the
area of a circle, or any sector of it exactly, in a series of rational
numbers? Or, when in the October following, October 26, 1674,
he wrote to say that he had found the circumference of a circle in a
series of very simple numbers; and also by the same "method" (a
favorite expression of Leibniz) any arc whose sine was given?
It was impossible that Leibniz could have had the two things that
I have italicized; or at least, the latter was impossible to him, be-
LEIBNIZ IN LONDON. 541
cause the only way for him to obtain it exactly, i. e., to know the
law of his series, was as yet unknown to him ; unless we are to as-
sume, contrary to his assertion, that the' binomial theorem was
known to him, which would involve his also having seen or been
told about other parts of Newton's work. The only way open to
Leibniz was to find the square root of I-*2, and then its reciprocal
by division ; and this would not give him the law of the series, even
if we assume that his knowledge of integration was sufficient to
enable him to proceed any further. From his manuscripts it does
not seem that even up to Nov. 1675 he had any further knowledge
of integrations than that omn.x = x2/2, and omn.jir2 = x3/3 ; but as
he says that he knows the latter from the quadrature of the parabola,
there is some possibility that he might have been able to integrate
every integral power of the variable from his reading of Wallis and
Mercator.
However, there is the strongest probability that he had not got
any proof for the two things italicized, and that the quadrature was
in the same category. Where then had he obtained it ? We find that
in December, 1670, Gregory had found out for himself Newton's
method of series; and two months later, February 15, 1671, sent
several theorems to Collins, one of which was that now known as
"Gregory's series." "And Mr. Collins was very free in communi-
cating what he had received both from Mr. Newton and Mr. Greg-
ory, as appears by his letters printed in the Commercium" ( from the
Recensio). One can imagine that Oldenburg would be one of the
first to receive the information, and that for a certainty it would be
passed on to Leibniz. I think then that Leibniz perceived that by
putting x= 1 in Gregory's series, and making the radius of the circle
equal to unity, he could get an arithmetical quadrature; from that
time onward he looked for a proof by pure geometry, and found it
after reading Barrow's proposition referred to above; if we assume
the possibility of integration of integral powers, it was an easy step
to find that the series he had to integrate was y2/(l+y2), and all
he had to look for on his figure was a line of this length. This very
well accords with the description of the way in which he found his
demonstration, as given in the manuscript which follows under the
heading II.
Lastly, in connection with the suggestion that I have made
above, namely, that Leibniz had another method for his arithmetical
quadrature than those he has given, there is one method that is
bound up with the change that he made from the Pascalian char-
acteristic triangle which he used at first, to the Barrovian differential
triangle (see my note in The Monist, Oct., 1916, p. 615). In Example
5 of the method of the differential triangle (see my Barrow, p. 123),
Barrow has found the subtangent for the curve y = tznx, from a
consideration of the figures on next page, and finds that
>= — rL— ™. = ^.BG
rr+mm CG2
542
THE MONIST.
where r is the radius of the circle, m is the ordinate MP, which is
equal to BG, and t is the subtangent TP.
Fig. 5.
Now if we put the radius equal to unity, and for the ratio t/m
substitute what was known by Leibniz to be equal to it, namely,
QP/RM or EF/GH (by construction), we have the sum of all the
EF's is equal to the sum of ordinates equal to CK2 ( radius =1)
applied to G at right angles to BG. Analytically, calling BG z, we
have
arc BE=sum.omn.
1 +
applied to the line z \
hence by division
arc BE=sum.omn.(l— z* + z*— z* + etc. )
etc.
I can hardly see how Leibniz could have missed this with his
analytical mind, even although Barrow has missed it; but there is
a strong probability that at the time of writing, Barrow had not seen
the quadrature of the hyperbola by Mercator, and, if he had, such
algebraical work would not have appealed to him at all.
As far as I can make out, there is only one other alternative,
which involves a direct contradiction of Leibniz's own statement ;
that is that his proof was not by the transmutation of figures in the
first instance. Color is lent to this view by a letter of Leibniz and
other papers, quoted by Sloman (pp. 131ff, in the English edition
of the work referred to in footnote 23) ; also even by a passage in
the Historia (see Monist, Oct., 1916, p. 599), where, while giving
the story of the discovery of the arithmetical tetragonism, Leibniz
distinctly hints at an algebraical method; for he says immediately
afterwards, "The author obtained the same result by the method of
transmutations, of which he sent an account to England." This
reads as if he had another method in addition to the method by
transmutations.
Let us consider this algebraical method. To square the circle,
Leibniz has to integrate ^/(l-x2)=y, say; let y=\-xz, then
y= (l-22)/(l+22), which is rational; moreover, he would also have
been able to have substantiated his statement that at this time he
LEIBNIZ IN LONDON. 543
also had a proof of the series for the arc whose sine was given, for
which he would only have had to integrate \/^(\-xz). But one
cannot conceive that Leibniz had any means of expressing the ele-
ment of z in terms of the element of x. Geometrically, he was in-
capable of it, without using Barrow's infinitesimal method ; and of
this we find the first instance in a manuscript dated November 1,
1675. Algebraically, he could not, for at this same date he could
not differentiate a product. How then are we to account for the
fact that he says he has a method for demonstrating both series
for the arc, given the sine or the tangent ? I think I can answer this.
Many times we find assertions made, not only by Leibniz in those
times, but by others in other times, of the possession of discoveries,
when all that the assertor has is the idea of how they may be ob-
tained. Thus, in the passage quoted, the concluding statement is,
"and thus again all that remains to be done is the summation of
rationals." So that if we accept this alternative we are bound to
come to the conclusion that Leibniz did not yet recognize, what he
ought to have done from the work of Pascal, that an area was not
a mere summation of lines, but of rectangles formed by these lines
ordinated at certain definite points along a straight line. That is to
say, he did not recognize the fundamental principle of integration,
namely, the importance of the factor dx or ds. When he had to
write out his proof he found that the summation of (l-2*)/(l+.sr2)
or its reciprocal was beyond him ; or rather that the series he found
by Mercator's method was not correct ; he had to resort to the geo-
metrical proof, of which he got the idea by digging in Barrow's
mine, as above; he found that this would not work for the other
series ; and consequently he dropped all claim to the second series.
In his letters of 1676, therefore, we find him offering to send New-
ton the proof of his quadrature in return for the method of proof
of the series for the arc when the sine is given.
Thus I come to the conclusion that Leibniz obtained these series
in some way by correspondence, thought he had got a proof of his
own, (which turned out to be incorrect), and much later did obtain
a proof of his arithmetical quadrature by the transmutation of
figures, after obtaining the idea from Barrow. As the special case,
when x = l\\z radius, had not been specifically mentioned by Gregory,
Leibniz considered that he had a right to claim it, more particularly
as he thought he had devised a proof for it, if it was necessary to
produce one ; for of course, Gregory had given no proof according
to the usual custom of the time. Then, when he did find a proof,
after having found that his original idea was hopeless, one can
hardly blame him for sticking to his claim.
NOTE 2. On the introduction of the Leibnizian algorithm.
(Referred to in footnote 21.)
The two passages in which the signs for integration and dif-
544 THE MONIST.
ferentiation are respectively introduced occur in the manuscript of
October 26, 1675.
i. "It will be useful to write / for omn., so that // = omn. /, or the
sum of the I's."
ii. Not for some time is the sign for differentiation introduced,
and then in these words : "I propose to return to former considera-
tions. Given / and its relation to x, to find //. Now this comes
from the contrary calculus, that is to say if / l = ya. Let us assume
that l = ya/d, or as / increases, so d will diminish the dimensions.
But / means a sum, and d a difference. From the given y, we can
always find ya/d or /, or the difference of the y's. Hence one
equation may be changed into the other, "
Now of these the introduction of the symbol for integration
can no more be called an invention than the use of 2 to stand for
"the sum of alj such terms as." It was simply, as Leibniz himself
says, a convenient and useful abbreviation for sum.omn. or omn.
It is nothing more or less than the long s then in general use ; indeed
it was so thought of by contemporary mathematicians, Newton for
one at any rate, for we find in the Recensio the passage, "Mr. Leib-
niz has used the symbols sx, sy, ss for the sums of ordinates ever
since the year 1686." This may have been an instance of prejudice,
or perhaps the printers of the Phil. Trans, may not have had an
integral sign in their fonts of type; but it shows up the fact that
the English accepted it as the initial letter of the \yord "summa."
Now let us consider the introduction of the letter d. Gerhardt
says that it resulted as antithesis to the sign /. How he can possibly
derive this from the context I cannot surmise. I am well aware
that in another passage he was unable to assign a meaning to the
introduction of a letter, which was, to me, clearly used for the simple
purpose of keeping the dimensions correct. We have this use again
in the present passage. Leibniz knows that the sum of the lengths,
/ /. is an area : hence taking y to represent a length, given in terms
of x, he introduces the length denoted by a to give with y the area
of a rectangle. Therefore he argues that / must be an area divided
by a length, and he writes I = ya/d, where d is another length, intro-.
duced to keep the dimensions correct. This is clear from the sen-
tence that follows next : "so will d diminish the dimensions."
So far the sequence of ideas is easy to follow, and there is not
the slightest trace of any concept of differentiation, nor, if the /'s
are ordinated to any axis, any trace of a connection between d and
an element of that axis. The difficulty begins with the next sentence :
"But / means a sum, and d a difference." The first idea that strikes
one is that this was added later, after that he had found out the
connection between the inverse-tangent problem and quadratures.
Gerhardt gives no suggestion on the point, so until the paper can be
reexamined for small details like differences in the ink or character
of the writing this idea will be disregarded. The next is that about
this time he was reading Barrow, and then one is at once reminded
LEIBNIZ IN LONDON.
545
of Lect. X, Prop. 1 1 ; this is the proposition in which Barrow proves
that differentiation is the inverse of integration. If we consider
this in the manner of Leibniz, we get the equivalent that is set down
on the right-hand side below:
BARROW
Let ZGE be any curve of
which the axis is VD; and let
ordinates applied to this axis,
VZ, PG, DE, continually in-
crease from the initial ordinate
VZ ; also let VIF be a line such
that if any straight line EDF is
drawn perpendicular to VD, cut-
ting the curves in the points E.
F, and VD in D, the rectangle
contained by DF and a given
length R is equal to the inter-
cepted space VDEZ; also let
DE:DF = R:DT, ^nd join DT.
Then TF will touch the curve
VIF.
Cor. It should be observed that
VDEZ.
LEIBNIZ
Let AC be a curve, whose axis
is AB, and let the ordinate AB
be/;
let AD be another curve, having
the same axis, and let its ordinate
DB be called y ;
let this curve AD be such that
the area ABC, i. e., all the /'s or
//. is equal to the product of
BD and a fixed line, i. e., equal
to ay;
then, taking B(B) equal to unity,
we have / = aw, where w : B ( B )
= DB:BT, orw = y/d, i. e.,
l = ay/d.
We thus see that the d that results as the "antithesis to the
integral sign" (als Gegensatz sich ergab), is not a difference
at all, but the subtangent ; it is y/d or w (on account of B(B) being
taken as unity) that is the difference between the ordinates y. But
there is not the slightest trace of the idea of differentiation : this
is made more manifest by the work which follows, which is based
on his idea of obtaining independent equations, and eliminating all
variables but one and thus reducing the problem to a quadrature.
And yet he seems to perceive from the equation that gives the dif-
ference of the y's as a quotient, that in some unintelligible way a
division means a difference. Later therefore we find him trying
to find an interpretation of d as an operator, whether he writes it
546 THE MONIST.
in front of his y, or as a denominator ; namely, when he considers
what value he is to assign to d(xy}. I venture to assert, unless we
assume that Leibniz is considering this proposition of Barrow's,
that there is no possible connection to be made out between the
several sentences of this passage. Also that in no sense can this
introduction of the letter d be looked on as an algorithm with any
idea in it of differentiation.
I am well aware that in the above I have adduced no positive
proof that my idea is correct; I have not had the advantage of
Gerhardt in seeing these manuscripts. But I have honestly tried to
find other ways of explaining the circumstances that lead from y/d
as a quotient to dy as a difference, and I can find none other that
is feasible than that given above, namely, that, perhaps by accident,
Leibniz uses d for the subtangent (instead of the usual t), and per-
ceives from such a figure as the above (which of course I do not
intend to say he has given) that y/d (where d is the subtangent)
works out the same as dy (when d.v is taken to be unity) ; in other
words the subtangent d is equal to y/(dy/dx).
NOTE 3. On the progress made by Leibniz before November, 1676.
(Referred to in footnote 30.)
The remark made by Gerhardt that Leibniz "had made a
progress, by the introduction of his algorithm into the higher anal-
ysis, beyond anything that came to his knowledge in London," is,
to say the least of it, a matter of opinion. From a study of the six
manuscripts, that Gerhardt has given us, that bear dates between
that of the introduction of the integral and differential symbols
(Oct. 26, 1675) and that of his return to Germany, via Amsterdam
(after Nov., 1676), I fail to see that there is very much occasion
for the main part of the above statement, namely, that the progress
made by Leibniz was at all greater than anything that came to his
knowledge in London ; as for this progress, if for a moment we
assume its superiority, being due to the reason set in ftalics, I fail
to see that Gerhardt has any grounds whatever for such a state-
ment.
The six manuscripts in question have been given, translated
into English and annotated in The Monist, April, 1917; but for con-
venience I here add a precis of them.
i. Nov. 1, 1675. A continuation of the work on moments about
axes ; the new symbols do not occur, omn. being still used.
He has now read Wallis, Gregory and Barrow, in addition
to Cavalieri and St. Vincent; he speaks of his theorem of
breaking up a figure into triangles as bringing out something
new ; the whole tone of this manuscript is in the main Pas-
calian.
ii. Nov. 11, 1675. He successfully obtains a solution of the prob-
lem of finding a curve such that the rectangle contained by
LEIBNIZ IN LONDON. 547
the subnormal and ordinate is constant. This he considers
to be "one of the most difficult things in the whole of geom-
etry." He uses the integral sign, and the denominator d;
but neither integration nor differentiation, the fact that
yz/2d = y, being taken from the "quadrature of the triangle."
In verifying his result he quotes Slusius's Rule of Tangents.
Further on, he has the note that x/d and dx are the same
thing, though there is nothing to show why he comes to this
conclusion ; see the last critical note. He also comes to the
conclusion that d(xy) is not the same as dx.dy; but in the
last bit of work in this manuscript he uses special letters
for the infinitesimals, showing that he has been trying to
find the effect of d as an operator, or perhaps trying to find
the reason of the equality x/d and dx. He has failed to
solve a problem, which results in the differential equation, as
we should now write it, x + y.dy/dx = az/y, or as Leibniz
has it x + w=az/y; although he gives an incorrect solution,
which he asserts to be true. This time he does not attempt
to verify his solution, the reason being obviously that he is
unable to do so, because one side of his equation is a product.
As a matter of fact, I have it on the authority of Professor
Forsyth that there is no solution of this equation in elemen-
tary functions ; or at least he says that he has been unable
to find one, which I take it comes to the same thing. The one
advance that can be found here is the appreciation that squares
and products of infinitesimals can be neglected, as he has
doubtless found in reading Barrow. It is worth noting that
he now uses the differential triangle in Barrow's form instead
of the form he says he got from Pascal.
iii. Nov. 21, 1675. In this manuscript he sets himself another
problem, which he fails to solve; the curve required is log-
arithmic, and this fact even he fails to bring out. In gen-
eralizations that arise from the consideration of his problem
he obtains dxy-xy — xdy, in a more or less analytical man-
ner; but immediately afterward states that nothing new can
be obtained from it ; he has already obtained this formula by
his consideration of moments, geometrically; and he does
not appreciate the advance there is in obtaining it algebra-
ically. The manuscript concludes with a consideration of the
figure by means of which it is generally supposed that he
affected his arithmetical quadrature. This is very remarkable
on account of the heading, which reads, "A new kind of
Trigonometry of indivisibles, by the help of ordinates that
are not parallel but converge." What I refer to is the use of
the word new, which I have here italicized. It is to be ob-
served that the diagram and the results are almost identical
with those of Barrow, Lect. XI, Prop. 22-24 (see the first
critical note). He concludes by a reference to the trochoids,
548 THE MONIST.
which shows that he is still under the influence of Pascal, if
indeed he is not still studying his works.
iv. Nov. 22, 1675. He returns to the subjects of the previous day.
But there is here no mention of the signs of integration or
differentiation.
v. June 28, 1676. Here we have a certain advance, for there
occurs the statement: "The true general method of tangents
is by means of differences." While he uses dy and dz for the
elements of 3; and z, he uses ft for the element of x ; the rest
of the work is merely Harrovian in principle. This mere
substitution of dy and dz for the special letters used by Bar-
row for the same things can hardly be called progress. What
progress there might be is barred by the use of equations
with three or more variables in them.
vi. July, 1676. The remark on the last manuscript is corroborated
by the contents of this manuscript. Leibniz asserts that
he has solved two problems, of which Descartes had alone
solved one, and owned that he could not solve the other. The
truth is that he has not solved the former, which was fairly
easy, only given an alternative construction which is, if any-
thing, more difficult to carry out than a construction from the
original data for the curve. The latter he gets out in a hazy
fashion (". . . .which belongs to a logarithmic curve"). This
conclusion he comes to after several erroneous steps of rea-
soning; whereas the solution stared him in the face about a
quarter of the way through the work, where he has the
equation cdy = ydx, if he could have integrated dy/y with
certainty. The failure I think arises from the study of Pas-
cal, who lays it down that only one of the variables can in-
crease arithmetically, and Mercator's work has been with y
increasing arithmetically, and Leibniz has already considered
that the x is increasing arithmetically (See my notes on this
manuscript in The Monist for July, 1917).
Throughout the whole of these manuscripts, he makes no prog-
ress, because he is hampered by the idea of keeping one of his
variables increasing uniformly; he seldom uses his algorithm for
differentiation ; and when he does do so, it is merely a substitution
of dx, etc. for the special letters used by Barrow. In fact these
manuscripts appear to me to be the records of his work on the text-
books of his study, Pascal, Wallis, Gregory, and Barrow ; and we
see him trying to fit the matter and methods found in them into his
own ideas and notations. It is not until November, 1676, when he
has arrived on the Continent, after having seen Newton's paper,
that we have any Differential Calculus ; even then some of the
standard forms that he gives are not quite correct ; on the other hand,
he gives the method of substitution to differentiate an irrational,
though he uses the Barrovian method to differentiate the general
equation of the second degree, merely using dy and dx instead of
LEIBNIZ IN LONDON. 549
Barrow's special letters. It is not until July, 1677, that he is able
to give anything like an intelligible account of the differentiation of
products, powers, quotients and roots. Lastly I doubt if Leibniz
ever did really appreciate the Newtonian idea that dy/dx was a rate,
or else the example he gives of the use of the second and third
differentials in his answer to Nieuwentiit would not have contained
so many ridiculous errors.
TRANSLATIONS OF THE MANUSCRIPTS
Alluded to by Dr. Gerhardt.
I.
Scientific memoranda of the visit to England at the beginning of
the year 1673.
When at the beginning of the year 1673, I accompanied his
Excellency the Ambassador of Mainz, Baron Schornborn, a nephew
(on his father's side) of the Elector, from Paris to London, although
I stayed in England scarcely a month, among various distractions,
I still gave attention to increasing my knowledge of philosophy;
for at that time the English held a high reputation in this subject.
To set out a long minute record of daily happenings is useless
on account of its inequality; for the fortune of all the days was
not the same ; indeed the points worth remarking heaped themselves
up one day, and the next gaped with emptiness. For this reason
perhaps it will be more satisfactory to go by heading of subjects,
one remark recalling another as it were.
The principal heads for the subjects noted may be taken as
Arithmetic, Geometry, Music, Optics, Astronomy, Mechanics, Bot-
any, Anatomy, Chemistry, Medicine, and Miscellaneous.
ARITHMETIC. The line of proportions or Gunter's lines or the
double scale. Logarithmotechnia or compendium for calculating
logarithms. To recognize square numbers from non-squares by their
end figures. Morland's machine.
ALGEBRA. Substance of English algebraical work of 27 years. Al-
gebra of Pell. At first few rules, but lots of selected examples.
Renaldinus not thought much of in England.
GEOMETRY. Tangents to all curves. Development of geometrical
figures by the motion of a point in a moving line.
550
THE MONIST.
Music. Its universal character. System of Birthincha. Vossius
will publish Music.
OPTICS. They told me of a certain phenomenon that Barrow con-
fessed that he was unable to solve. The difficulty of Newton hitherto
unsolved, Father Pardies giving it up. Hook adheres to a cata-
dioptric instrument of 9 feet, because for another of 50 feet move-
ment inconveniences them. The secret of the largest aperture which
can be given to microscopes is primarily as great as the distance of
the object.
ASTRONOMY. Arrangement of Hook for
observing whether the earth at any time
sensibly approaches or recedes from the
fixed stars, from which it can be judged
that it is not in the center of the uni-
verse; he erected it in a fine tube set
perpendicularly, and observed the stars
that are vertically overhead. He, lying
flat on his back, observed their dimen-
sions most exactly.
CHEMISTRY.
MECHANICS.
PNEUMATICS.
METEOROLOGY.
HYDROSTATICS.
NAVIGATION.
MAGNETISM.
PHYSICS.
BOTANY.
ANATOMY.
MEDICINE.
MISCELLANEOUS.
ii.
[This manuscript is very lengthy, the translation running to
LEIBNIZ IN LONDON. 551
about 6000 words, of which the first 5000 are written as a concise
history of all the great geometers and their works, that are antece-
dent to Leibniz himself. This part is quite unimportant for the
purpose of estimating the part that was played by Leibniz, and it
passes my comprehension why Gerhardt should give it at length,
while he has condensed the other two, which are really important.
Hence, in what follows, I have given a precis of the first 5000 words,
with here and there quotations, in which Leibniz has something to
say that is either critical of the work of others, or a claim to superior
knowledge or better method of his own. The last part, which pur-
ports to be the history of his arithmetical quadrature, together with
his claim to the surpassing value of his achievement, I have given
in full.]
(Precis). Geometry is a modern thing, probably due to the Greeks.
The great name among the Ancients is that of Archimedes, who
first used indivisibles; this use was more profound than that of
Cavalieri, but the method became lost. The name of Apollonius
must not be altogether omitted.
The learning of the Greeks passed on to the Arabs, who con-
quered them ; among these we have Alhazen, and a certain Mahomet,
who gave the formula for the general quadratic.
This brings us to the cubic and biquadratic equations, which
were solved in the sixteenth century. The cubic is due to one Scipio
Ferreus of Bologna ; one of his pupils set the solution as a challenge
after Scipio's death ; Tartalea took up the challenge, found a solu-
tion and told his friend Cardan ; the latter extended it and published
it without the consent of Tartalea. Vieta, Descartes, and Ferrarius
gave the solution of the biquadratic. But even Descartes and Vieta
failed at equations of higher degrees. With regard to the work
of Descartes, Leibniz remarks that "its origin [that is, of the method
of solution] was a widely different and more fertile spring; and if
Descartes had only recognized this, he would have rendered the
discovery of Scipio more general and carried it to further heights.
But what has befallen me in this connection I will say in another
place." Leibniz further remarks that the method of Descartes
fails to give the roots of equations of higher degree, although the
quality of the roots may be learned through it. "I will show in
another place that the reason for this is clearly known to me from
the most fundamental principles of the art, and that I have estab-
lished an extremely easy method, and one that is adapted too for
enlarging science, by the many things that follow from it."
In the seventeenth century, Leibniz goes on to say, after Archi-
552 THE MONIST.
mtdes and Galileo's several times and influence are gone by, there
is no writer from whom more is to be learned than from Descartes ;
and yet he is "unable to pass over certain boastful remarks that he
makes, by which the less experienced among us may be led into
error." Descartes had said that by his method every geometrical
problem could be reduced to the finding of the roots of equations.
Leibniz remarks that this shows Dzscartes's ignorance of the matter.
"For when the magnitude of curved lines or the space enclosed by
such is required (which happen more frequently than perhaps Des-
cartes thought, since he had not applied himself sufficiently to the
'mechanics' of Galileo), neither equations nor Cartesian curves can
help us, and there is need of equations of a totally new kind, of
constructions and new curves, and finally of a new calculus, given
so far by nobody, of ivhich, if nothing else, I can now give certain
examples at least, which are remarkable enough." . ... 7 have men-
tioned these things so that men may understand that there are cer-
tain methods in Geometry, for which they may look in vain in the
works of Descartes."
Returning to geometry purely, Leibniz next mentions the work
of Galileo, Cavalieri (whose method he considers is rough and lim-
ited in extent), Torricelli, Roberval, Pascal, Wallis, Huygens, and
Slusius, as contributors to the new geometry. He considers that
a new epoch opens with the work of Neil and van Huraet (on recti-
fication of curves), James Gregory, and Brouncker. "Finally Mer-
cator gave a general formula for the area under a hyperbola." He
claims Mercator as "an eminent German geometer" ; but rather
decries his discovery as being an easy one, on account of the
ordinates working out as rational in terms of the abscissa. "But it
was not so easy to give the magnitude of the circle, and its parts,
expressed as an infinite series of rational numbers ; . . . . for the
circle, however you treat it, has ordinates that are irrational. How-
ever I, as soon as I had found a certain very general theorem, by
means of which any figure whatever could be converted into an-
other that is quite different from it, but yet of equivalent area, set
to work to try whether the circle could not be converted in some way
into a rational figure; and the thing came out beautifully;.... it
will be worth while here to give a short account of the matter."
(In full). Nearly everybody who has up to now treated of the
geometry of indivisibles has been accustomed to break up their
figures into rectangles or parallelograms only by means of ordinates
LEIBNIZ IN LONDON.
553
parallel to one another. But the reasoning of Desargues and Pascal
always pleased me very much ; these in Conies, as we can call them
in general, include under the name of ordinates not only parallels,
but also straight lines meeting in or converging to a point, especially
when parallels are included under the name of converging, by
saying that the point of convergence goes off to an infinite distance.
Thus while others only consider parallel ordinates, and have broken
up their figures into parallelograms AB(B)(A), (A)(B)((B))
((A)), in the way that Cavalieri does, I employ converging lines
n.nd resolve the given figure into triangles CD(D), C(D)((D)), and
at once draw another figure of which the ordinates AB, (A)(B),
etc., are proportional to these triangles.
(El ((£))
Now this is the case if the AB's are equal to the CE's where
it is supposed that the straight lines DE are tangents to the given
curve ; for in that case, as I will show below, it will come out that
the space B(B)(A)A will be double of the segment C(D)DC, and
for any figure such as C(D)DC another that is equivalent to it can
be drawn. Now, supposing that the curve D(D)((D)) is circular
and that CA is a part of the diameter, then, calling CA or FB x,
and CF or AB y, and the radius of the circle unity, calculation
will show that the value of x is 2y-/(\±yz). Thus the ordinate
FB or x can be expressed rationally in terms of the given abscissa
CF or 3'. Such figures as these, in which the ordinates can be ex-
pressed rationally in terms of the abscissae, I call rational. Thus
we have drawn a rational figure equivalent to the circle, and this
will be soon seen to be sufficient to give the arithmetical quadrature
of the latter. For, from the sum of a geometric series of an infinite
number of decreasing terms that is well known to all geometers, it
follows that y*-y* + y* - y* + y10 - y12 + etc. to infinity is the same as
3'V(1+3|2)» i- e-» tne same as \x, if only we understand that y is a
quantity that is less than the radius, or unity. Now, since we have
to collect together the infinite number of $x's into one sum, in order
554 THE MONIST.
to obtain the quadrature of half the figure C(F)(B)BC and what
it comes to, namely, that of the circle ; so also have we to collect
together the infinite number of series y*-y* + ys-y* + ylo-yl2 + etc.,
into one sum, and this by the method of indivisibles and infinites
can be done without difficulty. For, suppose that the last y, which
in general is taken as C(F), to be b, then the sum of every y2 will
be &3/3, and of every y* will be b5/S, and of every y* will be b1/?,
and so on ; hence, the sum of the infinite number of £-*"'s, or of the
series yz-y* + ys-y6 + yl°-yl2+ etc., i. e., the area of half the space
C(F)(B)BC, will be b*/3-b«/5 + b7/7-b»/9 etc. From which,
by the help of ordinary geometry, it can be easily deduced that the
square on the diameter is to the area of the circle as 1 is to 1/1 -
1/3 +1/5 -1/7 + etc. ; also speaking in general, supposing b to be
the tangent, then the arc is b/l-b3/3 + b*/S-b'/7 + &9/9 -&"/!! +
etc. Hence it now follows that any one without the help of tables
and continual bisections of angles and extractions of roots can ap-
proximate to the magnitude of the arc to any degree of accuracy
desired, so long as the tangent & is a little less than the radius ;
so that if we take the tangent to be a little less than the tenth part
of the radius, the arc may be obtained with sufficient accuracy.
Let us take the tangent to be a tenth part of the radius, then if we
want the arc, it will be
11 1 1 , 1
1 1 = f»fr> •
10 3000 500000 70000000 9000000000
and reducing all to a common denominator, and adding the numbers
into one sum ( for it is not worth while going any further) , then the
arc will be a little greater than 518027821302775/5197500000000000,
and the defect of this value from the true value will be less than the
1/1000000000000 part of the radius. For if we do not subtract the
last term, 1/1100000000000, the value would be too great, and if
we do subtract it, the value is less than the true value, there-
fore the error is less than 1/1100000000000, and thus is less than
1/1000000000000.
It is seen how exactly it comes out with such easy calculation
involving only additions, subtractions and multiplications, to an ex-
tent that is not obtainable with tables. Also if the ratio of the tan-
gent to the radius is anything else, the arc can similarly be found,
and this is especially easy when it can be expressed in decimal parts.
Again, since now the ratio of the circumference to the radius is
given in numbers of any required degree of accuracy, by this also
LEIBNIZ IN LONDON. 555
the ratio of a given arc to the circumference is given, and thus
also the quantity of angle for a given tangent will appear with any
required degree of accuracy. In this way tables may be corrected,
supplemented, or, if need be, enlarged, with no great trouble. Any
one who will just remember this fairly easy rule will be able without
tables to attain to any required degree of accuracy with very little
labor. How great an acquisition this is to geometry, I leave it to
those who understand to estimate.
CRITICAL NOTE.
It is difficult to see the object that Leibniz had in writing this
long historical prelude to an imperfect proof of his arithmetical
quadrature, unless it can be ascribed to a motive of self-praise.
This suggestion would seem to be corroborated by the claims that
Leibniz makes in the parts where I have quoted his own words in
italics in the precis, and by the concluding sentence of the trans-
lation given in full. Even if this is so, there may be some plea of
justification put forward ; for Leibniz appears to have been a man
impelled by many contradictory motives, but these I think can all
be traced back to one origin. The time in which he lived was a time
of great discoveries in geometry; Leibniz knew in his soul that he
had it in him to be one of the great men in this branch of learning,
but as truly recognized his great disability due to his lateness in
starting, and felt that his only chance was to belong to the very
exclusive set who corresponded with one another; he saw that the
only way of entering this set was to do something brilliant. This
may be taken as some excuse for any self-praise that we find, and to
a less extent for his, to my mind, undoubted plagiarisms. With
regard to the behavior of Leibniz, when charged with these plagiar-
isms, Sloman is not beyond calling Leibniz a liar point-blank: I
prefer to call his statements perversions of the truth, made under
stress of circumstances, so that his reputation as a great and original
thinker should not suffer. For instance, to explain what I mean, I
will take the statement of Leibniz to de 1'Hospital that he owed
nothing to Barrow. As I have said in another place, from one point
of view, the point of view that Leibniz would take for the purposes
of this letter, Barrow would be a hindrance rather than a help to
Leibniz, in the formulation of his algebraical calculus, after he had
once absorbed all the fundamental ideas. That is, it would seem
that Leibniz always tries to tell the truth, but to put it in a form
that to the uninformed reader will convey quite a wrong impression.
Another example of this juggling with words and phrases is given
by Sloman, in the shape of a letter from Leibniz, dated August 27,
1676, and the first draft of the same ; these two read together are
very much the same, but read apart convey a totally different im-
pression.
A second characteristic of Leibniz may also be traced back to
556 THE MONIST.
his desire to make up for his lateness in starting ; that is, the some-
times ridiculous claims that Leibniz makes to discoveries, or rather
hints at having made them. An instance is given in the Historia
(see Monist, Oct., 1916, p. 599). "It is required to form the sum
of all the ordinates V (\-xx} =y; suppose y = ±\+.-xz, from which
x -=-2z / (\ + zz) , and 3;= (±sz+ \~}/(zz-\- 1) ; and thus again all that
remains to be done is the summation of rationals." Unless we
assume that Leibniz never understood in all his life what we now
call the change of the variable in integration, which to me seems
rather far-fetched, the only reason why this should have been
allowed to appear in a tract that was certainly written after 1712,
is that Leibniz had never attempted this summation ; he had set this
down in 1674 and 1675 as a method of quadrature for the circle,
not at that time having perceived the importance of the factor dz,
or, in other words, the way in which the ordinates should be ordi-
nated ; for as I have already pointed out, at that time Leibniz could
not have found dz, since he could not differentiate a product. This
?oes to prove that his reading of Pascal was not of the profoundest ;
or Pascal is very careful over this point, going to the trouble of
calling the y's ordinates when drawn through the points of equal
division of the base, and sines when they are drawn through the
points of equal division of the arc. Probably to this characteristic
is due the claim, xset in italics in the manuscript above, with respect
to equations of higher degrees. He thought he had a general
rrrthod, which he had not time to verify by particular examples,
and so find that his claim was erroneous. For surely this cannot
be read as a claim to the Tschirnhausian transformation and the
expression of a quintic in the canonical form x5 + fix + q = 0.
The date of the above manuscript is almost certainly antecedent
to the manuscript that Leibniz got ready for the press, De Quadra-
tura ; hence his claim to be able to give examples of the calculus,
except for integral powers which had already been done by Wallis,
is without foundation.
With regard to the arithmetical quadrature itself, the great
importance of it in the estimation of Leibniz is apparently in the
correction and enlargement of tables ; this claim, as Leibniz puts it,
is ridiculous, although it could be so used by first constructing tables
for angles whose tangents are given. But Leibniz, after giving a
calculation true to twelve places of decimals, states that "the ratio
of the circumference to the radius is now known," and proposes
to use that. Apparently he does not see that to calculate this ratio
from the series he gives, it will be necessary to take a billion or so
of terms ! For he does not give any hint of any modification of the
series, or the use of the value obtained for some small angle.
Lastly, with regard to the calculation, it is strange that the
denominator chosen as a common denominator is 15 times what it
need have been ; also it is a matter of wonder, considering that tables
of logarithms were known to Leibniz, as a reader of Mercator and
LEIBNIZ IN LONDON.
557
others, that Leibniz puts the matter in fractional form instead of
working in decimals ; thus, the arc whose tangent is 0. 1 is equal to
0.1 - 0.00033333333333333
0.000002 0.00000001428571428
0 . 000000000 1111 0^000000000000909
0 . 1000020001 11111 0 . 000333347619956
= 0.99966865249
Finally, note that while Wallis and Brouncker are mentioned,
Barrow is not. This is all part and parcel of his successful attempt
to conceal, from all but Oldenburg, the fact that he had a copy of
Barrow in his possession, right from the commencement of his
studies.
in.
Transcribed from a manuscript tract of Newton on "Analysis by
means of equations zvith an infinite number of terms."
ABnjr, BDny, a, b, c given quantities,
m, n whole numbers. If then
n y
__ _ _
x " n [ \ y] n area of ABD.
m + n
In connection with this the following ex-
ample is to be noted :
If -3 ( n^r2) r\y, that is to say, if o=l,
X
n = -\, and m = -2, then we shall have
r/T n ] -x-1 (or — ) naBD,
1 / \ x I
produced indefinitely in the direction of a;
the calculation makes this negative because
it lies on the other side of BD.
Again, if ^ (or x~l) n y , then ^x\ n ^x° n |*1 (* this
ought to be written — 1*) n — n infinity, which is the area of the
hyperbola on either side.
If - — ~2 ny, on division we obtain
ABCD
1 + x*
ynl- .v-
x
., and then
.,
lf
_ £ i
3 '*" 5 " " 7 " etc- ;
the term xz is the first in the division, the value
of y will be x~z - x~* + x~* - etc.,
and hence BD a n — -
3 -5 +etC'
558 THE MONIST.
The first method is to be used when x is small enough, and the
second when x is large enough.
Gerhardt then remarks that Leibniz has noted completely the
following two cases of extractions of roots:
Hy.
Gerhardt further notifies the reader that he has omitted everything
that he has found Leibniz to have copied out word for word, on
comparison with Biot's edition of the Commercium Epistolicum
(1856).
In the above, Leibniz marks interpolated remarks of his own
with either [ ] or (* *).
In the same manner, Leibniz has written out word for word
the part of the manuscript dealing with the solution of adfected
equations (against this he has put the final observation: "And these
things that have been given will be sufficient for the investigation
of areas of curves"), in addition to the part which follows, "the
application of what has been given to other problems of the same
kind," which, as being already known to him, he has not copied out.
He goes straight on to the next section, "To find the converse of the
foregoing, that is, to find the base when given the area, and to find
the base when given the length of the curve." He has written this
out word for word ; also he has noted fully to the end the "proof of
the method of solution of adfected equations."
At the end of these extracts from Newton's tract follow the
words, "I extracted these things from the letter of Newton 20 Aug.
to Newton." Gerhardt states that he has already said all that is
necessary about the contents of these extracts.
SECOND SHEET.
Extracts from the correspondence between Collins and Gregory.
Among a number of partly illegible and unintelligible notes the fol-
lowing were to be noticed.
Gregory, January, 1670: Barrow shows himself to be most
subtle in the geometry of optics. I think that he is superior to all
whose works I have looked into, and I esteem this author beyond
anything that can be imagined.
Sept., 1670: I think that Barrow has gone infinitely further
than all those who have written before him. From his method of
drawing tangents, combined with certain meditations of my own,
I found a general geometrical method of drawing tangents, without
calculation, to all curves, which not only contain his particular
LEIBNIZ IN LONDON. 559
methods, but the general method as well. This is shown in 12
propositions.
Letter of Newton, 1672: ABC is any
angle, ABn^r, BCny. Take, for example,
the equation,
0 10023
Xs -2 Jy + bx* -&x -by* -/ n 0.
DAB 3 22100
Multiply the equation by an arithmetical pro-
gression, both for the second dimension y and for x\ the first
product will be the numerator, and the other divided by x will be
the denominator of a fraction which will express BD, thus.
-2*»y + 2fr» ~3/
3 # -4 xy +2 bx -#'
Moreover that this is only a corollary or a case of a general method
for both mechanical and geometrical lines, whether the curve is
referred to a straight line, or to another curve, without the trouble
of calculation, and other abstruse problems about curves, etc. This
method differs from that of Hudde and also from that of Sluse,
in that it is not necessary to eliminate irrationals.
NOTE.
It is almost useless trying to write a critical note on the above
in such an incomplete state. But I may remark that Leibniz appar-
ently was at the time quite ignorant of what we now term "putting
in the limits for a definite integral."
Gerhardt considers that the existence of this extract proves
conclusively that Leibniz did not see the letter of Newton so often
referred to ; forgetting, as Sloman remarks, that Leibniz ought not
to have seen the tract at all !
P. S. In allusion to footnotes 3 and 18, with regard to the use
of the word "moment" or "momentum" in the sense used by Leibniz,
I have found (since the above was written) that Cavalieri, in his
Exercitationes Sex, defines the term in the mechanical sense and
gives much of the matter of Pascal on Centers of Gravity, as it
appears in the "Letters of Dettonville." I suggest that Leibniz saw
it in Cavalieri, and that its origin is to be traced to Galileo. J. M. C.
OUR MUSICAL IDIOM.
WITH AN INTRODUCTION BY GLENN DILLARD GUNN.
INTRODUCTION.
The effort to expand the means of musical expression is as old
as the art itself. It is recorded in each chapter of musical history ;
it has been interrupted only during those periods of the art's devel-
opment wherein the composer has been concerned with the com-
pletion of art-types already defined.
Every advance in the art has been prefaced by a period of ex-
perimental effort, which has sought new modes of expression. So
soon as these modes of expression have been defined and their
tendencies and the laws governing them have been apprehended,
experiment has been replaced by careful conformance to law and
tradition, which has operated to the perfection of the new art type.
The present is preeminently an epoch of experimentation. The
old art types have been completed. The harmonic vocabulary based
upon the sequences of tonality established in these completed art
forms also has been exhausted and for the past half century com-
posers have been concerned with the development of new harmonic
idioms. (Viz., Liszt, Wagner, Strauss, Franck, D'Indy, Debussy,
Ravel, Shoenberg, Busoni.) As these composers have discovered
and employed new harmonies, new scales and new sequences of
tonality with their resultant new harmonic progressions, the theo-
retician has endeavored to classify their discoveries according to
his established system with results weirdly confusing. The crying
need of the moment seems to be a new system for the naming and
classifying of all possible tonal combinations.
OUR MUSICAL IDIOM. $l
"Harmony is that which sounds together," wrote Bernard Ziehn
twenty-five years ago. But the average theoretician comprehends
only those simultaneously produced sounds which may be arranged
in series of superimposed thirds. In the meantime the composer
has consciously employed many harmonies which are not formed of
superimposed thirds. (Viz., Debussy's major second as the first
interval of the tonic chord, or Shoenberg's combinations of super-
imposed fourths, to cite familiar examples.) The executive artist,
upon whom the composer is dependent for the delivery of his mes-
sage, is, in turn, dependent upon the theoretician for a logical
classification of the new harmonies. The composer of the present
is almost equally dependent upon some scientific classification of his
material. Naturally the public has first looked to him for this classi-
fication* But he seems able to make it only for himself, as Reger
and Shoenberg have done. In any event the world has been slow
to adopt these special classifications and is still seeking a general
system that will include all possible harmonies in logical order.
That system has been evolved by Mr. Ernst Lecher Bacon of
Chicago, who by applying the principles of algebraic permutations
to the problem has succeeded in formulating all harmonies that may
possibly exist in the present system of twelve tones (of itself a most
important service) and having formulated them, has found a system
of nomenclature which actually describes any possible combination
of tones and makes a general or special classification possible.
The value of this new system of nomenclature to the executive
artist is immediately apparent. That puzzled individual may name
and classify the new tonal combinations which he is required to
memorize and present convincingly to the public. The composer is
even more importantly served by Mr. Bacon's researches. For he
is shown at a glance all possible harmonies (th°re are but 350) and
all possible scales (of which there are about 1490). He may select
from the clear and concise tables placed at his disposal those har-
monies and scales which seem to him useful and beautiful and having
familiarized himself with their color and feeling, in short, made
them a part of his own consciousness, may employ them subjectively
to the expression of feeling and sensibility, to the building up of
his own especial harmonic idiom. For though the composer's work
is best when it is most subjective, he is constantly obliged to concern
himself with the facts of his art and out of these facts to fashion
562 THE MONIST.
that delicate fabric of feeling and fantasy which is to give freer
and fuller powers of expression to the music of the future.
GLENN DILLARD GUNN.
THE FORMATION OF SCALES.
The chromatic scale has become established as the basis of
modern harmony. Though the major and minor modes are still
accorded that recognition which the printed key signature seems to
imply, modern compositions so bristle with accidentals that even to
the eye, and still more to the listening ear, is it evident that the
restrictions of the major and minor system have been destroyed.
However, few of our modern composers treat the chromatic
scale purely in itself; the intrusion of other scales which are syn-
thetically formed from the chromatic scale is always felt. The
chromatic scale is only the analytic product of others, which consist
of combinations of its semitones, wherein intervals are found which
are collections of semitones. Of these synthetically formed scales
there are many in number but few in use; and each may form a
separate harmonic basis. Beethoven and Liszt, the latter more
notably, occasionally used scales differing markedly from the major
and minor ; but their appearance was only incidental, and the scales
were rarely made use of as bases for harmonic systems. Busoni
created, as he says, 113 different scales through rearrangements or
permutations of the intervals of the major and minor scales.1 De-
bussy has used a few unfamiliar scales, notably the whole tone scale,
for a thorough harmonic basis. However, as will be shown, a vast
number of scales that have never before been conceived are opened
to discovery through the application of the principles of algebraic
permutation to arrangements of tonal sequences.
A scale is a series of ascending or descending tones. Such a
series may conform to a pattern, a regularly recurring succession
of intervals in certain order, bounded by a fixed interval ; or it may
not conform to pattern. The pattern may or may not conform to
the duodecimal system. If the scale conform to pattern it must be
jf bounded by a fixed interval. Intervals of simple physical ratio are
preferred, and these are found, slightly tempered, in the duodecimal
system.
Until now, the octave only has been consciously used as a fixed
1 His figures are incorrect as, mathematically computed, the number of
permutations of the combination of intervals is : (a) of the major scale, 21 ;
(&) of the minor scale, 140.
OUR MUSICAL IDIOM. 563
interval, but there is no reason why, in specialized cases, other inter-
vals could not exist between corresponding tones in recurrences of
the pattern. We may have scales repeating at each fourth, fifth,
sixth, seventh, ninth, tenth, etc. But because we are at present
engaged in a classification of scales which incidentally involves the
discovery of a multitude of unheard-of ones, and because a classi-
fication of such scales as these would be of formidable length, we
must be content to study that most important class of scales in
which each succeeding repetition begins an octave above or below
the preceding one.
Again we must distinguish between two classes of scales whose
basis is the octave. The first class is that one in which the smallest
scale-units in the octave number 12 ; this is our duodecimal system.
The second class contains many systems, in each of which the num-
ber of the smallest units is either greater or less than 12. We will
consider both of these classes, for in the consideration of the first
class we can enlarge considerably the present scope of the duo-
decimal system, while in the consideration of the second class we
may discern dimly certain possibilities of the future. First we will
discuss the scale possibilities of the duodecimal system.
By a division of the octave into twelve parts the common
chromatic scale is formed. Now by grouping together certain of
these twelfths of an octave, the so-called "semitones," we may form
scales whose gradation is uneven and less refined than that of the
chromatic scale. If we are given a certain combination of intervals
which, added together, give the octave, we can permutate these in
a number of different ways ; that is, we can rearrange the given
intervals to form different scales. We also may have combinations
in which the same intervals occur more than once. If n is the num-
ber of intervals between octaves in the scale and «x of them are
alike, and n2 others are alike, etc., the number of scales (P) that
can be formed by permuting the given combination of intervals is:
(The exclamation point, read "factorial," denotes that the
number which it follows is a product of all integers less than and
including itself, each integer being a factor only once.)
For example, we desire to find the number of scales that can
be formed with the intervals of the major scale. The major scale
consists of 5 whole tones and 2 half tones, making a total of 7
intervals.
5^4
THE MONIST.
Thus P=(n!/«1!n2!) = (7!/5!.2!)
= [1.2. 3. 4. 5.6.7/(1.2.3.4. 5.) (1.2)]
P = 42/2=21.
As an example of the way in which all possible scales can be
formed out of a certain combination of intervals, 20 scales will be
formed out of the combination (three minor thirds and three minor
seconds) as follows:
The symbol ( — 3) will be written below respective minor thirds.
123
-3-3-3
-4
-3 -3 -3
-3
io&E
-3-3
-3 -3 -3-3
8
-3 -3 -3
9
-3 -3 -3 -3 -3 -3 -3 -3 -3
10 11 12
o
-3
13
-3-3 -3-3-3
14
-3-3 -3
'
15
^
feb
-3 -3 -3 -3 -3 -3
16 17
• • • o
18
o
-3 -3 -3
«.
fey
>o*»
o^±
-3 -3-3
-3 -3 -3
-3 -3 -3
19
20
" n i Ip °1T
torfeo-^1-0^
-3 -3
-3 -3 -3
OUR MUSICAL IDIOM. 565
It will be observed that a new series begins with each double
or triple bar.
A triple bar is written before each chromatic elevation of the
lowest minor third.
A double bar is written before each chromatic elevation of the
middle minor third.
The uppermost minor third always starts at its lowest possible
position and is raised successively to its highest possible one, after
which a change is made in the relative position of the lower minor
thirds.
This method of forming all scales from a certain combination
of intervals is purely arbitrary.
Now it is possible to form a great number of combinations in
which the sum of the intervals is an octave. Moreover, as we have
seen, usually a number of scales can be formed out of each combina-
tion. Each scale is to be considered a permutation of the combina-
tion's intervals.
In the following table will be found every combination possible
with intervals as small as the minor second and not greater than
the major third. Intervals larger than the major third are not
used because in the formation of scales they would make gradation
too abrupt and uneven. The table will also include calculations
of the number of permutations (to be regarded as the number of
scales) possible with each respective combination, according to the
formula. The vertical columns contain the intervals minor 2d,
major 2d, minor 3d, major 3d, respectively. The horizontal rows
of numbers are the combinations. A number (n) falling in a
vertical column (v) means that the interval (v) is repeated n times
in the combination in which n lies (see Table I).
To make the function and construction of the table more plain
two of the combinations may be explained. Combination 1 indi-
cated by the number in the extreme left-hand column, consists of
twelve semitones. It is therefore a formula of the chromatic scale,
and has therefore only one permutation. Combination 21 contains
two minor seconds, two major seconds and two minor thirds. From
it may be formed fifteen scales or permutations.
Means have now been shown to find all possible scales in the
twelve-tone system, scales which have intervals exceeding the major
third in size being omitted. Adding the number of permutations
formed with all combinations a total of 1490 scales is found.
566
THE MONIST.
A systematic study of these 1490 new scales would lead to the
discovery of many valuable scales. I have found many that are
interesting by this method, but will mention only a certain class of
these scales, which I will call equipartite for want of a better name.
TABLE I.
I
COMBINATIONS
PERMUTATIONS
18
COMBINATIONS
PERMUTATIONS
ii
O H
MINOR ]
SRCONDS
MAJOR
SECONDS
8
ta
O
ta
in
PERMU-
TATIONS
gS
*£
s£
MAJOR
THIRDS
J
1%
<2
0 H
PERMU-
TATIONS
£8
a£
go
<«
Sw
II
II
12
Pl2!/I2!
I
3
3
6!/3l3l
2O
2
10
I
n!/io!
II
19
2
5
7!/2!5!
21
3
9
I
101/9!
IO
20
2
3
I
61/213!
60
4
8
2
io!/2!8!
45
21
2
2
2
6!/2! 2! 2!
go
5
8
I
9!/8!
9
22
2
I
2
51/2! 2!
30
6
7
I
I
9!/7!
72
23
2
2
I
51/2! 2!
30
7
6
3
9!/3!6!
84
24
I
4
I
6l/4i
30
8
6
i
I
8!/6!
56
25
I
2
I
I
Si/2!
60
9
6
2
8 !/2! 6!
28
26
I
I
3
573!
20
10
5
2
I
81/215!
1 68
27
I
i
2
4!/2!
12
ii
5
I
I
71/5'
42
28
6
6!/6!
I
12
4
4
8!/4!4!
70
29
4
I
§1/41
5
13
4
2
I
7!/2U!
105
30
3
2
5!/2!3!
10
M
4
I
2
7!/2!4!
i°5
31
2
2
41/2! 2!
6
15
4
2
61/214!
15
32
I
2
I
4l/a!
12
16
3
3
I
71/513!
140
33
4
474!
I
J7
3
i
I
I
61/3]
120
34
3
31/3!
I
Total 1490
An equipartite scale is one in which the same pattern of inter-
vals is repeated an integral number of times within the octave.
If a scale is bipartite a group of intervals will appear twice within
the octave with no remainder; if the scale is to begin on F its two
parts begin, respectively, on F and B. As a result in this case it is
immaterial whether the tonic is B or F, for the scale sounds alike
either way, except for the transposition.
OUR MUSICAL IDIOM.
567
We may split the sum of twelve semitones (semitones being
regarded as intervals) into two parts or three. Dividing it into two
parts, each part containing six semitones, allows us again to divide
this semi-octave into two or three parts. Dividing the octave into
TABLE II (FOR BIPARTITE SCALES).
COMBINATIONS
PERMUTATIONS
No.
I
MINOR
SECOND
2
MAJOR
SECOND
3
MINOR
THIRD
4
MAJOR
THIRD
CALCULATIONS
No. OF
PERM .
I
6
P=6!/6!
I
2
4
I
P==5!/4'
5
3
3
I
P=4'/$l
4
4
2
2
P=4!/2! 2!
6
5
2
I
P=3!/2!
3
6
I
I
I
P=*i
6
7
3
P=3'/3>
i
8
I
I
P=2!
2
9
2
P=2!/2!
I
Total 29
TABLE III (FOR TRIPARTITE SCALES).
COMBINATIONS
PERMUTATIONS
No.
I
MINOR
SECOND
MAJOR
SECOND
3
MINOR
THIRD
4
MAJOR
THIRD
CALCULATIONS
No. OF
PERM.
I
^4
P=4!/4!
L_J
I
2
2
I
P=3!/2!
3
3
I
I
P=2!
2
4
2
P=2!/2!
I
5
I
P=i!
I
Total 8
three parts, each part has four semitones, which may again be
divided by two. Thus we may split the octave into 2, 3, 4, and 6
equal parts. Scales formed by such divisions may be called, respec-
tively, bipartite, tripartite, quadripartite, and sexpartite. As the
568
THE MONIST.
last two types may be classed under the first and second they
do not require a separate classification. In Tables II and III
the combinations in the bipartite and tripartite types are given; in
other words, the possibilities of combinations with six and four
semitones, respectively, are shown. Each arrangement of a com-
bination is then repeated in the remaining half or two-thirds of the
octave.
A few interesting equipartite scales are herewith shown:
1 2
-o-
(1, 2 and 3 are from Table 2, combination No. 4; 4, 5 and 6 are from
Table 3, combination No. 3.)
Scales formed by permutating combination No. 4 in Table II,
and combinations Nos. 2 and 3 in Table III are especially interesting.
No. 1 is formed by alternating major and minor seconds, while No.
3 is formed in the same way, except that in it the order of the inter-
vals of No. 1 is reversed. Even such a mechanically formed scale
as this sounds beautiful and original. It is a noteworthy fact that
in scales 1 and 3 the chords formed on every degree are diminished.
Scales Nos. 4 and 5 are built similarly; only a minor third and a
minor second alternate. Chords formed on every degree of these
scales are augmented.
SCALES FORMED FROM SYSTEMS OTHER THAN THE DUO-
DECIMAL.
Although to-day the importance of systems containing other
intervals than multiples of semitones is questionable, it is neverthe-
less interesting to know that such systems may be exploited for
scale and harmonic possibilities in the same manner as our present
system. Busoni has already experimented with the tripartite tone
scale ; that is, a scale in which each whole tone is divided into three
instead of two whole parts. The physicist may scorn the idea of a
OUR MUSICAL IDIOM. 569
new system, knowing that the duodecimal system contains the sim-
plest physical intervals, yet it must be remembered that the perfect
intervals are also not found in the 12-tone system, because of "tem-
pering." Moreover in the other systems many of the most impor-
tant intervals of the duodecimal system will be duplicated. Al-
though probably no system will ever be of equal importance with
the duodecimal, it is not inconceivable that, just as certain new
scales within our present system have been chosen by recent com-
posers as harmonic and melodic idioms of expression, so certain
"foreign" systems may once be chosen for similar purposes.
Accordingly, we are to consider any equal divisions of the
octave. However, certain divisions, as for example into 11 or 13
equal parts, are not of importance, since the intervals formed in this
way would only be confounded with poorly tuned intervals of the
12-tone scale. In order to discriminate in the selection of numbers
with which to divide the octave it is well to choose only those num-
bers which are multiples of the smallest prime numbers, 2, 3, and
5. We may call each of these systems an "N-tone chromatic sys-
tem." If the system is one in which the number of smallest intervals
is 9, we may call it a 9-tone chromatic system. We are not bound
to confine the use of the term "chromatic" to our duodecimal system,
since in its musical application the word is used to describe a suc-
cession of the smallest possible intervals.
In considering the N-tone chromatic systems we may go through
the same steps through which we have passed in considering the
duodecimal system. In each of these unfamiliar systems there are
chromatic intervals which may be combined and permutated to form
scales of more rapid and uneven gradation. Just as before, we have
to set a certain limit to the size of an interval employed in one of
these scales. In the five-tone system a coupling of only two chro-
matic intervals produces an interval almost too great to exist in a-
scale of moderately refined gradation. In the 24-tonal system a
coupling of as many as 6 intervals into 1 is acceptable. It will readily
be seen that to construct tables for all N-tonal systems through
which an infinite number of gradations is possible, would require
much space. It has already been stated that those systems having
numbers of chromatic intervals equal to multiples of 2, 3 and 5, are
most important. They are systems of 4, 5, 6, 8, 9, 10, 12, 14, '15,
16, 18, 20, 21, 22, 24, etc., chromatic tones ; for demonstration I will
select only 2 of these; namely 8- and 9-tone systems.
570
THE MONIST.
In the tables that follow I will give the number of combinations
and calculate the number of permutations for each combination with
the selected N-tonal systems. In other words, scale possibilities
with systems having 8 and 9 tones will be shown in each respective
table.
TABLE II, N=8 TABLE III, N— 9
I
COMBINATIONS
PERMUTATIONS
i
COMBINATIONS
PERMUTATIONS
a
PERMU-
TATIONS
H
-*?
U
O
8
OCTAVE
1
OCTAVE
<
h
<°
O H
PERMU-
TATIONS
•AVXDQ
?
i
OCTAVK
1 OCTAVK
i
= »
31
U H
81/8!
I
9
9!/9!
I
2
I
7!/6|
7
2
7
I
81/71
8
3
5
I
61/51
6
3
6
I
71/01
7
4
4
2
61/214!
15
4
5
2
71/2! 5!
21
•5
3
I
I
5l/3l
20
5
4
I
I
6!/4!
30
6
2
3
5!/2!3!
10
6
3
3
6!/3!3!
2O
7
2
2
41/2! 2!
6
7
3
2
5V2!3!
10
8
I
2
I
4'/2!
12
8
2
2
I
51/2! 2!
30
9
4
4'/4!
I
9
I
4
5»/4l
5
10
i
2
3!/2!
3
10
ii
I
I
3
2
I
4!/2!
4!/3!
12
4
Total 81
Intervals used do not exceed | octave.
12
3
3!/3l
I
Duodecimal System^.
Total 149
Intervals used do not exceed | octave
or a major third, as translated.
Number of intervals corresponding
to Duodecimal System = 3.
The Numbers of Tones and Intervals Found Correspondingly in
Any Two N -Tonal Systems.
If we choose a common tonic for all N-tone chromatic scales
we will find certain other tones which are common to two or more
of these scales. For example, if we form both a 9-tone chromatic
and a duodecimal scale upon C, we will expect to find two tones in
common besides the C and its octave. They will be E and G sharp ;
for each of these tones marks the partition of the octave into three
equal parts. This means that certain intervals in one system are
OUR MUSICAL IDIOM. 571
the same as intervals of another. But an interval common to two
systems cannot be the same multiple of the smallest unit in each sys-
tem. If we desire to find the number of intervals which are found
correspondingly in each of the two systems, we need merely to
find the largest factor common to the number of chromatic divisions
of both systems. For example, to find the number of intervals
which are common to the 18-tone and the 12-tone chromatic scales
we find the G. C. F. of 18 and 12, which is 6. This is the desired
number. Of course, intervals which are multiples of this common
interval (the whole-tone, in this case) are also common to both
systems.
Intervals of N-Tone Chromatic Scales.
Throughout our entire treatment of scale possibilities there is
one interval which remains constant ; namely, the octave. The ratio
of this interval, that is the ratio2 of the vibration frequency of the
higher tone to that of the lower tone is always 2. If N is the fre-
quency of the lower tone, its octave is 2 N. Now N and 2 N may
be written as 2° N and 21 N respectively, since any quantity with an
exponent 0 equals unity. It is evident that the frequencies of any
tones between NX 2° and Nx2* can be expressed as N times the
coefficient 2 with an exponent varying between 0 and 1.
If the octave contains r equal intervals, the difference between
0 and 1 of the exponent of 2 will be divided into r parts. This is
true because (a) equal intervals form equal ratios of vibration; and
(&) equal ratios may be expressed as the quotients of a constant
in which the difference of the constant's exponents in the numerators
and respective denominators remains constant. To illustrate:
21/2° = 21-° = 2
26/25 = 26-6 = 2.
Hence 21/2° = 26/2*.
Thus
2^ or v 2
expresses the ratio of any interval formed by two adjacent tones
in an equally tempered scale of r intervals. Moreover the intervals
which any tonic (arbitrarily chosen in the case of the equally tem-
pered scales) forms with the successive ascending tones above it,
are, respectively:
2iA 22A 2»A 2*A 2<'-1>A 2'/' or 2.
2 This ratio is physically defined as the interval itself.
572 THE MONIST.
From these facts we derive two general formulas: (A) ex-
pressing the physical interval- or vibration ratio between 2 tones
and (B) the vibration frequency of any tone lying above a given
tone, N.
(A) I (interval }=2c/r,
J*
where r = the number of chromatic intervals per octave, in the given
system; and c - the number of chromatic intervals separating the
two tones whose physical ratio is to be found.
With these formulas we can express the various intervals of
any equally tempered scale.
NOTATION OF SCALES.
In considering the great number of scales of which we have
learned in the previous section we are confronted with the problem
of their notation. Our present notation is really suited for seven
scales only ; namely, the major scale and the scales formed by
cyclically rotating the permutation of the intervals of the major
scale, that is, the Dorian, Phrygian, Lydian, Mixolydian, etc. We
cannot write even a minor scale without the use of an accidental.
Then with regard to the 1483 other scales, because of this great
number and variety, we cannot do more than make general state-
ments.
We realize, to begin with, that the ideal notation of our present-
day music should be one which is designed to eliminate the incon-
veniences of accidentals. Such a notation would be naturally one
designed from the chromatic scale; and because the chromatic
scale contains all of the 1490 other scales of the duodecimal system,
it would be adaptable, in a perfect sense, to all of these scales. We
could accomplish the notation of the chromatic 12-tone scale with a
six-line staff giving each degree a separate line or space, as shown :
f^_ O N
-<3-
O * '
The major scale on this staff would be:
do re mi fa sol la ti do
The minor scale would be:
OUR MUSICAL IDIOM. 573
The mental picture we obtain of the relation of the intervals of
these two scales in this manner is alone an advantage. Furthermore,
in the six-line staff notation we are less bound to avoid deviations
from our chosen scale ; we are freer to escape from the tyranny of
sharps and flats. An abhorrence of accidentals has always tied us
to our chosen scales. Other advantages of this notation could be
cited, but the chief one is, of course, that merely through the addi-
tion of another line (which does not confuse us optically) we are
able entirely to avoid accidentals.
However, the difficulty of introducing this system into common
use would be almost too great to be overcome. An attempt at this
could be likened to the recent attempts at introducing a universal
language; for were we all to learn a universal language we would
still have to retain a knowledge of the old for its literature. We
are therefore compelled to adjust our new scales to the common
notation of the five-line staff.
We may eliminate from consideration not only the major scale
and those scales formed by a cyclic rotation5 of its permutation of
intervals, but also the minor scale with its corresponding scales
formed by a similar cyclic rotation. This suggests to us a process
that will greatly simplify our whole problem. We see that the
notation for one scale is suitable for all other scales formed by a
cyclic rotation of the permutation of its intervals. The number of
these scales will depend upon the number of tones or intervals in
the original one. The notation for a scale of n tones or n intervals
will serve for (n-1) cyclically related scales. Thus one notation
serves for n scales.
We realize that out of a certain combination of intervals we
may form more than one cyclic group, for some combinations have
as many as 168 scales while in no cyclic group can there be more
than 11 different scales. A formula with which we may calculate
the number of cyclic groups in each combination is :
G* («- !)!/%! *, in,!....
where n is the number of intervals in a combination, and nlt n.,, n3,
3 The term defines itself. A cyclic rotation of a permutation is one in
which the terms are always written in the same order, but each successive
permutation begins with the second term of the preceding one. The following
is a cyclic group of permutations : ABCD, BCD A, CDAB, DABC.
574 THE MONIST.
etc. are the numbers of times respectively which certain intervals
are repeated in the combination. There are few exceptions to this
formula, all of which are of one type. The erroneous type is that
in which («1 + M2 + w3 + n4. . . ) exceeds (w-1). These exceptions
often cause fractions which cannot be integrally expressed. In cases
of this exception we must find our number of cyclic groups by actual
trial. But if we have found one signature suitable for each whole
• cyclic group of scales we have, in general, shown only one-twelfth
of possible signatures, for in most cases a different signature is
necessary for each chromatic degree. Only in equipartite scales
% are fewer signatures than twelve necessary to each group. If a
scale is bipartite only six signatures are necessary; if tripartite,
four; if quadripartite, three; and if sexpartite, two.
As we are considering these 227 scales representing cyclic
groups primarily for their notation, we are confronted with the
question, what signature shall we give to a work based on a scale
like the following?
oP"
None of our conventional signatures for major scales will apply
to this scale ; for we see the three essential signatures are :
d flat being unnecessary as a signature because it is cancelled im-
mediately, the scale being an 8-tone scale, which necessitates the
repetition of one note.
We will find that most of the scales, like this one, will require
signatures other than those which we have employed for our major
and minor modes. Consequently we will not try to reconcile our
customary signatures with those natural to the new scales. There-
fore, in order to make a signature for any scale on any degree, write
down those accidentals which appear in the notation of the scale,
omitting those accidentals only which are cancelled as the scale con-
tinues. We may rightly call this a natural system of signatures.
Concerning the method of finding each scale representative of
a cyclic group for a given scale degree, the following means are
perhaps the simplest:
OUR MUSICAL IDIOM. 575
1. Choose an interval which occurs singly in the combination
and place it in the lowest position in the scale.
2. Permutate the other intervals above it in every possible way.
3. Each permutation, with the first interval remaining in a
fixed position, will form a desired scale.
4. When no interval occurs singly in the combination there is
no rule which applies generally; but because of the small number
of combinations of this character the desired scales can be easily
found by trial.
There is little need for investigating the problems of notation
of N-tonal systems until such systems come into use. Solutions to
such problems are really simple and arbitrary. Suffice it to say,
there is no need of retaining the five-line staff for N-tonal notation.
It would be unfortunate if one were compelled to read a totally
new system of intervals from a staff with which one would con-
stantly associate accustomed intervals.
Although it may seem strange that so much attention is paid
a subject like the formation of scales, there is nevertheless justi-
fication in an investigation of this sort. A scale has far greater im-
portance than the mere sequence of tones comprising it would imply.
Practically all of the hundreds of melodies we know can be formed,
almost without accidentals, from the major and minor scales. Vir-
tually all of the common harmonies can be constructed from these
modes. The vast amount of musical thought and feeling has until
recently expressed itself in major and minor. But the chromatic
scale offers a much wider field of expression ; for it contains not
only the major ^nd minor scales, but over fourteen hundred others.
Nevertheless, although the chromatic scale has become the basis for
modern harmony, melody does not seem to flow freely chromatically.
Our musical speech continually demands some simple group of tones
and larger intervals. Without some limitation more binding than
the chromatic scale, we are helplessly confused with the wealth of
possibilities. Such limitations are found among the multitude of
scales derived synthetically from the chromatic.
Debussy and some of his colleagues have made their idiom or
"dialect," as it were, the whole-tone scale. This one scale, because
of its uncertain "tonality," and its "color," has been the outstanding
characteristic of the French impressionists.
A few other scales, such as the Greek "modes," which are all
cyclically related to the major scale, have been the basis for numer-
576 THE MONIST.
cms works. On the whole, there is no reason why other scales,
among the vast number shown to exist, should not become equally
important idioms of expression.
Limitations of space unfortunately prevent me from tabulating
completely the fourteen hundred and ninety scales of the duodecimal
system.
Concerning the N-tone scales, it is well to consider for illus-
trative purposes the words of Busoni in regard to his tripartite tone
scale:4 "The tripartite tone," says he, "has for some time been de-
manding admittance, and we have left the call unheeded." With
the tripartite tone he encounters a difficulty which will be found
also in considering other N-tone scales. He says we would lose
through the tripartite tone the minor third and the perfect fifth.
Now a chromatic scale in which the most important intervals do not
occur (intervals whose ratios are expressed as quotients of the
smaller integers) will never form quite as valuable a system as a
chromatic scale that contains them. Realizing this, Busoni has at-
tempted to reconcile the 12-tone with the 18-tone system; that is, a
system of bipartite tones with one of tripartite tones. His solution
is naturally a 36-tone scale involving the sexpartite tone. To enter-
tain any hopes for a system of sexpartite tones seems to me futile.
A system of 24-tone chromatics might be better reconciled with our
duodecimal system. This example merely shows us that we cannot
attempt to reconcile the N-tonal systems with each other or with
the duodecimal system. An 18-tone chromatic system is perhaps
next in importance to the duodecimal system, but it is comprehensive
and important enough in itself, even though it does not contain
minor thirds and perfect fifths.
Again we must consider how we are to produce these tones,
as Btisoni has mentioned in regard to his tripartite tone scale. For
experiment and a training of the ear to the tripartite tone Busoni
recommends Dr. Thaddeus Cahill's dynamophone, an instrument
which would, however, be very difficult to obtain or to construct.
A Seebeck's siren with a special disk for each system would be a
good substitute. The number of holes in each circular row could
be mathematically computed with the help of the formulas:
4 For Busoni's statements read his Sketch of a New Esthetic of Music,
New York, 1911.
OUR MUSICAL IDIOM. 577
which are explained in previous pages. A motor to Devolve the disk
would furnish a constant speed of rotation.
Such experiments would furnish means of acquiring a sense of
intervals other than those to which we are accustomed ; but, in
Busoni's words, "only a long and careful series of experiments and
a continued training of the ear can render this material approach-
able and plastic for the coming generation, and for art."
A NEW HARMONY.
Our present system of harmony, the system of chords (har-
monies formed of superimposed thirds), is deficient in two important
respects. First, it is often unwieldy ; and second, it is not fully com-
prehensive. This latter shortcoming is partly responsible for the
former, since it is true that we may represent certain harmonies,
seemingly not within the scope of our system, in complex ways. To
illustrate: let us examine the various unsatisfactory ways of de-
scribing the simple harmony:
If the harmony is a chord, we must be able to build it up by
superimposing thirds. But no complete chord exists that contains
each of these and only four notes.
But there are chords containing more than four notes which
contain the notes of the harmony, such as the following:
A B C^ D
tl
From these chords we may strike out those notes foreign to
the harmony and derive what we call an incomplete chord. The
harmony :
may therefore be termed an incomplete nth chord (as in A, B, or
D) or an incomplete ijth chord (as in C). If we are willing to
recognize an incompleteness of this sort as mathematically rigid,
we must still admit that such a naming of the harmony as an in-
complete llth does nothing more than justify its existence among
chords. It does not name the harmony for there are innumerable
578 THE MONIST.
incomplete llths. To name the llth chord in each case is difficult,
the general method being that of determining upon what degree of
the major or minor scale it is built. But, again, must we consider
all harmonies in the light of the major or minor scales to-day when
many other scales are being used? Furthermore we must find where
the incompleteness lies. Lastly one would suppose that every har-
mony has one fundamental position, but here are four. One should
be able to tell what sort of inversion of the fundamental harmony
the one in question is. How is this possible when the harmony in
its position is a different inversion with each fundamental?
If we allow ourselves the latitude of recognizing diminished
thirds, we may say the harmony is composed of two diminished
thirds separated by a minor third. Taking this liberty we might
have a specialised chord or so-called altered chord, but how shall
we describe any particular one ? Moreover, it is false to assume that
diminished thirds are thirds at all; for they are seconds.
Sometimes, if the harmony is preceded or followed by others,
we may analyze it under our present system by considering certain
tones as "passing tones," "suspensions," "afterbeats," "syncops,"
"organ-point," etc. The awkward system of figured bases sometimes
affords a means for expressing simpler chords.
If such is the fate that a simple harmony like the above suffers
in analysis, what lot befalls the multitiude of more complex har-
monies ? The best that modern analysis can do for them is to treat
them in relation to surrounding harmonies. Even then, "unresolved
suspensions," etc. are continually met with in modern music. If
harmony is "that which sounds together," we should be able to
define any combination of simultaneously sounding tones, whether
this combination is surrounded by others or not. A note suspended
from a consonant to a dissonant chord is sounding in the second as
well as in the first harmony. Does not an organ-point form a
separate harmony with each of a series of chords "moving through
it," even though these chords are dissonant with the organ-point?
A harmony is a harmony whether dissonant or consonant. Yet of
the vast majority of dissonant harmonies few can be adequately
named and classified in themselves.
The chord system is adequate in analysis of older works only.
It can give only a superficial analysis of modern works.
A more important objection even than that of inadequate
nomenclature is that by reason of our use of the chord system we
OUR MUSICAL IDIOM. 579
are hindered in enlarging our scope of harmonies. The conception
of harmonies given us by this system restrains us from enlarging
our harmonic vocabulary. Bred in the chord system, we are prone
to regard any harmony which is not chordic in construction as a
mere variance of some "simple" chordic form. Many a stereotyped
theorist would shudder at the notion of giving the above harmony
a prolonged and separate existence. It must be immediately re-
solved into a stable form ; the tonic triad of G major, etc., etc. Are
we blind to the existence of harmonies not made up of superimposed
thirds? Shall we refuse to recognize non-chordic harmonies merely
for the technical reason that we employ a system of superimposed
thirds, which was an expedient solution to theoretical problems two
centuries ago? Because of the limitations of our present system, a
vast number of harmonies remain to-day virtually undiscovered.
Although many have been employed passingly and subconsciously,
few have been employed deliberately, few are spoken of as a part
of the composer's vocabulary.
Fully realizing the importance of the chord system in the anal-
ysis of older works (for these were conceived in the spirit of the
chord system), I believe it is important that a new and fully com-
prehensive system should supplement it, a system that would prove
adequate for the analysis of modern writings. Whereas the old
system embraces chords only, the new should embrace all harmonies.
The chord scheme would then take its place as a sub-system of the
more general and all-inclusive system.
Just as the present method is more than a mere scheme of
nomenclature, so the one which I propose should be considered as
affecting more than the mere naming of harmonies. The chord
system teaches us that all harmonies are chords, are built by imposing
major and minor thirds upon each other. The proposed system
should, as will be shown, teach us to recognize harmonies which are
built by superimposing any intervals. It should teach us a broader
conception of harmonies and, as I believe, a more valuable one,
since the importance of a notion usually depends upon its generality.
To-day the modern composer habitually employs the twelve-
tone scale as the source of his harmonic invention. The abundance
of accidentals in our modern composition is superficial but none the
less accurate evidence of the passing of the feeling for the diatonic
modes. To-day there are also a few scales which are formed of
new arrangements in the intervals of our duodecimal system.
580 THE MONIST.
But the octave still remains the common basis for all scales now
used ; each scale repeats itself at successive octaves.
It seems only natural that we make this interval which is of
the greatest importance because it has the simplest ratio, the basis
for our harmony. We may therefore call its interval unity.5
Having established the octave as the basic interval, and having
assigned to it the number one, we turn our attention to the lesser
intervals. The semitone, since it is one-twelfth the gradation of
the octave, will be known as the interval, one-twelfth. The ''whole
tone" becomes two-twelfths. Tabulating all of our intervals in their
old and new nomenclature we have:
DIATONIC NAME NATURAL OR CHROMATIC NAME
Minor Second One Twelfth
Major Second Two Twelfths
Minor Third . Three "
Major Third Four
Perfect Fourth Five
Augmented Fourth Six
Perfect Fifth Seven
Minor Sixth Eight
Major Sixth . . . . Nine
Minor Seventh Ten
Major Seventh . . . .'". . . . . Eleven "
Octave One.
Although many of these fractions expressing intervals are not
reduced to their simplest form, it is of advantage to retain the com-
mon denominator, twelve ; for if all intervals can be expressed as
quotients of variable integers and the constant twelve, we need con-
sider only the numerators and eliminate the common denominator.
Thus the intervals, one-twelfth, two-twelfths, three-twelfths etc.,
may be called respectively, one, two, three, etc. It is clear that this
nomenclature is founded entirely upon the chromatic scale since
every interval is measured as a multiple of the intervals comprising
the chromatic scale.
In naming harmonies having more than one interval, the ad-
vantages of the chromatic nomenclature are immediately apparent.
For instance, the major triad is said to be formed of a minor third
placed above a major third. In other words the interval 3 is placed
above the interval 4. Thus the major triad in fundamental position
5 For simplicity we call the interval unity, although the physical interval of
the octave is 2.
OUR MUSICAL IDIOM. 581
is a 34 or 4-3 harmony. Likewise, the minor triad is a 3-4 harmony ;
the dominant sept chord in fundamental position, a 4-3-3 harmony ;
and the dominant sept chord in its first inversion is a 3-3-2 harmony.
The harmony under previous discussion is, in the form
a 2-3-2 harmony.
This change in nomenclature means more than is at first ap-
parent. It means an expansion of our conception of harmonies
which may, perhaps, offset the limitations felt in the minds of many
who can think of music only as varying arrangements of groups of
superimposed thirds. We may freely think of any harmonies as
being composed of superimposed intervals of any sort, instead of
being shackled by considering every harmony made up of super-
imposed thirds, or inversions of these. Our vocabulary of har-
monies, instead of embracing only chords, will embrace all har-
monies.
It may be well here to forestall a possible objection that the
chord system is the nearest to the ideal from the physical point of
view. It is true that chords are physically the "cleanest" harmonies,
i. e., their tones have the simplest vibration ratios. It may be said
from this that the system of building up thirds is founded not upon
an arbitrary choice, but upon an acoustic basis. But I answer that
from this point of view it is immaterial whether we think of major
and minor thirds or of 4s and 3s in building up the most important
harmonies. If thirds and sums of thirds are the most important
intervals, then, after we have learned the chromatic nomenclature,
3s, 4s, 6s and 7s will come to be recognized as the most important
intervals. There is no ground for any charge that the chromatic
nomenclature is more empiric and less scientific than the chord
system.
Again, is not the chromatic nomenclature a simple and an ac-
curate method for naming and classifying any harmony? Instead
of grappling with such harmonies as these:
ft r: \ jf — ** i
considering them in relation to surrounding harmonies, and in
themselves by devious ways, we simply name them chromatically
582 THE MONIST.
as consisting of superimposed intervals. They are respectively:
6-4-6-4 and 5-5-4-4 harmonies.
Our next task is to study more closely the nature of harmonies
and to discover suitable means for systematically finding all of
these harmonies.
First we will recognize a tone essentially the same whether its
vibration frequency is increased or diminished by a power of two;
for the ordinary ears hear it as essentially the same. For this reason,
we are compelled to regard an inversion of a harmony only as a
different position of the harmony and not as a different harmony.
In the chord system we decide upon one position of a harmony
which we call close fundamenal, namely that position in which the
harmony's root is in the base. (There is, however, no distinction
made between "open" and "close" fundamental positions.) Like-
wise, and for purpose of classification, we should decide upon a
fundamental position of a harmony in our chromatic system of
nomenclature. The choice of a fundamental position, though arbi-
trary, will be made later in the discussion.
The relations between the tones comprising a harmony are
intervals. A harmony may be thought of as a combination of inter-
vals. However, a combination of intervals may form more than
one harmony. The major triad is a 4-3 harmony, that is it is made
up of the intervals 4 and 3 ; yet if we merely reverse the order of
these intervals we have another harmony, the minor triad. It is
clear then that a harmony is one permutation only of a given com-
bination of intervals.
Now it might be thought that one could form all harmonies
existing in our duodecimal system by permutating all possible com-
binations of intervals in all possible ways. However, by doing this
we would calculate an immense number of harmonies, very many of
which would only be inversions of each other. To avoid the repe-
titions occasioned by inversions, in classifying all harmonies, we
come to the consideration of an extra interval with each harmony.
If we invert the major triad or the 4-3 harmony we have first a
3-5 harmony (commonly called the first inversion) and then a 5-4
harmony (called the second inversion). The extra interval to be
considered in this case is the interval 5, which is bounded by the
highest tone of the triad and the note an octave above the triad's
lowest tone, the triad standing in fundamental position — an interval
which will be defined as the complement. If the triad is in 4-3
position of f, the complement is the interval c-f :
OUR MUSICAL IDIOM. $83
^complement s 5
»
it
If the triad is in 5-3 position on c, the complement becomes the
interval a-c or 3.
zn complement = 3
The complement will be denoted by the letter R in the following
paragraphs. The term will be employed in the discussion of har-
monies having any number of tones or intervals.
Now if A represents the lowest interval of a harmony in close
position, that is, such a position in which all tones are within the
compass of an octave ; B the interval above A ; C the interval above
B, etc., and R the complement, a harmony could be represented thus :
R
or ABC R
C
B
A
The first inversion of this harmony would be represented thus :
A
R
or BC..RA
C
B
the second inversion thus:
B
A
R
or C...RAB8
C
• Each different letter does not necessarily denote a different sized interval
584 THE MONIST.
We can think of all the different positions of the same harmony
as arranged clockwise in a circle. Taking the intervals in clockwise
order we find a different inversion taking each letter as a starting-
point. It is clear that the inversions of a harmony are cyclic rota-
tions of the permutation of intervals forming the fundamental
position, whichever it may be. Therefore if we desire to form all
possible harmonies from a given combination of intervals, we employ
only those permutations of the given combination which are not
cyclically related ; that is, out of each group of permutations which
are cyclic rotations of each other we select only one permutation
as representative of a single harmony.
The Number of Harmonies with a Given Combination.
Assume we are given a harmony A-B, in which A and B are of
different magnitude. Furthermore let us make A and B such inter-
vals that the complement (R) of A-B, is unequal in magnitude to
either A or B. Including R in our letter representation, we denote
the harmony as A-B-R. The minor (3-4-5) and major (4-3-5)
triads are good examples of such a harmony. We have already
seen that the inversions of a harmony are cyclic rotations of its
arrangement of intervals. Conversely, if two or more harmonies
are cyclically related they are only different positions or inversions
of one and the same harmony.
Since we are now engaged in finding how many harmonies can
be formed from a given combination of intervals, our problem be-
comes one of finding the number of cyclically unrelated permuta-
tions of the given combination.
Let us experiment with our harmony A-B-R. The permuta-
tions of the combination are:
ABR BRA RAB
ARB BAR RBA
The arrangements in the upper row are cyclically related ; likewise
those in the lower; yet no one of the permutations in the upper
row is related cyclically to any one in the lower row. Hence, only
two harmonies : A-B-R and A-R-B, can be formed of the combina-
tion A, B, and R. We observe that both harmonies are represented
in each vertical group; moreover in each vertical group one letter
occupies the same position (i. e., first in this case) in both harmonies
while the other two letters are permutated.
Let us experiment in like manner with a combination of 4
intervals (R included as one), in which all the intervals are of
BCRA
CRAB
RABC
ERCA
CABR
RCAB
BRAC
CBRA
RACE
BACR
CRBA
RBAC
BCAR
CARB
RBCA
BARC
CBAR
RCBA
OUR MUSICAL IDIOM. 585
different magnitudes. We represent its parts as A, B, C, and R.
Its permutations are :
ABCR
ABRC
ACER
ACRE
ARBC
ARCB
Here, too, the permutations have been so arranged that those
in any one horizontal row are cyclically related, while no permuta-
tion in one horizontal row is cyclically related to any one in any
other horizontal row. Thus any one of the vertical columns contains
all possible harmonies : in this case 3 ! or 6. It will be noticed here
as before, that by retaining one letter in a stationary position
throughout while permutating the remaining letters, all possible har-
monies are formed from the given combination ; for, although many
of the permutations of the remaining letters are cyclically related,
the stationary letter will bear a different relation to the other letters
in each case. In each vertical column one letter is held in the same
position, allowing the remaining 3 letters to be permutated in 3!
different ways.
With a combination of two intervals (R included) we form
1 ! or 1 harmony; with three intervals (R included) we form 2!
or 2, with four intervals we form 3 ! or 6, etc. Thus with n different
intervals we form (n-1) ! different harmonies.
But this formula does not apply to combinations in which two
or more intervals are alike ; and such combinations are by far more
numerous than the others. In fact, no harmony can have as many
as five or more different intervals, provided of course that all its
tones are bounded by the octave. To illustrate, we find the sum
of the five smallest intervals (1, 2, 3, 4, 5) to be 15, which exceeds
the octave 12 by 3.
To experiment with harmonies in which two or more intervals
are alike let us take the combination : A, B, B, and R. Its permu-
tations are:
ABBR BBRA BRAB RABB
ABRB BRBA BABR RBAB
ARBB BARB BEAR RBBA
Here again the horizontal rows contain cyclically related permuta-
tions while the vertical columns do not. We find all three of the
586 THE MONIST.
possible harmonies represented in any one vertical group (in each
of which one letter has a stationary position throughout). How-
ever, since the combination contains two B's there are two "B rows,"
each of which contains the three harmonies. Thus if one B is held
stationary while the remaining intervals are permutated, we obtain
six instead of three harmonies ; showing that our rule of stationaries7
holds only for singly occurring intervals. But if we hold A sta-
tionary, we permutate B-B-R in 3 !/2 ! different ways forming 3
different harmonies.
With the combination A, B, B, R and R we can form the fol-
lowing harmonies: ABBRR, ABRBR, ABRRB, ARBRB,
A R B B R, A R R B B, by holding A in the same position and per-
mutating the remaining letters in 4!/2!2! (=6) different ways.
The general formula then for the number of harmonies to a
given combination in which at least one interval occurs singly is:
H (harmonies) = (n-1) !/w1!n2!n3!
where n is the total number of intervals (R included) in the com-
bination, and nlf n2, ns, etc., are the respective number of times
which certain intervals are found. This is the general formula for
the number of harmonies to a combination, as the large majority of
combinations contain singly occurring intervals. To find the num-
ber of harmonies in a combination containing no singly occurring
intervals actual trial must be resorted to; any formula for this
would be beyond the scope of this work. As an example of the
error to which the formula leads if it is applied to combinations
having no singly occurring interval, we will apply it to the com-
bination 3 A's and 3 B's. Here
(n-l)!/w1!n2!....=5!/3!3! = 1.2.3.4.5/(1.2.3) (1.2.3) = 10/3
The result is an impossibility, since a fractional number of harmonies
cannot exist.8
The Number of Possible Combinations.
We will henceforth regard a harmony as in a prime position
if its tones are reduced to within the compass of an octave.
7 That is, the method of holding one interval stationary and permutating the
remaining intervals in all possible ways to obtain the several different har-
monies. '
8 We remember that throughout the last pages we have considered the
complement R as one of the n intervals of a combination. It might be sup-
posed that we could now eliminate R from consideration and thereby make our
formula, n !/MI \nz\. .. , This could not be done generally, since R might be an
interval having a magnitude equal to that of another interval (i. e., A, B, or
C, etc.) in the combination; in that case it would necessarily figure in the
denominator of the fraction.
OUR MUSICAL IDIOM.
587
No harmonies in prime position can consist of less than two
or more than twelve tones, in our duodecimal system. Hence no
prime harmony can have less than two (R included) or more than
twelve intervals. Now the number of combinations possible within
the octave could be computed, but the result would be of no value,
since in finding the number of harmonies we must treat each com-
bination separately. Furthermore, mere numbers interest us only
speculatively while a concrete method of obtaining all harmonies
is of real value.
To simplify the task of finding the combinations possible with
the twelve units within the octave, we will treat separately those
groups of combinations having a different number of tones or
intervals. A separate table will be made for each group of com-
binations having a certain fixed number of intervals. It is, of
course, evident that the sum of the intervals of each combination
equals the octave 12, since R is always one of the combination's
intervals. In the tables the vertical columns contain the respective
intervals, varying in magnitude as the number of intervals of the
combination permit. In the horizontal rows are found the com-
binations. If a number greater than 1 is found in a combination,
it indicates how many times the interval in whose vertical column
it lies is repeated in the combination. Thus a number 3 lying in
a column headed by the number 2, indicates that the combination
contains the interval 2 (or the whole tone) thrice. The tables follow.9
TABLE I; N (NUMBER OF INTERVALS INCLUDING R)=2
VARIOUS INTERVALS
fNo. OF
I
2
3
4
5
6
7
8
9
10
II
HARMONIES
H
I
I
I
H=(«— 1)!=(2_ i)!
I
2
I
I
«
I
3
i
i
«
I
4
i
i
«
I
5
i
i
«
I
6
2
u
I
Total t>
9 The combination numbers found to the left of the respective combinations
are only for future reference.
DC
_„„„„-„„--„-
IH
~
k.
04
Z
O
^>
r- '
«: «
Z X
^~^ ^~N i-
s ;
- M ~ ~ *-
P 0
« 3 >"^7,'-^-~7i>^
— X
3 <
^ co pi N pa
DM
J*
II
CO
II
y
E
II
Z
s
—
t—t
o
-
w
ON
M
s
oo
M
t^
~
0
IH M IH
VI
M PI w
rh
« rr.
CO >H M CS IH
n M <N « M -
>H
pj M |_ M M
IH N co^iOvOt^.iO QiO IH c<
s
>H CO CO COW COVOVD COCOIH POO) coin
5
CALCULATIONS OF
HARMONIES
_. _. .2 — _. — — ."« — 75
CO CO .^ CO "co rO CO >, CO >,
|| || « PQ W
"i
o
H
Oi
M
00
~
r^
M
o
IH
«0
Ol IH IH IH
•«*•
IH IH M P) IH
co
C4 M -H M rf-
d
IH dWIH POOICSlH
« W fO^-u^vO t^oo 0>0 « <N rOTl-u^
OUR MUSICAL IDIOM.
TAUI E IV; N=5
I
2
3
4
5
6
7
8
CALCULATIONS OF
HARMONIKS
H
I
4
i
H=4!/4!
i
1
3
I
i
473!
4
3
3
i
i
"
4
4
3
i
i
"
4
5
2
2
i
4!/2l2!
6
6
2
I
i
i
4!/2!
12
7
2
I
2
4!/2! 2!
6
8
2
2
I
"
6
9
I
3
i
4W
4
10
I
2
I
I
4!/2!
12
ii
I
I
3
473!
4
12
4
I
4!/4!
i
13
3
2
Bv trial
2
Toial oo
TAB.E V; N=6
i
2
3
4
5
6
7
CALCULATI--NS OK
HARMONICS
H
I
5
i
H=5!/5!
i
2
4
I
i
5V4!
5
3
4
i
i
n
5
4
4
2
By trial
3
5
3
2
i
5!/2!3!
10
6
3
I
i
I
5V3!
20
7
3
3
By trial
4
8
2
3
I
SVa'3'
10
9
2
2
2
By trial
16
10
I
4
I
574!
5
ii
6
By trial
i
Total 80
590
THE MONIST.
TABLE VI; N=
i
2
3
4
5
6
CALCULATIONS OF
HAKMONIES
H
I
6
i
H=6!/6!
i
2
5
I
i
61/51
6
3
5
i
i
"
6
4
4
2
i
6!/2! 4!
i5
5
4
I
2
"
15
6
3
3
I
6!/3!3!
20
7
2
5
By trial
1
Total 66
TABLE VII; N=
i
2
3
4
5
CALCULATIONS OF
HAKMONIES
H
I
7
i
H=7!/7!
i
2
6
I
i
7!/6!
7
3
6
2
By trial
4
4
5
2
I
7!/2! 5!
21
5
4
4
By trial
IO
Total 43
TABLE VIII; N=
i
2
3
4
CALCULATIONS OF
HARMONIES
H
I
8
i
H=8!/8!
i
2
7
I
i
81/71
8
3
6
3
By trial
to
Total 19
TABLE IX; N=no
I
2
3
CALCULATIONS OF
HARMONIES
H
I
9
I
H=9!/9!
i
2
8
2
By trial
5
Total 6
OUR MUSICAL IDIOM. 59!
TABLE X;N=n TABLE XI; N=i2
i
i
2
CALCULATIONS OF
HARMONIES
H
10
I
H=IO!/IO!
i
I
CALCULATIONS or
HARMONIES
H
12
By trial
i
Let us tabulate the number of harmonies found in each respective
table :
Table No. 1
10
11
Intervals 2
3
4
5
6
7
8
9
" 10
" 11
" 12
Total No. H 6
Total 350
We notice that as the number of intervals increases to 6 the number
of harmonies increases, while as the number of intervals increases
beyond 6 the number of harmonies decreases. Thus, six-tone
harmonies (or harmonies of six 'intervals including R) are most
numerous ; five- and seven-tone harmonies next in number ; four- and
eight-tone harmonies next ; etc. More harmonies can be formed
from combination 4 Table VII than from any other combination.
From it we obtain 21 harmonies. Finally we see that the total num-
ber of harmonies in the duodecimal system is 350.
The harmonies of least dissonance will be those having the
fewest small intervals. There are 9 harmonies having intervals no
smaller than 3 (minor third) ; there are 28 having intervals no
smaller than 2 (major second) ; and there are 55 harmonies having
only one interval, 1 (minor second).
So far we have only shown the number of harmonies with
each separate combination. Now it remains to show every har-
mony on the staff. Means have been shown for finding the number
of harmonies from each combination. We merely retain a singly
occurring interval in one position (preferably the lowest) and per-
mutate the remaining intervals. But in notating harmonies we
592
THE MONIST.
should at least represent them in some standard form; and thus
we arrive at the long-delayed decision about fundamental position.
Fundamental Position.
Among the 350 complex harmonies which we have found, there
are many — nay, a large majority — which could not be represented
as plain chords. Furthermore, since our vocabulary of intervals
and harmonies has become a chromatic one, we will no longer at-
tempt to reconcile the limited number of harmonies known as chords,
with all of 350 harmonies ; hence no attempt to make a chordic posi-
tion the fundamental form. Arbitrarily we might choose as our
fundamental a form in which the span, or the interval between the
extreme tones of a harmony, is smallest. Or, since we have found
it convenient to place any singly occurring interval in the lowest
position in forming harmonies from combinations, we might call
such a position fundamental. The question is difficult, and although
my solution is only arbitrary I believe the fundamental position
should be one which satisfies the following conditions:
I. The harmony is prime.
II. The smallest interval (it may be R) occupies the lowest
position, and thereby becomes A.
III. In case there exist two or more smallest intervals, the one
or more other smallest intervals are nearest A.
IV. In case the two or more smallest intervals are spaced regu-
larly apart, the next smallest interval is nearest A.
Illustrations follow in order respective to the conditions of the
definition.
ANY POSITION FUNDAMENTAL POSITION
4 Q-
Condhionl]! -i
2- 2 -4 -(4)
OUR MUSICAL IDIOM. 593
Condition IV
3-7-6-5
The first example illustrates how a harmony in more or less
spread-out position (left-hand column) is reduced to prime position.
The second illustrates how another harmony (9-8-5) is reduced
to prime position, following which it is placed so that the smallest
interval (1) occupies the lowest position.
The third illustrates a harmony which, in any prime position,
contains two smallest intervals. We are not satisfied with making
either of these A ; the equivalent of A must be nearest A. Thus in
this position:
2-2-4- (4)
A's equivalent is nearer A than in the position:
2-4-4-12)
The fourth illustrates a harmony containing two smallest inter-
vals (=1) which are equally separated in any prime position of the
harmony. Thus the two intervals (1) are always separated by the
interval 5. But because the next smallest interval
lies nearer -/v — than
we consider the interval e-f as A.
Thus in classifying any harmony, only three short steps are
necessary. First we reduce its tones to within the compass of the
octave ; second, we select from the prime positions the fundamental
position ; third, we name the harmony according to the chromatic
nomenclature.
Having determined the fundamental position, we are prepared
to write out on the staff all existing harmonies with the help of
the previous tables. Every harmony will appear in fundamental
form; while each will be respectively named. The harmonies fol-
low:
594
THE MONIST.
ALL EXISTING HARMONIES OF THE DUODECIMAL SYSTEM.
LISTED IN FUNDAMENTAL POSITIONS AND BY TABLES.
(The names of the harmonies are written respectively below; the number of
combination in which each is found appears above the staff.)
Table I
A. C.I C. 2 C. 3 C.4 C.5 C.6
Table H
C.2
3 -(9) etc. 4
C.3
6
C.4
1-1 ' 1-2- (9)
C.5
1-3
C.6
1-8
C.7
1-4
-o-
1-7
C.8
c>y»
1-5 1-6 2-2 2-3 2-7
C.9 C.10 C.ll C.12
2-4 2-6 2-5
Table III
C.2
3-3 3-4 3-5 4-4
C.3
1-3-6 1-6-2 1-6-3 1-2-4 1-2-5 1-4-2 1-4-5
OUR MUSICAL IDIOM.
C.9
595
C.10
«
^t
s
o*» i -&*.
-e^t
C»E*
1-5-2 1-5-4 1-3-3 1-3-5 1-5-3 1-3-4
C.ll C.12
*
^
Ho
1-4-3 1-4-4 2-2-2 2-2-3 2-2-5 2-3-2
C.13 C.14 C.15
S;
2-2-4 2-4-2 2-3-3 2-3-4 2-4-3 3-3-3
Table IV
C.I
C.2
^S
k£§£
ti:
1-1-1-1 1-1-1-2 1-1-1-7
C.3
1-1-2-1
1-1-7-1 1-1-1-3
C.4
1-1-1-6 1-1-3-1
1-1-6-1
C.5
ft» M'l'g
-gT «»
1-1-1-4 1-1-1-5. 1-1-4-1 1-1-5-1
1-1-2-2
-o-
s
tfc
1-1-2-6 l-l-6-2-(2) l-2-2-l-(G) 1-2-1-2-16) 1-2-1-6 -(2)
C.6
1-1-2-3 1-2-1-3 1-1-3-5 1-1-3-2
1-3-1-2
1-1-2-5 1-2-3-1 -1-3-2-1 1-3-1-5 1-1-5-3
596
THE MONIST.
C 7
3X
Btt
1-2-1-5 1-1-5-3 1-1-2-4 l-4-l-4-(2) 1-1-4-2
0.8
*
1-2-1-4 1-4-1-2- (4) 1-1-4-4- (2) l-l-3-3-(4) 1-3-1-3
C.9
l-4-l-3-(3) 1-1-3-4 1-3-1-4 1-1-4-3 1-2-2- 2- (5)
C.1(T
1-2-2-5 1-2-5-2 1-5-2-2 1-2-2-3 -(4) 1-2-2-4 1-2-3-2
i±
1-2-3-4 1-2-4-2 1-2-4-3 1-3-2-2 1-3-2-4 1-3-4-2
C.ll
-G
O
1-4-2-2 1-4-2-3
1-4-3-2 l-2-3-3-(3) 1-3-2-3
CA <) f* i\ Q-
• i.4t v. lu
1-3-3-2 1-3-3-3 2-2-2-2-(4) 2-2-2-3-(3) 2-2-3-2
Table V
C.2
[r^^Mmygf- Y3-
1-1- 1-1-1- (7) l-l-l-l-2-(6) 1-1-1-2-1 1-1-2-1-1
C.3
\f\ b*w
1-1-1-6-1 1-1-1-1-6 M-l-l-3-(5) 1-1-1-3-1
OUR MUSICAL IDIOM. 597
C.4
> Ujft^!
1-1-3-1-1 1-1-1-5-1
1-1-1-1-5 1-1-1-1 -4 -(4)
C.5
1-1-1-4-1 1-1-4-1-1 l-l-l-2-2-(5) 1-1-2-1-2
1-1-2-2-1 1-1-2-5-1 1-2-1-2-1 1-1-5-1-2
1-1-1-2-5
C.6
1-1-2-1-5 1-1-5-2-1 1-1-1-5-2
»,-H»ai Ibi'^au Ib'^gt Mq8j
1-l-l-2-3-(4) 1-1-1-3-2 1-1-2-1-3 1-1-2-3-1
1-1-3-1-2 1-1-3-2-1 1-1-3-4-1 1-2-1-3-1
*
1-1-4-1-2 1-1-2-4-1 1-2-1-4-1 1-1-4-1-3
1-1-1-3-4 1-1-3-1-4 1-1-4-2-1 1-1-1-4-2
^%Ul!>!'
<^
1-1-1-2-4 1-1-2-1-4 1-1-4-3-1 1-1-1-4-3
598
THE MONIST.
C.7
Fte
S!^=H
1-1-1-3-3- (3) l-l-3-l-3-(3) 1-1-3-3-1 1-3-1-3-1
C78
l-l-2-2-2-(4) l-2-l-2-2-(4) -1-2-2-1-2 1-4-1-2-2 1-1-2-2-4
go:
1-2-1-2-4
"C.9
1-2-2-1-4 1-1-2-4-2 1-2-1-4-2 1-1-4-2-2
-ft
1-1 -2-2 -3 -(3) 1-1-2-3-2 -(3) l-l-2-3-3-(2) 1-1-3-2-2
y | 'Vf -T-
^~1>"g» £E
&^
-vt®
1-1-3-2-3 1-1-3-3-2 1-2-1-2-3 1-2-1- 3-2 » 1-2-1-3-3
4g$n
-4*— e
1-3-1-2-2 1-3-1-2-3
-V V '. ^>i»
1-3-1-3-2 1-2-2-1-3 1-2-3-1-2
C.10
1-2-3-1-3 1-3-2-1-3 1-2-2-2-2-C3) 1-2-2-2-3
C.ll
— o»j
1-2-2-3-2 1-2-3-2-2 1-3-2-2-2 2-2-2-2-2-(2)
Table VI
C.I
C.2
1-1-1-1-1-1- (6) 1-1-1-1-1 -2 -(5) 1-1-1-1-2-1 -(5) l-l-l-2-l-l-(5)
OUR MUSICAL IDIOM.
599
C.3
\r 5^C"% I f
<£|
111131(4) 111311(4) 111411(3) 111141(3)
C.4
111114(3) 111182(4) 111212(4) 112112(4)
112141(2) 111214(2) 112211(4) 114121(2)
112114(2) 111412(2) 111421(2)
C.5
111142(2)
111123(3) 111321(3) 113121(3) 112131(3)
11213(3) 111132(3) 111312(3) 113112(3)
lAuE;
111231(3) 111133(2) 111313(2) 113113(2)
C.6
111331(2) 113131(2) 112113(3) 111222(3) •
6oo
THE MONIST.
fife
112122(3) 112231(2) 111223(2) 112212(3)
S
121212(3) 112123(2) 112312(2) 112321(2)
111232(2) 112221(3) 121221(3) 112213(2)
&
&
121312 (2)
121213 (2) 112132 (2)
113122(2) 113212(2) 113221(2)
C.7
111322(2) 112222(2) 121222(2) 122122(2)
C.2
Table VII
C.I
1111111(5)
1 1 1 1 1 1 2 (4) 1 1 1 1 1 2 1 (4)
gs
£
11211(4) 1112111(4)
±ka
1111411(2)
C.3
5
1111141(2) 1111114(2) 1111113(3)
m^
B
£
1111131(3) 1111311(3) 1113111(3)
OUR MUSICAL IDIOM.
601
C.4
}>™ \1\
HH-
111 1122(3)
.
,1111212(3) 1112113 (3)
1112311(2) 1111231(2) 1111123(2)
s*
1111221(3) 1112121(3) 1121121(3)
1112131(2) 1111213(2) 1112211(3)
&
1121211(3)
1121131(2) 1112113(2)
1113112(2) 1113121(2) 1113211(2)
1111 31 2 (2)
C.5
111^1 321(2) 11111.32 (2)
ffi
1111222(2) 1112122(2) 1121122(2)
fcs
1112221(2) 1112212(2) 1122112(2)
v^ai u
1122121<2) 1121221(2) 1121212(2) 1212121(2)
602
Table Vffl
.C.I
THE MONIST.
C.2
0-
3
11111111(4) 11111112(3) 11111121(3)11111211(3)
f^Mi A i *l> L&M AH?, i al" 8n i. H?i i *l* lAl.&i
p^lJy %^{ IJr N* &§}{ Ibw ^ aifS{
11112111(3)
11113111(2)
11111311(2)
C.3
11111131(2) 11111113(2) 11111122(2)
11111212(2)
krr^rrStth-^
11112112(2) 11121112(2)
•M«^
11112211(2) 11111221(2) 11112121(2)
11121121(2) 11121211(2) 11211211(2)
Table IX
C.I
C.2
111111111(3) 111111112(2) 111111121(2)
P B»
111111211(2) 111112111(2) 111121111(2)
Table X
Table XI
C.I
\\r-j? *\> 41?-^N~I
U [J[J ^l^ >j'- : — I
-- *^ J -.--r- . C»
1111111111
111111 11111
OUR MUSICAL IDIOM. 603
NOTES ON THE NOTATION OF THE ABOVE HARMONIES.
1. R is represented in parentheses where it is included in the notation.
2. The degree most convenient for representation is chosen for the base note.
3. The Number Names under each respective harmony have their integers
separated at first by dashes. Later these dashes are omitted.
4. When the number of sharps and flats becomes excessively great, it is written
n b or n Jt. Thus 8 8 $ # becomes 4 8.
5. There are other means of notating some of these denser harmonies. For
example, it would be possible to employ two staffs, or double stems. Our
present system of notation allows of no better methods.
Inversions of the Harmonies.
Many of these harmonies, especially those of many tones, may
sound unesthetic in their fundamental form because of their dis-
sonance, even to an ear trained to an appreciation of the most "ultra-
modern" music. A conglomeration of slow beats caused by adjacent
tones will, indeed, almost approach a common noise. However,
such dissonance can be largely reduced in the same harmony when
the tones of the fundamental position are scattered by octaves.
Thus many harmonies, seemingly obscure in their fundamental posi-
tion, become more appreciable to us by inverting them or spreading
them out. The different forms and inversions of almost every har-
mony (made possible by the range of modern instruments) allow
of the greatest variety of effects. The number of inversions and
positions of most harmonies is astounding. Now, in making our
rather superficial study of inversions, we will be obliged to use a few
technical expressions ; which are enumerated below.
A prime position of a harmony has been previously defined.
Any prime inversion of a fundamental10 harmony will be known
as primary inversion.
An inversion not prime, but containing no interval as great
as the octave will be considered a secondary inversion.
An inversion containing one or more intervals exceeding the
octave in magnitude will be known as a tertiary inversion.
An example of each type is given, respectively, below ; the three
inversions are in the same harmony. Although a form like (3)
would, according to the chord system, be considered a fundamental
position since the root (e) occupies the lowest position, we will
10 That is, a harmony in fundaemntal position, according to definition.
604
THE MONIST.
find it more convenient to regard any position which is not prime,
even though it have the root in lowest position, as an inversion.
Al A 2 A3 A 4 JO
9) tr «; «;
Fundamental Pos. Primary Inv. Secondary Inv. Tertiary Inv.
In the following paragraphs, a few general principles are dis-
cussed in the form of propositions.
Proposition I. A harmony of n tones has (w-1) inversions of
the first degree.
Since a prime inversion can be formed with each tone as a
lowest tone, and since (n-1) tones are available as lowest tones
(one tone being employed as the lowest tone for the fundamental)
it follows that there are (»-l) inversions possible.
In other words, an w-tone harmony has n prime positions.
Proposition II. The number of secondary inversions of a har-
many of n tones is (n\-n).
Let us experiment with two-, three- and four-tone harmonies,
as follows, allowing no interval between adjacent notes to be as
great as 12.
With two-tone harmonies we can form only two or 2 ! positions
which conform to our limitations.
For example :
-o-
With three-tone harmonies only 6 or 3! such positions are
possible; as for example:
-o — &-
-o-
o
With four-tone harmonies only 24 or 4 ! are possible :
OUR MUSICAL IDIOM. 605
With 5-tone harmonies 5! or 120 such inversions are possible,
etc., etc. With n tones n! positions of this type are possible. But
since these positions in which the intervals are less than 12 include
n prime positions, the secondary inversions number (n!-n).
It is evident that the number of tertiary inversions is entirely
dependent upon the range of the instrument employed to represent
them.
By the span of a harmony is meant the interval of its extreme
tones or the sum of its intervals. Thus the span of
5-9
is (9 + 5) equals 14.11
Proposition III. In a harmony of n tones the sum of the spans
of its prime positions is (w-1) octaves. Let us take an example
from a 4-tone harmony. Adding its prime positions we have
A+ B+ C
B+ C+ R
A + C+ R
A+ B + R
= 3(A + B + C-tR)=3 octaves
= (4-1) octaves.
Progressions of Harmonies,
This subject interests us more from a speculative than a prac-
tical standpoint, since the possibilities in this direction are well-nigh
unlimited, as will be shown. However, if the few suggestions that
follow are carried out in limited form practical ends are attainable.
Any harmony may of course be preceded or followed by any
other harmonies. Whether the progressions between harmonies
sound abrupt or smooth depends partly upon the harmonies in ques-
tion, partly upon the positions chosen, and partly upon the degree
of broad appreciation to which we have been trained. However,
smoothness or abruptness of progression does not concern us here,
for either may be more desirable according to the character of a
composition. The inclination of a composer with ideas certainly
11 In finding the span of a harmony, R is evidently not included to make
the sum 12, since the span of all harmonies would consequently be multiples
of 12.
606 THE MONIST.
deserves more consideration than the ever weakening law of the-
orists.
Let us now calculate the number of progressions of two suc-
cessive harmonies which can be made with a given number of har-
monies, let us say n harmonies. One of these n harmonies placed
upon one degree of the chromatic scale can progress to the same
harmony placed on 11 other degrees. The same harmony can form
12 progressions with any of the other harmonies given, since any
other harmony can be placed on 12 different degrees. And since
there are (w-1) such other harmonies, the first harmony can form
(n-l)12 progressions with the remaining harmonies. Thus the
total number of progressions possible between a single harmony
and the remaining harmonies is: ll + 12-(w-l).
But as many progressions are possible with each of the (n- 1)
remaining harmonies. Hence the total number of progressions of
two successive harmonies possible with n harmonies is:
n[ll + 12(w-l)] = (12n-l)n or 12w2-n.
Thus, with only 2 harmonies we can form (24-1)2, or 46
progressions. With 5 harmonies (which is the limit of vocabulary
with many persons, and in which may be included the major triad,
minor triad, dominant sept, supertonic sept and leading-tone sept),
(60—1)5 or 295 progressions of only two successive ones are pos-
sible. With the 350 existing harmonies, the possibilities of pro-
gression of two at a time are: (350x12-1)350=1,469,650.
The number of possibilities of progressions of three at a time
will be (\2nz-n)\2n, since each progression of two harmonies may
be followed by one of n harmonies placed on any one of 12 different
degrees of the scale. Thus with 350 harmonies 6,172,530,000 pro-
gressions of three are possible.
The general formula expressing the number of progressions
possible is:
where n is the number of harmonies among which the progressions
are to be made, while s is the number of harmonies to be used at
a time in a progression.
As mentioned before, the enormous figures just given mean
little practically, yet they serve to emphasize the fact that variety
is not only desirable but possible; and this is only variety of one
kind, harmonic variety.
Melody, rhythm and form are quite as variable as harmony,
OUR MUSICAL IDIOM. 607
and the variability of music is measured as the product of these
respective elements of it and is therefore quite beyond the bounds
of comprehension. Formerly I shared the foolish and common fear
that as more music is written the possibilities of future invention
narrow. I actually felt that the field for contemporary composition
is narrower than it was a century ago. To-day it seems to me that
every great musical work enlarges the field of the future.
"And myriad strains are there since the beginning still waiting
for manifestation."12
ERNST LECHER BACON.
CHICAGO, ILLINOIS.
12 Busoni, A New Esthetic of Music.
CRITICISMS AND DISCUSSIONS.
THE PRIMITIVE AND THE MODERN CONCEPTIONS OF
PERSONAL IMMORTALITY.
In an interesting review of my recent book (The Beliefs in God and Im-
mortality: an Anthropological, Psychological, and Statistical Study) in the
April number of this journal, it is inadvertently written that the book is
"simply a statistical investigation." This statement is true of Part II only.
It is not applicable to Part I, for it treats exclusively of "The two conceptions
of immortality: their origins, their different characteristics, and the attempted
demonstration of the truth of the modern conception." No more does the
statement apply to Part III, which discusses "The present utility of the beliefs
in personal immortality and in a personal God."
In the first half of the present paper, I set forth very briefly that which I
consider the main contribution contained in Part I of my book. In the second
half, I give some information concerning the statistics (Part II). J. H. L.
A curious contradiction seems to exist with regard to the
origin of the belief in survival after death. It is authoritatively
affirmed by anthropologists that that belief is to be found in every
tribe now extant. Frazer, less dogmatic, writes that "it might be
hard to point to a single tribe of men, however savage, of whom
one could say with certainty that the faith is totally wanting among
them."1 And yet historians no less competent in their field than
the anthropologists to whom we refer state with disconcerting una-
nimity that the belief in immortality appeared late among the people
from whom Europe drew its civilization. We are told, for instance,
that the Israelites' belief in immortality cannot be traced much
further back than the beginning of the Christian era. The covenant
Yahveh made with his people does not allude to a future life. The
nation alone was an object of his care. The great prophets them-
*J. G. Frazer, The Belief in Immortality, pp. 25, 33.
CRITICISMS AND DISCUSSIONS. 609
selves, when they inveigh against sin, care only for the danger there-
from to the existence of the nation. Among the Greeks also the
belief in immortality is said to have app ared late. Pythagoras,
the Mysteries, and Plato are named as marking the rise of the faith.
Of the Romans, Carter says that they did not have the idea of a
personal soul: "It was not present at the time of the Punic wars.
We see only scanty traces of it in the literature of the Ciceronean
age."2
These apparently contradictory affirmations may be explained
in two ways: either the particular survival-idea expressed in the
belief in ghosts, universal among primitive p ople, had at the be-
ginning of the historical period disappeared from among the nations
just mentioned ; or the immortality which the historians of these
nations have in mind is so different from the primary conception
of continuation after death that they disregard that belief.
The first of these two suppositions is not tenable. When the
historical period opens, a belief in survival was incontrovertibly
present among the peoples of whom the historians we have quoted
speak. In the Old Testament traces of polydemonistic belief are
definite enough to preclude divergence of opinion. The evidence is
just as clear in the case of the Greeks and of the Jews. The
Homeric conception of man is of a dual personality composed of a
visible earthly being and of its shadow or copy, which manifests
its presence in dreams and continues to live in Hades after the
severance of death. Jane Harrison has conclusively demonstrated
that while the religion of the Olympic gods was in process of forma-
tion, and even much later, the Greeks practised rites clearly indic-
ative of the belief in human ghosts.3
The idea of manes, essential to the religion of the old Romans,
is a "vague conception of shades of the dead dwelling bdow the
earth."4 If one is to believe Lucretius, and there seems to be no
reason why he should not be credited in this particular, the Romans
were haunted by a dread of the judgment to come.
If the presence at the beginning of the historical period of
practices indicative of a belief in survival, in the very people among
whom the idea of immortality is said to have appeared late, is no
longer a moot point, shall we hold that the kind of continuation
2 J. B. Carter, The Religious Life in Ancient Rome, p. 72.
3 Jane Harrison, Prolegomena to the Study of Greek Religion, 1st ed., p. 11.
4 W. Ward Fowler, The Religious Experience of the Roman People, p. 386
6lO THE MONIST.
after death which our historians have in mind when they deny the
existence of the belief in immortality at the beginning of the histor-
ical period is so different from the idea entertained by the savage
that they do not take that belief into account? The present paper
will show that the early conception of survival after death — let it
be called the primitive conception — is, as a matter of fact, radically
different from the modern conception in point of origin, of nature,
and of function.
What was the nature of the primitive belief in the countries
bordering the eastern end of the Mediterranean Sea at the period
to which it is customary to trace the rise of the belief in immortality ?
Let us begin with Egypt, the land of the "religion of eternal life."
The oldest historical documents we possess, the inscriptions in the
passages and chambers of the great pyramids, called the Pyramid
texts, belong to an already complex civilization although they date
bask to about 3400 B. C. The glimpses of earlier belief found in
these texts suffice, however, to indicate the presence of a religion
of the underworld according to which the dead continued in unhappy
existence under the earth. "The prehistoric Osiris faith," writes
Breasted, "involved a forbidding hereafter which was dreaded."
In an inscription on a stela addressed by a dead wife to her husband
we read: "Oh my comrade, my husband. Cease not to eat and
drink, to be drunken, to enjoy the love of women, to hold festivals.
Follow thy longing by day and by night. Give care no room in thy
heart. For the West Land (a domain of the dead) is a land of
sleep and darkness, a dwelling-place wherein those who are there
remain."8
The Babylonian dead were supposed to dwell in a great cave
underneath the earth, the most common name of which was Arttla.
It "was pictured as a vast place, dark and gloomy .... surrounded by
seven walls and strongly guarded, it was a place to which no living
person could go and from which no mortal could ever depart after
once entering it."6 For the Babylonians death made all men equal.
There were no distinctions of rank in the underworld ; kings, priests,
conjurers, magicians, and common people, all found themselves to-
gether in the dry and dusty kurnugea. Everything one touched
was dusty. Dust and earth were the food, the muddy water the
B A. Wiedemann, The Realms of the Egyptian Dead, p. 28.
6 Morris Jastrow, Aspects of Religious Belief and Practice in Babylonia
and Assyria, pp. 353, 356, 358.
CRITICISMS AND DISCUSSIONS. 6l I
drink of those living the shadowy life of the underworld.7 They
lived an ineffective, drowsy, starved existence.
Sheol of the Hebrews, like the underworld of the Babylonians,
was a place of dread. The shades were forgotten of God. Yahveh
was the God of the living, not of the dead. "Go thy way," says
Ecclesiastes, "eat thy bread with joy, and drink thy wine with a
merry heart .'. . . Let thy garments be always white ; and let not thy
head lack oil. Live joyfully with the wife thou lovest all the days
of thy life of vanity. .. .for there is no work, nor device, nor
knowledge, nor wisdom, in Sheol whither thou goest."
In Greece also the souls went to the land of the d ad bemoaning
their lot, for it was wretched. From that dark country souls never
returned. Homer draws a repulsive picture of the dead hovering
in the dark realm of Acheron, hazily conscious, hollow voiced, weak,
and indifferent.
Neither the Egyptians, nor the Babylonians, nor the Hebrews,
nor the Greeks could, it seems, think of beings deprived of a vig-
orous, effective body as enjoying a happy life. The few fortunate
individuals who were translated to Elysium or elsewhere without
passing through death and lived on happily, had retained their
body. The knowledge of the decomposition of the body after death
and of t-he tenuous unsubstantial nature of ghostly apparitions,
account naturally enough for the weakness and ineffectiveness at-
tributed to ghosts.
For centuries this repulsive and hopeless belief oppressed the
millions from among whom was to rise European civilization. A
turning point had, however, been reached at the dawn of the his-
torical period. The primitive, belief was apparently doomed, for
the leaders in those nations had not only felt the social danger it
threatened, and had in consequence begun to deprecate as evil the
cult addressed to ghosts, but they had also become clearly conscious
of moral cravings, the satisfaction of which death seemed to make
impossible.
Regarding the opposition that had arisen to the primitive belief,
we may recall that in Israel, the religion of Yahveh was the deter-
mined enemy of the cult of the dead in all its forms. And of the
Greeks we are told by Jane Harrison that "that which was in the
7 Friedrich Delitzsch, Das Land ohne Heimkehr, die Gedanken der Baby-
lonier-Assyrer iiber Tod und Jenseits, p. 16. He thinks, however, that as early
as the thirtieth century B. C. a distinction in the abode of the shades made its
appearance. Some of them lived in peace and comfort in a country provided
with water (pp. 18-22).
612 THE MONIST.
sixth and even in the fifth century before the Christian era the
real religion of the main bulk of the (Hellenic) people, a religion
not of cheerful tendance but of fear and deprecation," was the
same that Plutarch centuries later, and with him most of his great
contemporaries, regarded as superstition. Among the Romans,
ghosts had so far lost individuality as to be regarded by modern
historians as impersonal forces. The cult had become'to an amazing
degree a matter of mere conventional behavior.8 Thus a period of
greatly decreased influence among the people of the primitive belief
in immortality and of definite antagonism to it by the leaders had
arrived.
Simultaneously with this opposition to the old belief, the con-
sciousness of the insufficiency of this life to satisfy the cravings
of the heart and the demands of conscience manifested itself in
various and increasingly significant ways. One notes precursory
signs : for instance, the averred translation of Menelaus to Elysium ;
of Ganymede to Olympus ; of Parnapishtim to an earthly paradise
somewhere in Mesopotamia ; of Enoch, who was taken up unto
his Lord ; and of Elijah, who was carried in a chariot of fire by a
whirlwind into Heaven. One notes also the appearance among the
ancient Hebrews of Messianic hopes ; in particular, of the belief in
the day of Yahveh when the righteous who had descended to Sheol
would arise and participate in the triumph of the nation. The faith--
ful were to be resurrected, not in order to live a blessed, independent
existence elsewhere than on this earth, but in order to be reincorpo-
rated in the earthly life of the nation. These were preliminary
manifestations of needs which found their full expression in the
modern conception of immortality.
The formation of that conception, as it took place among the
Hebrews, is exceedingly interesting. Lack of space forbids any-
thing more than a passing reference to some of the main facts. Job
is an early shining instance among the Hebrews of a clear con-
sciousness of the moral incompleteness involved in the limitation
of human existence to earthly life. Yet he died without the hope
of a blessed immortality. His nearest approach to it is a fleeting
persuasion or hope that after death he would enjoy for a moment
a vision of God, who would then vindicate his mysterious ways.
The transformation of Yahveh, the God of the nation, into a
God maintaining individual converse with the members thereof, and
holding each individual, and no longer the nation alone, as morally
8 W. Ward Fowler, loc. at., pp. 386-388.
CRITICISMS AND DISCUSSIONS. 613
responsible to him, is intimately connected with the establishment
among the Jews of the modern belief in immortality. The tragic
inner life of Jer. miah shows us how circumstances forced him into
individual relationship with Yahveh (chapters xv-xvii). Ezekiel
continued the development of Jeremiah's thought. From the exist-
ence of an individual relationship with a just God, he drew the
unavoidable conclusion that each individual is to be rewarded or
punished according to his desert. This new doctrine permeates the
Psalms and the book of Proverbs. But when limited to earthly
existence, the doctrine is obviously false. Job and the author of
Ecclesiastes are up in arms against this truncated truth : "All things
come alike to all, there is one event to the righteous and to the
wicked : to the good and to the clean, and to the unclean ; to him
that sacrificeth and to him that sacrificeth not ; as is the good, so is
the sinner; and he that sweareth, as he that feareth an oath."
Ezekiel's doctrine could be made true only by positing another life
after death in which the injustice of this life would be repaired.
This has remained a chief argument of those believers in immor-
tality who also believe in a benevolent and righteous Creator.
The conception of and the belief in a blessed future existence
in which man's deepest and noblest yearnings are to be realized,
followed upon the appearance of a deep sense of the worth of these
cravings. Whenever, among peoples already familiar with the idea
of soul or ghost, these cravings were sufficiently keenly felt, they
seemed to have given rise to a belief similar to the Christian belief
in immortality.
In Egypt in the religion of the sun-god, lort£ before the book
of Job was written, a glorious existence with the god had been con-
ceived. In Greece, Plato taught a lofty doctrine of successive
earthly incarnations for the gradual purification of souls from the
pollution which comes to them from their association with matter.
Ultimately souls entered the glorious world of pure spirits. But
this doctrine did not originate with the Greek philosopher. He
tells us himself that he got it from the Orphic priests. The Orphic
cult was addressed to Dionysos by a sect that had evolved a definite
system of religio-philosophic belief, the chief article of which was
the double composition of man: one part mortal, coming from the
Titans, the other divine. Man's task was to rid himself of the
Titanic ebment, which corresponds to the body, in order to return
pure to God. The deliverance of the soul could not be achieved
614 THE MONIST.
suddenly nor without the helping mediation of Orpheus, who, let
it be noted, demanded a pure life as condition of salvation from
rebirth.
The nature of the primary conception of continuation after
death gives proof that, unlike the modern conception, it was not
born of desires for the fulfilment in another existence of hopes
frustrated on this earth. Had it had that origin, it would neces-
sarily have been conceived of in a form designed to satisfy these
desires. The nature of the belief and its universality among sav-
ages show it to have been imposed, regardless of man's feeling
toward it, as irresistibly as the belief in the existence of any object
present to the senses.
Differences in origin lead to differences in function. In the
primary belief, the ghosts, even those of friends, are on the whole
sources of anxiety and fear, and the relations maintained with them
aim almost exclusively at warding off their interferences in human
affairs. No one loves a ghost and, speaking generally, no one de-
sires to become one. The modern belief is, on the contrary, a vivi-
fying conviction or hope, calling forth the best that is in one's per-
sonality.
To consider these two conceptions as bearing to each other the
relation of the seed to the fruit, is, therefore, to disregard their
respective nature and function as well as their origin. In none of
these respects have these conceptions anything essential in common.
That is why the primary conception had to be discredited and dis-
carded before the modern one of a glorious life, fulfilling the noblest
human demands, could be formed and entertained.
STATISTICS OF CONTEMPORARY RELIGIOUS BELIEFS.
' In Part II of my book, I attempted to discover what propor-
tions of the members of a number of influential classes (physical
scientists, biological scientists, historians, sociologists, psychologists,
and college students of non-technical departments) believe in per-
sonal immortality and in the God whose existence is presupposed by
all the organized religions, i. e., a God conceived of as acting upon the
physical world or at least upon man, at man's request, desire, or desert.
It appeared to me of great interest both practically and scientifically
to find out definitely the percentages of believers, disbelievers, and
doubters among these classes, and to correlate eminence in them
and the special kinds of knowledge possessed by their members with
these percentages.
CRITICISMS AND DISCUSSIONS. 615
I was aware that the statistics of belief so far gathered have
little or no statistical value. When, as in the case of the extensive
inquiry of the Society for Psychical Research, less than one-third
of those who were solicited answered, no particular meaning attaches
to the discovery that two-thirds of that one-third believe in immor-
tality. In order to obtain statistics valid for the whole of a group,
it is not necessary, it is true, to poll every member of the group.
It is sufficient to consider a part of that group, provided that every
member of that part or a very high percentage, answer the inquiry,
and that the selection of the part investigated be made according to
chance. The statistics of that part may then, according to the law
of probability, be held valid for the whole group.
The statistical defect from which the inquiry of the Society
for Psychical Research suffers, is often combined with an insufficient
definition of the belief under investigation. Not long ago some rash
person affirmed in the English press that "it is extremely doubtful
whether any scientist or philosopher really holds the doctrine of a
personal God." Thereupon a Mr. Tabrum collected from among
English scientists 140 expressions of opinion on the question, "Is
there any real conflict between the facts of science and the funda-
mentals of Christianity?" But the author did not define what he
meant by "fundamentals," neither did he ask his correspondents to
state the meaning they attached to that expression. Strange to say,
very few thought it necessary to be explicit. Lord Rayleigh wrote,
for instance, "I may say that in my opinion true science and true
religion neither are nor could be opposed." This has the appear-
ance of a misplaced pleasantry. Any one may make that statement ;
its significance depends altogether upon what is meant by "true
religion." You may have in mind some conception of religion
which would tolerate neither the Apostles' nor the Nicean creed,
nor even a personal God!
In my own investigation I endeavored to avoid the two major
defects illustrated above, and succeeded, I think, in establishing
statistics of belief valid for the entire classes named above, so far
as the United States is concerned.
The student of human development will be interested in the
possibility now opened to ascertain the statistical history of re-
ligious beliefs. By instituting at some future time an investigation
similar to mine, it would become possible to express with a high
degree of exactness the changes that have taken place in the spread
of the beliefs here considered.
6l6 THE MONIST.
If I cannot enter here into details as to the statistical method
I have followed, the results secured, and their interpretation, I may
at 1 ast add in conclusion the following figures and some brief in-
formation.9 •
PHYSICAL BIOLO- HISTORI- SOCIOLO- PSVCHOL-
BKLIBVKRS IN Goo SCIKMISIS GISTS ANS GISTS OGISTS
Lesser Men 49.7 39.1 63. 29.2 32.1
Greater Men . . 34.8 16.9 32.9 19.4 13.2
Lesser Men 57.1 45.1 67.6 52.2 26.9
Greater Men 40. 25.4 35.3 27.1 8.8
These figures show that in every class of persons investigated
the number of believers in God is less, and in most classes very
much less, than the number of non-believers, and that the number
of believers in immortality is somewhat larger than in a personal
God ; that among the more distinguished, unbelief is very much more
frequent than among the less distinguished ; and finally that not only
the degree of ability, but also the kind of knowledge possessed is
significantly relat d to the rejection of these beliefs.
"The correlation shown, without exception in every one of our
groups, between eminence and disbelief appears to me of mom3ntous
significance. In three of these groups (biologists, historians and
psychologists) the number of belkvers among the men of greater
distinction is only half, or less than half the number of believers
among the less distinguished men. I do not see any way of avoiding
the conclusion that disbelief in a personal God and in personal im-
mortality is directly proportional to abilities making for success in
the sciences in question.10
"With regard to the kinds of knowledge which favor disbelief,
the figures show that the historians and the physical scientists pro-
vide the greater ; and the psychologists, the sociologists and the
biologists the smaller number of believers. The explanation is, I
think, that psychologists, sociologists and biologists in very large
numbers have come to recognize fixed orderliness in organic and
psychic life, and not merely in inorganic existence; while frequently
physical scientists have recognized the presence of invariable law
in the inorganic world only. The belief in a personal God as defined
for the purpose of our investigation is, therefore, less often pos-
9 These figures are percentages of the number of persons who answered the
questionnaire.
10 Concerning these abilities and their influence, see Chapter X.
CRITICISMS AND DISCUSSIONS. 617
sible to students of psychic and of organic life than to physical
scientists.
"The place occupi d by the historians next to the physical
scientists would indicate that for the present the reign of law is
not so clearly revealed in the events with which history deals as
in biology, economics, and psychology. A large number of his-
torians continue to see the hand of God in human affairs. The in-
fluence, destructive of Christian beliefs, attributed in this inter-
pretation to more intimate knowledge of organic and psychic life,
appears incontrovertibly, as far as psychic life is concerned, in the
remarkable fact that whereas in every other group the number of
believers in immortality is greater than that in God, among the
psychologists the reverse is true; the numbT of believers in im-
mortaHtv among the greater psychologists sinks to 8.8 percent.
"One may affirm, it seems, that in general the greater the ability
of the psychologist, the more difficult it becomes for him to believe
in thp continuation of individual life aft~r bodily death.
"The students' statistics show that young people enter college
possessed of the beliefs still aorpted, more or less perfunctorily,
in the average home of the land, and that as their mental powers
mature and their horizon widens a large prec^ntage of them aban-
don the cardinal Christian beliefs. It seems probable that on leaving
coMege, from 40 to 45 percent of the students with whom we are
concerned deny or doubt the fundamental dogmas of the Christian
religion. The marked decrease in belief that takes place during
the later adolescent years in those who spend those years in study
under the influence of persons of high culture, is a portentous indi-
cation of the fate which, according to our statistics, increased knowl-
edge and the possession of certain capacities leading to eminence
reserve to the beliefs in a personal God and in personal immor-
tality."11
To the statistical data are added a large number of letters from
my correspondents and a somewhat full study of the religious ideas
of students. These together with the statistics make a picture of the
present religious situation both vivid and relatively exact.
J. H. LEUBA.
BRYN MAWR COLLEGE.
11 The Belief in God and Immortality, pp. 277-281.
6l8 THE MONIST.
NOTES ON RECENT WORK IN THE PHILOSOPHY OF
SCIENCE.
Federigo Enriques ("Sur quelques questions soulevees par 1'in-
fini mathematique," Rev. de metaphysique et de morale, March,
1917, Vol. XXIV, pp. 149-164) points out that experience, when
it is idealized by reason, puts before us two kinds of infinity, the
actually and the potentially infinite ; suppose then "that we are
potentially given by thought an infinity of objects, the question
arises as to whether there is any reason to consider as logically de-
fined a new object of thought which expresses the totality or the
limit of these objects even when they are not constructed with
respect to a concept of the kind which we suppose to be given
a priori." The answer to this question depends on a fundamental
tendency of the mind ; it will be negative or in some degree positive
according as we feel ourselves borne toward nominalism or toward
realism. Realist doctrine — at least in mathematics — in its first his-
torical form rested on the assumption that a simple passage to the
limit was always possible, and this was gradually destroyed by the
progress of the infinitesimal calculus. The realist doctrine in its
second historical form rests on the principle that (p. 159) "every
infinity of virtually defined objects may be considered as a totality
forming a class and constituting a new logical object." As dis-
tinguished from realism in its first form, in this new realism we
conceive that the properties of the new object are absolutely new
and that thus we cannot state them a priori by an induction extended
from the finite to the infinite. This new form of realism is due, in
its mathematical form, to Georg Cantor, but (p. 159) "the philos-
opher B. Russell has developed in the widest sense the philosophical
consequences of the realism thus introduced into mathematics."
The various paradoxes of mathematical logic have led to the con-
clusion that there are,, in certain cases, no such things as classes of
perfectly definite objects ; and this realism, in its second form, is
partially unsuccessful. On p. 163 we read that "the principle of an
infinite number of choices is adopted by Russell and by Zermelo,"
and so we 'are apparently again forced to the conclusion that En-
riques is quite unaware of the tendency shown by Russell's work
published since 1905. Since the question is rather important, per-
CRITICISMS AND DISCUSSIONS. 619
haps the present reviewer may be forgiven for dwelling on some
work on the paradoxes in question since 1903.
In Russell's Principles of Mathematics (Cambridge, 1903)
there was not any definite suggestion that the concept of class should
be restricted, though there was certainly a more or less vague feeling
that some classes should be excluded (see Monist, January, 1912.
Vol. XXII, pp. 153, 157-158; and January, 1917, Vol. XXVII, p.
144). The merit of perceiving that a restriction was necessary and
of attempting to give a criterion to decide which classes were legit-
imate seems due to Jourdain, in a paper published in 1904 (see
Monist, January, 1917, Vol. XXVII, pp. 148-150). The question
as to the being or not-being of a class is totally different from the
question of the possibility of an infinite series of acts of selection, —
which, by the way, neither was nor is assumed by Russell, though
it is believed in by Zermelo and many others. The merit of being
the first to publish an explicit recognition of the postulate involved
in this assumption is due to Zermelo in 1904, and the questions re-
lating to Zermelo's axiom have been frequently confused with, for
example, Jonrdain's "proof" by even eminent people; though both
Russell and Jourdain pointed out repeatedly that the questions
involved are quite different. In 1905 and later Russell published
papers gradually showing how it was possible to work through a
great deal of Cantor's theory without assuming that there are such
things as classes at all, and a thorough exposition of this theory is
one of the most important parts of Whitehead and Russell's Prin-
cipla Mathcmatica (Vol. I, Cambridge, 1910). Thus it is obvious
that this theory of Russell's not only makes the considerations of
Jourdain and some others quite superfluous, but also makes such
criticisms as that of Enriques entirely off the point (cf. Monist,
January, 1917, Vol. XXVII, pp. 145-148).
This historical sketch is also relevant to our consideration of
a recent paper by Dmitry Mirimanoff ("Les antimonies de Russell
et de Burali-Forti et le probleme fondamental de la theorie des
ensembles," in L'enseignemcnt maihcmatique, 1917, Vol. XIX, pp.
37-52). The author remarks (p. 48) that we can find in the works
of Bertrand Russell, Henri Poincare, and Julius Konig (Neue
Grundlagen der Logik. Arithmctik und Mengcnlchre, Leipsic, 1914)
"a profound logical and psychological analysis of the Cantorian
antinomies and of the notion of class," but that he "will not have
any need of this analysis for the end which" he lias in view. His
62O THE MONIST
article is characterized by the following quotation from p. 33:
"People believed, and it seemed quite evident, that the existence of
individuals necessarily implies that of the class of them, but Burali-
Forti and Russell showed by different examples that a class of
individuals may be non-existent, although the individuals exist. As
we cannot refuse to accept this new fact, we are obliged to conclude
from it that the proposition which seemed evident and which was
believed always to be true is only true under certain conditions.
And then arises the problem which we may regard as the funda-
mental problem of the theory of aggregates : What are the necessary
and sufficient conditions for the existence of a class of individuals?"
On this confusion between the ideas of existence and entity, see the
article quoted above in, The Monist for January, 1917. The author
gives a solution of this problem for the particular case of classes
that he calls "ordinary" classes, and his deductions rest on three
postulates (p. 49) which are applied by some in the study of prob-
1 ms of the theory of aggregates. Further, the examples of Russell
and Burali-Forti are modified (pp. 39-48) in a way that seems
advantageous to the author, and the author announces (pp. 39, 52)
his intention to give in a future article the reasons which determined
him not to adopt in this paper the theory of Konig. The criterion
which the author arrives at (pp. 48-52) for deciding whether indi-
viduals have a class or not is practically that suggested by Jourdain
in 1904: individuals have a class if they can be arranged in a seg-
ment of the series W of all ordinal numbers and not if they cannot
be so arranged. It is not worth while to enter into a criticism of
this suggested criterion, which in any case has become quite super-
fluous through the work of Russell referred to above. Mirimanoff
(p. 52) regrets that it has been impossible for him to become ac-
quainted with work that has appeared since the beginning of the
war. If he had read — which he nowhere gives any sign of having
done — the works referred to above of 1904 to 1910, we do not think
that it would have been necessary to write this paper.
In the Revue de metaphysique et de morale for January, 1917,
there is an address given by the late Victor Delbos on the general
characteristics of French philosophy. A part of the late Louis
Couturat's Manuel de logislique, which he wrote about 1906 but
which is not yet published, is printed. These extracts form the
greater part of the second chapter of this book, and are on the
CRITICISMS AND DISCUSSIONS, 621
logical relations of concepts and propositions. An interesting fact
about them is that the work of Frege seems to have influenced
Couturat to a greater extent than was the case with Couturat's
Principes of 1905. This contribution is fairly elementary, and does
not deal with those paradoxes which are, perhaps, of the greatest
interest to logicians, although it just mentions them. F. Colonna
D'Istria writes on the logic of medicine according to Cabanis's
Rapports dit physique et du moral de I'homme. Arnold Reymond
makes a critical study of the new and recast edition of Edouard
Claparede's Psychologic dc I enfant et pedagoyie experimentale
(Geneva, 1916). Thomas Ruyssen writes on "an idea in peril:
humanity, humanitarianism, humanism."' Finally there are obituary
notices of Theodule Ribot (1839-1916), the eminent psychologist,
and Henri Dufumief.
* * *
In the number of the Revue de metaphysique et de morale for
May, 1917, A. Espinas deals with the initial idea of the philosophy
of Descartes, and the late Victor Delbos's lecture on method in the
history of philosophy forms the second of his three lectures on the
history of philosophy. A manuscript by the late Louis Couturat,
which was certainly written before 1902 and which will not form
part of the projected Manuel dc logistique, is printed here and is on
the algebra of logic and the calculus of probabilities. Finally Ales-
sandro Padoa has a paper on the consequences of a change of
primitive ideas in any deductive theory whatever.
* * *
In a paper on the ''infinite numbers" which Bernard le Bovier
Fontenelle tried to introduce in his Elements de la geometric de
I'infini (Paris, 1727). Branislav Petronievics ("Stir les nombres in-
finis de Fontenelle." Rcndiconti dclla R. Accademia dei Lincei
[CJasse di science fisiche. iiiatcmatiche e naturali], Vol. XXVI,
1917, pp. 309-316) tries to show that this "first attempt at a rational
theory of infinite numbers," although it is obviously full of contra-
dictions which were at once pointed out by MacLaurin and others,
possesses a historical value when compared with the theories of
Cantor and Veronese. Cantor's theory has, says Petronievics, an
arithmetical starting-point, while that of Veronese has a geometrical
one ; and Veronese establishes that there is no point on an infinite
straight line which corresponds to the first transfinite ordinal num-
ber of Cantor, so that "geometrical application of the transfinite
622 THE MONIST.
numbers of Cantor is not possible." In spite of the fact that Fon-
tenelle introduced his "infinite number of all finite numbers" as
"the last one" of this series itself, Petronievics maintains that the
theory of Fontenelle has a historical value in that it "may be re-
garded as the common source of the theories of Cantor and Vero-
nese," and that "it is not impossible to suppose, in view of the
likenesses between the theories, that Cantor and Veronese both
arrived at establishing the principles of their theories when trying
to avoid the flagrant contradictions into which Fontenelle fell."
There does not seem to the reviewer to be the smallest ground
for supposing that Cantor was led to his theory either by reading
Fontenelle or by setting out deliberately to generalize arithmetic.
Indeed, one of the points of the long introduction to the trans-
lation of Cantor's later papers published under the title of Con-
tributions to the Founding of the Theory of Transfinite Num-
bers (Chicago and London, 1915) is to show that Cantor was com-
pelled to generalize the idea of number as a consequence of the
natural development of his process of "derivation" of geometrical
point-sets. In this extension it appeared clearly that the transfinite
numbers began beyond the whole series of finite numbers in oppo-
sition to Fontenelle's notion mentioned above. Fontenelle says on
page 30 of his book : "We must not be frightened at the words 'last
term' in this connection. It is a last finite term that the natural
series of numbers has not, but not to have a last finite term is the
same thing as to have a last infinite term." This is a charming way
of turning a universal negative proposition into a particular affirma-
tive one. It seems that the first time that Cantor spoke more or less
publicly of Fontenelle's theory was in a letter of 1886 (cf. Zur
Lehre vom Transfiniten, Halle, 1890, p. 50), and therefore long
after he had founded his absolutely different theory. That hearing
of one theory may have been the psychological cause of Cantor's
thinking about a fundamentally different theory is of course both
possibly and probably irrelevant, but there is no ground for sup-
posing that even this happened. It is a mistake to say that ordinal
numbers cannot have geometrical applications : an illustration of
the way such numbers can appear is given by this: To the series
on the .r-axis formed by the points 1, 1/2, 1/3, . . . ., 1/n,. . . ., the
point 0 bears exactly the same relation as the first transfinite ordinal
does to the finite ordinals in order of magnitude. For Cantor's
remarks on Veronese and Veronese's reply, we may quote the above
CRITICISMS AND DISCUSSIONS. 623
Contributions (pp. 117-118). A purely analytical exposition of the
infinite and infinitesimal numbers of Veronese was given by T.
Levi-Civita ("Infiniti e Infinitesimi attnali," Atti R. Istituto Veneto,
1892).
* * *
Florian Cajori ("The Zero and Principle of Local Value used
by the Maya of Central America," Science, Vol. XLIV, 1916, pp.
714-717) draws attention to the fact, hitherto apparently unnoticed
by mathematicians, that the Maya of Central America and southern
Mexico seem to have used a symbol for zero and the principle of
local value much earlier than any one else. The material for Cajori's
remarks is furnished by An Introduction to the Study of the Maya
Hieroglyphs, by Sylvanus GriswoFd Morley (Bulletin 57 of the
Bureau of American Ethnology, Washington, 1915). The early
Babylonians possessed the principle of local value, but so far as we
know did not possess a zero. About 200 B. C. they did have a
symbol for zero, which, as Smith and Karpinski (Hindu- Arabic
Numerals, Boston, 1911, p. 51) say, was "not used in calculation,
nor does it always occur when units of any order are lacking."
They did not employ it systematically in writing numbers and not
at all in performing computations. The Hindus certainly did not
use their symbol for zero systematically before probably the sixth
century A. D., and the earliest undoubted occurrence of zero in
Indian numerals is A. D. 876 (cf. also G. R. Kaye, Indian Mathe-
matics, Calcutta, and Simla, 1915, p. 31, for the date of the appear-
ance of the principle of local value in India). Now, it seems that
the Maya used the zero and the principle of local value at the begin-
ning of the Christian era if not much earlier. "As far as is known,
the Maya used their numeral systems only in the counting of time
as it arose in their calendar, ritual, and astronomy." Of the several
Maya numeral notations the one which is of greatest interest as
embodying the principle of local value and the symbol for zero is
found in Maya codices but not in their inscriptions. The number
system was vigesimal, with the solitary break that 18 (and not 20)
uinals make 1 tun, and the symbols 1 to 19, both inclusive, are ex-
pressed by bars and dots. Each bar stands for five units and each
dot for one unit, and the dots are written above the bars. Thus
19 is written as three bars above one another and four dots on the
top. "The values of the bars and dots are added in each case. The
zero, which plays a leading part in the notations found on inscrip-
624 THE MONIST.
tions as well as those on codices, is represented in the codices by
a symbol that looks roughly like a half-closed eye.... In writing
20. .. the principle of local value enters for the first time. It is ex-
pressed by a dot placed over the symbol for zero. The numerals
are written, not horizontally, but vertically, the unit of the lowest
order or value being assigned the lowest position. Accordingly,
37 was expressed by the symbols for 17 (three bars and two dots)
in the kin [units] place and one dot, representing 20, placed above
the 17, in the uinal place. The number 300 is expressed by three
bars drawn above the symbol for zero (3 x 5 x 20 = 300). The largest
number which can be written by the use of only two places or posi-
tions is 17x20+19 = 359. To write 360, the Maya drew two zeros,
one above the other, with one dot higher up, in third place. Using
three places to represent kins, uinals, and tuns, they could write any
number not larger than 7199. Proceeding in this way the Maya
wrote numbers in very compact form. The highest number found
in codices is 12,489,781. . . . The symbols representing this number
occupy six different places, one above the other. . . . The second
numeral notation that was fully developed and employed by the
Maya is found in their inscriptions. It employs the zero, but not
the principle of local value. Special glyphs are employed to desig-
nate the different units. It is as if we were to write 1203 as: '1
thousand, 2 hundreds, 0 tens, 3 ones.' "
The question as to the origin of the arithmetical notation
that we call "Hindu-Arabic" has received a new and unexpected
contribution from Carra de Vaux ("Sur 1'origine des chiffres,"
Scientia, Vol. XXI, 1917, pp. 273-282.) With the Arabs these figures -
are called hindi and the usual meaning of this word is "Indian."
Xow. the Arabian historian Masoudi, writing in 943 A. D., said that
the Hindu numerals were discovered by a congress of wise men
gathered together by the powerful and wise king Brahman under
whom arts and sciences flourished. "People who are even slightly
familiar with the history of philosophy will recognize this at once
as a neo-Platonic legend," and a mention of the "Era of the
Creation" allowed de Vaux to conclude that this legend is Persian,
for that is a Persian era. Also in the work of the other Arabic
historian, Albirouni, we have a remark that the numerals came from
India, that is vague and contrasts strongly with his usual exactness.
This seems to show a lack of definite knowledge on Albirouni's part.
The author then examines the word hindi, and comes to the
CRITICISMS AND DISCUSSIONS. 625
conclusion that it is a form of h'.ndasi whose root is the Persian
hid and which means metrical or arithmetical. Thus "signs of
h'nd" means "arithmitical signs" and not "signs, of India." Con-
sider this example: Apollonius of Perga, who was not an Indian,
was said to be el-hindi in some Arabic manuscripts, thus this word
must evidently be translated as if it were cl-h'ndasi, the geometer
or <: ngineer. It is to be noticed that in Arabian treatises the abacus
is called taklit which is a Persian name. Thus de Vaux concluded
that the numerals originated in the Gre k world, and the history
of their slow diffusion is easier to explain if we admit that they
are a neo-Platonic "or (so:t) neo- Pythagorean" invention, for the
Pythagoreans are well known to have had a taste for secrecy. From
Greece the numerals passed to Persia and the Latin world, and from
Persia to India and afterward to Arabia.
The figures themselves were not formed from letters of the
alphabet, but directly by means of very simple conventions. These
characters are due to the neo-Platonists and were known in the
schools of Persia before they were in Islam, and it is there that the
Arabs found them. From Persia also they passed into India.
In the first number (January, 1917) of Vol. XXIV of the Amer-
ican Mathematical Monthly, the official journal of the Mathematical
Association of America, there is an important paper by Edward
V. Huntington on "The Logical Skeleton of Elementary Dynam-
ics" (pp. 1-16). The object of the artic1e is to outline the logical
structure of elementary dynamics. Any logically d veloped science
must begin with undefined concepts, in terms of which all the other
concepts of the science are expressed ; and in this case Huntington
takes them to be: (1) Space and time, with the derived concepts of
velocity and acceleration; (2) Forces, "as sugg sted by the tension
and compression in our own muscles" (p. 1) and "as measured
by a spring balance" (p. 3) ; and (3) Inert material bodies, on
which our forces act. The unproved propositions, from which all
the other propositions of the science are derivd, are only four in
number. The first is (p. 4) that "a free particle, when acted on
by a force, acquires an acceleration in the direction of the force;
furthermore, if a given particle is acted on at different times by two
forces F and F', and if a and a' are the corresponding accelerations,
then F/F' = a/a'; that is, the accelerations are proportional to the
forces." This principle "is best regarded as a scientific hypothesis,
626 THE MONIST.
the truth of which has been abundantly verified by experiment."
It contains the answer to the fundamental question of dynamics:
"If a force gets hold of a free particle, and proceeds to act on it,
what happens to the particle?" The second and third principles,
which cover the case of the particle being acted on by several forces
simultaneously, are the principle of the vector addition of forces
arid the principle of the independence of two perpendicular forces
(p. 5). For any given body the ratio of the force to the acceleration
produced is constant, and the value of this ratio "is a characteristic
of the body, which may be called its inertia" (p. 4). "The weight
of a body in a given locality with respect to a given frame of
reference is best defined as the force required to support the body
at rest with respect to that frame in the given locality" (p. 5). We
then have the theorem that "if W is the weight of a body in a given
locality and g is the falling acceleration of that body in the same
locality, then the ratio W/g is independent of the locality, and is a
correct expression for the inertia of the body" (p. 6). The proof
for the case of fixed axes follows immediately from the first two
principles, and a proof for the case of moving axes "belongs later
in the course." The theorem (p. 6) that, in any given locality, the
falling accelerations of all bodies are equal, "can be proved from
general considerations ; or, if preferred, it may be accepted as an
empirical fact." The words "mass of three pounds" are taken
(p. 7) as meaning the same thing as that the body in question "has
a weight of 3 Ibs." in the standard locality ; the weight being a
multiple of the unit of force (Ib. in the British system). The
fourth and last fundamental principle is the principle of action and
reaction (p. 8) : "When two particles are in contact with each other,
or attract or repel each other according to any law like that of
gravitation or magnetism, the interaction between them may be
represented by a pair of twin forces, equal in magnitude and oppo-
site in direction — one of the twins acting on one particle and one
on the other, along their joining line." The definition of the
"centroid or center of mass" is as a "weighted average" (p. 8),
and the theorem on the motion of the center of mass is then proved.
It is emphasized (p. 11) that the difficulties outside the four funda-
mental principles are of a mathematical sort. It will be seen that
the system is based on fundamental units of force, length, and time
instead of on units of mass, length, and time, and the author shows
by tables (pp. 15, 16) the higher practical value of the system of
CRITICISMS AND DISCUSSIONS. 627
derived units advocated by him. This is "one of the best arguments
in favor of the use of force rather than mass as the principal unde-
fined concept of dynamics. The only reason why the text-books
so insistently base their derived units on mass instead of on force
is apparently that a standard lump of metal is easier to preserve in
a museum than a standard spring balance. But this is no argument
for the logical priority of mass over force. As a matter of fact,
the fundamental unit of force is as easy to preserve as the funda-
mental unit of mass, though the method of doing so does not consist
in simply storing away a spring balance" (p. 16). The name of the
unit of force, in the British system, is "pound" (lb.) and is defined
as "the force required to support a carefully preserved lump of
metal, called the 'standard pound avoirdupois,' in vacuo, in the
standard locality" (p. 14).
The reviewer would remark that, though mass under the name
"inertia" is a derived unit in the system advocated by Huntington,
we have to use the unit of mass as a practical means of preserving
the unit of force. It is quite true that this fact is no argument for
the logical priority of mass: it is merely a question of practical
convenience. But in either of the two systems there seem to be, at
first sight, three fundamental und fined units, and so, from a logical
point of view, nothing is gained by replacing mass by force as a
fundamental unit. But let us look at the matter more closely. As
we have learned from the work of Mach (see, e. g., his Mechanics,
3d edition, Chicago and London, 1907, p. 243), "mass-ratio" can
be defined in terms of the mutual accelerations of bodies, and so
there seems to be a logical advantage in the system in which force
is not regarded as fundamental, but is defined. Further, even in
Huntington's system, "force" can be defined by the property that F/a
is constant, and then his system and Mach's seem to be identical.
By the way, the forces we use in dynamics are not all "suggested
by the tension and compression in our own muscles" : the attraction
of the sun is not : and it is both logically objectionable and rather
confusing to a student to have various concepts with a single name.
It must be added that stress is (pp. 7-8) rightly laid on the
difficulty which beginners have in realizing that, when a particle
describes a curve, there is actually an acceleration along the normal.
There is an interesting review of Florian Cajori's William
Oughtred on pp. 29-30 written by Louis C. Karpinski, where it is
stated that Cajori's remark that Napier was the first to use a decimal
628 THE MONIST.
point (1616 and 1617) is an error: it was first used by Pitiscus
in 1612.
In the February number is a short account (pp. 54-55) of
Cajori's presidential address to the annual meeting of the Mathemat-
ical Association of America in New York City at the end of 1916
entitled "Discussions of Fluxions from Berkeley to Woodhouse."
This address was a shortened account of a book by Cajori which
will appear before very long in the "Open Court Classics of Science
and Philosophy." At a meeting of the Council it was decided,
among other things, to consider the question of possible assistance
for the Revue semestriclle and the Jahrbuch iiber die Fortschritte
der Mathematik in view of the difficulties that must attend publi-
cation owing to the war (p. 64). Very much the same discussion
was held by the Chicago Section of the American Mathematical
Society (p. 97). David Eugene Smith, in "On the Origin of Certain
Typical Problems" (pp. 65-71), has a very learned article on the
history of the problems of (1) filling a cistern of water, (2) the
Josephspiel, or the problem of the Turks and Christians, (3) the
testament of a man about to die, dividing his estate, (4) the problem
of pursuit.
* * *
Frank Egleston Robbins (Amer. Math. Monthly, March, 1917,
Vol. XXIV, pp. 121-123) gives an interesting and critical review
of George Johnson's partial translation of and commentary on the
Introduction to Arithmetic of Nicomachus, in his dissertation on
The Arithmetical Philosophy of Nicomachus of Gerasa (Lancaster,
Pa., 1916). The essay of Nicomachus is of course the earliest ex-
tant attempt at a systematization of the Greek science of theoret-
ical, as distinguished from practical, arithmetic.
* * *
In the American Mathematical Monthly for May, 1917, W. H.
Bussey ("The Origin of Mathematical Induction," Vol. XXIV, pp.
199-207) points out that Moritz Cantor is mistaken about the use
of complete induction both in his Geschichte der Mathematik (Vol.
II, 2d ed., 1900, p. 749) and his note, correcting this mistake,
on Maurolycns in the Zeitschrift filr mathematischen und natur-
wisscnschaftlichen Unterricht (Vol. XXXIII, 1902, p. 536). In
the note Cantor said that he had found that Maurolycus described
and used the method in his Arithmetic or um libri duo (Venice, 1575),
and that Pascal had expressly borrowed the method from Mauroly-
CRITICISMS AND DISCUSSIONS. 629
cus; Bussey shows that there is an error, in a minor respect, in
Cantor's references. Bussey quotes a number of Maurolycus's
theorems. In his sixth proposition, that any integer (n) added to
the preceding one is equal to the "collateral odd number" (the nth
odd number, 2n- 1), Maurolycus's proof, freely translated is: "The
integer 2 added to unity makes the integer 3, but when added to 3
it makes an amount greater by 2 and this .... is the next odd integer,
namely 5. Again, since the integer 3 added to 2 makes 5, which is
the collateral [third] odd integer, when it is added to 4 the result
will be greater by 2, that is.... it will be the next odd integer,
which is 7. And in like manner to infinity as the proposition states."
On this proof Bussey remarks (p. 201) : "This is not a very clear
statement of a proof by mathematical induction but the idea is
there." The eleventh proposition, that every triangular number added
to the preceding triangular number is equal to the collateral square
number, or, in modern notation, n(n+ l)/2+ (n- l)n/2 = n2, is the
one which Cantor said, in the above note, Pascal got from Mauro-
lycus and which Matirolycus proved by complete induction. But
Cantor is mistaken in saying that this theorem is proved by com-
plete induction: the first undoubted case of a proof by complete
induction is the fifteen proposition, that the sum of the first n odd
integers is equal to the nth square number. Maurolycus's proof is
(p. 203) : "By a previous proposition the first square number
(unity) added to the following odd number (3) makes the following
square number (4) ; and this second square number (4) added to
the third odd number (5) makes the third square number (9) ;
and likewise the third square number (9) added to the fourth odd
number (7) makes the fourth square number (16) ; and so succes-
sively to infinity. ..." Pascal mentioned in a letter to Carcavi the
fact that he borrowed from Maurolycus, and he repeatedly used the
method of complete induction in connection with his arithmetical
triangle and its applications. Bussey then gives two interesting
examples of Pascal's use of the method of complete induction, and
finally gives some other and more recent uses of it.
* * *
In the same number of the Monthly, David Eugene Smith
("Mathematical Problems in Relation to the History of Economics
and Commerce," pp. 221-223) maintains that "a very good history
of civilization could be written from the wide range of problems
of mathematics." In the subject of commercial and economic his-
630 THE MONIST.
tory, for example, he mentions that the problems in the manuscripts
and early printed books on arithmetic in the fifteenth century tell
us that Venice was then the center of the silk trade, although Bo-
logna, Genoa, and Florence were then prominent ; the problems also
tell us the cost of the luxuries and necessities of life; the rent of
houses ; the changes in commercial customs and the rise in standards
of business integrity. "Not only to the economist and the student
of commerce is the field a rich one, but it is well worth the study
of any one who may be possessed of doubt as to the relation of
mathematics to the daily life of the race. Not only can the history
of the problem easily be made the history of commerce and econom-
ics, but the history of mathematics can easily be made the history
of civilization."
In the number of the Bulletin of the American Mathematical
Society for March, 1917 (Vol. XXIII), there are two interesting
papers by Edward V. Huntington on the logical postulates for
order. In "Complete Existential Theory of the Postulates for Serial
Order" (pp. 276-280) Huntington establishes the "complete inde-
pendence"— in the sense defined by E. H. Moore of Chicago in his
Introduction to a Form of General Analysis of 1910 — of each of
three different sets of postulates for serial order. The first set is
new and very convenient for many purposes ; the second set dates
back to Vailati (1892) ; the third set is a modification of the second
set and was introduced in Huntington's well-known paper on "The
Continuum as a Type of Order" in the Annals of Mathematics for
1905. In "Complete Existential Theory of the Postulates for Well
Ordered Sets" (pp. 280-282) Huntington gives three sets of in-
dependent postulates for well-ordered systems, each of these three
sets, being "completely independent" in the above sense. R. L.
Borger ("A Theorem in the Analysis of Real Variables," pp. 287-
290) gives a theorem on two real functions of two real variables
which is derived from a theorem in Kowalewski's Die komplexen
V eranderlichen and ihre Funktionen, and deduces from it the ex-
ceedingly fundamental and important theorem that if any function
of a complex variable possesses a finite derivative at each point of a
simply connected closed region, then this derivative is continuous,
all the derivatives of the function exist, and the function may be
represented by a power-series. Mathematicians who are acquainted
with the nature of the progress brought about by Goursat's proof
CRITICISMS AND DISCUSSIONS. 63!
of Cauchy's theorem will at once see how important this note is.
J. R. Kline ("Concerning the Complement of a Countable Infinity
of Point Sets of a Certain Type," pp. 290-292) proves a theorem
which is a general case of the theorem proved by Hausdorff in his
Grundziige der Mengenlchre of 1914 that, if E denotes a Euclidean
space of two or more dimensions while R is an enumerable set of
points belonging to E, then E - R is a connected set. Kline's theorem
was proved by Robert L. Moore (Trans. Amer. Math. Soc. for
1916) on the basis of a system of axioms proposed by him.
The number of the Bulletin of the American Mathematical
Society for May, 1917 (Vol. XXIII, No. 8), contains several articles
of interest to those who cultivate the philosophical and historical
aspects of mathematics. Samuel Beatty ("The Inversion of an
Analytic Function," pp. 347-353) proves the existence of the inverse
of an analytic function when the conception of an analytic function
which is due to Goursat is the starting-point. In the theory of
Weierstrass this proof is made to depend on the representation by a
series of powers and in Cauchy's theory on the Jacobian of the real
and imaginary parts of the function with reference to the real and
imaginary parts of the variable. It is well known that Goursat
showed in 1900 how the fundamental proposition on complex inte-
gration in Cauchy's theory could be proved merely from the assump-
tion that the function in question has a finite derivative at each point
of a simply connected domain, without any assumption of the con-
tinuity of this derivative. This continuity was then proved as a
consequence of the Cauchy-Goursat theorem. The method of
Beatty 's proof makes use of the theory of sets of points. Thomas
S. Fiske ("Emory McClintock," pp. 353-357) gives a biography of
Emory McClintock (1840-1916). McClintock 's first paper on pure
mathematics entitled "An Essay on the Calculus of Enlargement,"
in the American Journal of Mathematics for 1879 "was an effort
to present the theory of finite differences and the differential cal-
culus from a unified point of view. The paper may be regarded as
a precursor of recent attempts to consider difference equations as
differential equations of infinite order. His other more important
papers were a series of researches on solvable quintic equations
published in the American Journal of Mathematics [for 1884, 1885
and 1898] and a paper on the theory of numbers ['On the Nature
and Use of the Functions Employed in the Recognition of Quad-
632 THE MONIST.
ratic Residues'] published in the third volume [1902] of" the Trans-
actions of the American Mathematical Society (p. 355). "When
one considers that McClintock made no use of the powerful labor-
saving machinery which has revolutionized modern analysis, the
results obtained by him in his researches on quintic equations, as
well as some of his other achievements, appear to indicate a truly
wonderful power of manipulation and clearness of vision" (p. 356).
A list of McClintock's mathematical publications is given.
J. H. Weaver ("On Foci of Conies," pp. 357-365) gives (1)
a short historical sketch of the development of the properties of
conies connected with the foci, and (2) some of the theorems from
Pappus which have a bearing on foci and tangents. According to
Zeuthen (Geschichte der Mathematlk im Alterthum und Mittelalter,
Copenhagen, 1896, p. 211) it seems that the focus for the parabola
may have b:en known to Euclid. However, we have no mention
of such points or of any of their properties until the work of Apol-
lonius on conies (Book III, Probs. 45-52), but Apollonius did not
use or mention in any way a focus for the parabola. Pappus gave
the first recorded use and proofs of the focus-directrix definition
of conies. Johann Kepler named the points in question in a work
of 1604, and part of the short account of the conic sections that he
gave (Opera Omnla, ed. Frisch, Frankfort, 1859, Vol. II, p. 185)
is freely translated by Weaver (p. 359) as follows: "There are
among these curves certain points of especial consideration, which
have a certain definition but no name, unless they usurp for name
the definition of some property. For if from these points lines are
drawn to the points of contact of tangents to the section, th°s? lines
make equal angles with the tangents. . . .We, because of the prop-
erties of light and the eye, from the viewpoint of mechanics shall
call these points foci. We might have called them centers, because
they are on the axis of the s'ction, if authors, in the hyperbola and
ellipse, were not accustomed to calling another point the center. In
the circle there is one focus, the center. In the ellipse there are
two foci equally distant from the center, and more removed in the
more acute. In the parabola, one focus is within the section and
the other may be considered either within or without the section
and removed to an infinite distance from the first focus, so that if
a line drawn from this caecus [blind] focus to a point of the section
will be parallel to the axis. In the hyperbola, the external focus
becomes nearer the internal focus as the hyperbola becomes more
CRITICISMS AND DISCUSSIONS. 633
obtuse." The method of the work of Kepler was developed and
added to by Desargues, and important work on foci was done by
Maclaurin, Poncelet, Pliicker, and many others.
Finally, there is a short and extremely interesting paper by
Jekuthial Ginsburg ("New Light on Our Numerals," with an in-
troductory note by David Eugene Smith, pp. 366-369). That our
common numerals are of Hindu origin seems to the author to be a
well-established fact, and that Europe received them from the Arabs
seems equally certain, but how and when these numerals reached the
Arabs is a question that has never been satisfactorily answered.
The article calls attention to a paper by the French orientalist F.
Nau in the Journal asiatique for 1910 (Series X, Vol. XVI) showing
that the Hindu numerals were known to and appreciated by the
Syrian writer Severus Sebokht who lived in the second half of the
seventh century ; that is, about one hundred years before the date
of the first definite trace that we have hitherto had of the introduc-
tion of the system into Bagdad. Sebokht says, after asserting that
the Greeks, in astronomy, were merely the pupils of the Babylonians :
"I will omit all discussion of the science of the Hindus, a people
not the same as the Syrians ; their subtle discoveries in this science
of astronomy, discoveries that are more ingenious than those of
the Greeks and the Babylonians ; their valuable methods of calcula-
tion ; and their computing that surpasses description. I wish only to
say that this computation is done by means of nine signs. If those
who belr.ve, because they speak Greek, that they have reached the
limits of science should know these things they would be convinced
that there are also others who know something" (p. 363). On this
fragment Ginsburg remarks (pp. 363-369) that it "clearly shows
that not only did Sebokht know something of the numerals, but
that he understood their full significance, and may even have known
the zero as Rabbi ben Esra did, in spite of the fact that he, too,
speaks of nine numerals." However, Smith (pp. 366-367) remarks
that the artich "shows that the zero was probably not in the system
as then mentioned, showing at least that its value was not generally
comprehended in the seventh century and possibly confirming the
impression that the symbol had not yet been invented." With re-
gard to the question as to how Sebokht could have obtained in-
formation about the Hindu numerals, Ginsburg remarks (p. 369)
that the city where Severus lived, in the northeast part of Meso-
potamia, "was situated in a rich and fruitful country, was long the
634 THE MONIST.
center of a very extensive trade, and was the great northern em-
porium for the merchandise of the east and west ;" and "the ex-
change of goods is always accompanied by the exchange of ideas."
Further, the weight of the evidence is (p. 369) in favor of Sebokht's
work being at least one of the agencies by means of which the
knowledge of the numerals was transmitted to the Arabs.
Raphael Demos ("A Discussion of a Certain Type of Negative
Proposition," Mind, Vol. XXVI, 1917, pp. 188-196) applies to par-
ticular negative propositions the treatment which Bertrand Russell
(cf. Russell and Whitehead's Principia Mathematica, Vol. I, Cam-
bridge, 1910) has applied to "descriptive phrases" or "incomplete
symbols." Russell "found himself confronted with the fact that to
accept descriptive phrases as significant in their given form would
be to people the world of things with the apparent objects of such
self -contradictory and fantastic descriptions as 'round-square,' 'cen-
taur/ etc." ; and Demos "was faced with the fact that to accept
negative propositions at their face value would be to people the
world of objects with negative facts, a type of objects which ex-
perience fails to disclose." Demos, somewhat like Russell, by view-
ing the negative proposition as an incomplete symbol, was led to
declare it meaningless in its apparent form, and its apparent object
— the negative fact — to be nothing. In this article he stated, first,
that a particular simple negative proposition is an objective entity
whose peculiarity as negative is not dependent upon the mind's
attitude toward it. He then argued that the negative proposition
cannot be construed in the form which it apparently possesses, inas-
much as such construction would make it formally different from
positive propositions and would endow it with purely negative ob-
jects, which are, it seems, nowhere to be found in experience. He
concluded that some special interpretation must be given to the
negative proposition, and showed that its negative element is a
modification, not of any distinct constituent (such as the predicate)
in the proposition, but of the whole content of it. Thus any nega-
tive proposition is a modification, in terms of "not," of the rest
of its content, and — since the latter is positive — a modification of
some particular positive proposition. He stated the meaning of
"not" to be "opposite" — a relational qualification in terms of the
familiar relation of opposition or contrariety among positive propo-
sitions— and hence the meaning of the whole proposition "not-/>" to
CRITICISMS AND DISCUSSIONS. 635
*
be "opposite of p." He argued that, so stated, a negative proposi-
tion is an ambiguous description of some positive proposition, and
that, completely stated, it is of the form "an opposite of p is true,"
or "some q is true which is an opposite of />." Thus he defined a
particular simple negative proposition as an ambiguous description
of some true positive proposition in terms of the latter's relation
of opposition to a certain other positive proposition, such that, in
terms of the former, reference is achieved to the latter. Lastly, he
explained that negative knowledge is knowledge of a true positive
proposition by description in terms of its opposition to some other
proposition, and hence must be characterized as positive in reference
but not in content, inasmuch as the proposition referred to is not a
constituent of the complex of assertion or knowledge. "Substan-
tially the above definition of simple negative propositions applies
to double and 'w-ple' negatives as well ; the latter, too, are descrip-
tions of positive propositions which are true in terms of what they
oppose. There is this difference, however, that whereas simple nega-
tives are functions of a positive content, double and other negatives
are functions of a negative content, such that any negative propo-
sition in the nth power is a function of a content which is negative
in the («— l)th power." 4>
C. E. Hooper publishes "The Meaning of the Universe" (Mind,
April, 1917), the first instalment of an article of massive appearance.
The definition is as follows : the Universe means the totality of real
thought-objects (or object-matters) considered under four related
aspects: (1) space, (2) time, (3) the variety in unity of natural
characters, i. e., real thought-objects as particulars having natures
of their own, but natures agreeing in various specific and generic
respects with the natures of other particulars, (4) unity in variety
of natural causation. Time and space are both objective. Mr.
Hooper goes on to define thought-object, reality and aspect. A
thought-object is apparently an intended object, whether or not a
reality corresponds to the intention (e. g.. Kant's noumenoti). Real-
ity is contrasted not with appearance but with "mental figment,"
and includes subsistent as well as existent objects. It is difficult to
tell how far the term "thought-object" has an idealistic bias in Mr.
Hooper's mind, but reality, at all events, seems to be merely a sum
or system of objects which are severally real. The universe is thus
a real thought which contains all other real thought-objects in their
636 THE MO1STIST.
manifold relations. Symbolic entities (ideas, signs) are compre-
hended, but whether per se or only as reflected upon (made objects
of thought) is not stated. It would seem that imaginary or incon-
ceivable thought-objects, such as Meinong's pets, the golden moun-
tain and the round square, are to have no place in the universe, but
are discarded as "figments." While the universe contains finite
thought-objects and symbols, it does so only in fact, not in nature.
Of the four modes or aspects, space and time may be classed to-
gether as coincidentals, while the systems of natural characters and
natural causation may be termed co-essentials. On the other hand
space and nature may be classed together as static, time and causa-
tion as dynamic. We find some difficulty in understanding how
Mr. Hooper accounts for the universe's being known at all. "It
cannot, like a finite object, be actually related to some fellow object.
It is as related to the mind or system of subjective ideas that we
know all that is possible to know about it." But the "mind" if
genuinely symbolic is a thought-object; and if the universe cannot
be related to a finite object which is part of itself we do not know
how it can be related to the mind. But criticism of so substantial
an article should be deferred until its completion in succeeding
issues. 17
* * *
Agnes Cuming ("Lotze, Bradley, and Bosanquet," Mind, April,
1917) declares Lotze's logic to be a partial revolt against the in-
tellectualism of Hegel. Our intelligent experience, according to
Lotze, is only a small part of the real world and thought is only a
small part of our intelligent experience. Thought is a tool, a sub-
stitute for adequate perceptive intuition. Bradley 's and Bosanquet's
logic are similar in so far as each is influenced by Lotze. They hold
an almost identical definition of "idea," and agree in their theories
of judgment. Bradley however arrives at reality ontologically and
Bosanquet epistemologically. "Knowledge for Bosanquet is the
system of reality progressively demonstrated before our eyes....
In this emphasis on system as the postulate of knowledge. . . .Bo-
sanquet is in advance of Bradley." Lotze insists on feeling as a
criterion, and is thus very far from Bosanquet with his conception
of system, but he admits the essential of Bosanquet's position, which
is the inadequacy of feeling. Lotze is a dualist : he divides sharply
the feeling which supplies the material from thought, exercising a
formal activity upon it. In Bradley the dualism becomes a gloomy
CRITICISMS AND DISCUSSIONS. 637
scepticism ; thought and its object are forever sundered. But Bo-
sanquet bridges the gulf. In both Bosanquet and Bradley the sepa-
ration of thought and reality is inherited from Lotze with his idea
of the "scaffold" of thought. The only possible criterion of knowl-
edge is immanent — a criticism of a lower from a higher point of
view. Miss Cuming considers that Bosanquet has improved upon
both Lotze and Bradley ; the direction which she believes to be
progress seems to be almost a return to more orthodox H.gelianism.
In the same number B. M. Laing ("Schopenhauer and Individ-
uality") considers that Schopenhauer fails to appreciate the meta-
physical claims of individuality. He dethrones reason, making it a
mere temporary organ of the will. He interprets Kant in such a
way as to make Kant assert that the mind creates the world of
things, instead of merely conditioning it. This perversion of the
Kantian doctrine leads Schopenhauer to hold (in contrast to Kant)
that the world of space and time is an illusion. Hence he is unable
to conserve individuality, and tends to confuse individuality with
(temporal and spatial) individnation. Schopenhauer's monism is a
mere prejudice against multiplicity, and his will a purs abstraction.
Furthermore, he confuses the will with bodily wants and cravings.
Schopenhauer exposes himself on every side to such destructive
criticism: but while Mr. Laing seizes upon some of his weakest
points in his interpretation of Kant, the view of individuality which
Schopenhauer represents, and which is more abiding then Schopen-
hauer, cannot be said to be demolished. ij
* * *
Scientia for February, 1917, opens with an article by Gino
Loria on the history of imaginary numbers. He takes as his text
Kronecker's aphorism "Die ganzen Zahlen hat der Hebe Gott ge-
macht, alles andere ist Menschenwerk." There is no square root of
a negative quantity, said the Viga Ganita, for it is not a square, and
the mysterious quantities remained an enigma and defied concrete
interpretation until a memoir appeared about the end of the eight-
eenth century, written by an unknown Danish land-surveyor, Caspar
Wessel by name. It seems to have suffered a fate like that of Swin-
burne's Queen Rosamund, of which not a copy was asked for or
sold. Mendel's discovery remained unheeded for forty years, but
Wessel's was not unearthed until a century after his death. But in
638 THE MONIST.
this summary is no more room for comment on Loria's charming
paper. P. Zeeman writes upon the hypothesis of the immovable
ether, describing the experiments of Fizeau, Michelsen and Morley,
Eichenwald, and himself, and concluding that it seems impossible
by any imaginable means to measure absolute velocities. He points
the lesson that the most important scientific principles are the results
of boldly generalized experiments. J. R. Carracido discusses the
foundations of biochemistry, and the lines research must follow in
the pursuit of organic synthesis. He does not see why success
should not be reaped before the end of the century, and success it
will be, even if limited to the synthesis of the most rudimentary
form of living matter.
The number for March, 1917, contains an admirable article by
Gaston Milhaud, in which he attacks the very difficult problem as
to the extent to which Descartes was influenced by Bacon. M.
Cantone discusses the present trend of physical research in a paper
surveying the work of those whose discoveries have in thirty or
forty years revolutionized our outlook on the world of matter.
Etienne Rabaud writes on the life and death of species. An analysis
of the current doctrine of "means of defense" prepares for the
question as to how it is that species persist in spite of the daily
hecatombs of individuals. y
In Scicntia for April, 1917, we have, from the pen of Francisco
Iniguez, of the Observatory of Madrid, a slight but interesting
sketch of what we know about stellar spectra, their classification,
and the light they throw on the subject of the evolution of the stars.
We are warned of the limits we must set to our inferences in con-
sidering the nebulae, for what we know as yet of these celestial
bodies does not justify our indulging in theories on the subject. The
author then indicates how we may infer the existence of dark stars,
how their evolution still continues, and points out their connection
with meteorites and cosmic dust. Etienne Rabaud brings to a close
his paper on the life and death of species, of which this second
instalment deals with the conditions of the persistence and of the
disappearance of species. He finds the affinity of organisms to be
the crux of the problem. This leads to the consideration of the
conditions of attraction and repulsion, and to a short discussion of
parasitism and symbiosis, with the cosmic influences which, often
of great complexity, determine the life of a species. The whole
BOOK REVIEWS AND NOTES. 639
forms a graphic picture of the variations of the relative proportion
of individuals and species. The slightest change in the conditions
of normal life may lead to the disappearance of the last member of
a species, or on the contrary the species may thrive and continue to
thrive. Species persist, they increase immeasurably in numbers, or
their numbers fall off, and they disappear, subject but to the inter-
vention of two sets of influences, affinity and the circumstances that
determine their displacements in space. "These modifications, no
doubt, sometimes involve other important modifications in the con-
ditions of life ; variations may ensue which find their repercussion
in the aggregate of the interaction. Thus, linked to one another
and to the world from which they come, the life, the transforma-
tions, and the death of organisms are functions of their interde-
pendence." y
BOOK REVIEWS AND NOTES.
DIDEROT'S EARLY PHILOSOPHICAL WORKS. Translated and edited by Margaret
Jourdain. "The Open Court Classics of Science and Philosophy," No. 4.
Chicago and London : The Open Court Publishing Co., 1916. Pp. vi, 246.
Price $1.25 or 4s. 6d. net.
Among the documents of the renaissance of the eighteenth century, none
are of more interest than the early informal contributions to ethics and philos-
ophy of Diderot, written with much of the incoherence of the epistolary form.
They are, as he claims again and again, Letters; and they are letters written in
a hurry. The Philosophic Thoughts, which is the only one of Diderot's works
in this selection not in the epistolary form, is said to have been thrown to-
gether between Good Friday and Easter Monday of 1746. Yet they are not
philosophic journalism, no mechanical transmission of the current philosophical
coin of the day. It is for their originality of outlook that they have been
closely studied in Germany, while in England there is Lord Morley's study of
Diderot in relation to the movement centered in the Encyclopedic, Diderot and
the Encyclopedists.
This selection includes the Philosophic Thoughts, a breviary of eighteenth-
century scepticism, a copy of which was found in the possession of the un-
fortunate La Barre, and in which Diderot appears as a Deist, to whom the
argument from design (Thought XX, pp. 56-58) is still of weight: "I am
greatly deceived (he writes) if this proof is not well worth the best that has
ever issued from the schools." That very argument is very differently treated
in the Letter on the Blind (p. 109) by Diderot's mouthpiece, the blind mathe-
matician, Nicholas Sattnderson, who conjectures a world in its early stages
"in a state of ferment," without any vestiges of that "intelligent Being whose
wisdom fills you with such wonder and admiration here.... What is our
world, but a complex, subject to cycles of change, all of which show a continual
640 THE MONISt.
tendency to destruction ; a rapid succession of beings that appear one by one,
flourish and disappear; a merely transitory symmetry and momentary appear-
ance of order?"
In the brilliant passages in which Diderot sketches the probability of evo-
lution he appears as a forerunner of thinkers such as Erasmus Darwin in
England and Lamarck in France. Transformism only needed the partial
scientific confirmation it received from Lamarck and Geoffrey St. Hilaire in
the early decades of the nineteenth century, "to pass from the realm of sys-
tematic philosophy into that of scientific controversy."
The Letter on the Deaf and Dumb, a criticism addressed to the Abbe
Batteaux, author of the Fine Arts Reduced to a Single Principle, has its in-
terest as a forerunner of Lessing's Laokoon, in esthetics. It also contains the
idea of a muet de convention (theoretical mute), which is closely paralleled by
Condillac's Statue in the Treatise on the Sensations, published three years after
Diderot's Letter. Condillac's treatment of the idea, however, was far more
systematic and detailed than Diderot's, and he did not by his own account owe
the suggestion of his statue to Diderot.
Diderot, the most German of French authors, as far as his style is con-
cerned, bears translation well. He has been neglected by translators, however,
until this edition, which includes all that is of permanent value in his early
works of 1751, the date of the Letter on the Deaf and Dumb, excluding the
relatively uninteresting Sceptic's Walk. ft
THE NEW PHILOSOPHY OF HENRI BERGSON. By Edouard Le Roy. Translated
from the French by Vincent Benson, M.A. New York: Holt. Pp. 235.
Price $1.25 net; by mail $1.35.
This interpreter of Bergson's philosophy is also the author of the article
"What is a Dogma?" in the body of this issue of The Monist. He is particu-
larly fitted for the present task because though not a pupil of Bergson's he
had followed much the same trains of thought quite independently so that
when he became acquainted with Bergson he recognized in his work, as he
himself says, "the striking realization of a presentiment and a desire." That
M. Le Roy has comprehensively grasped Bergson's spirit and conclusions so
that the present volume furnishes a valuable prolegomenon to the study of the
famous Frenchman^ thought is attested by the following lines in the Preface
in which Bergson himself has set the seal of his approval on the task. M.
Bergson wrote to M. Le Roy: "Underneath and beyond the method you have
caught the intention and the spirit.... Your study could not be more con-
scientious or true to the original. As it advances, condensation increases in
a marked degree : the reader becomes aware that the explanation is undergoing
a progressive involution similar to the involution by which we determine the
reality of Time. To produce this feeling, much more has been necessary than
a close study of my works : it has required deep sympathy of thought, the
power, in fact, of rethinking the subject in a personal and original manner.
Nowhere is this sympathy more in evidence than in your concluding pages,
where in a few words you point out the possibilities of further developments
of the doctrine. In this direction I should myself say exactly what you have
said." f
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