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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 



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 




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 





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 


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. Engel 1 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. 


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. Engel 4 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 


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 


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 


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. 


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 result 6 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. 


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 

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 Musebeck 8 is inclined to attribute their small 
effect on each other. Victor Schlegel's 9 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. 


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- 


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 spoke 10 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. 


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, 


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. 


Without entering in detail into a discussion of the 
causes of this neglect of Grassmann's work 13 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 essay 15 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 abstract 16 of the Aus- 
dehnungslehre, intended for mathematicians. Thirty years 
later Grassmann spoke to Delbriick 17 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. 


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 foreseen 19 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. 


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, an d 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- 

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. 


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 

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. Delbriick 21 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. 


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- 


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. Clebsch 22 also 
shortly afterward accorded him a full measure of admira- 
tion. Grassmann 23 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. 


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. 


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. 


own period. Indications are indeed not wanting that in 
the modern theory of transformation-groups 26 lies the cri- 
terion for a final judgment. 

A. E. HEATH." 

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. 


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. 


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 


drew back in terror from deeper study of his early work. 
He says 2 : "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. Whitehead 3 in England and from G. Peano 4 and C. 
Burali-Forti 5 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. Merz 6 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, 

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. 


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. 


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- 


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 


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. 


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 known 11 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." 


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. 


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. 


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, 


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 warning 14 
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 

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. 


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. 


out of reach of those who have never been freed from the 
confines of the existent world. 

Cajori 17 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 account 18 of the essen- 
tial features of Grassmann's Ausdehnungslehre. 



17 Teaching and History of Mathematics in the United States. 

18 Translated by W. W. Beman of the University of Michigan. 



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, 


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 an d 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 


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 
writes 1 : "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 Grassmann 2 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 logic 3 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. 


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. 


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.). 


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. 


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. 


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 logic 7 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 

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.) 


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. 


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. 


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 Analyse 13 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. 


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 


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. 


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." 


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." 


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 mobility 21 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- 

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. 


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. 


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. 


'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 relations 24 but by reducing figures 
to their projective properties and relations, at least a real 
geometry of position 25 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. 


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. 


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 

Grassmann rightly regarded the fact that substitution 
was not possible as a serious defect in the calculus. So he 


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(d f e,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 


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. 


It is impossible to follow Grassmann's development 26 
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- 

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. 


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). 


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." 


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- 

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; 


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 Branford 29 : "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. 


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 out 30 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 sciences 31 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 
play 88 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." 


(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 remarked 34 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 


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. 



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 


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 


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 


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 


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 


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 


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- 

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, 


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 


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 


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 ? 


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. 


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 


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. 


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 

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. 


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 


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- 


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- 

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 


to accept such a justification as a statement of antecedent 

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- 

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. 


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- 

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- 


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 

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 


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 

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 


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- 


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 ygean 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 

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 


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 


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; 


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. 




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. 


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. 


was appointed professor of the philosophic theory of re- 

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 people 2 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. 


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). 


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 manuscript 8 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. 


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 
council 7 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." 


citizen does not work against the government, for the 
government's object is to work for the good of the whole 

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 


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 

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- 


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. 


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. 


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 fundament ! 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 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 

Sulzbach, 1837. 

13 For a criticism of Bolzano's theories see M. Palagyi, Kant und Bolzano. 
Halle, 1905. 


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 

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, 


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. 


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>anX n 
as x tends to unity through values less than unity. This definition has the 


subsequent mathematical developments of the theory of 
infinite series. However it was left to the genius of Bol- 
zano 19 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 
^a n x n 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. 


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. 


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 u n be such a num- 
ber that the property M holds for all values of x which 
are less than u n , and if the property does not hold for 
all values of x without exception, then of all the num- 
bers u n 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- 


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 u n -)-D is an unsuitable 
number. Then, bisecting the interval between u n and 
u n -f- D, we get the number u n -\- D/2 ; bisecting the inter- 
val between u n and u n -f- D/2 the number u n -\- D/2 2 ; and 
so on. When either all the numbers u n -\- D/2 r for r = 
i, 2, 3. .. are unsuitable or there is a number R such 
that u n -f D/2 R is an unsuitable and u n -f D/2 R l a suit- 
able number. In the first case the existence of U is estab- 
lished, U being equal to u n . In the second case we repeat 
the process, dividing the interval between u n -\- D/2 R l 
and u n -f D/2 R . Again, either all the numbers u n + D/2 R 
+ D/2 R + J , s i, 2. . . . are unsuitable or there is a num- 
ber S such that u n -f- D/2 R -f- D/2 R s is an unsuitable 
and u n + D/2 R -f D/2 R + S i a suitable number. We 
continue the same process : if it does not terminate we get 
finally to an infinite series 

U H + D/2 R + 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. 


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 

a + bx m + cx n + . . . + px r , 

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- 


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. 


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. 


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 




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 <?.) 


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. 


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 


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 

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 


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 boycott 8 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). 


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. 


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," 


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 


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. 




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. 


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 Zahlenf 3 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 sentence 8 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 


different from a is not an element of S." This is after- 
ward 7 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 logic 8 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 system 9 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- 

9 [Op. cit., p. 46:] 10 [Ibid., pp. 45-46.] 


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 Grundlagen 11 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- 
larity 18 (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. 

l2 Sit2ungsberichte der Jenaischen Gesellschaft fur Medicin und Natur- 
wissenschaft, July 17, 1885. 

13 Cf. 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).] 


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. 


If 17 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. 


the Greek letter "" will be used 18 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, "( 2 H~3- 12 ) 1 " is a name 
of the number 5, composed of the function-name "(2 + 
3-? 2 )l" an d "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. 


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: "o 2 = 4", "i 2 = 4", "2 2 = 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 "2 2 " 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 "2 2 " denotes the number 4. Accordingly I say that 
the number 4 is the "denotation" of "4" and of "2 2 ", 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 "2 2 " and "2 + 2" 
have not the same meaning, nor have "2 2 = 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 

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). 


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 


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 "" an d "" 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. 


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. 


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" P re ~ 
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 "2 2 = 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 

Thus the function "$(!)" preceded by a horizontal 
line, denotes a concept and the function "*?(!,)" pre- 


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 2 2 = $" 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 


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. 




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 


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." 


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 


V ^t a * ** 




<$ 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). 


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 


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 




*' J * 

' i 


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! 


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! 

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. 


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. 


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 


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: 


"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. 


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. 


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- 


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 book 1 to refute 

1 New Essays Concerning Human Understanding. Translated by A. G. 
Langley. 2d ed., Chicago and London, 1916. 


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 


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 


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- 

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- 


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. 


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. 


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 member 3 of N, and use 
some such word as "entity" or "being" instead. 

Mr. B. Russell 4 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 

f Cf., 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. 


leads to non-existential propositions, and so contradicts the the- 

This doctrine was repeated in Mr. Russell's Principles of 
Mathematics; 5 the existence-theorems of mathematics were said 8 
to be "proofs that the various classes defined are not null," and the 
earlier statement 7 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. MacColl 8 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 e lt e 2 , e 3 ,. . . ., he denoted "our universe of real existences," 
and by o i} o z , o 3 ,. . . ., "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. 


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. Shearman 10 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 

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. 


The last sentence refers to Frege's 12 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, o 1} o z , o 3 , etc., 
just as I break up the former into separate individuals e l} e 2 , e 3 , 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. Russell 1 * 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. 


contains an apparent variable must not be one of the possible 
values of this variable. 16 But Poincare did notperceive 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. Russell 19 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 

M We 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. 


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 1904 21 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. 


"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 


existence." But, as Dr. Whitehead and Mr. Russell 23 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 before 24 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;.... A n ,. . . .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 

"Monist, Jan. 1910, Vol. XX, pp. 113-116. 


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. 




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. 


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- 

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 


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 


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- 

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 


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 

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 


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 

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 

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 


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- 

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 

In the second condition it is essential that certain agreements 


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 

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. 




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. 


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 

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. 


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 

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- 


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. 

2 This is my translation. Legge's rendering is not correct. 







~^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) : 


II. And they went out and fted from the sepulcher, for 


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 : 


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 


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 


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 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 


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 


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 


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 

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 


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 


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 


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 

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. 



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 


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 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] : 


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 


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- 39 1 -) 

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 


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 


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 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- 

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- 


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 


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. 


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. 




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 


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. 


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 


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- 


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 


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 


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 

"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. 


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. 


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 

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, 


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. 


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 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- 

"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 


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. 


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. 


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- 


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 


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. 


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. 


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 


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. 


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 


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 


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 


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. 


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 

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. 


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. 




^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 


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 Miinsterberg 1 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. 


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. 


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- 


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. 


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- 

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 

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. 


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. 


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. 


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 


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- 

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. 


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. 


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- 


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. 


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. 


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 

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. 


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 

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. 


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. 


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- 


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- 


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- 

2S Journ. Abnorm. Psychol., II, 1907, p. 30. 
29 Op. cit., p. 35. 


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. 


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 

Instances of the conservation of forgotten experiences 
abound both in normal and pathological life. They are 

32 Ibid., pp. 4Sff. 


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- 

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. 


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. 


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. 


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 

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. 


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. 


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- 


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 

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. 





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. 


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 


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. 


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. 


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. 




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. 


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 tables 2 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- 

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. 


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 : 

h z w d z h x 

- + - = xy --, e _ k = w , 

* w 2 

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. 


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 


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. 



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. 


Moment of DE (= x) about AD is x z /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- 


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- 

The only remark that seems to be necessary with regard 
to the second part of this manuscript is that Weissenborn 7 
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 

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 

7 Weissenborn, Principien der hoheren Analysis, Halle, 1856. 



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. 


For, the moment of the space ABCEA about AD is made up 
of rectangles contained by BC (= y) and AB (= x} ; also the moment 


about AD of the space ADCEA, the complement of the former 

/ x 2 \ 
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 


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 = 


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 = . 



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.*." 


*+/-0. (6) 

X" X 

and, on removing the fractions, 

az 2 + bx* + cx 2 z + dx s + exz + fx 2 = (7) 

Again, let x 2 = 2w (8) ; then, substituting this value in 

equation (3), we have 

and therefore 

x _-a*-2bw -ey-f (10) 

= V2w; (11) 

and, squaring each side, we have 10 

a z y 2 + 4aby 2 w + 2aey a + 2afy 2 + 4b 2 w 2 + 4bewy + 4bfw 

+ e z y 2 + 2fey + f 2 - 2c 2 y 2 w - 4cdyw - 2d 2 w = 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. 



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 ca 2 . Call the area 
of the figure v\ then the distance of the center of gravity from the 

ba 2 
straight line BC, namely CG, is equal to , and its distance from 



the straight line DE, namely EH, is equal to - ; therefore CG is to 

EH as b is to c, or they are in a given ratio. 11 



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. 


tional to these, that is, when a straight line passes through F, 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 

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. 


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 

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 

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. 


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, 


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 

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 


Let xw n az, then w ""> , and we have 


az az " ult..r. omn. omn.omn. : 

x x ' 

az az az 

hence omn. "" ult.^r omn. = - omn.omn. 5 ; 

X X 2 X 2 ' 

inserting this value in the preceding equation, we have 

10 az i, az 

omn.a^r *~> ult.-r 2 omn. -= - ult.^r. omn.omn. -= , 
x 2 x 2 

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. 


ii: az 

omn. ult.jr. own.: omn.omn. 7 ; 
x 2 x 2 

and this can proceed in this manner indefinitely. 

a a a 

Again, omn. " x omn. -=- omn.omn. z- 
x x* x f 

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 


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. 


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. 


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. I 2 

= omn. omn. / . 

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 

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 


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 
x 2 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 


of x are in arithmetical progression; hence, substituting 
x for / in the equation 

omn.xl = omn./ omn. omn./, 

and remembering that omn.jir == x 2 /?, 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 


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 be 17 

v = b + cx + dy + ex 2 + fy 2 + gyx + hy s + lx* + mxyy + yxx + etc. ; ... (i) 
let it be set in order for tangents, as follows : 
-dy- 2fy 2 - gyx - 3hy 3 - 2mxy 2 - mx z y - etc. 


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 y 5 -f- by* = 2qqv 3 yyv 3 , then the equation must be written y 5 -f- by 4 -f- 
yy 3 = 2qqv 3 yyv 3 ; 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. 


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, 

<y 2 = x/( 1 + x*) = x - x z + x 3 - x* + x* - x 6 + 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 



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- 


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. 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 


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 

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 


omn.p= -, i. e., = ir~'> therefore we have a theorem that to me 


seems admirable, and one that will be of great service to this new 
calculus, namely, 

omn. I 2 - , / , , 

- = 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 

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. 


From this it will appear that a law of things of the same kind 


should always be noted, as it is useful in obviating errors of cal- 

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 



/3 _f_ I /3 _ ^3. 
S a ~ J 

we have 

j CJC^ j 9 ^ o 

ar<? = + barx + T~ + ** 

5 TC 

For indeed Cl 3 = x, because / is supposed to be equal 19 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 / I 3 = fix a z x = 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? 



Also fcJp = C -?~, that is = C JJ- f ba 2 = 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 I 2 = c , we can ob- 

J 3a 3 

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 


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 z 2 /2a = y, or by summing ^/2ay = z, and 
then z z /t = p = l=ya/d. Hence, if in an indefinite equation, we 
eliminate y by substituting in its place z z /2a, and investigate the 
t of this new equation which is indefinite like the first, and 
then by the help of the value z z /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 



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 



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; 


, AI t-x 
but = 
v t 

therefore AI = 


and QI = v AI = v v + , i. e. QI = , 

* 2* t 


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 

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. 



consist of the same letters and signs; and whether this is possible, 
will immediately appear on being worked out analytically. 


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, 






X B 








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 



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 

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. 



= *, then HL = - = 


But, on account of 

the similar triangles, ^^r = ^^7 T\ > 
ri L> r r/ (^ =/ ) 


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. 


Hence, EN = DE(=0 -HD(=*) -NH ( = -^ =<-*-^. 

Now NL= 

and =77- = TTT or MN = rrj ; thus we have 



hence, since ^= -<ffid 2 , 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.y 2 ; 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. y 2 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 : e z = fj -\- d 2 and not 
f 2 d 2 ; 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. 


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 + y 2 /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 - d z z - 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 


parabolas, 24 one under the other, equal to a cylinder, is immediately 
evident by calculus; for the ordinate of a circle y=^/a 2 -x 2 , that is, 
the product of -\/a + x and ^a-x; and in the same way, -\/2av-v 2 

= y, which gives y=V^ m to ^2a-v; and these come to the same 

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 

x 2 x 2 


ax-b* , 

x 2 

then, multiplying by ifax + b t we have 


and, multiplying by ^ox b, we have 1= ~ 5 

\ax + b 

ax 2 b z x 

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. 



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 

/ 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- 

Note. The values f, g, a, h, are the lengths of TQ, QP, PT, and the per- 
pendicular from Q on PT. 


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. 


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," 


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 
(y 2 -(- x z ) (a* yx) = 2y 2 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 y 2 logy. 

The introduction of dx instead of x/d marks a further 
advance, more important perhaps than the use of fy dy ; 


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- 



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. 


y y2 27 

J wz= 2' rWZ 2d 


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 = ^ . 

/y 2 
-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* = y 3 ; taking c to be 
the latus rectum, and supposing that for c 2 we put 3ba, or c=^3ba, 
we have 3xba = y 3 . 

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. 

v 2 

y 9 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. 


The artifice of this analysis 28 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=a z /x; 
but w = y z /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- 

a 2 y2 f 

For then x + w= and wz=-. also \z = x, 
y 2<t J 

, = %', thus, ^ = |1, and ^w; 

y 2 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 

/ a 2 r Cc? / 

or I #= I , 

2d y J J y 2 



* 2 ^ (* v 

- + = I - or d 


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 

29 Required y = /(*), such that y dy/dx = a 2 /x; the solution is y 2 = 2a 2 
log e A^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 a z /x with the ordinate of the equilateral hyperbola xy = a 2 , 
and its integral with the quadrature of this curve. The omission of the a 2 
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. 


but, if we join AC, A(C), then these are equal to ^/* 2 + y 2 ; 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 = dx T Tf 

' 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 2a z /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 a 2 / 

x + = or xy + =a 2 ; 
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. 


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, 

by the hypothesis of the question. 

Now, if we take the y's to be in arithmetical progression, we 

r 2 .x 2 

~ + 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 axy 

and now we have 


dx- - --- 
* Logy 

Hence at length we obtain an equation, in which x and y alone 
remain, and unshackled, namely 

y z + x z , a?-yx=2y 2 L,ogy ; 

and this equation, since it is determinate, will give the required 

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. 



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 = y 2 /2, or the moment of all the differences FC 
will be equal to the moment of the last term, and ydy = d(y z /2), or 
y*dy = y dy 2 /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 y 2 + x* = 2 I , as before; i. e., dx 3 + 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(a z /y) -dy 2 ; for dy 2 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 y 2 , must be such that the 
x f s 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. 


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 



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 



x+ = 

Hence, ~\/x 2 + y z = 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/x z + y z = w } and w = dy z /2dx. Let the 

^s be in arithmetical progression then (f^/* 2 + y 2 =) JAC = y 2 ; 
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+ (/) = (y 2 ), or 1 + (y 2 ) = (?*). 
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 v z =.x(x -}-v), and the solution is a pair of straight 


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, 


and whether dx/dy is the same thing as d\ it may be seen that 
if y = z 2 + bz, and x = cz + d; then 

dy = z 2 + 2pz + p 2 , + bz + bp, -z 2 -bz, 
and this becomes dy = 2z + b/3. 

In the same way dx = + cf$, and hence 
dx dy = 2z + b eft 2 , 

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 = x z /2, hence xy = x z /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 ! 


Suppose J zy to be the sum that is required. Let J zy = w t 

dw r [ dw . 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: 

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 ~ 

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 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 p 2 . 
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, 



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 + dx 2 + ey 2 + fyx = 0. Making use of 
'the method of tangents, we have 

bt + 2dxt + fyt = -cy - 2ey 2 - 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 - 2x 
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 y 3 = bw yw would 
first of all be written 

y 3 -\-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, 

3y a -}- 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. 


hence t= 7- ^-r- P From this it is quite evident that t can 
b + 2dx+fy 

always be divided by y (and by x), and since w = fiy/t, therefore 
we have 

$b + 2dx +fy -w c+fx,^Pb + 2 dx 

w = - 5 -- ,- , and y = -- f - , 
c 2ey jx j+2e 

but from just above \= - , . hence we have 

c + ey +/x 



we +/x, P6 + 2dx 


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. 


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. 


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- 



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- 


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 = a 2 /(*-*). If this is multi- 
plied by t, we have 

hence. ta 2 = ty 2 - xy 2 , 

or t = xy z /(a z -y 2 ) ; 45 and therefore t/x=y 2 /(a 2 -y 2 ), or all the f's 

together equal the moment about the vertex of every y 2 /(a 2 -y 2 ). 

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) 


Further, wx= , and J wx=-yx J yft, 

/Pa r ~y* 

n 2 _ 2 

Also w=dy, dy - *-, and therefore 


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), /?^- 


hence, ft =dyx. 

40 There is a mistake in sign ; a 2 y z should be y z a 2 ; hence the work 
that follows is also wrong. 


We have thus obtained two equations that are mutually inde- 

d* vie (46 ' 

pendent, the first f = - ............... (1) 

ay a + y, ay 

and the second dyx= ....................... (2) 

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 = BC 2 /2; that is to 

say, BP= , /= & = y, and therefore 

tx 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 dxydy 2 -dydy 2 x=2a?dy. Hence, 

2c?dy + dy dy^x 

y dy 2 

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 y z a 2 = v 2 . 
x y 2 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 Cx z y z = a 2 , which is the gen- 
eral solution of the equation. Hence the problem is impossible. 


the y's to be in arithmetical progression, that is that dy = fi a con- 
stant, and dy 2 = 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 

48 This is tantamount to a confession by Leibniz that he cannot explicitly 
integrate fa 2 /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. 



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 

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. 


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. 


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 


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 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- + cy 2 + etc. 
= 0, and the other to the straight HneTC(C) which, on account of the 
relation TA/AE = TB/BC. will beb/e=(bx)/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 

s 2 = y- + v 2 + 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 y 2 , 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 


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 hy 2 + y 3 = cx s + dx* + ex + f is the equa- 
tion to a curve of which the tangents are required, assume a curve 
of which the equation is hy 2 + y z = gx + q, for that of which the 
tangents are already known ; eliminating y, we have an equation 
such as gx + q = cx 3 + dx 2 + 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. 


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 

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- 


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? 





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. 


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. 


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. 


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 

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. 


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- 
or y, PP- 299ff; James, Human Immortality, pp. 7-29. 

8 Cf. Bergson, Matter and Memory, Chapter II. 


examination of the "law of substance" the second main support 
of Haeckel's system. 


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 


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. 


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. 


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. 


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. 


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. 


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. 


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. 

22 For a clear description of the distinction between personal and mechan- 
ical determination, cf. Ward, Realm of Ends, pp. 179ff. 


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. 




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. 


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 


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 


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 

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 

The universe has certainly to be explained from the highest 


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 

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- 


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. 


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 


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 

"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 


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 

"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 


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 


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 

"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, 

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 


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 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 book 1 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 


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 


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 


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. 


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 


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 



"Wic Alles sich zum Ganzen webt ! 
Eins in dem Andern wirkt und lebt." 


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. 


\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 


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>at 2 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 

hence ^ +* V^(c/m} . (l z /v z \ whence c/m *= (2i^// 2 X), 
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 


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 .7Xio 7 ). 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/ 2 /\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 

By electrometric methods one may find the total charge 


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/io 12 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 times 2 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 (1.63/10 24 ) of a gram. Hence the mass of an electron 
would be 0.8/10 27 gram. 


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 HMf 2 . 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' V 2 , 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')V 2 -. Now all 
motions take place in the all-pervading ether. It follows 
then that, if the ether itself has mass, when put into move- 


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)]v 2 , and the ordinary 
mass m is thus increased by %k(e 2 /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 


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 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 

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 


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. 


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- 


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 J 3 13 13 H 14 J 5 15 
8 8 8 9 9 10 10 10 id 10 10 

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. 


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 w r e 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 


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 


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- 


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- 


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 


of mathematics was pointed at it steadily for twenty-eight 

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 

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- 


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 


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 


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 


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 

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 


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 


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- 


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, 


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 


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 


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. 


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 


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." 


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 


3 Adapted from Maurice Guerin's Lc Ccntaure. 



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- 


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 


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. 


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 


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. 


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, 


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- 

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 


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 


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. 


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- 


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. . . . 


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. 


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. 


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. 


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. 


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. 


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 

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 


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. 


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 


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 

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 


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 

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; 


which might, of course j 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. 


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. 


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 


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. 


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.) 


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 

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 


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. 


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 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. 


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 


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 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. 


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 expressions 8 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 

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. 


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. 


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. 


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- 


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. 


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- 

The contrast between the two men was really antip- 


odal 18 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." 


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- 


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 

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. 


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. 


20 Quoted from Patmore by Underbill (Mysticism, 4th ed., 1912, p. 29). 


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- 

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. 


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- 


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. 


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 Neuburger 7 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. 


"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. 


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- 

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 

"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 


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. 


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- 

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. 


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. 


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.) 


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. 


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 


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 

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. 




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 an d 1851). The pres- 
ence of a nucleus within the cells began to be recognized. 
Hereditary problems were almost only discussed by breed- 

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- 



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 


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 


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 


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 

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- 


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 


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- 

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. 


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. 






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 


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. 

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- 


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 b z y = 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\Jr z -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 b 2 y = cx 2 + 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. 


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 

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. 

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, 


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 

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 yx x 

hence , ,__, _. _ ml __ t 

hence, = '-. bu , < J; 

y t y dy 

therefore fe^-JL. or dx y x dx = dy ; 

dy yx 

hence dxy- $xdx=-n$dy. 



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= (AC 2 /2) +BC B3 

Let this .r 2 /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 == 

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 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^jy 1 dz, therefore 

>' 2 a (* 

~o j \ y 1 dz- Hence we have both the area of the figure and the 

' 2 

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 curve 5 " 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." 


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/y 2 
dy/y 3 , 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 


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. 


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. 

x 3 + px* + qx =0 x*+ xy + y 1 + x + y+ *= Q 

y ' 2 y , 2xdx + xdy + 2ydy+dx+dy = 

y 3 ydx 


2yx* + yx y dy y~+2x+l 

y z x 

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, 

-3-1 1 3 6 10 15 
-4-1 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 


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, dx 3 = 3;r 2 , etc. 
U l' ;; Jl 2 ,1 _ 3 

a -- 9 , a o = -- n , o , etc. 

x x l x x x 6 x 1 


d V*= /- , etc. 


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 ., 
x 3 X s 

and d Jx or dx* will be \x~' A or 4/1. 


Let y = X*, then dy = 2* dx or ^ = 2 x . 



This reasoning is general, and it does not depend on what the pro- 
gression for the .r's may be. " By the same method, the general 
rule is established as: 


- = ^ x , am 

Suppose that we have any equation whatever, say, 

ay 2 + byx + cs 2 + f 2 x + g-y + h 3 = 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 

ay z + byx + ex 2 + f 2 x + g 2 y + h 3 = ~* 

a2dyy + by dx + 2cxdx + f 2 d.v + g-dy 

> = 

ady 2 -f bdxdy + cdx 2 = 

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, 

ay 2 bx 2 cs 2 fyx gyx hxs ly mx nz. p = 0. 
From this we get another equation 

ay 2 bx 2 cz~ fyx simi- ly mx simi- p 

2adyy 2bdxx 2cdzz fydx larly Idy mdx larly 

a dy 2 bdx 2 cdz 2 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. 


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~ > an d ~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, s d.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. 


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. = 

then byx -\- { etc. = 0, and so on for the rest. It is to be seen what is to be 
done about the A 3 . For the purpose of making these calculations better, the 
equation ay 2 -\- byx + ex 2 -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. 


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- 


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 



[General method for drawing tangents to curves without cal- 
culation,and without reducing irrational or fractional quan- 

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 + hxy 2 + kx 3 + ly* + etc. = d 
One has only to write 

-i- t> + c v + dxv + 2?* + 2fyv + gx'*y + hy 1 ^ f 3&* 2 + 3fy'' 2 t> 
dy 2s x )' 2hxyv 




that is to say, if the equation is changed to a proportion, 
$ _ c + dx + 2fy -H KX* + 2hxy + 3()' 2 + 2ntx l y + etc. 
v b + dy + 2fx + 2gxy + hy* + 3^' 2 + etc. 

f TR 

and, supposing that - expresses the ratio - 

v BC =x 


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. 


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 gentlemen 06 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* + fy 2 + 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. 



Again, let w be equal to j/ <*> , then d< z.\- l /a> ; and this 

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.) 



Let there be two or more other curves, AF, AH, and suppose 


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 = x 2 , 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 



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 

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 


toward A, above B, will be drawn In the contrary direction, be- 
low B. 

Addition and Subtraction. Let y = vw()a, then dy will be 
equal to dvdw(}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. 


Roots or extractions. If y = */, then dz= z *~ / . 


Equations expressed in rational integral terms. 

a H- fo> + c v 4- toy + ez/ 2 + /y 2 + gv 2 y + hvy 2 + kv 3 + ly 3 

4 ntv-y 2 + nv*y 4- pvy 3 + qv* + ry* = 0, 

supposing that a, b, c, t, e, etc. are magnitudes that are known and 
determined ; then we should have 
= bdv + cdy + tvdy 4- 2cvdv + 2fydy + gv 2 dy + h\ 2 dv 
tydv +2gvydy +2hvydy 

4- Zly-dy + 2mv 2 ydy 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 ev z 4 fy z 4- gv*y + etc. = 0, 

then da 4 dbv + dcy + tdvy 4 edv 2 + /rfv 2 4 ^rf?' 2 v + etc. will also be equal 
to 0. Now da = 0, dbv = bdv, dcy - cdy, di'y = vdy + ydv\ also dv 2 = 
2vdv, since dv* is equal to s,v s ~ l ,dv, that is to say (by substituting 
2 for z] 2vdv\ and dv 3 y = v 2 dy 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 


for them, we shall have dv 2 y = 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- 

[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- 


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. 


fundamental rules for the first time; the figure notation 
is changed from the clumsy C, (C), ((C)) to the neat 
iC, 2 C, 3 C; 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 


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 (a 2 -f- x z y/ 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 

(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- 

[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.] 



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 t C X D, 2 C,D, 3 C 3 D, etc. Also the straight lines jD 2 C, 
2 D 3 C, 3 D 4 C, 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 iD 2 C, 2 D 3 C, etc. are con- 
sidered as the momentaneous increments 68 of the line BC, increas- 
ing continuously as it descends along AB, then it is plain that the 
straight line joining these two points, 2 C X C 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 

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 


AB the abscissa along an asymptote, and a for the side of the 
power, or of the area of the rectangle AB.BC; then 


dy ax. 

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. 



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 

Moreover, differences are the opposite to sums; thus 4 B 4 C is 
the sum of all the differences such as 3 D 4 C, 2 D 3 C, 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 t B jD + 2 B 2 D + 3 B 3 D + etc. For the nar- 
row triangles jC ^ 2 C, 2 C 2 D 3 C, 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 a B jE : 2 B 2 E : : jD 2 C : 2 D 3 C, 
and so on; or again, let AjBijBjC, 1 C 1 D: 1 D 2 C, etc., or dx:dy 
be in the ratio of a constant or never-varying straight line a to t B 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 3 B 4 E, which is formed from 3 B 3 E and 3 B 4 B ; 
hence, fed* is the sum of all such rectangles, 3 B 4 E + 2 B !E + 3 B 2 E 
+ etc., and this sum is the figure A 4 B 4 EA, if it is supposed that the 



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 3 D 4 C and the constant length a, and the sum of 


"v f 



\ .a 


"\ B 

1 \ 

c ,\* 


these rectangles, namely fady, or 3 D 4 C.a + 2 D 3 C.a + 1 D 2 C. 
is the same as gD^ + aDgC + ^oC + etc. into a, that is, the same 
as 4 B 4 C.a; therefore we have fady = afdy = ay. Therefore edx 
= ay, that is, the area A 4 B 4 EA will be equal to the rectangle con- 
tained by 4 B 4 C 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 2 C 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 




U---- R . 




S x 


E. x 



B | 



at once a method for finding the length of a curve by means of 


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 V aa + *'*" 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, 1 F 1 G = 1 B 1 T, then ftdy, or the sum of the rectangles 
contained by t (e.g., 4 F 4 G or 4 B 4 T) and dy ( 3 F 4 F or 3 D 4 C) is 
equal to the rectangles 4 F 3 G + 3 F 2 G + 2 F jG + etc., or the area of the 

70 All the things given are to be found in Barrow, but his name is not even 

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? 


figure A 4 F 4 GA is equal to fy dx, that is, to the figure A 4 B 4 CA ; 
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 2 C will be applied perpendicular to the 
axis AB at 2 B), 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 = x z , and the figure referred to by Leibniz is that which Wallis 
calls the complement of the semiparabola. 


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, 

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 1 3 6 10 15 x 

Sums 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. 


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. 


then dy = 0, and the result becomes , which is the value we 


used a little while before in the case of the tangent to the hyper- 

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 

d v = x d yy dx . 


hence, putting zv for x, and sdv + vdz for dx or dzv in the above, 
we obtain what was stated. 


Powers follow: dx z = 2xdx, dx* = Zx*dx, and so on. For, putting 
y = x, and v=x, we can write dx z for dxy, and this is (from above) 
equal to xdy + ydx, or (if x = y, and consequently dx = dy) equal 
to 2xdx. Similarly, for dx s 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. d t \/x h dx'' r , 
(where by h:r I mean h/r, or h divided by r), or dx e (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>\/x H . 

Moreover, conversely, we have 



f. t *Wxm~, f- e <tx==-l , f<i 

J e + V J x e e-1.*- 1 J 


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. 


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 


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- 


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 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 




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 


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 evanescent 73 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." 


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 


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 

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 


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 1 X l Y = y 

and from the point jY let fall a perpendicular X YD to some greater 
ordinate 2 X 2 Y that follows, and let jX 2 X, the difference between 
A X X and A 2 X, be called dx ; and similarly, let D 2 Y, the difference 
between t X X Y and 2 X ,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 t Y 2 Y 
is produced until it meets the axis in T, then the ratio of the ordinate 
X X jY to T t X, 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 2 X 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 t Y is the tangent, ,X ^Y is to T X X 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 ; 



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 

x 3 -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 = X X t Y : T t X ; 

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 a X t Y to a XT be 
called (dy) ; in this way dy and dx will always be assignables 
bearing to one another the ratio of D 2 Y to D 1 Y, 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 t X (=*) and A 2 X (=x + dx) let there cor- 
respond the ordinates 1 X 1 'Y(=y) and 2 X 2 Y (=y + dy), and also 
the ordinates 1 X l V(=v) and 2 X 2 V (=v + dv), and the ordinates 


l X 1 Z(=rr) and 2 X 8 Z(=s + <k). Let the chords /Y-jY, 1 V 2 V, ^Z, 
when produced meet the axis AXX in T, U, W. Take any straight 
line you will as (d)x, and, while the point 4 X remains fixed and 
the point 2 X 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 t XW 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 t X 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, 


a dy - xdv + v dx + dx dv, 

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 X X coincides with 
2 X, 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.y n -^' 


and this I will prove in a manner a little more detailed than those 
above, thus: 

MX JL - /_ 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 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 is reached) ; 
take away from the one side a"- f x e , and from the other side v n , 
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 2 X 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 2 X, 2 Y 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 X X ,Y, to the subtangent iXT, 
where it is supposed that Tj Y touches the curve in t Y. 

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 



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, 2 X 2 Y, 3 X 3 Y, of which jX/Y 
remains constant, but ,X 2 Y and 3 X 3 Y continually approach jX.^Y 
until finally they both coincide with it simultaneously ; which will 
happen if the speed with which 3 X approaches jX is to the speed 
with which 2 X approaches ,X is in the ratio of !X 3 X to t X,,X. 
Also let two straight lines be assigned, (d).v always constant for 
any position of 2 X, and . 2 (d}x for any position of 3 X ; also let (d)y 
always be to (rf).r as D 2 Y is to jX 2 X, or as v (i. e., jX^) is to 
,XT; thus, while (d}x remains always the same, (d)y will be 
altered as 2 X approaches jX ; similarly, let 2 (d)v be to *(d}x as 
2 D 3 Y to 2 X 3 X or as y+ dy (i. e., 2 X 2 Y) to 2 X 2 T ; thus while 2 (</).r 
remains constant, 2 (rf)v will be altered as 3 X approaches t X. 

Also let (d)y be always taken in the varying line ,X 2 Y, and 
let 2 X !<w be equal to (d) y, and similarly take 2 (d}y in the line 3 X 3 Y, 
and let 3 X .,w be equal to 2 (d}y. Thus, while 2 X and 3 X continually 
approach to the straight line jX jY, 2 X t w and 3 X 2 o> continually 
approach it also, and finally coincide with it at the same time as 

2 X and 3 X. Further, let the point in the ordinate jX 1 Y, 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 X X touches the curve in X Y, because 
then indeed X Y and 2 Y coincide. Now, since all this can be done, 


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 2 Y, 3 X :! Y be three ordinates, of which the values 
are v, y + dy, y + dy + ddy, and let t X 2 X (dx) and 2 X 3 X (dx + ddx) 
be any distances, and D.,Y(dy) and 2 D 3 Y (dy+ddy) the differ- 
ences. Now the difference between (d)y and 2 (d)y, or between 
,X and ,X 2 n is 8,Q, and that between iXoX and 2 X 3 X is ddx; 
also let 

(d)dx: (d)x = dx: .,(d).v, 74 and similarly let 
(d)dv:(rf).v = a n8: 1 X 2 X 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- 

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 2 X and 3 X approach ,X. Also, since (d)dx, (since it bears an 
assignable ratio to (d)x, however nearly 2 X approaches to X X, 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 S X S Y 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. 


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, 

(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 2 X 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 2 (d*)dv 
(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 t X and 2 X the 

ddv x ddy v 2 dx dy 

ddx a ddx a a ddx 




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. 


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. 


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. 


Tt is the wave with endless rise and fall, 


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. 


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. 


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. 


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. 


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; 


Sees them with Beauty clothed or Ugliness, 

And names them from their power to bless or ban 
God and Devil, Ormuzd and Ahriman. 


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. 


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. 


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. 



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. 



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. 


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. 


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. 


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. 


Yet what avails my battle for the Right, 


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. 


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. 


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! 





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- 


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. 


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 

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 


numbers was just as fine as a very striking result in mathematical 

'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 


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- 


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, 


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 


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/" 




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 

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- 


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" 

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. 


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. 


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 


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- 

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 


(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 



6. The methods of training were two, at least only two were ac- 


Oldest and best known of all was the method derived from the An- 

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 


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 

Straight-edged still was the sword ; with it blocks were sliced into 

Shavings were sliced into threads, and threads were chopped into 


Parts of ineffable smallness, divisible reckoned no further. 
Masonry part of the course, in which arches with bricks were 


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 


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- 

(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 ; 


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- 


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 


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 


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- 

11. Some of the records of giants the youth Gen-Tan-Agg had de- 



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 


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 


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 


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- 

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- 

"Lives there a mortal in Ris-Pah, who long has courted my favor; 

Often of late have I thought that at last I'd rew r ard his devotion. 

Lacks he but one little thing, only one thing to render him fitting 


Trainer of offspring of mine; but the lack mean I now to forgive 


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 


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 


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 

Famed through the kingdoms of mortals, became a renown for the 

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- 


(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, 


Shun-Fluk to accuse him of stealing and sending him forth as his 

None seemed to have guessed the truth, save a man by the name of 


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- 

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. 




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 


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 
3 1( %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. 


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 


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. 




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- 


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? 

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. 



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.'' 





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. 


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 


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. 


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 


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. 


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 

"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 


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 

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 


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 

Perhaps some one will ask, why then do I take the 


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 


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 dogma 1 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. 


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