Mathematical essays and recreations

LIBRARY OF
WELLESLEY COLLEGE

PRESENTED BY

Charlotte Smith Tuton '58

m c->'^^^

MATHEMATICAL ESSAYS

RECREATIONS

HERMANN SCHUBERT

PROFESSOR OF MATHEMATICS IN THE JOHANNEUM, HAMBURG, GERMANY

FROM THE GERMAN BY

THOMAS J. McCORMACK

FOURTH EDITION

CHICAGO ::: LONDON
THE OPEN COURT PUBLISHING COMPANY

1917

COPYRIGHT BY

THE OPEN COURT PUBLISHING CO.

/^ /./'

( ' "

PRINTED IN THE UNITED STATES OF AMERICA

TRANSLATOR'S NOTE.

'l '^HE essays and recreations constituting this volume are by one of the foremost
mathematicians and text-book writers of Germany. The monistic construc-
tion of arithmetic, the systematic and organic development of all its consequences
from a few thoroughly established principles, is quite foreign to the general run of
American and English elementary text-booksi and the first three essays of Professor
Schubert will, therefore, from a logical and esthetic side, be full of suggestions for
elementary mathematical teachers and students, as well as for non-mathematical
readers. For the actual detailed development of the system of ari*bmetic here
sketched, we may refer the reader to Professor Schubert's volume Arithmetik
und Algebra, recently published in the Göschen-Sammlung (Göschen, Leipsic), —
an extraordinarily cheap series containing many other unique and valuable text-
books in mathematics and the sciences.

The remaining essays on "Magic Squares," "The Fourth Dimension," and
"The History of the Squaring of the Circle," will be found to be the most com-
plete generally accessible accounts in English, and to have, one and all, a distinct
educational and ethical lesson.

In all these essays, which are of a simple and popular character, and designed
for the general public, Professor Schubert has incorporated much of his original
research.

Thomas J. McCormack.
La Salle, 111., December, 1898.

CONTENTS.

PAGE

Notion and Definition of Number i

Monism in Arithmetic 8

On the Nature of Mathematical Knowledge 27

The Magic Square 39

The Fourth Dimension 64

The Squaring of the Circle 112

NOTION AND DEFINITION OF NUMBER.

li /TANY essays have been written on the definition of number.
■i-^-^ But most of them contain too many technical expressions,
both philosophical and mathematical, to suit the non-mathemati-
cian. The clearest idea of what counting and numbers mean may
be gained from the observation of children and of nations in the
childhood of civilisation. When children count or add, they use
either their fingers, or small sticks of wood, or pebbles, or similar
things, which they adjoin singly to the things to be counted or
otherwise ordinally associate with them. As we know from history,
the Romans and Greeks employed their fingers when they counted
or added. And even to-day we frequently meet with people to whom
the use of the fingers is absolutely indispensable for computation.

Still better proof that the accurate association of such "other"
things with the things to be counted is the essential element of nu-
meration are the tales of travellers in Africa, telling us how African
tribes sometimes inform friendly nations of the number of the enemies
who have invaded their domain. The conveyance of the informa-
tion is effected not by messengers, but simply by placing at spots
selected for the purpose a number of stones exactly equal to the
number of the invaders. No one will deny that the number of the
tribe's foes is thus communicated, even though no name exists for
this number in the languages of the tribes. The reason wh}' the
fingers are so universally employed as a means of numeration is,
that every one possesses a definite number of fingers, sufficiently
large for purposes of computation and that they are always at hand.

Besides this first and chief element of numeration which, as we

2 NOTION AND DEFINITION OF NUMBER.

have seen, is the exact, individual conjunction or association of other
things with the things to be counted, is to be mentioned a second
important element, which in some respects perhaps is not so abso-
lutel}^ essential; namely, that the things to be counted shall be re-
garded as of the same kind. Thus, any one who subjects apples and
nuts collectively to a process of numeration will regard them for the
time being as objects of the same kind, perhaps by subsuming them
under the common notion of fruit. Wc may therefore lay down pro-
visionally the following as a definition of counting : to count a group
of things is to regard the things as the same in kind and to associate
ordinall}'-, accurately, and singly with them other things. In writing,
we associate with the things to be counted simple signs, like points,
strokes, or circles. The form of the symbols we use is indifferent.
Neither need they be uniform. It is also indifferent what the spa-
tial relations or dispositions of these symbols are. Although, of
course, it is much more convenient and simpler to fashion symbols
growing out of operations of counting on principles of uniformity
and to place them spatially near each other. In this manner are
produced what I have called * natural number-pictures ; for ex-
ample,

etc.

Now-a-days such natural number-pictures are rarel}' employed, and
are to be seeir only on dominoes, dice, and sometimes, also, on play-
ing-cards.

It can be shown by archaeological evidence that originally nu-
meral writing was made up wholly of natural number-pictures. For
example, the Romans in early times represented all numbers, w^hich
were written at all, by assemblages of strokes. We have remnants
of this writing in the first three numerals of the modern Roman sys-
tem. If we needed additional evidence that the Romans originally
employed natural number-signs, we might cite the passage in Livy,
Vn. 3, where we are told, that, in accordance with a very ancient
law, a nail was annually driven into a certain spot in the sanctuary of

* System der Arithmetik. (Potsdam : Aug. Stein. 1885.)

NOTION AND DEFINITION OF NUMBER. 3

Minerva, the ' ' inventrix " cf counting, for the purpose of showing the
number of 3'ears which had elapsed since the building of the edifice.
We learn from the same source that also in the temple at Volsinii
nails were shown which the Etruscans had placed there as marks for
the number of years.

Also recent researches in the civilisation of ancient Mexico show
that natural number-pictures were the first stage of numeral nota-
tion. Whosoever has carefully studied in any large ethnographical
collection the monuments of ancient Mexico, will surely have re-
marked that the nations which inhabited Mexico before its conquest
by the Spaniards, possessed natural number-signs for all numbers
from one to nineteen, which they formed by combinations of circles.
If in our studies of the past of modern civilised peoples, we meet
with natural number-pictures only among the Greeks or Romans,
and some Oriental nations, the reason is that the other nations, as the
Germans, before they came into contact with the Romans and adopted
the more highl}^ developed notation of the latter, were not yet suffi-
ciently advanced in civilisation to feel any need of expressing num-
bers symbolically. But since the most perfect of all systems of nu-
meration, the Hindu system of "local value," was introduced and
adopted in Europe in the twelfth centur}', the Roman numeral sys-
tem gradually disappeared, at least from practical computation, and
at present we are only reminded by the Roman characters of inscrip-
tions of the first and primitive stage of all numeral notation. To-
day we see natural number-pictures, except in the above-mentioned
games, only very rarely, as where the tally-men of wharves or ware-
houses make single strokes with a pencil or a piece of chalk, one for
each bale or sack which is counted.

As in writing it is of consequence to associate with each of the
things to be counted some simple sign, so in speaking it is of con
sequence to utter for each single thing counted some short sound.
It is quite indifferent here what this sound is called, also whether
the sounds which are associated with the things to be counted are
the same in kind or not, and finally, whether they are uttered at
equal or unequal intervals of time. Yet it is more convenient and
simpler to employ the same sound and to observe equal intervals in

(j. NOTION AND DEFINITION OF NUMBER.

their utterance. We arrive thus at natural number-words. For ex-
ample, utterances like,

oh, oh-oh, oh-oh-oh, oh-oh-oh-oh, oh-oh-oh-oh-oh,
are natural number-words for the numbers from one to five. Num-
ber-words of this description are not now to be found in any known
language. And yet we hear such natural number-words constantly,
every day and night of our lives ; the only difference being that the
speakers are not human beings but machines — namely, the striking-
apparatus of our clocks.

Word-forms of the kind described are too inconvenient, how-
ever, for use in language, not only for the speaker, on account of
their ultimate length, but also for the hearer, who must be constantly
on the qui vive lest he misunderstand a numeral word so formed. It
has thus come about that the languages of men from time imme-
morial have possessed numeral words which exhibit no trace of the
original idea of single association. But if we should always select
for every new numeral word some new and special verbal root, we
should find ourselves in possession of an inordinately large number
of roots, and too severely tax our powers of memory. Accordingly,
the languages of both civilised and uncivilised peoples always con-
struct their words for larger numbers from words for smaller num-
bers. What number we shall begin with in the formation of com
pound numeral words is quite indifferent, so far as the idea of num-
ber itself is concerned. Yet we find, nevertheless, in nearly all
languages one and the same number taken as the first station in the
formation of compound numeral words, and this number is ten.
Chinese and Latins, Fins and Malays, that is, peoples who have no
linguistic relationship, all display in the formation of numeral words
the similarity of beginning with the number ten the formation of
compound numerals. No other reason can be found for this striking
agreement than the fact that all the forefathers of these nations pos-
sessed ten fingers.

Granting it were impossible to prove in an}^ other way that
people originally used their fingers in reckoning, the conclusion
could be inferred with sufficient certainty solely from this agreement
with regard to the first resting-point in the formation of compound

NOTION AND DEFINITION OF NUMBER. 5

numerals among the most various races. In the Indo-Germanic
tongues the numeral words from ten to ninety-nine are formed by
composition from smaller numeral words. Two methods remain
for continuing the formation of the numerals : either to take a new
root as our basis of composition (hundred) or to go on counting
from ninety-nine, saying tenty, eleventy, etc. If we were logically
to follow out this second method we should get tenty-ty for a thou-
sand, tenty-ty-ty for ten thousand, etc. But in the utterance of such
words, the S3dlable ty would be so frequently repeated that the same
inconvenience would be produced as above in our individual num-
ber-pictures. For this reason the genius which controls the for-
mation of speech took the first course.

But this course is only logically carried out in the old Indian
numeral words. In Sanskrit we not only have for ten, hundred,
and thousand a new root, but new bases of composition also exist
for ten thousand, one hundred thousand, ten millions, etc., which
are in no wise related with the words for smaller numbers. Such
roots exist among the Hindus for all numerals up to the number ex-
pressed by a one and fifty-four appended naughts. In no other lan-
guage do we find this principle carried so far. In most languages the
numeral words for the numbers consisting of a one with four and
five appended naughts are compounded, and in further formations
use is made of the words million, billion, trillion, etc., which really
exhibit only one root, before which numeral words of the Latin
tongue are placed.

Besides numeral word-systems based on the number ten, only lo-
gical systems are found based on the number five and on the number
twenty. Systems of numeral words which have the basis five occur
in equatorial Africa. (See the language-tables of Stanley's books
on Africa.) The Aztecs and Mayas of ancient Mexico had the base
twenty. In Europe it was mainly the Celts who reckoned with
twenty as base, The French language still shows some few traces
of the Celtic vicenary system, as in its word for eighty, quaire-vingt.
The choice of five and of twenty as bases is explained simply enough
by the fact that each hand has five fingers, and that hands and feet
together have twenty fingers and toes.

b NOTION AND DEFINITION OF NUMBER.

As we see, the languages of humanity nov/ no longer possess
natural number-signs and number-words, but employ names and
systems of notation adopted subsequently to this first stage. . Ac-
cordingly, we must add to the definition of counting above given a
third^ factor or element which, though not absolutely necessary, is
yet important, namely, that we must be able to express the results
of the above-defined associating of certain other things with the
things to be counted, by some conventional sign or numeral word.

Having thus established what counting or numbering means,
we are in a position to define also the notion of uiuiiber, which we
do by simply saying that by number we understand the results of
counting or numeration. These are naturally composed of two ele-
ments. First, of the ordinary number-word or number-sign ; and
secondly, of the word standing for the specific things counted. For
example, eight men, seven trees, five cities. When, now, we have
counted one group of things, and subsequently also counted another
group of things of the same kind, and thereupon we conceive the
two groups of things combined into a single group, we can save
ourselves the labor of counting the things a third time by blending
the number-pictures belonging to the two groups into a single num-
ber-picture belonging to the whole. In this way we arrive on the
one hand at the idea of addition, and on l-he other, at the notion of
"unnamed " number. Since we have no means of tellfng from the
two original number-pictures and the third one which is produced
from these, the kind or character of the things counted, we are ulti-
mately led in our conception of number to abstract wholly from the
nature of the things counted, and to form the definition of unnamed
number.

We thus see that to ascend from the notion of named number
to the notion of unnamed number, the notion of addition, joined to
a high power of abstraction, is necessary. Here again our theory
is best verified by observations of children learning to count and
add. A child, in beginning arithmetic, can well understand what
five pens or five chairs are, but he cannot be made to understand
from this alone what five abstractly is. But if we put beside the
first five pens three other pens, or beside the five chairs three other

NOTION AND DEFINITION OF NUMBER. 7

chairs, we can usually bring the child to see that five things plus
three things are always eight things, no matter of what nature the
things are, and that accordingly we need not always specify in
counting what kind of things we mean. At first we always make
the answer to our question of what five plus three is, easy for the
child, by relieving him of the process of abstraction, which is neces-
sary to ascend from the named to the unnamed number, an end
•^'.^lich we accomplish by not asking first what five plus three is, but
uy associating with the numbers words designating things within
the sphere of the child's experience, for example, by asking how
many five pens plus three pens are.

The preceding reflexions have led us to the notion of unnamed
or abstract numbers. The arithmetician calls these numbers posi-
tive whole numbers, or positive integers, as he knows of other kinds
of numbers, for example, negative numbers, irrational numbers, etc.
Still, observation of the world of actual facts, as revealed to us by our
senses, can naturally lead us only to positive whole numbers, such
only, and no others, being results of actual counting. All other kinds
of numbers are nothing but artificial inventions of mathematicians
created for the purpose of giving to the chief tool of the mathema-
tician, namely, arithmetical notation, a more convenient and more
practical form, so that the solution of the problems which arise in
mathematics may be simplified. All numbers, excepting the results
of counting above defined, are and remain mere symbols, which,
although they are of incalculable value in mathematics, and, there-
fore, can scarcely be dispensed with, yet could, if it were a ques-
tion of principle, be avoided. Kronecker has shown that any prob-
lem in which positive whole numbers are given, and only such are
sought, always admits of solution without the help of other kinds of
numlters, although the employment of the latter wonderfully sim-
plifies the solution.

How these derived species of numbers, by the logical applica-
tion of a single principle, flow naturally from the notion of number
and of addition above deduced, I shall show in the next article en-
titled ''Monism in Arithmetic."

MONISM IN ARITHMETIC.

TN HIS Primer of Philosophy, Dr. Paul Carus defines monism
^ as a "unitary conception of the world." Similarly, we shall
understand by monism in a science the unitary conception of that
science. The more a science advances the more does monism domi-
nate it. An example of this is furnished by physics. Whereas
formerly physics was made up of wholly isolated branches, like
Mechanics, Heat, Optics, Electricity, and so forth, each of which
received independent explanations, physics has now donned an al-
most absolute monistic form, by the reduction of all phenomena to
the motions of molecules. For example, optical and electrical phe-
nomena, we now know, are caused by the undulatory movements
of the ether, and the length of the ether-waves constitutes the sole
difference between light and electricity.

Still more distinctly than in physics is the monistic element
displayed in pure arithmetic, by which we understand the theory of
the combination of two numbers into a third by addition and the
direct and indirect operations springing out of addition. Pure arith-
metic is a science which has completely attained its goal, and which
can prove that it has, exclusively by internal evidence. For it may
be shown on the one hand that besides the seven familiar operations
of addition, subtraction, multiplication, division, involution, Solu-
tion, and the finding of logarithms, no other operations are defin-
able which present anything essentially new; and on the other hand
that fresh extensions of the domain of numbers beyond irrational,
imaginary, and complex numbers are arithmetically impossible.
Arithmetic may be compared to a tree that has completed its growth.

MONISM IN ARITHMETIC. g

the boughs and branches of which ma}^ still increase in size or even
give forth fresh sprouts, but whose main trunk has attained its full-
est development.

Since arithmetic has arrived at its maturity, the more profound
investigation of the nature of numbers and their combinations shows
that a unitary-conception of arithmetic is not only possible but also
necessary. If we logically abide by this unitary conception, we ar-
rive, starting from the notion of counting and the allied notion of
addition, at all conceivable operations and at all possible extensions of
the notion of number. Although previously expressed by Grassmann,
Hankel, E. Schroder, and Kronecker, the author of the present ar-
ticle, in his "System of Arithmetic," Potsdam, 1885, was the first
to work out the idea referred to, fully and logically and in a form
comprehensible for beginners. This book, which Kronecker in his
"Notion of Number," an essay published in Zeller's jubilee work,
makes special mention of, is intended for persons proposing to learn
arithmetic. As that cannot be the object of the readers of these es-
says, whose purpose will rather be the study of the logical construc-
tion of the science from some single fundamental principle, the fol-
lowing pages will simply give of the notions and laws of arithmetic
what is absolutely necessary for an understanding of its develop-
ment.

The starting-point of arithmetic is the idea of counting and of
number as the result of counting. On this subject, the reader is re-
quested to read the first essay of this collection. It is there shown
that the idea of addition springs immediately from the idea of count-
ing. As in counting it is indifferent in what order we count, so in
addition it is indifferent, for the sum, or the result of the addition,
whether we add the first number to the second or the second to the
first. This law, which in the symbolic language of arithmetic, is
expressed by the formula

a-}- d = ^-\- a,

is called the cotnmuiative lazü of addition. Notwithstanding this law,
however, it is evidently desirable to distinguish the two quantities
which are to be summed, and out of which the sum is produced, by
special names. As a fact, the two summands usually are distin-

lO MONISM IN ARITHMETIC.

guished in some way, for example, by saying a is to be increased by
b, or b is to be added to a, and so forth. Here, it is plain, a is al-
ways something that is to be increased, b the increase. Accordingly
it has been proposed to call the number which is regarded in addi-
tion as the passive number or the one to be changed, the augend,
and the other which plays the active part, which accomplishes the
change, so to speak, the increynent. Both words are derived from
the Latin and are appropriately chosen. Augend is derived from
augere, to increase, and signifies that which is to be increased ; in-
crement comes from increscere, to grow, and signifies as in its ordi-
nary meaning what is added.

Besides the commutative law one other follows from the idea ot
counting — the associative law of addition. This law, which has ref-
erence not to two but to three numbers, states that having a certain
sum, a -\-b, it is indifferent for the result whether we increase the
increment b of that sum by a number, or whether we increase the
sum itself by the same number. Expressed in the symbolic lan-
guage of arithmetic this law reads,

^ + (^+0 = (^ + ^)+^-
To obtain now all the rules of addition we have only to apply the
two laws of commutation and association above stated, though fre-
quently, in the deduction of the same rule, each must be applied
many times. I may pass over here both the rules and their estab-
lishment.

In addition, two numbers, the augend a and the increment b
are combined into a third number c, the sum. From this operation
spring necessarily two inverse operations, the common feature of
which is, that the sum sought in addition is regarded in both as
known, and the difference that in the one the augend also is regarded
as known, and in the other the increment. If we ask what number
added to a gives c, we seek the increment. If we ask what number
increased by b gives c, we seek the augend. As a matter of reckon-
ing, the solution of the two questions is the same, since by the com-
mutative law of addition a-\- b = b -\- a. Consequently, only one
common name is in use for the two inverses of addition, namely,
subtraction. But with respect to the notions involved, the two oper-

MONISM IN ARITHMETIC. II

ations do differ, and it is accordingly desirable in a logical investi-
gation of the structure of arithmetic, to distinguish the two by dif-
ferent names. As in all probability no terms have yet been sug-
gested for these two kinds of subtraction, I propose here for the
first time the following words for the two operations, namely, de-
traction to denote the finding of the augend, and siibtertraction to
denote the finding of the increment. We obtain these terms simply
enough by thinking of the augmentation of some object already ex-
isting. For example, the cathedral at Cologne had in its tower an
augend that waited centuries for its increment, which was only
supplied a few decades ago. As the cathedral had originally a
height of one hundred and thirty metres, but after completion was
increased in height twenty-six metres, of the total height of one
hundred and fifty-six metres one hundred and thirty metres is clearly
the augend and twenty-six metres the increment. If, now, we wished
to recover the augend we should have to pull down (Latin, detrahere)
the upper part along the whole height. Accordingly, the finding of
the augend is called detraction. If we sought the increment, we
should have to pull out the original part from beneath (Latin, subter-
trahere). For this reason, the finding of the increment is called siib-
tertraction. Owing to the commutative law, the two inverse opera-
tions, as matters of computation, become one, which bears the name
of subtraction. The sign of this operation is the minus sign, a hori-
zontal stroke. The number which originally was sum, is called in
subtraction minuend ; the number which in addition was increment
is now called detractor; the number which in addition was augend
is now called subtertractor. Comprising the two conceptually dif-
ferent operations in one single operation, subtraction, we employ
for the number which before was increment or augend, the term sub-
trahend, a word which on account of its passive ending is not very
good, and for which, accordingly, E. Schröder proposes to substi-
tute the word subtrahent, having an active ending. The result of
subtraction, or what is the same thing, the number sought, is called
the difference. The definition-formula of subtraction reads

a — b -{- b=:a,
that is, a minus b is the »lumber which increased by b gives a, or

12 MONISM IN ARITHMETIC.

the number which added to b gives a, according as the one or the
other of the two operations inverse to addition is meant. From the
formula for subtraction, and from the rules which hold for addition,
follow now at once the rules which refer to both addition and sub-
^•"action. These rules we here omit.

From the foregoing it is plain that the minuend is necessarily
iarger than the subtrahent. For in the process of addition the minu-
end was the sum, and the sum grew out of the union of two natural
number-pictures.* Thus 5 minus g, or 11 minus 12, or 8 minus 8,
are combinations of numbers wholly destitute of meaning; for no
number, that. is, no result of counting, exists that added to 9 gives
the sum 5, or added to 12 gives the sum 11, or added to 8 gives 8.
What, then, is to be done? Shall we banish entirely from arith-
metic such meaningless combinations of numbers ; or, since they
have no meaning, shall we rather invest them with one? If we do
the first, arithmetic will still be confined in the strait-jacket into
which it was forced by the original definition of number as the re-
sult of counting. If we adopt the latter alternative we are forced
to extend our notion of number. But in doing this, we sow the
first seeds of the science of pure arithmetic, an organic body of
knowledge which fructifies all other provinces of science.

What significance, then, shall we impart to the symbol

5-9?
Since 5 minus 9 possesses no significance whatever, we may, of
course, impart to it any significance we wish. But as a matter
of practical convenience it should be invested with no meaning
that is likely to render it subject to exceptions. As the form of the
symbol 5 — 9 is the form of a difference, it will be obviously con-
venient to give it a meaning which will allow us to reckon with it as
we reckon with every other real difference, that is, with a difference
in which the minuend is larger than the subtrahent. This being
agreed upon, it follows at once that all such symbols in which the
number before the minus sign is less than the number behind it by
the same amount may be put equal to one another. It is practical,

* See page 2, supra.

MONISM IN ARITHMETIC. I3

therefore, to comprise all these sN'mbols under some one single sym-
bol, and to construct this latter symbol so that it will appear un-
equivocally from it by how much the number before the minus sign
is less than the number behind it. This difference, accordingly, is
written down and the minus sign placed before it.

If the two numbers of such a differential yi^r/;/ are equal, a totally
new sign must be invented for the expression of the fact, having
no relation to the signs which state results of counting. This in-
vention Vvas not made by the ancient Greeks, as one might naturally
suppose from the high mathematical attainments of that people, but
by Hindu Brahman priests at the end of the fourth century after
Christ. The symbol which they invented they called isiphra, empty,
whence is derived the English cipJicr. The form of this sign has been
dirferent in different times and with different peoples. But for the
last two or three centuries, since the symbolic language of arith-
metic has become thoroughly established as an international char-
acter, the form of the sign has been 0 (French zero, German nuU^.

In calling this symbol and the symbols formed of a minus sign
followed by a result of counting, uunihcrs, we widen the province of
numbers, which before was wholly limited to results of counting.
In no other way can zero and the negative numbers be introduced
into arithmetic. No man can prove that 7 minus 1 1 is equal to i
minus 5. Originally, both are meaningless symbols. And not until
we agree to impart to them a significance which allows us to reckon
with them as we reckon Avith real differences are we led to a state-
ment of identity between 7 minus 11 and i minus 5. It was a long
time before the negative numbers mentioned acquired the full rights
of citizenship in arithmetic. Cardan called them, in his Ars Magna,
1545, ;///'W(?/''/_/fr/'/ (imaginary numbers), as distinguished from minieri
veri (real numbers). Not until Descartes, in the first half of the
seventeenth century, was any one bold enough to substitute numeri
ficti and niinicri veri indiscriminately for the same letter of algebraic
expressions.

We have invested, thus, combinations of signs originally mean-
ingless, in which a smaller number stood before than after a minus
sign, with a meaning which enables us to reckon with such apparent

14 MONISM IN ARITHMETIC.

differences exactly as we do with ordinary differences. Now it is
just this practical shift of imparting meanings to combinations, which
logically applied deduces naturally the whole system of arithmetic
from the idea of counting and of addition, and which we may char-
acterise, therefore, as the foundation-principle of its whole construc-
tion. This principle, which Hankel once called the principle of per-
manence, but which I prefer to call the principle of no exception,
may be stated in general terms as follows :

In the construction of arithmetic every combination of tivo previously
defined numbers by a sign for a previously defined operation {plus, minus,
tifnes, etc. ) shall be invested with meaning, even where the original defi-
nition of the operation used excludes such a co7nbination; and the mean-
ing imparted is to be such that the combination considered shall obey the
same formula of definition as a combination having from the outset a sig-
nification, so that the old laws of reckoning shall still hold good and may
still be applied to it.

A person who is competent to apply this principle rigorously
and logically will arrive at combinations of numbers whose results
are termed irrational or imaginary with the same necessity and fa-
cility as at the combinations above discussed, whose results are
termed negative numbers and zero. To think of such combinations
as results and to call the products reached also " numbers " is a mis-
use of language. It were better if we used the phrase /^;'wj- of num-
bers for all numbers that are not the results of counting. But usus
tyr annus!

It will now be my task to show how all numbers at which arith-
metic ever has arrived or ever can arrive naturally flow from the
simple application of the principle of no exception.

Owing to the commutative and associative laws for addition it
is wholly indifferent for the result of a series of additive processes
in what order the numbers to be summed are added. For example,

« + (/. + r + ./) + (^ +/) == (a + ^ + 0 + (^+ ^) +/
The necessary consequence of this is that we may neglect the con-
sideration of the order of the numbers and give heed only to what
the quantities are that are to be summed, and, when they are equal,
take note of only two things, namely, of what the quantity which is

MONISM IN ARITHMETIC. I5

to be repeatedly summed is called and how often it occurs. We
thus reach the notion of multiplication. To multiply a by b means
to form the sum of h numbers each of which is called a. The num-
ber conceived summed is called the multiplicand, the number which
indicates or counts how often the first is conceived summed is called
the multiplier.

It appears hence, that the multiplier must be a result of count-
ing, or a number in the original sense of the word, but that the mul-
tiplicand may be any number hitherto defined, that is, may also be
zero or negative. It also follows from this definition that though
the multiplicand may be a concrete number the multiplier cannot.
Therefore, the commutative law of multiplication does not hold
when the multiplicand is concrete. For, to take an example, though
there is sense in requiring four trees to be summed three times,
there is no sense in conceiving the number three summed "four
trees times." When, however, multiplicand and multiplier are un-
named results of counting, (abstract numbers,) two fundamental
laws hold in multiplication, exactly analogous to the fundamental
laws of addition, namely, the law of commutation and the law of
association. Thus,

a times b^^b times a,
and, a times {b times e) = {a times b) times c.
The truth and correctness of these laws will be evident, if keeping
to the definition of multiplication as an abbreviated addition of equal
summands, we go back to the laws of addition. Owing to the com-
mutative law it is unnecessary, for purposes of practical reckoning,
to distinguish multiplicand and multiplier. Both have, therefore, a
common name : factor. The result of the multiplication is called the
product; the symbol of multiplication is a dot (.) or a cross (X)>
which is read " times." Joined with the fundamental formula above
written are a group of subsidiary formulae which give directions how
a sum or difference is multiplied and how multiplication is performed
with a sum or difference. I need not enter, however, into any dis-
cussion of these rules here.

As the combination of two numbers by a sign of multiplication
has no significance according to our definition of multiplication.

l6 MONISM IN ARITHMETIC.

ivhen the multiplier is zero or a negative number, it will be seen
that we are again in a position where it is necessary to apply the
above explained principle of no exception. We revert, therefore, to
what we above established, that zero and negative numbers are sym-
bols which have the form of differences, and lay down the rule that
multiplications with zero and negative numbers shall be performed
exactly as with real differences. Why, then, is minus one times
minus one, for example, equal to plus one? For no other reason
than that minus one can be multiplied with an ordinary difference,
as, for example, 8 minus 5, by first multiplying by 8, then multiply-
ing by 5, and subtracting the differences obtained, and because
agreeably to the principle of no exception we must say that the mul-
tiplication must be performed according to exactly the same rule
with a s3^mbol which has the form of a difference whose minuend is
less by one than its subtrahent.

As from addition two inverse operations, detraction and subter-
traction, spring, so also from multiplication two inverse operations
must proceed which differ from each other simply in the respect that
in the one the multiplicand is sought and in the other the multiplier.
As matters of computation, these two inverse operations coalesce
in a single operation, namely, division, owing to the validity of the
commutative law in multiplication. But in so far as they are differ-
ent ideas, they must be distinguished. As most civilised languages
distinguish the two inverse processes of multiplication in the case
in which the multiplicand is a line, we will adopt for arithmetic a
name which is used in this exception. Let us take this example,

4 yards X 3 = 1 2 yards.
If twelve yards and four yards are given, and the multiplier 3 is
sought, I ask, how many summands, each equal to four yards, give
twelve 3^ards, or, what is the same thing, how. often I can lay a
length of four yards on a length of twelve yards? But this is measur-
ing. Secondly, if twelve yards and the number 3 are given, and the
multiplicand four yards is sought, I ask what summand it is which
taken three times gives twelve yards, or, what is the same thing,
what part I shall obtain if I cut up twelve yards into three equal
parts? But this is partition, or parting. If, therefore, the multi-

MONISM IN ARITHMETIC. 17

plier is sought we call the division measuring, and if the multipli-
cand is sought, we call it pariing. In both cases the number which
was originally the product is called the dividend, and the result the
quotient. The number which originally was multiplicand is called
the measure ; the number which originally was multiplier is called
the parter. The common name for measure and parter is divisor.
The common symbol for both kinds of division is a colon, a hori-
zontal stroke, or a combination of both. Its definitional formula

reads,

a ,
(a-^ l>) . 0 = a, or, -- . 0=1 a.

Accordingly, dividing a by b means, to find the number which mul-
tiplied by b gives a, or to find the number 7vith which /' must be
multipHed to produce a. From this formula, together with the
formulae relative to multiplication, the well-known rules of division
^re derived, which I here pass over.

In the dividend of a quotient only such numbers can have a
place which are the product of the divisor with some previously de-
fined number. For example, if the divisor is 5 the dividend can
only be 5, 10, 15, and so forth, and o, — 5, — 10 and so forth. Ac-
cordingly, a stroke of division having underneath it 5 and above it
a number different from the numbers just named is a combination
of symbols having no meaning. For example, f or -U- are meaning-
less symbols. Now, conformably to the principle of no exception
we must invest such symbols which have the form of a quotient
without their dividend being the product of the divisor with any
number yet defined, with a meaning such that we shall be able to
reckon with such apparent quotients as with ordinary quotients.
This is done by our agreeing always to put the product of such a
quotient form with its divisor equal to its dividend. In this way we
reach the definition of broken numbers ox fractions, which by the
application of the principle of no exception spring from division ex-
actly as zero and negative numbers sprang from subtraction. The
latter had their origin in the impossibility of the subtraction ; the
former have their origin in the impossibility of the division. Putting

1 8 MONISM IN ARITHMETIC.

together now both these extensions of the domain of numbers, we
arrive at negative fractional numbers.

We pass over the easily deduced rules of computation for frac-
tions and shall only direct the reader's attention to tne connexion
which exists between fractional and non-fractional or, as we usually
say, whole numbers. Since the number 12 lies between the num-
bers 10 and 15, or, what is the same thing, 10 < 12 < 15, and. since
10:5 = 2, 15:5 = 3, we say also that 12:5 lies between 2 and 3, or

that

2<-V-<3-

In itself, the notion of ''less than " has significance only for results
of counting. Consequently, it must first be stated what is meant
when it is said that 2 is less than ^^-. Plainly, nothing is meant by
this except that 2 times 5 is less than 12. We thus see that every
broken number can be so interpolated between two whole numbers
differing from each other only by i that the one shall be smaller
and the other greater, where smaller and greater have the meaning
above given.

From the above definitions and the laws of commutation and
association all possible rules of computation follow, which in virtue
ot tne principle of no exception now hold indiscriminately for all
numbers hitherto defined. It is a consequence of these rules, again,
that the combination of two such numbers by means of any of the
operations defined must in every case lead to a number which has
been already defined, that is, to a positive or negative whole or frac-
tional number, or to zero. The sole exception is the case where
such a number is to be divided by zero. If the dividend also is
zero, that is, if we have the combination ^, the expression is one
which stands for any number whatsoever, because any number what-
soever, no matter what it is, if multiplied by zero gives zero. But
if the dividend is not zero but some other number a, be it what it
will, we get a quotient form to which no number hitherto defined
can be equated. But we discover that if we apply the ordinary arith-
metical rule5 to ^ -^ 0 all such forms may be equated to one another
both .vhcn a is positive and also when a is negative. We may there-
fore invent two new signs for such quotient forms, namely + co and

MONISM IN ARITHMETIC. I9

— 00. We find, further, that in transferring the notions greater and
less to these symbols, -j- oo is greater than any positive number,
however great, and — oo is smaller than any negative number, how-
ever small. We read these new signs, accordingly, "plus infinitely
great" and ''minus infinitely great."

But even here arithmetic has not reached its completion, al-
though the combination of as many previously defined numbers as
we please by as many previously defined operations as we please
will still lead necessarily to some previously defined number. Every
science must make every possible advance, and still one step in ad-
vance is possible in arithmetic. For in virtue of the laws of com-
mutation and association, which also fortunately obtain in multipli-
cation, just as we advance from addition to multiplication, so here
again we may ascend from multiplication to an operation of the third
degree. For, just as for <z -f ^ + ^ + '^'we read 4.«, so with the same
reason we may introduce some more abbreviated designation for
a. a- a. a. The introduction of this new operation is in itself simply
a matter of convenience and not an extension of the ideas of arith-
metic. But if after having introduced this operation we repeatedly
apply the monistic principle of arithmetic, the principle of no ex-
ception, we reach new means of computation which have led to un«
dreamt of advances not only in the hands of mathematicians but
also in the hands of natural scientists. The abbreviated designation
mentioned, which, fructified by the principle of no exception, can
render science such incalculable services, is simply that of writing
for a product of h factors of which each is called a, a^ , which we
read a to the //'' power. Here a new direct operation, that of invo-
lution, is defined, and from now on we are justified in distinguishing
operations which are not inverses of others, as addition, multiplica-
tion, and involution, by nii7nbers of degree. Addition is the direct
operation of the first degree, multiplication that of the second de-
gree, and involution that of the third degree. In the expression
a'' the passive number a is called the case, the active number b the
exponent, the result, the power.

But whilst in the direct operations of the first and second de-
gree, the laws of commutation and asscciation hold, here in involu

20 MONISM IN ARITHMETIC.

tion, the operation of third degree, the two laws are inappHcable,
and the result of their inapplicability is that operations of a still
higher degree than the third form no possible advancement of pure
arithmetic. The product of b factors a is not equal to the product
of a factors h\ that is, the law of commutation does not hold. The
only two different integers for which a to the b"' power is equal to b
to the a^'' power are 2 and 4, for 2 to the 4"' power is 16, and 4 to
the second power also is 16. So, too, the law of association as a
general rule does not hold. For it is hardly the same thing whether
we take the (Z^")''^' power of a or the c"' power of a''.

From the definition of involution follow the usual rules for reck-
oning with powers, of which we shall only mention one, namely,
that the {b — r)'^' power of a is equal to the result of the division of
a to the ¥'' power by a to the c"' power. If we put here c equal to
b, we are obliged, by the principle of no exception, to put a to the
0''' power equal to i; a new result not contained in the original no-
tion of involution, for that implied necessarily that the exponent
should be a result of counting. Again, if we make b smaller than c
we obtain a negative exponent, which we should not know how to
dispose of if we did not follow our monistic law of arithmetic. Ac-
cording to the latter, a to the {b — cy'' power must still remain equal
to a^ divided by a!' even when b is smaller than e. Whence follows
that a to the minus d'^' power is equal to i divided by a to the
d"'^ power, or to take specific numbers, that 3 to the minus 2'"^ power
is equal to ^.

At this point, perhaps, the reader will inquire what a raised to a
fractional power is. But this can be explained only when we have
discussed the inverse processes of involution, to which we now pass.

If a'' = c, we may ask two questions: first, what the base is
which raised to the b"' power gives e; the second, what the exponent
of the power is to which a must be raised to produce e. In the first
case we seek the base, and term the operation which yields this re-
sult; evolution; in the second case we seek the exponent and call the
operation which yields this exponent, the finding of the logarithm.
In the first case, we write ^/ e = a (which we read, the b"' root of c is
equal to «•), and call c the radieand, b the exponent of the root, and a

MONISM IN ARITHMETIC. 21

the 7-oot. In the second case, we write log^<r = ^ (which we read, the
logarithm of c to the base a is equal to I)), and call c the logaritJimand
or number, a the base of the logarithm, and b the logarithm.

While, owing to the validity of the law of commutation in addi-
tion and multiplication, the two inverse processes of those opera-
tions are identical so far as computation is concerned, here in the
case of involution the two inverse operations are in this regard es-
sentially different, for in this case the law of commutation does not
hold.

From the definitional formulae for evolution and the finding of
logarithms, namely,

(l*/7)* =-- c, and {a) '°k«^ = c,
follow, by the application of the laws of involution, the rules for
computation with roots and logarithms. These rules we pass over
here, only remarking, first> that for the present V c has meaning
only when c is the b""- power of some number already defined ; and,
secondly, that for the present also log^rhas meaning only when c
can be produced by raising the number a to some power which is a
number already defined. In the phrase "has only meaning for the
present" is contained a possibility of new extensions of the domain
of number. But before we pass to those extensions we shall first
make use of the idea of evolution just defined to extend the notion
of power also to cases in which the exponent is a fractional number.

According to the original definition of involution, a'' was mean-
mgless except where b was a result of counting. But afterwards,
even powers which had for their exponents zero or a negative integer
could be invested with meaning. Now we have to consider the
arithmetical combination "« raised to the fractional power—." The
principle of no exception compels us to give to the arithmetical com-
bination " tz to the --""■ power " a significance such that all the rules
of computation will hold with respect to it. Now, one rule that
holds is, that the m"' power of the ;/'''' power of a is equal to the
{jnY.li)''' power of a. Consequently, the q"' power of a raised to the
--"'■ power must be equal to a raised to a power whose exponent i*
equal to ^ times ./. But the last-mentioned product gives, according
to the definition cf division, the number p. Consequently the sym

22 MONISM IN ARITHMETIC.

bol a to the — '''' power is so constituted that its q'^ power is equal to
a to the/'''' power; i. e., it is equal to the q"' root of a^. Similarly,
we find that the symbol ^'ato the minus — '^' power" must be put
equal to i divided by the q'^ root of a to the p"' power, if we are to
reckon with this symbol as we do with real powers. Again, just as
a to the b"' power is invested with meaning when d^ is a fractional
number, so some meaning harmonious with the principle of no ex-
ception must be imparted to the b'^' root of c where h is a positive or
negative fractional number. For example, the three-fourths'^' root
of 8 is equal to 8 to the A power, that is, to the cube root of 8 to the
4"' power, or 16.

The principle underlying arithmetic now also compels us to
give to the symbol the ^'b"' root of <r " a meaning when c is not the
b"' power of any number yet defined. First, let c be any positive
integer or fraction. Then always to be able to reckon with the
b"' root of c in the same way that we do with extractible roots, we
must agree always to put the b"' power of the b"' root of c equal to
c — for example, (1/3)'^ always exactly equal to 3. A careful inspec-
tion of the new symbols, which we will also call numbers, shows, that
though no one of them is exactly equal to a number hitherto defined,
yet by a certain extension of the notions greater and less, two num-
bers of the character of numbers already defined may be found for
each such new number, such that the new number is greater than the
one and less than the other of the two, and that further, these two
numbers may be made to differ from each other by as small a quan-
tity as we please. For example,

/'7^3 343 993 ^^^^9 27 C3\3

The number 3, as we see, is here included between two limits which
are the third powers of two numbers | and | whose difference is -^0-
We could also have arranged it so that the difference should be
equal to yi^^, or to any specified number, however small. Now, in-
stead of putting the symbol " less than " between (|)^ and 3, and
between 3 and (|)^, let us put it between their third roots ; for ex-
ample^ let us say :

^ < ^3"< f , meaning by this that (|)3 < 3 < (|)3.
In this sense we may say that the new numbers always lie between

MONISM IN ARITHMETIC. 23

f.v/o old numbers whose difference may be made as small as we
please. Numbers possessing this property are called irraiiotial num-
bers, in contradistinction to the numbers hitherto defined, which are
termed rational. The considerations which before led us to negative
rational numbers, now also lead us to negative irrational numbers.
The repeated application of addition and multiplication as of their
inverse processes to irrational numbers, (numbers which though not
exactly equal to previously defined rational numbers may yet be
brought as near to them as we please,) again simply leads to num-
bers of the same class.

A totally new domain of numbers is reached, however, when we
attempt to impart meaning to the square roots of negative numbers.
The square root of minus 9 is neither equal to plus 3 nor to minus
3, since each multiplied by itself gives plus g, nor is it equal to any
other number hitherto defined. Accordingly, the square root of minus
9 is a new number-form, to which, harmoniously with the principle
of no exception, we may give the definition that (i^ — g)^ shall al-
ways be put equal to minus g.* Keeping to this definition we see
at once that V — a, where a is any positive rational or irrational
number, is a symbol which can be put equal to the product of V -\- a
by v' — I. In extending to these new numbers the rights of arith-
metical citizenship, in calling them also ''numbers," and so shaping
their definition that we can reckon with them by the same rules as
with already defined numbers, we obtain a fourth extension of the
domain of numbers which has become of the greatest importance
for the progress of all branches of mathematics. The newly defined
numbers are called imaginary, in contradistinction to all heretofore
defined, which are called real. Since all imaginary numbers can be
represented as products of real numbers with the square root of
minus one, it is convenient to introduce for this one imaginary num-
ber some concise symbol. This symbol is the first letter of the word
imaginary, namely, /; so that we can always put for such an ex-
pression as 1/ — 9, 3 . /'.

If we combine real and imaginary numbers by operations of the

* Henceforward we shall use the simpler sign ^ ior f .

24 MONISM IN ARITHMETIC.

fiist and second degree, always supposing that we follow in our
reckoning with imaginary numbers the same rules that we do in
reckoning with real numbers, we always arrive again at real or
imaginary numbers, excepting when we join together a real and an
imaginary number by addition or its inverse operations. In this
case we reach the symbol a -\- i . b, where a and b stand for real num-
bers. Agreeably to the principle of no exception we are permitted
to reckon with a -\- ib according to the same rules of computation as
with symbols previously defined, if for the second power of / we
always substitute minus i.

In the numerical combination a -{- ib, which we also call num-
ber, we have found the most general numerical form to which the
laws of arithmetic can lead, even though we wished to extend the
limits of arithmetic still further. Of course, we must represent to
ourselves here by a and b either zero or positive or negative rational
or irrational numbers. If b is zero, a -\- ib represents all real num-
bers ; if d! is zero, it stands for all purely imaginary numbers. This
general number a -)- ib is called a complex number, so that the com-
plex number includes in itself as special cases all numbers hereto-
fore defined. By the introduction of irrational, purely imaginär}^
and the still more general complex numbers, all combinations be-
come invested with meaning which the operations of the third de-
gree can produce. For example, the fifth root of 5 is an- irrational
number, the logarithm of 2 to the base 10 is an irrational number.
The logarithm of minus i to the base 2 is a purely imaginary num-
ber ; the fourth root of minus i is a complex number. Indeed, we
may recognise, proceeding still further, that every co7nbination of two
complex numbers, by meafis of any of the operations of the first, second,
or third degree will lead in turn to a complex 7iumbcr, that is to say,
never furnishes occasion, by application of the principle of no ex-
ception, for inventing new forms of numbers.

A certain limit is thus reached in the construction of arithmetic.
But such a limit was also twice previously reached. After the in-
vestigation of addition and its inverse operations, we reached no
other numbers except zero and positive and negative whole num-
bers, and every combination of such numbers by operations of the

MONISM IN ARITHMETIC. 2^

first degree led to no new numbers. After the investigation of mul-
tiplication and its inverse operations, the positive or negative frac-
tional numbers and "infinitely great" were added, and again we
could say that the combination of two already defined numbers
by operations of the first and second degree in turn also always
led to numbers already defined. Now we have reached a point at
which we can say that the combination of two complex numbers by
all operations of the first, second, and third degree must again
always lead to complex numbers ; only that now such a combina-
tion does not necessarily always lead to a single number, but may
lead to many regularly arranged numbers. For example, the com-
bination "logarithm of minus one to a positive base" furnishes a
countless number of results which form an arithmetical series of
purely imaginary numbers. S/i7/, in no case now do we arrive at new
classes of numbers. But just as before the ascent from multiplication
to involution brought in its train the definition of new numbers, so
it is also possible that some new operation springing out of involution
as involution sprang from multiplication /night furnish the germ of other
new numbers which are not reducible to a -\- i b. As a matter of fact,
mathematicians have asked themselves this question and investi-
gated the direct operation of the fourth degree, together with its
inverse processes. The result of their investigations was, that an
operation which springs from involution as involution sprang from
multiplication is incapable of performing any real mathematical ser-
vice ; the reason of which is, that in involution the laws of commu-
tation and association do not hold. It also further appeared that
the operations of the fourth degree could not give rise to new num-
bers. No more so can operations of still higher degrees. With
respect to quaternions, which many might be disposed to regard as
new numbers, it will be evident that though quaternions are valu-
able means of investigation in geometry and mechanics they are not
numbers of arithmetic, because the rules of arithmetic are not un-
conditionalh' applicable to them.

The building up of arithmetic is thus completed. The exten-
sions of the domain of number are ended. It only remains to be
asked why the science of arithmetic appears in its structure so logi-

20 MONISM IN ARITHMETIC.

cal, natural, and unarbitrary; why zero, negative, and fractional
numbers appear as much derived and as little original as irrational,
imaginary, and complex numbers? We answer, wholly and alone in
virtue of the logical application of the monistic principle of arith-
metic, the principle of no exception.

ON THE NATURE OF MATHEMATICAL
KNOWLEDGE.

"1\ /f ATHEMATICALLY certain and unequivocal" is a phrase
-i-^-^ which is often heard in the sciences and in common life,
to express the idea that the seal of truth is more deeply imprinted
upon a proposition than is the case with ordinary acts of knowledge.
We propose to investigate in this article the extent to which math-
ematical knowledge really is more certain and unequivocal than
other knowledge.

The intrinsic character of mathematical research and know-
ledge is based essentially on three properties : first, on its conserv-
ative attitude towards the old truths and discoveries of mathematics ;
secondl}^, on its progressive mode of development, due to the inces-
sant acquisition of new kijowledge on the basis of the old ; and thirdly,
on its self-sufficiency and its consequent absolute independence.

That mathematics is the most conservative of all the sciences
is apparent from the incontestability of its propositions. This last
character bestows on mathematics the enviable superiority that no
new development can undo the work of previous developments or
substitute new in the place of old results. The discoveries that
Pythagoras, Archimedes, and Apollonius made are as valid to-day
as they were two thousand years ago. This is a trait which no
other science possesses. The notions of previous centuries regard-
ing the nature of heat have been disproved. Goethe's theory of
colors is now antiquated. The theory of the binary combination of
salts was supplanted by the theory of substitution, and this, in its
turn, has also given wa}' to new^er conceptions. Think of the pro-

28 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.

found changes which the conceptions of theoretical medicine, zool-
ogy, botany, mineralogy, and geology have undergone. It is the
same, too, in the other sciences. In philology, comparative linguis-
tics, and history our ideas are quite different from what they formerly
were.

In no other science is it so indispensable a condition that what-
ever is asserted must be true, as it is in mathematics. Whenever,
therefore, a controversy arises in mathematics, the issue is not
whether a thing is true or not, but whether the proof might not be
conducted more simply in some other way, or whether the proposi-
tion demonstrated is sufficiently important for the advancement of the
science as to deserve especial enunciation and emphasis, or finally,
whether the proposition is not a special case of some other and
more general truth which is just as easily discovered.

Let me recall the controversy which has been waged in this
century regarding the eleventh axiom of Euclid, that only one line
can be drawn through a point parallel to another straight line. This
discussion impugned in no wise the truth of the proposition ; for
that things are true in mathematics is so much a matter of course
that on this point it is impossible for a controversy to arise. The
discussion merely touched the question whether the axiom was
capable of demonstration solely by means pf the other propositions,
or whether it was not a special property, apprehensible only by
sense-experience, of that space of three dimensions in which the
organic world has been produced and which therefore is of all others
alone within the reach of our powers of representation. The truth
of the last supposition affects in no respect the correctness of the
axiom but simply assigns to it, in an epistemological regard, a dif-
ferent status from what it would have if it were demonstrable, as
was at one time thought^ without the aid of the senses, and solely by
the other propositions of mathematics.

I may recall also a second controversy which arose a few de-
cades ago as to whether all continuous functions were differentiable.
In the outcome, continuous functions were defined that possessed
no differential coefficient, and it was thus learned that certain truths
which were enunciated unconditionally by Newton, Leibnitz, and

ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 29

their mathematical successors, required qualification. But this did
not invalidate at all the correctness of the method of differentiation,
nor its application in all practical cases; the theoretical specula-
tions pursued on this subject simply clarified ideas and sifted out
the conditions upon which differentiability depended. Happily the
gifted minds who invent the new methods and open up the new
paths of research in mathematics, are not deterred by the fear that
a subsequent generation gifted with unusual acumen will spy out
isolated cases in whicli their methods fail. Happily the creators
of the differential calculus pushed onward without a thought that a
critical posterity would discover exceptions to their results. In
every great advance that mathematics makes, the clarification and
scrutinisation of the results reached are reserved necessaril\' for a
subsequent period, but with it the demonstration of those results is
more rigorously established. Despite all this, however, in no sci-
ence does cognition bear so unmistakably the imprint of truth as in
pure mathematics. And this fact bestows on mathematics its con-
servative character.

This conservative character again is displayed in the objects of
mathematical research. The physician, the historian, the geogra-
pher, and the philologist have to-day quite different fields of inves-
tigation from what they had centuries ago. In mathematics, too,
every new age gives birth to new problems, arising partly from the
advance of the science itself, and partly also from the advance of
civilisation, where improvements in the other sciences bring in their
train new problems that are constantly taxing afresh the resources
of mathematics. But despite all this, in mathematics more than in
any other science problems exist that have played a role for hun-
dreds, nay, for thousands of years.

In the oldest mathematical manuscript which we possess, the
Rhind Papyrus of the British Museum, which dates back to the
eighteenth century before Christ, and whose decipherment we owe
to the industry of Eisenlohr, we find an attempt to solve the prob-
lem of converting a circle into a square of equal area, a problem
whose history covers a period of three and a half thousand years.
For it was not until 1882 that a rigorous proof was given of the im-

30 ON THE NATURE OF MATHEMATICAL KNOWLEDGE-

possibility of solving this problem exactly by the use of straight
edge and compasses alone. (See pp, ii6, 141-143.)

It is, of course, the insoluble problems that have the longest
history; partly because it is harder to show that a thing is impos-
sible than that it is possible, and, on the other hand, because prob-
lems that have long defied solution are ever evoking anew the spirit
of inquiry and the ambition of mathematicians, and because the un-
certainty of insolubility lends to such problems a peculiar charm.
Of the geometrical problems that have occupied competent and in-
competent minds from the time of the ancient Greeks to the present
may be mentioned in addition to the squaring of the circle two
others that are also perhaps well-known to educated readers, at least
by name : the trisection of the angle and the Delic problem of the
duplication of the cube. All three problems involve the condition,
which is often overlooked by lay readers, that only straight edge
and compasses shall be employed in the constructions. In the tri-
section of the angle any angle is assigned, and it is required to find
the two straight lines which divide the angle into three equal parts.
In the Delic problem the edge of a cube is given and the edge of a
second cube is sought, containing twice the volume of the first cube.
In Greece, in the golden age of the sciences, when all scholars had
to understand mathematics, it was a fashionable requisite almost to
have employed oneself on these famous problems.

Fortunately for us, these problems were insoluble. For in their
ambition to conquer them it came to pass that men busied themselves
more and more with geometry, and in this way kept constantly dis-
covering new truths and developing new theories, all of which per-
haps might never have been done if the problems had been soluble
and had early received their solutions. Thus is the struggle after
truth often more fruitful than the actual discovery of truth. So, too,
although in a slightly different sense, the apophthegm of Lessing is
confirmed here, that the search for truth is to be preferred to its
possession.

Whilst the three above-named problems are now acknowledged
to be insoluble, and have ceased, therefore, to stimulate mathemat-
ical inquiry, there are of course other problems in mathematics

ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 31

whose solution has been sought for a long time, but not yet reached,
and in the case of which there is no reason for supposing that they
are insoluble. Of such problems the two following perhaps have
found their way out of the isolated circles of mathematicians and
have become more or less known to other scholars. I refer to the
astronomical Problem of Three Bodies and to the problem of the
frequency of prime numbers. The first of these two problems as-
sumes three or more heavenly bodies whose movements are mutually
influenced by one another according to Newton's law of gravitation,
and requires the exact determination of the path which each body
describes. The second problem requires the construction of a for-
mula which shall tell how many prime numbers there are below a
certain given number. So far these two problems have been solved
only approximately, and not with absolute mathematical exactness.
If the eternal and inviolable correctness of its truths lends to
mathematical researci, and therefore also to mathematical knowl-
edge, a conservative Cj aracter, on the other hand, by the continuous
outgrowth of new tru;hs and methods from the old, progressivcness
is also one of its chare cteristics. In marvellous profusion old knowl-
edge is augmented by new, which has the old as its necessary con-
dition, and, therefore, could not have arisen had not the old pre-
ceded it. The indestructibility of the edifice of mathematics renders
it possible that the work can be carried to ever loftier and loftier
heights without fear that the highest stories shall be less solid and
safe than the foundations, which are the axioms, or the lower sto-
ries, which are the elementar}' propositions. But it is necessary for
this that all the stones should be properly fitted together; audit
would be idle labor to attempt to lay a stone that belonged above in
a place below. A good example of a stone of this character belong-
ing in what is now the uppermost layer of the edifice, is Linde-
mann's demonstration of the insolubility of the quadrature of the
circle, a demonstration of which interesting simplifications have
been given by several mathematicians, including Weierstrass and
Felix Klein. Lmdemann's demonstration could not have been pro-
duced in the preceding century, because it rests necessarily on theo-
ries whose development falls in the present century. It is true,

32 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.

Lambert succeeded in 1761 in demonstrating the irrationality of the
ratio of the circumference of a circle to its diameter, or, which is
the same thing, the irrationality of the ratio of the area of a circle
to the area of the square on its radius. Afterwards, Lambert also
supplied a proof that it was impossible for this ratio to be the square
root of a rational number. But this was the first step only in a long
journey. The attempt to prove that the old problem is insoluble
was still destined to fail. An astounding mass of mathematical in-
vestigations were necessary before the demonstration could be suc-
cessfull3' accomplished.

As we see, the majority of the mathematical truths now pos-
sessed by us presuppose the intellectual toil of many centuries. A
mathematician, therefore, who wishes to-day to acquire a thorough
understanding of modern research in this department, must think
over again in quickened tempo the mathematical labors of several
centuries. This constant dependence of new results on old ones
stamps mathematics as a science of uncommon exclusiveness and
renders it generally impossible to lay open to vminitiated readers a
speedy path to the apprehension of the higher mathematical truths.
For this reason, too, the theories and results of mathematics are
rarely adapted for popular presentation. There is no royal road to
the knowledge of mathematics, as Euclid once said to the first
Egyptian Ptolemy. This same inaccessibility of mathematics, al-
though it secures for it a lofty and aristocratic place among the sci-
ences, also renders it odious to those who have never learned it, and
who dread the great labor involved in acquiring an understanding
of the questions of modern mathematics. Neither in the languages
nor in the natural sciences are the investigations and results so
closely interdependent as to make it impossible to acquaint the un-
initiated student with single branches or with particular results of
these sciences, without causing him to go through a long course of
preliminary study.

The third trait which distinguishes mathematical research is its
self-sufficiency. In philology the field of inquiry is the organic one
of languages, and philology, therefore, is dependent in its investiga-
tions on the mode of development of languages, which is more or

ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 33

less accidental. Its task is connected with something which is given
to it from without and which it cannot alter. It is much the same
with the science of history, which must contemplate the history of
mankind as it has actually occurred. Also zoölog}', botany, mineral-
og}', geology, and chemistry work with given data. In order not
to become involved in futile speculations the last-mentioned sci-
ences are constantly and inevitably obliged to rever-t to observa-
tions by the senses. It is then their task to link together these in-
dividual observations b}' bonds of causality and in this way to
erect from the single stones an edifice, the view of which will render
it easier for limited human intelligence to comprehend nature.
Physics of all sciences stands nearest to mathematics in this respect,
because unlike the other sciences she is generally in need of only a
few observations in order to proceed deductively. But ph3'sics,
too, must resort to observations of nature, and could not do without
them for any length of time.

Mathematics alone, after certain premises have been perma-
nently established, is able to pursue its course of development in-
dependently and unmindful of things outside of it. It can leave
entirely unnoticed, questions and influences emanating from the
outer world, and continue nevertheless its development.

As regards geometry, the first beginnings of this science did
indeed take their origin in the requirements of practical life. But
it was not long before it freed itself from the restrictions of the prac-
tical art to which it owed its birth. Herodotus recounts that geom-
etry had its origin in Egypt where the inundations of the Nile ob-
literated the boundaries of the riparian estates, and by giving rise
to frequent disputes constantly compelled the inhabitants to compare
the areas of fields of different shapes. But with the early Greek
mathematicians, who were the heirs of the Egyptian art of measure-
ment, geometry appeared as a science which men pursued for its
own sake without a thought of how their intellectual discoveries
could be turned to practical account.

Nevertheless, although the workers in the domain of pure math-
ematics are not stimulated by the thought that their researches are
likely to be of practical value, yet that result is still frequently real-

34 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.

ised, often after the lapse of centuries. The history of mathe-
matics shows numerous instances of mathematical results which
were originally the outcome of a mere desire to extend the science,
suddenly receiving in astronomy, mechanics, or in physics practical
applications which their originators could scarce have dreamt of.
Thus Apollonius erected in ancient times the stately edifice of the
properties of conic sections, without having any idea that the plan-
ets moved about the sun in conic sections, and that a Kepler and a
Newton were one day to come who should apply these properties to
explaining and calculating the motions of the planets about the
sun. The question of the practical availability of its results in other
fields has at no period exercised more than a subordinate influence
on mathematical inquiry. Particularly is this true of j/iodern math-
ematical research, whether the same consist in the extended de-
velopment of isolated theories or in uniting under a higher point of
view theories heretofore regarded as different.*

This independence of its character has rendered the results of
pure mathematics independent also of the accidental direction which
the development of civilisation has taken on our planet ; so that
the remark is not altogether without justification, that if beings en-
dowed with intelligence existed on other planets, the truths of math-
ematics would afford the only basis of an understanding with them.
Uninterruptedly and wholly from its own resources mathematics has
built itself up. It is scarcely credible to a person not versed in the
science, that mathematicians can derive satisfaction from the com-
fortless and wearisome operation of heaping up demonstration on
demonstration, of rivetting truth on truth, and of tormenting them-
selves with self-imposed problems, whose solution stands no one in
stead, and affords satisfaction to no one but the solver himself. Yet
this self-sufficiency of mathematicians becomes a little more intel-
ligible when we reflect that the progress which has been made, par-
ticularly in the last few decades, and which is uninfluenced from
without, does not consist solely in the accumulation of new truths

*Cf. Felix Klein, "Remarks Given at the Opening of the Mathematical and
Astronomical Congress at Chicago." Tlie Monist (Vol. IV. No. i, October, 1893).

ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 35

and in the enunciation of new problems, nor solely in deductions
and solutions, but culminates rather in the discovery of new meth-
ods and points of view in which the old disconnected and isolated
results appear suddenly in a new connexion or as different interpre-
tations of a common fundamental truth, or finally, as a single or-
ganic whole.

Thus, for example, the idea of representing imaginary and com-
plex numbers in a plane, two apparently different branches, the theory
of dividing the circumference of a circle into any given number of
equal parts, and the theory of the solutions of the equation .y" = i,
have been made to exhibit an extremely simple connexion with one
another which enables us to express many a truth of algebra in a
corresponding truth of geometry and vice versa. Another example is
afforded by the discovery which we chiefly owe to Alfred Clebsch,
of the relation which subsists between the higher theory of func-
tions and the theory of algebraic curves, a relation which led to the
discovery of the condition under which two curves can be co-ordi-
nated to each other, point for point, and hence also adequately rep-
resented on each other. Of course such combinations and exten-
sions of view possess a much greater charm for the mathematician
than the mere accumulation of truths and solutions, whose fascina-
tion consists entirely in their truth or correcrness.

From these three cardinal characteristics, now, which distin-
guish mathematical research from research in other fields, we may
gather at once the three positive characteristics that distinguish
mathematical knowledge from other knowledge. They may be briefly
expressed as follows; first, mathematical knowledge bears more
distinctly the imprint of truth on all its results than any other
kind of knowledge ; secondly, it is alwa3's a sure preliminary step to
the attainment of other correct knowledge ; thirdly, it has no need
of other knowledge. Naturally, however, there are associated with
these characteristics which place mathematical knowledge high
above all other knowledge, other characteristics which somewhat
counterbalance the great superiority which mathematics thus ap-
pears to have over the other sciences. In order to show more dis-
tmctly the nature of these characteristics, which we prefer to call

36 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.

negative, we shall select and confine our remarks to a branch which
is commonly taken to be synonymous with mathematics, namc.j, to
arithmetic in the broadest sense of the word.

The subject of inquiry in arithmetic is numbers and their com-
binations. On this account arithmetic is, of all sciences, most free
from what lies outside its boundaries. Perception by the senses is
necessary only in an extremely insignificant measure for the under-
standing of its definitions and premises. It is possible to acquaint
a person who lacks both sight and hearing with the fundamental
principles of arithmetic solely by the medium of "time." Such a
person needs only the sense of feeling. By slight excitations of his
skin, induced at equal or unequal intervals of time, he can be led to
the notion of differences of time and hence also to the notion of dif-
ferences of number. Uninfluenced by matter and force, independ-
ently, too, of the properties of geometrical magnitudes, arithmetic
could be conducted solely by its own intrinsic potencies to its high-
est goals, drawing deductively truth from truth, without a break.

But what sort of a science should we arrive at by this method
of procedure? Nothing but a gigantic web of self-evident truths.
For, once we admit the first notions and premises to which a man
thus bereft of his senses can be led, we are compelled of necessity
also to admit the derivative results of arithmetic. If the beginnings
of arithmetic appear self-evident, the rest of it, .too, bears this
character. Owing to this deductive character of arithmetic, and to
its exemption from influence from without, this science appears to
one person extremely attractive, while to another it appears ex-
tremely repulsive, according as each is constituted. Be that as it
may, however, a finished and complete science of this character
subserves no purpose in the comprehension of the world, or in the
advancement of civilisation. Hence, an arithmetic which heaps up
theorem on theorem with never a thought of how its results are to
be turned to practical account in the acquisition of knowledge in
other fields, resembles an inquisitive physician, who, taking up his
abode in a desert, should arrive there at momentous results in
bacteriology, but should bear them with him to his grave, without
their ever redounding to the benefit of humanity. The value of

ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 37

arithmetical knowledge lies entirely in its applications. But this
constitutes no reason why many mathematicians, pursuing their
purely deductive bent of mind, should not devote themselves ex-
clusively to pure arithmetical developments and leave it to others
a1 the proper time to turn to the material profit of the world the
capital which they have garnered.

Geometry, on the other hand, must have recourse in a much
higher degree than arithmetic to the outside world for its first notions
and premises. The axioms of geometry are nothing but facts of ex-
perience perceived by our senses. The geometry which Bol3-ai, Lo-
bachevski, Gauss, Riemann, and Helmholtz created and which is
both independent of the eleventh axiom of Euclid and perfectly
free from self-contradictions, has supplied an epistemological dem-
onstration that geometry is a science that rests on the observation
of nature, and therefore in the correct sense of the word, is a natu-
ral science.

Yet what a difference there is, for instance, between geometry
and chemistry! Both derive their constructive materials from sense-
perception. But whilst geometry is compelled to draw only its first
results from observation and is then in a position to move forward
deductively to other results without being under the necessity of
making fresh observations, chemistry on the other hand is still
compelled to make observations and to have recourse to nature.

It follows, therefore, that a given act of geometrical knowledge
and a given act of chemical knowledge are with respect to the cer-
tainty of the truth they contain not qualitatively but only quantita-
tively different. In chemistry the probability of error is greater
than in geometry, because more numerous and more difficult ob-
servations have to be made there than in geometry, where only the
very first premises, which no man with sound senses could ever im-
pugn, rest on observation.

The preceding reflexions deprive mathematical knowledge of
that degree of certainty and incontestability which is commonly
attributed to it when we say a thing is " mathematically certain. "
This certainty is lessened still more as we pass to the semi-math-
ematical sciences, where mechanics has the first claim to our at-

38 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.

tention. All the notions of mechanics, and consequently of all
the other departments of physics, are composed, by multiplication
or division, of three fundamental notions — length, time, and mass.
That is to say, to the notions of geometry resting on length and its
powers, two other fundamental notions, time and mass, are added,
which, joined to that of length, lead to the notions of force, work,
horse-power, atmospheric pressure, etc. The knowledge of me-
chanics, thus, highly certain though it be, is rendered less certain
than that of geometry and a fortiori than that of arithmetic. The
uncertainty of knowledge continues to increase in branches which
are still more remote from mathematics, owing to the increasing com-
plexity of the observational material which must here be put to the
test.

Still, although mathematical knowledge does not lead to abso-
lutely certain results, it yet invests known results with incomparably
greater trustworthiness than does the knowledge of the other sci-
ences. But after all, it remains a useless accumulation of capital
so long as it is not turned to practical account in other sciences,
such as metaphysics, physics, chemistry, biology, political economy,
etc. Hence also arises an obligation on the part of the other sci-
ences, so to shape their problems and investigations that they can
be made susceptible of mathematical treatment. Then will mathe-
matics gladly perform her duty. The moment a science has ad-
vanced far enough to permit of the mathematical formulation of its
problems, mathematics will -not be slow to treat and to solve these
problems. Mathematical knowledge, aristocratic as it may appear
by the greater certainty of its results, will, so far as the advance-
ment of human kind is concerned, never be more than a useless
mass of self-evident truths, unless it constantly places itself in the
service of the other sciences.

THE MAGIC SQUARE.

INTRODUCTORY.

/~\P ALL the philosophies of modern times tliere is none that
^-^ has emphasised so much the importance of form and formal
thought as the monism of T/ie Mofiist. An expression of this phi-
losophy is found in the following passages :

"The order that prevails among the facts of reality is due to the laws of form.
Upon the order of the world depends its cognisability.

" . . . . The laws of form are no less eternal than are matter and energy and
' Verily I say unto you, till heaven and earth pass, one jot or one tittle shall in no
wise pass from the law ! '

"The laws of form and their origin have been a puzzle to all philosophers.
• Ay, there's the rub ! ' The difficulties of Hume's problem of causation, of Kant's
a priori, of Plato's ideas, of Mill's method of deduction, etc., etc., all arise from a
one-sided view of form and the laws of form and formal thought. "

Considering the stupendous results achieved by engineering
and the other applied sciences with the assistance of mathematics,
we must confess that the forms of thought are wonderful indeed,
and it is not at all astonishing that the primitive thinkers of man-
kind when the importance of the laws of formal thought in some
way or another first dawned on their minds, attributed magic powers
to numbers and geometrical figures.

We shall devote the following pages to a brief review of
magic squares, the consideration of which has led many a man to
believe in mysticism. And yet there is no mysticism about thein
tmless we either consider everything mystical, even that twice two
IS four, or join the sceptic in his exclamation that we can truly not

ALBERT DURER S ENGRAVING

MELANCHOLY

GENIUS OF THE INDUSTRIAL SCIENCE OF MECHANICS

THE MAGIC SQUARE. 4I

know whether twice two might not be five in other spheres of thft •
universe.

The author of the short article on "Magic Squares " in the Eng-
lish Cyclopaedia (Vol. Ill, p. 415), presumably Prof. De Morgan,

says :

" Though the question of magic squares be in itself of no use, yet it belongs to
a class of problems which call into action a beneficial species of investigation. With-
out laying down any rules for their construction, we shall content ourselves with
destroying their magic quality, and showing that the non-existence of such squares
would be much more surprising than their existence."

This is the point. There obtains a symphonic harmony in
mathematics which is the more startling the more obvious and self-
evident it appears to him who understands the laws that produce this

symphonic harmony.

* *

On the wood-cut named "Melancholia"* of the famous Nurem-
berg painter, Albrecht Dürer, is found among a number of other

* The term melancholy meant in Dürer's time, as it did also in Shakespeare's
and Milton's, " thought or thoughtfulness." Says Milton in // Penser oso :

" Hail, thou Goddess, sage and holy,
Hail divinest melancholy
Whose saintly visage is too bright
To hit the sense of human sight,
And therefore to our weaker view
O'erlaid with black, staid Wisdom's hue. — I, la.

'^'jought that does not lead to action produces a gloomy state of mind. Thought-
rjlness which cannot find a way out of itself is that melancholy which engenders
weakness, — a truth which is illustrated in Hamlet. Shakespeare still uses the words
thought and melancholy as synonyms, saying :

" The native hue of resolution
Is sicklied o'er with the pale cast of thought."

Dürer's melancholy does not represent the gloominess of thought, but the power
of invention. Soberness and even a certain sadness are considered only as an ele-
ment of this melancholy, but on the whole the genius of thought appears bright,
self-possessed, and strong.

Dürer represents the Science of Mechanical Invention as a winged female figure
musing over some problem. Scattered on the floor around her lie some of the sim-
ple tools used in the sixteenth century. The ladder leaning against the house as-
sists in climbing otherwise inaccessible heights. A scale, an hour-glass, a bell, and
the magic square are hanging on the wall behind her.

At a distance a bat-like creature, being the gloom of melancholy, hovers in the
air like a dark cloud, but the sun rises above the horizon, and at the happy middle
between these two extremes stands the rainbow of serene hope and cheerful confi-
dence.

42

THE MAGIC SQUARE.

emblems, which the reader will notice in our reproduction of the cut,
the subjoined square. This arrangement of the sixteen natural num

I

14

15

4

12

7

6

9

8

II

10

5

13

2

3

16

Fig. I.

bers from i to 16 possesses the remarkable property that the same
sum 34 will always be obtained whether we add together the four
figures of any of the horizontal rows or the four of any vertical row
or the four which lie in either of the two diagonals. Such an ar-
rangement of numbers is termed a magic square, and the square
which we have reproduced above is tlie first magic square which is
met with in the Christian Occident.

Like chess and many of the problems founded on the figure of
the chess-board, the problem of constructing a magic square also
probably traces its origin to Indian soil. From there the problem
found its way among the Arabs, and by them it was brought to the
Roman Orient. Finally, since Albrecht Diirer's time, the scholars of
Western Europe also have occupied themselves with methods for
the construction of squares of this character.

The oldest and the simplest magic square consists of the quad-
ratic arrangement of the nine numbers from i to g in such a man
ner that the sum of each horizontal, vertical, or diagonal row, al-
ways remains the same, namely 15. This square is the adjoined.

276
9 5 I
4 3 8

Fig. 2.

Here, we will find, 15 always comes out whether we add 2 and 7 and
6, or 9 and 5 and i, or 4 and 3 and 8, or 2 and 9 and 4, or 7 and 5
and 3, or 6 and i and 8, or 2 and 5 and 8, or 6 and 5 and 4.

The question naturally presents itself, whether this condition
of the constant equality of the added sum also remains fulfilled when
the numbers are assigned different places. It may be easily shown

THE MAGIC SQUARE.

43

however that 5 necessarily must occupy the middle place, and that
the even numbers must stand in the corners. This being so, there
are but 7 additional arrangements possible, which differ from the
arrangement above given and from one another only in the respect
that the rows at the top, at the left, at the bottom, and at the right,
exchange places with one another and that in addition a mirror be
imagined present with each arrangement. So too from Diirer's
square of 4 times 4 places, by transpositions, a whole set of new
correct squares may be formed. A magic square of the 4 times 4
numbers from i to 16 is formed in the simplest manner as foUo'vs.
We inscribe the numbers from i to 16 in their natural order in the
squares, thus :

I

2

3

4

5

6

7

8

9

10

II

12

13

14

15

16

Fig. 3.

We then leave the numbers in the four corner-squares, viz. i, 4, 13,
16, as well also as the numbers in the four middle-squares, viz. 6,
. 7, 10, II, in their original places ; and in the place of the remaining
eight numbers, we write the complements of the same with respect
to 17 : thus 15 instead of 2, 14 instead of 3, 12 instead of 5, 9 in-
stead of 8, 8 instead of g, 5 instead of 12, 3 instead of 14, and 2 in-
stead of 15. We obtain thus the magic square

I

15

14

4

=34

12

6

7

9

=34

8

10

II

5

=34

13

3

2

16

=34

34

34 34

34

Fi^

!• 4-

from which the same sum 34 always results. It is an interesting
property of this square that any four numbers which form a rectangle
or square about the centre also always give the same sum 34 ; for
example, i, 4, 13, 16, or 6, 7, 10, 11, or 15, 14, 3, 2, or 12, g, 5, 8,

44

THE MAGIC SQUARE.

or 15, 8, 2, 9, or 14, 12^ 3, 5. We may easily convince ourselves
that this square is obtainable from the square of Dürer by inter-
changing with one another the two middle vertical rows.

EARLY METHODS FOR THE CONSTRUCTION OF ODD-NUMBERED

SQUARES.

Since early times rules have also been known for the construc-
tion of magic squares of more than 3 times 3, or 4 times 4 spaces.
In the first place, it is easy to calculate the sum which in the case
of any given number of cells must result from the addition of each
row. We take the determinate number of cells in each side of the
square which we have to fill, multiply that number by itself, add i,
again multiply the number thus obtained by the number of the cells
in each side, and, finally, divide the product by 2. Thus, with 4
times 4 cells or squares, we get : 4 times 4 are 16, 16 and i are 17,
and one half of 17 times 4 is 34. Similarly, with 5 times 5 squares,
we get : 5 times 5 are 25, and i makes 26, and the half of 26 times
5 is 65. Analogously, for 6 times 6 squares the summation iii is
obtained, for 7 times 7 squares 175, for 8 times 8 squares 260, for 9
times 9 squares 36g, for 10 times 10 squares 505, and so on. The
Hindu rule for the construction of magic squares whose roots are
odd, may be enunciated as follows : To start with, write i in the
centre of the topmost row, then write 2 in the lowest space of the

//'

30

39

48

I

10

19

28

=175

38

47

7

9

18

27

29

=175

46

6

8

17

26

35

37

=175

5

14

16

25

34

36

45

=175

13

15

24

33

42

44

4

=175

21

23

32

41

43

3

12

=175

22

3^

40

49

2

II

20

=175

175

'75

175

175

175

175

175

Fig. 5.

vertical column next adjacent to the right, and then so inscribe the
remaining numbers in their natural order in the squares diagonally
upwards towards the right, that on reaching the right-hand margin

THE MAGIC SQUARE.

45

the inscription shall be continued from the left-hand margin in the
row just above, and on reaching the upper margin shall be continued
from the lower margin in the column next adjacent to the right,
noting that whenever we are arrested in our progress by a square
already occupied we are to fill out the square next beneath the one
we have last filled. In this manner, for example, the last preced-
ing square of 7 times 7 cells is formed, in which the reader is re»
quested to follow the numbers in their natural sequence (Fig. 5).

For the next further advancements of the theory of magic
squares and of the methods for their construction we are indebted
to the Byzantian Greek, Moschopulus, who lived in the fourteenth
century ; also, after Albrecht Dürer who lived about the year 1500,
to the celebrated arithmetician Adam Riese, and to the mathemati-
cian Michael Stifel, which two last lived about 1550. In the seven-
teenth century Bachet de Meziriac, and Athanasius Kircher em-
ployed-themselves on magic squares. About 1700, finally, the
French mathematicians De la Hire and Sauveur made considerable
contributions to the theor}'. In recent times mathematicians have
concerned themselves much less about magic squares, as they have
indeed about mathematical recreations generally. But quite recently
the Brunswick mathematician Scheffler has put forth his own and
other's studies on this subject in an elegant form.

6

7

14

5

13

21

4

12

20

27

28

3

II

19

35

2

10

18

26

34

42

I 9

17

25

33

41 49

8

16

24

32

40

48

15

23

31

39

47

22

30

38

46

29

36

37
43

44

45

Fig. 6.

46 THE MAGIC SQUARE.

The best known of the various methods of constructing magic
squares of an odd number of cells is the following. First write the
numbers in diagonal succession as in the preceding diagram (Fig. 6).
After 25 cells of the square of 49 cells which we have to fill out,
have thus been occupied, transfer the six figures found outside each
side of the square, without changing their configuration, into the
empty cells of the side directly opposite. By this method, which
we owe to Bachet de Meziriac, we obtain the following magic square
of the numbers from i to 49 :

4

35
10

41
16

47
22

29
II
42
17

23

5

12
36
18
49
24
6

30

37
19

43
25
7
31
13

20

44
26

I

32
14
38

45
27
2

33
8

39
21

28

3
34

9
40

15
46

Fig. 7

III.

MODERN MODES OF CONSTRUCTION OF ODD-NUMBERED
SQUARES.

The reader will justly ask whether there do not exist other cor-
rect magic squares which are constructed after a different method
from that just given, and whether there do not exist modes of con-
struction which will lead to all the imaginable and possible magic
squares of a definite number of cells. A general mode of construc-
tion of this character was first given for odd-numbered squares by
De la Hire, and recently perfected by Professor Scheffler.

To acquaint ourselves with this general method, let us select
as our example a square of 5. First we form two auxiliary squares.
In the first we write the numbers from i to 5 five times ; and in the
second, five times, the following multiples of five, viz.: o, 5, 10, 15,
20. It is clear now that by adding each of the numbers of the series
from I to 5 to each of the numbers o, 5, 10, 15, 20, we shall get
all the 25 numerals from i to 25. All that additionally remains to
be done therefore, is, so to inscribe the numbers that by the addition

THE MAGIC SQUARE,

47

of the two numbers in any two corresponding cells each combina-
tion shall come out once and only once ; and further that in each
horizontal, vertical, and diagonal row in each auxiliary square each
number shall once appear. Then the required sum of 65 must
necessarily result in every case, because the numbers from i to 5
added together make 15, and the numbers o, 5, 10, 15, 20 make 50.
We effect the required method of inscription by imagining the
numbers i, 2, 3, 4, 5 (or o, 5, 10, 15, 20) arranged in cyclical suc-
cession, that is I immediately following upon 5, and, starting from
any number whatsoever, by skipping each time either none or one
or two or three etc. figures. Cycles are thus obtained of the first, the
second, the third etc. orders ; for example 3 4 5 i 2 is a cycle of the
first order, 241 3 5 is a cycle of the second order, i 5 4 3 2 is a
cycle of the fourth order, etc. The only thing then to be looked out
for in the two auxiliary squares is, that the same "cycle" order be
horizontally preserved in all the rows, that the same also happens for
the vertical rows, but that the cycle order in the horizontal and ver-
tical rows is different. Finally we have only additionally to take
care that to the same numbers of the one auxiliary square not like
numbers but different numbers correspond in the other auxiliary
square, that is lie in similarly situated cells. The following auxiliary
squares are, for example, thus possible :

3

4

5

I

2

0

10

20

5

15

5

I

2

3

4

5

15

0

10

20

2

3

4

5

I

and

10

20

5

15

0

4

5

I

2

3

15

0

10

20

5

I

2

3

4

5

20

5

15

0

10

Fig. 8.

Fig. 9.

Adding in pairs the numbers which occupy similarly situated
ceils, we obtain the following correct magic square :

3

10
12

19
21

14
16

23

5
7

25
2

9
II

18

6

13
20
22
4

17
24

I

8
15

Fig. 10.

48

THE MAGIC SQUARE.

It will be seen that we are able thus to construct a i iy large
number of magic squares of 5 times 5 spaces by varying in every
possible manner the numbers in the two auxiliary squares. Further-
more, the squares thus formed possess the additional peculiarity,
that every 5 numbers which fill out two rows that are parallel to a
diagonal and lie on different sides of the diagonal also give the con-
stant sum of 65. For example: 3 and 7, 11, 20, 24; or 10, 14 and 18,
22, I. Altogether then the sum 65 is produced out of 20 rows or
pairs of rows. On this peculiarity is dependent the fact that if we
imagine an unlimited number of such squares placed by the side of,
above, or beneath an initial one, we shall be able to obtain as many
quadratic cells as we choose, so arranged that the square composed
of any 25 of these cells will form a correct magic square, as the fol-
lowing figure will show :

2

13

24

10

16

2

13

24

10

16

2

9

20

I

12

23

9

20

I

12

23

9

II

22

8

19

5

II

22

8

19

5

II

18

4
6

13

15

21

7

18
25

4
6

13

15

21

3
10

7
14
16

18

25

17
24

3
10

14
16

17
24

25

2

2

2

9

20

I

12

23

9

20

I

12

23

9

II

22

8

19

5

II

22

8

19

5

II

18

4

15

21

7

18

4

15

21

7

18

25

6

17

3
10

14
16

25
2

6

17

24

3
10

14
16

25

2

13

24

13

2

9

20

I
8

12
19

23

5

9

20
22

I
8

12
19

23
5

9

II

22

II

II

Fig. II.

Every square of every 25 of these numbers, as for example the
two dark-bordered ones, possesses the property that the addition of
the horizontal, vertical, and diagonal rows gives each the same
sum, 65.

As an example of a higher number of cells we will append here
a magic square of 11 times 11 spaces formed by the general method
of De la Hire from the two auxiliary squares of Figs. 12 and 13.
From these two auxiliary squares we obtain by the addition of the

THE MAGIC SQUARE.

49

iwo numbers of every two similarly situated cells, the magic
sauare, exhibited in Diagram 14, in which each row gives the same
sum 071.

I

•2

3

4

5

6

7

8

9

10

II

0

II

22

33

44

55

66

77

88

99

no

3

4

5

6

7

8

9

10

II

I

2

33

44

55

66

77

88

99

no

0

II

22

5

6

7

8

9

10

II

I

2

3

4

66

77

88

99

no

0

II

22

33

44

55

7

8

9

10

II

I

2

3

4

5

6

99

no

0

II

22

33

44

55

66

77

88

9

10

II

I

2

3

4

5

6

7

8

II

22

33

44

55

66

77

88

99

no

0

II

I

2

3

4

5

6

7

8

9

10

44

55

66

77

88

99

no

0

II

22

33

2

3

4

5

6

7

8

9

10

II

I

77

88

99

no

0

II

22

33

44

55

66

4

5

6

7

8

9

10

II

I

2

3

no

0

II

22

33

44

55

66

77

88

99

6

7

8

9

10

II

I

2

3

4

5

22

33

44

55

66

77

88

99

no

0

II

8

9

10

II

I

2

3

4

5

6

7

55

66

77

88

99

no

0

II

22

33

44

10

II

I

2

3

4

5

6

7

8

9

88

99

no

0

II

22

33

44

55

66

77

Fig.

Fig. 13.

I

13

25

37

49

61

73

85

97

109

121

36

48

60

72

84

96

108

120

II

12

24

71

83

95

107

119

10

22

23

35

47

59

106

118

9

21

33

34

46

58

70

82

94

20

32

44

45

57

69

81

93

105

117

8

55

56

68

80

92

104

116

7

19

31

43

79

91

103

115

'6

18

30

42

54

66

67

114

5

17

29

41

53

65

77

78

90

102

28

40

52

64

76

88

89

lOI

113

4

16

63

75

87

99

100

112

3

15

27

39

51

98

no

III

2

14

26

38

50

62

74

86

Fig. 14

EVEN-NUMBERED SQUARES.
Of magic squares having an even number of places we have
hitherto had to deal only with the square of 4. To construct squares
of this. description having a higher even number of places, differ-
ent and more complicated methods must be employed than for
squares of odd numbers of places. However, in this case also, as
in dealing with the square of 4, we start with the natural sequence

5°

THE MAGIC SQUARE.

Df the numbers and must then find the complements of the numbers
with respect to some other certain number (as 17 in the square ot
^) and also effect certain exchanges of the numbers with one an-
other. To form, for example, a magic square of 6 times 6 places,
we inscribe in the 12 diagonal cells the numbers that in the natural
sequence of inscription fall into these places, then in the remaining
cells the complements of the numbers that belong therein with re-
spect to 37, and finally effect the following six exchanges, viz. of
the numbers 33 and 3, 25 and 7, 20 and 14, 18 and 13, 10 and 9,
and 5 and 2. In this way the following magic square is obtained.

I

30
24
13

12

31

35

8

23
17

2

34
28

15
21

9

4

3

16
22
10
33

32
11

14
20
29

5

6
7
19
18

25

Fig. 15.

This square may also be constructed by the method of De la
Hire, from two auxiliary squares with the numbers i, 2, 3, 4, 5, 6
and o, 6, 12, 18, 24, 30 respectively. In this case, however, the
vertical rows of the one square and the horizontal rows of the other
must each so contain two numbers three times repeated that the
summation shall always remain 21 and go respectively. In this
manner we get the magic square last given above from the two fol
lowing auxiliary squares :

I

5

4

3

2

6

6

2

4

3

5

I

6

5

3

4

2

I

I

5

3

4

2

6

6

2

3

4

5

I

I

2

4

3

5

6

0

30

30

0

30

0

24

6

24

24

6

6

18

18

12

12

12

18

12

12

18

18

18

12

6

24

6

6

24

24

30

0

0

30

0

30

Fig. 16.

Fig. 17.

It is to be noted in connection with this example that here also
as in the case of odd-numbered squares, it is possible so to inscribe

THE MAGIC SQUARE.

51

six rimes the numbers from i to 6 that each number shall appear once
and only once in each horizontal, vertical, and diagonal row; for
example, in the following manner :

123456
246135
365214
531642
654321
412563

Fig. 18.

But if we attempt so to insert, in a like manner, the other set of
numbers o, 6, 12, 18, 24, 30 in a second auxiliary square, that each
number of the first auxiliary square shall stand once and once only
in a corresponding cell with each number of the second square, all
the attempts we may make to fulfil coincidently the last named con-
dition will result in failure. It is therefore necessary to select aux-
iliary squares like the two given above. It is noteworthy, that the
fulfilment of the second condition is impossible only in the case of
the square of 6, but that in the case of the square of 4 or of the
squaie of 8, for example, two auxiliary squares, such as the method
of De la Hire requires, are possible. Thus, taking the square of 4
we get

I

2

3

4

0

4

8

12

4

3

2

I

and

8

12

0

4

2

I

4

3

12

8

4

0

3

4

I

2

4

0

12

8

Fig. 19.

Fig. 20.

The reader may form for himself the magic square which these
give.

The existence of these two auxiliary squares furnishes a key to
the solution of a pretty problem at cards. If we replace, namely,
the numbers i, 2, 3, 4 by the Ace, the King, the Queen, and the
Knave, and the numbers o, 4, 8, 12 by the four suits, clubs, spades,
hearts, and diamonds, we shall at once perceive that it is possible,
and must be so necessarily, quadratically to arrange in such a man-
ner the four Aces, the four Kings, the Four Queens, and the foui

52

THE MAGIC SQUARE.

Knaves, that in each horizontal, vertical, and diagonal row. each
one of the four suits and each one of the four denominations shall
appear once and once only. The auxiliary squares above given fur-
nish the appended solution of this problem :

CLUBS
ACE

SPADES
KING

HEARTS

QUEEN

DIAMONDS
KNAVE

HEARTS
KNAVE

DIAMONDS

QUEEN

CLUBS
KING

SPADES
ACE

DIAMONDS
KING

HEARTS
ACE

SPADES
KNAVE

CLUBS
QUEEN

SPADES
QUEEN

CLUBS
KNAVE

DIAMONDS
ACE

HEARTS
KING

Fig. 21.

To fix the solution of the problem in the memory, observe that,
starting from the several corners, each suit and each denomination
must be placed in the spots of the move of a Knight. If we fix the
positions of the four cards of any one row, there will be only two
possibilities left of so placing the other cards that the required con-
dition of having each suit and each denomination once and only
once in each row shall be fulfilled.

Of magic squares of an even number of places we have up to
this point examined only the squares of 4 and of 6. For the sake of
completeness we append here one of 8 and one of 10 places. The
mode of construction of these squares is similar to the method above
discussed for the lower even numbers.

I

56
48

25

33

24
16

57

63
10
18
39
31
42
50
7

62
II
19

30
43
51
6

4
53
45

28

21

13
60

5

52
44
29
37
20
12
61

59
14
22

35
27
46

54
3

58
15
23
34
26

47
55
2

8

49
41
32
40

17
9
64

Fig. 22.

THE MAGIC SQUARE.

33

I

go
80

31
60

50
61
21
20
91

99

12

79
69"
42
52
32
29
82
9

3

88

23

68

^
43
'38

73
18

93

97
14
77
34
57
47
64

27

4

96
86

25
66

45
55
36
75
15
6

5

85
26

^
46

56
35
'76
16

95

94
17
74
37
44
54
67

24

87
7

8

28
33
53
48
63
78
13
98

92

19
22
62
49
59

39

~~~
72

8,

2

10
11

71
40

51
41
70

30

81
100

Fig. 23.

The magic squares of even numbers thus constructed are not
the only possible ones. On the contrary, there are very many olhers
possible, which obey different laws of formation. It has been cal-
culated, for example, that with the square of 4 it is possible to con-
struct 880, and with the square of 6, several million, different magic
squares. The number of odd-numbered magic squares constructible
by the method of De la Hire is also very great. With the square of
7, the possible constructions amount to 363,916,800. With the
squares of higher numbers the multitude of the possibilities increases
in the same enormous ratio.

MAGIC SQUARES WHOSE SUMMATION GIVES THE NUMBER
OF A YEAR.

The magic squares which we have so far considered contain
only the natural numbers from i upwards. It is possible, however,
easily to deduce from a correct magic square other squares in which
a different law controls the sequence of the numbers to be inscribed.
Of the squares obtained in this manner, we shall devote our atten-
tion here only to such in which, although formed by the inscription
of successive numbers, the sum obtained from the addition of the
rows is a determinate number which we have fixed upon beforehand,
as the fjumber of a year. In such a case we have simply to add to
the numbers of the original square a determinate number so to be
calculated, that the required sum shall each time appear. If this

54

THE MAGIC SQUARE.

sum is divisible by 3, magic squares will always be obtainable with
3 times 3 spaces which shall give this sum. In such a case we di-
vide the sum required by 3 and subtract 5 from the result in order
to obtain the number which we have to add to each number of the
original square. If the sum desired is even but not divisible by 4,
we must then subtract from it 34 and take one fourth of the result,
to obtain the number which in this case is to be added in each
place. If, for example, we wish to obtain the number of the year
1890 as the resulting sum of each row, we shall have to add to each
of the numbers of an ordinary magic square of 4 times 4 spaces the
number 464 ; in other words, instead of the numbers from i to 16 we
have to insert in the squares the numbers from 465 to 480. As the
number of the year 1852 is divisible by eleven, it must be pos-
sible to deduce from the magic square constructed by us at the con-
clusion of Section III a second magic square in which each row of
"Tj cells will give the number of the year 1892. To do this, we sub
tract from 1892 the sum of the original square, namely 671, and di-
vide the remainder by 11, whereby we get iii and thus perceive
that the numbers from 112 to 232 are to be inscribed ir the cells of

112

124

136

148

160

171

184

rg6

208

220

232

=1

147

159

171

^

195

207

219

231

122

123

135

=1

182

194

206

218

230

121

133

^34

146

^

170

= 1

217

229

120

132

144

145

157

169

181

^

205

r:.

131

143

155

156

168

180

192

204

216

228

119

=1

166

167

179

191

203

215

227

118

130

142

154

=1

rgo

202

214

226

117

129

141

153

165

177

178

=1

225

116

128

140

152

164

176

188

189

201

213

=1

139

151

163

175

187

199

200

212

224

115

127

7:.

174

186

198

210

211

223

114

126

^

150

162

7.\

2og

221

222

113

125

137

149

161

173

^

197

=1

892
892
892
892

8f2

892

892
892

893

892

1892 1892 1892 1892 1892 1892 1892 1892 1892 1892 J892
Fig. 24.

the square required. We get in this way the preceding square, from
which of/e and tJie same sum, namely i8g2, can be obtained 44 times,
first from each of the 1 1 horizontal rows, secondly from each of the
II vertical rows, thirdly from each of the two diagonal rows, and

THE MAGIC SQUARE.

55

fourthly twenty additional times from each and every pair of any two
rows that lie parallel to a diagonal, have together 1 1 cells, and lie
on different sides of the diagonal, as for example, 196, 122, 158, 205,
131, 167, 214, 140, 187, 223, 149.

VI.

CONCENTRIC MAGIC SQUARES.

The acuteness of mathematicians has also discovered magic
squares which possess the peculiar property that if one row after an-
other be taken away from each side, the smaller inner squares re-
maining will still be magical squares, that is to say, all their rows
when added will give the same sum. It will be sufficient to give
two examples here of such squares, (the laws for their construction
being somewhat more complicated,) of which the first has 7 times 7
and the second 8 times 8 places. The numbers within each of the
dark-bordered frames form with respect to the centre smaller squares
which in their own turn are magical.

4

5

6

43

39

38

40

49

15

16

33

30

31

I

48

37

22

27

26

13

2

47

36

29

25

21

14

3

8

18

24

23

28

32

42

9

19

34

17

20

35

41

10

45

44

7

II

12

46

I

56

55

1 1

53

13

14

57

63

15

47

22

42

24

45

2

62

49

25

40

34

31

16

3

4

48

28

37

35

30

17

61

5

44

39

26

32

33

21

60

59

19

38

27

29

36

46

6

58

20

18

43

23

41

50

7

8

9

10

54

12

52

51

64

Fig- 25-

Fig. 26.

In the first of these two squares the internal square of 3 times 3
places contains the numbers from 21 to 29 in such a manner that
each row gives when added the sum of 75. This square lies within
a larger one of 5 times 5 spaces, which contains the numbers from
13 to 37 in such a manner that each row gives the sum of 125.
Finally, this last square forms part of a square of 7 times 7 places
which contains the numbers from i to 49 so that each row gives the
sum of 175.

In the second square the inner central square of 4 times 4 places
contains the cumbers from 25 to 40 in such a manner that each row

56

TKE MAGIC SQUARE.

gives the sum of 130. This square is the middle of a square of 6
times 6 places which so contains the numbers from 15 to 50 that
each row gives the sum 195. Finally, this last square is again the
middle of an ordinary magic square composed of the numbers from
I to 64.

VII.

MAGICAL SQUARES WITH MAGICAL PARTS.

If we divide a square of 8 times 8 places by means of the two
middle lines parallel to its sides into 4 parts containing each 4 times
4 spaces, we may propound the problem of so inserting the numbers
from I to 64 in these spaces that not only the whole shall form a
magic square, but also that each of the 4 parts individually shall be
magical, that is to say, give the same sum for each row. This prob-
lem also has been successfully solved, as the following diagram wiii
show.

I

4

63

62

5

8

59

58

6^

42

61
43

2
24

3

21

60

34

57
35

6
32

7
29

23

22

41

44

31

30

33

36

13

16

51

50

9

12

55

54

52
38
27

49
39
26

14
28

^11

15
25

56
46

53
47

10
20

II
17

40

19

18

45

48

Fig. 27

The 4 numbers in each row of any one of the sub-squares here, gives
130 ; so that the sum of each one of the rows of the large square
will be 260.

Finally, in further illustration of this idea, we will submit to
the consideration of our readers a very remarkable square of the
numbers from i to 81. This square, which will be found on the
following page (Fig. 28), is divided by parallel lines into 9 parts, of
which each contains 9 consecutive numbers tiiat severally make up
a magic square by themselves.

Wonderful as the properties of this square may appear, the law
by which the author constructed it is equally simple. We havs

THE Magic square.

bl

simply to regard the 9 parts as the 9 cells cf a magic square of the
numbers from I to IX, and then to inscribe by the magic prescript
in the square designated as I the numbers from i to 9, in the square

31

36

29

76

81

74

13

18

II

30

32

34

75

77

79

78

12

17

14
10

16

15

35

28

33

80

73

22

27

20

40

45

38

58

63

56

21

23

25

39

41

43

57

59

61

26

tg

24

44

37

42

62

55 60

67

72

65

4

9

2

49

54

47

66

68

70

3

5

7

48

50

52

71

64

6g

8

I

6

53

46

51

Fig. 28.

designated as 11 the numbers from 10 to 18, and so on. In this
way the square above given is obtained from the following base-
square :

IV

IX

II

III

V

VII

VIII

I

VI

Fig. 29.

VIII,

MAGIC SQUARES THAT INVOLVE THE MOVE OF THE
CHESS-KNIGHT.

What one of our readers does not know the problems contained
in the recreation columns of our magazines, the requirements of
which are to compose into a verse 8 times 8 quadratically arranged
syllables, of which every two successive syllables stand on spots so
situated with respect to each other that a chess-knight can move
from the one to the other? If we replace in such an arrangement
the 64 successive syllables by the 64 numbers from i to 64, we shall
obtain a knight-problem made up of numbers. Methods also exist
indeed for the construction of such dispositions of numbers, which
then form, the foundation of the construction of the problems in the
newspapers. But the majority of knight-problems of this class
are the outcome of experiment rather than the product of method-

58

THE MAGIC SQUARE.

ical creation. If however it is a severe test of patience to form a
knight-problem by experiment, it stands to reason that it is a still
severer trial to effect at the same time the additional result that the
64 numbers which form the knight-problem shall also form a magic
square.

This trial of endurance was undertaken several decades ago, by
a pensioned Moravian officer named Wenzelides, who was spending
the last days of his life in the country. After a series of trials which
lasted years he finally succeeded in so inscribing in the 64 squares
of the chess-board the numbers from i to 64 that successive num-
bers, as well also as the numbers 64 and i, were always removed
from one another in distance and direction by the move of a knight,
and that in addition thereto the summation of the horizontal and the
vertical rows always gave the same sum 260. Ultimately he dis-
covered several squares of this description, which were published in
the Berlin Chess Journal. One of these is here appended :

47
22
11
62

19
26

35

38

10

63
46
21
36
39
18
27

23

61
12

25
20

37

34

64
9
24
45
40

33
28

17

49
60

I

8
13
56
41
32

2
5

52
57
44
29
16
55

59

50

7

4

53

14

31'

42

6

3

58
51
30
43
54
15

Fig. 30.

The move of the knight and the equality of the summation of
the horizontal and vertical rows, therefore, are the facts to be noted
here. The diagonal rows do not give the sum 260. Perhaps some
one among our readers who possesses the time and patience will be
tempted to outdo Wenzelides, and to devise a numeral knight-prob-
lem of this kind which will give 260 not only in the horizontal and
vertical but also in the two diagonal rows.

THE MAGIC SQUARE.

59

MAGICAL POLYGONS.

So far we have considered only such extensions of the idea un-
derlying the construction of the magic square in which the figure of
the square was retained. We may however contrive extensions of
the idea in which instead of a square, a rectangle, a triangle, or a
pentagon, and the like, appear. Without entering into the con-
sideration of the methods for the construction of such figures, we
will give here of magical polygons simply a few examples, all sup-
plied by Professor Scheffler :

i) The numbers from i to 32 admit of being written in a rect-
angle of 4 X 8 in such a manner that the Ic ■ ^ horizontal rows give
the sum of 132 and the short vertical rows the sum -^r % ; thus :

I
9

32

10
2

31
23

II
30
3
22

29
12
21

4

28
20
13
5

19

27
6
14

18

7
26

15

16

25
8

17

Fig. 31.

2) The numbers from i to 27 admit of being so arranged in three
regular triangles about a point which forms a common centre, that
each side of the outermost triangle will present 6 numbers of the
total summation 96 and each side of the middle triangle 4 numbers
whose sum is 61 ; as the following figure shows :

24

27

,y; 20 9 II 21
16 17

15 8

12

7 13

4 23

19

25

Fig. 32.

6o THE MAGIC SQUARE.

3) The numbers from i to 80 admit of being formed about a
point as common centre into 4 pentagons, such that each side of
the iirst pentagon from within contains two numbers, each side of
the second pentagon four numbers, each of the third six numbers,
and each side of the fourth, outermost pentagon eight numbers.
The sum of the numbers of each side of the second pentagon is 122,
the sum of those of each side of the third pentagon is 248, and that
of those of each side of the fourth pentagon 254. Furthermore, the
sum of any four corner numbers lying in the same straight hne with
the centre, is also the same ; namely, 92.

25 54

31 49

15 80

70

36 44

5
30

50 70 72 32

55 71 '^ 66 27

45 "^ ^5 37

^^ 60 '4

40 56 _ 59

3-' 21 64

6p

6 62 57 58 ,3

75
77

41

46
51

7

14

53

43

4

3

7i

79

67

s

38

33

28

ig 2-2 63 18

12 39 68 74 42 13

4 29 34 7 78 47 52 3

Fig. 35-

4) The numbers from i to 73 admit of being arranged about a
centre, in which the number 37 is written, into three hexagons which
contain respectively 3, 5, and 7 numbers in each side and possess
the following pretty properties.- Each hexagon always gives the
öame sum, not only when the summation is made along its six sides,
but also when it is made along the six diameters that join its corners

THE MAGIC SQUARE. 6l

and along the six that are constructed at right angles to its sides ;
this sum, for the first hexagon from within, is iii, for the second
185, and for the third 25g.

I 5 6 70 60 59 58

63 8

62 19 53 46 22 45 9

61 20 24 64

2 48 31 42 38 49 57

;j 47 39 40 44 56

67 51 41 37 33 23 7

66 50 34 35 54 II

65 25 36 32 43 26 12

10 30 27 13

17 29 21 28 52 55 72

18 71

16 69 68 4 14 15 73
Fig- 34-

X.

MAGIC CUBES.
Several inquirers, particularly Kochansky (1686), Sauveur
(1710), Hugel (1859), and Scheffler (1882), have extended the prin-
ciple of the magic squares of the plane to three-dimensioned space.
Imagine a cube divided by planes parallel to its sides and equidistant
from one another, into cubical compartments. The problem is then,
so to insert in these compartments the successive natural numbers
that every row from the right to the left, every row from the front
to the back, every row from the top to the bottom, every diagonal
of a square, and every principal diagonal passing through the centre
of the cube shall contain numbers whose sum is always the same.
For 3 times 3 times 3 compartments, a magic cube of this descrip-
tion is not constructible. For 4 times, 4 times 4 compartments a
cube is constructible such that any row parallel to an edge of the
cube and every principal diagonal give the sum of 130. To obtain
a magic cube of 64 compartments, imagine the numbers which be-
long in the compartments written on the upper surface of the same

62

THE MAGIC SQUARE.

and the numbers then taken off in layers of i6 from the top down-
wards. We obtain thus 4 squares of 16 cells each, which together
make up the magic cube ; as the following diagrams \vi\\ show :

First Layer
from the Top.

Second Layer
from the Top.

Third Layer
from the Top.

Fourth Layer
from the Top.

I

48

32

49

63

18

34

15

62

19

35

14

4

45

29

52

60

21

37

12

6

43

27

54

7

42

26

55

57

24

40

9

56

25

41

8

10

39

23

58

II

38

22

59

53

28

44

5

13

36

20

61

51

30

46

3

50

31

47

2

16

33

17

64

The same sum 130 here comes out not less than 52 times ; viz.
in the first place from the 16 rows from left to right, secondly from
the 16 rows from the front to the back, thirdly from the 16 rows
counting from the top to the bottom, and lastly from the 4 rows
which join each two opposite corners of the cube, namely from the
rows: I, 43, 22, 64; 49, 27, 38, 16; 13, 39, 26, 52; 61, 23, 42, 4.

For a cube with 5 compartments in each edge the arrangement
of the figures can so be made that all the 75 rows parallel to any and
every edge, all the 30 rows lying in any diagonal of a square, and
all the 4 rows forming an}^ principal diagonal shall have one and the
same summation, 315.

Just as the magic squares of an odd number of cells could be
iormed with the aid of /zc^» auxiliary squares, so also odd-numbered
magic cubes can be constructed with the help of //iree auxiliary cubes.

First Layer from Top. Second Layer from Top. Third Layer from Top.

121

27

83

14

70

2

58

114

45

96

33

89

20

71

102

10

61

117

48

79

36

92

23

54

no

67

123

29

85

II

44

100

I

57

113

75

lOI

32

88

19

76

7

63

119

50

53

109

40

91

22

84

15

66

122

28

"5

41

97

3

59

87

18

74

105

ii

118

49

80

6

62

24

55

106

37

93

Fourth Layer

from

Top

Lowest Layer

64

120

46

77

8

95

21

52

108

39

98

4

60

III

42

104

35

86

17

73

107

38

94

25

51

13

69

125

26

82

16

72

103

34

90

47

78

9

65

116

30

81

12

68

124

56

112

43

99

5

THE MAGIC SQUARE. 63

In this manner the preceding magic cube of 5 times 5 times 5
compartments is formed, in which, it may be additionally noticed,
the middle number between i and 125, namely 63, is placed in the
central compartment ; by which arrangement the attainment of the
sum of 315 is assured in the four principal diagonals and the 30 sub-
diagonals. The condition attained in the magic squares, that the
diagonal-pairs parallel to the sub-diagonals also shall give the sum
315 is not attainable in this case but is so in the case of higher num-
bers of compartments.

CONCLUSION.

Musing on such problems as are the magic squares is fascinat-
ing to thinkers of a mathematical turn of mind. We take de-
light in discovering a harmony that abides as an intrinsic quality in
the forms of our thought. The problems of the magic squares are
playful puzzles, invented as it seems for mere pastime and sport.
But there is a deeper problem underlying all these little riddles, and
this deeper problem is of a sweeping significance. It is the phil-
osophical problem of the world-order.

The formal sciences are creations of the mind. We build the
sciences of mathematics, geometry, and algebra with our conception
of pure forms which are abstract ideas. And the same order that
prevails in these mental constructions permeates the universe, so that
an old philosopher, overwhelmed with the grandeur of law, imagined
r>e neard its rhythm in a cosmic harmony of the spheres.

THE FOURTH DIMENSION.

MATHEMATICAL AND SPIRITUALISTIC.

I.

INTRODUCTORY.

THE tendency to generalise long ago led mathematicians to
extend the notion of three-dimensional space, which is the
space of sensible representation, and to define aggregates of points,
or spaces, of more than three dimensions, with the view of employ-
ing these definitions as useful means of investigation. They had
no idea of requiring people to imagine four-dirnensional things and
worlds, and they were even still less remote from requiring them,
to believe in the real existence of a four-dimensioned space. In
the hands of mathematicians this extension of the notion of space
was a mere means devised for the discovery and expression, by
shorter and more convenient ways, of truths applicable to com-
mon geometry and to algebra operating with more than three un-
known quantities. At this stage, however, the spiritualists came in,
and coolly took possession of this private property of the mathenla-
ticians. They were in great perplexity as to where they should put
the spirits of the dead. To give them a place in the world acces-
sible to our senses was not exactly practicable. They were com-
pelled, therefore, to look around after some terra incognita, which
should oppose to the spirit of research inborn in humanity an insu-
perable barrier. The abiding-place of the spirits had perforce to be
inaccessible to the senses and full of mystery to the mind. This
property the four-dimensioned space of the mathematicians pos-
sessed. With an intellectual perversity which science has no idea of,
these spiritualists boldly asserted, first, that the whole world was

THE FOURTH DIMENSION. 65

situated in a four-dimensioned space as a plane might be situated in
the space familiar to us, secondly, that the spirits of the dead lived
in such a four-dimensioned space, thirdly, that these spirits could
accordingly act upon the world and, consequently, upon the human
beings resident in it, exactly as we three-dimensioned creatures can
produce effects upon things that are two-dimensioned ; for example,
such effects as that produced when we shatter a lamina of ice, and
so influence some possibly existing two-dimensioned ;V<r-world.

Since spiritualism, under the leadership of a Leipsic Professor,
Zöllner, thus proclaimed the existence of a four- dimensioned space,
this notion, which the mathematicians are thoroughly master of, —
for in all their operations with it, though they have forsaken the
path of actual representability, they have never left that of the truth,
— this notion has also passed into the heads of lay persons who have
used it as a catchword, ordinarily without having any clear idea of
what they or any one else mean by it. To clear up such ideas and
to correct the wrong impressions of cultured people who have not a
technical mathematical training, is the purpose of the following
pages. A similar elucidation was aimed at in the tracts which
Schlegel (Riemann, Berlin, 1888) and Cranz (Virchow-Holtzendorff's
Sammlung, Nos. 112 and 113) have published on the so-called fourth
dimension. Both treatises possess indubitable merits, but their
methods of presentation are in many respects too concise to give to
lay minds a profound comprehension of the subject. The author,
accordingly, has been able to add to the reflections which these ex-
cellent treatises offer, a great deal that appears to him necessary
for a thorough explanation in the minds of non-mathematicians of
the notion of the fourth dimension.

II.

THE CONCEPT OF DIMENSION.

Many text-books of stereometry begin with the words : "Every
body has three dimensions, length, breadth, and thickness." If we
should ask the author of a book of this description to tell us the
length, breadth, and thickness of an apple, of a sponge, or of a cloud
of tobacco smoke, he would be somewhat perplexed and would prob-

66 THE FOURTH DIMENSION.

ably say, that the definition in question referred to something dif-
ferent. A cubical box, or some similar structure, whose angles are
all right angles and whose bounding surfaces are consequently all
rectangles is the only body of which it can at all be unmistakablj»
asserted that there are three principal directions distinguishable in
it, of which any one can be called the length, any other the breadth,
and any third the thickness. We thus see that the notions of length,
breadth, and thickness are not sufficiently clear and universal to
enable us to derive from them any idea of what is meant when it is
said that every body possesses three dimensions, or that the space
of the world is three-dimensional.

This distinction may be made sharper and more evident by the
following considerations : We have, let us suppose, a straight line
on which a point is situated, and the problem is proposed to deter-
mine the position of the point on the line in an unequivocal manner.
The simplest way to solve this is, to state how far the point is re-
moved in the one or the other direction from some given fixed point;
just as in a thermometer the position of the surface of the mercury
is given by a statement of its distance in the direction of cold or heat
from a predetermined fixed point — the point of freezing water. To
state, therefore, the position of a point on a straight line, the sole
datum necessary is a single number, if beforehand we have fixed
upon some standard line, like the centimetre, and some definite
point to which we give the value zero, and have also previously de-
cided in what direction from the zero-point, points must be situated
whose position is expressed by positive numbers, and also in what
direction those must lie whose position is expressed by negative
numbers. This last-mentioned fact, that a single number is suffi-
cient to determine the place of a point in a straight line, is the real
reason why we attribute to the straight line or to any part of it a
single dimension.

More generally, we call every totality or system, of infinitely
numerous things, ßi^ij-dimensional, in which one number is all that
is requisite to determine and mark out any particular one of these
things from among the entire totality. Thus, time is one-dimen-
sional. We, as inhabitants of the earth, have naturally chosen as

THE FOURTH DIMENSION. 67

our unit of time, the period of the rotation of the earth about its
axis, namely, the day, or a definite portion of a day. The zero-point
of time is regarded in Christian countries as the year of the birth of
Christ, and the positive direction of time is the time subsequent to
the birth of Christ. These data fixed, all that is necessary to estab-
lish and distinguish any definite point of time amid the infinite to-
tality of all the points of time, is a single nufnber. Of course this
number need not be a whole number, but may be made up of the
sum of a whole number and a fraction in whose numerator and de-
nominator we may have numbers as great as we please. We may,
therefore, also say that the totality of all conceivable numerical
magnitudes, or of only such as are greater than one definite number
and smaller than some other definite number, is one-dimensional.

We shall add here a few additional examples of one-dimen-
sional magnitudes presented by geometry. First, the circumference
of a circle is a one-dimensional magnitude, as is every curved line,
whether it returns into itself or not. Further, the totality of all
equilateral triangles which stand on the same base is one-dimen-
sional, or the totality of all circles that can be described through
two fixed points. Also, the totality of all conceivable cubes will be
seen to be one-dimensional, provided they are distinguished, not
with respect to position, but with respect to naagnitude.

In conformity with the fundamental ideas by which we define
the notion of a one-dimensional manifoldness, it will be seen that
the attribute /w/?-dimensional must be applied to all totalities of
things in which two numbers are necessary (and sufficient) to dis-
tinguish any determinate individual thing amid the totality. The
simplest two-dimensioned complex which we know of is the plane.
To determine accurately the position of a point in a plane, the sim-
plest way is to take two axes at right angles to each other, that is,
fixed straight lines, and then to specify the distances by which the
point in question is removed from each of these axes.

This method of determining the position of a point in a plane
suggested to the celebrated philosopher and mathematician Des-
cartes the fundamental idea of analytical geometry, a branch of
mathematics in which by the simple artifice of ascribing to every

68 THE FOURTH DIMENSION.

point in a plane two numerical values, determined by its distances
from the two axes above referred to, planimetrical considerations
are transformed into algebraical. So, too, all kinds of curves that
graphically represent the dependence of things on time, make use
of the fact that the totality of the points in a plane is two-dimen-
sional. For example, to represent in a graphical form the increase
in the population of a city, we take a horizontal axis to represent
the time, and a perpendicular one to represent the numbers which
are the measures of the population. Any two lines, then, whose
lengths practical considerations determine, are taken as the unit of
time, which we may say is a year, and as the unit of population,
which we will say is one thousand. Some definite year, say 1850,
is fixed upon as the zero point. Then, from all the equally distant
points on the horizontal axis, which points stand for the years, we
proceed in directions parallel to the other axis, that is, in the per-
pendicular direction, just so much upwards as the numbers which
stand for the population of that year require. The terminal points
so reached, or the curve which runs through these terminal points,
will then present a graphic picture of the rates of increase of the
population of the town in the different years. The rectangular axes
of Descartes are employed in a similar way for the construction of
barometer curves, which specify for the different localities of a
country the amount of variation of the atmospheric pressure during
any period of time. Immediately next to the plane the surface of
the earth will be recognised as a two-dimensional aggregate of
points. In this case geographical latitude and longitude supply the
two numbers that are requisite accurately to determine the position
of a point. Also, the totality of all the possible straight lines that
can be drawn through any point in space is two-dimensional, as we
shall best understand if we picture to ourselves a plane which is
cut in a point by each of these straight lines and then remember
that by such a construction every point on the plane will belong to
some one line and, vice versa, a line to every point, whence it fol-
lows that the totality of all the straight lines, or, as we may call
them, rays, which pass through the point assigned are of the same
dimensions as the totality of the points of the imagined plane.

THE FOURTH DIMENSION. 69

The question might be asked, In what way and to what extent
in this case is the specification of two numbers requisite and suffi-
cient to determine amid all the rays which pass through the speci-
fied point a definite individual ray? To get a clear idea of the
problem here involved, let us imagine the ray produced far into the
heavens where some quite definite point will correspond to it. Now,
the position of a point in the heavens depends, as does the position
of a point on all spherical surfaces, on two numbers. In the heavens
these two numbers are ordinarily supplied by the two angles called
altitude, or the distance above the plane of the horizon, and azi-
muth, or the angular distance between the circle on which the alti-
tude is measured and the meridian of the observer. It will be seen
thus that the totality of all the luminous rays that an eye, conceived
as a point, can receive from the outer world is two-dimensional, and
also that a lumiiious point emits a two-dimensional group of luminous
rays. It will also be observed, in connexion with this example, that
the two-dimensional totality of all the rays that can be drawn through
a point in space is something different from the totality of the rays
that pass through a point but are required to lie in a given plane. Such
a group of objects as the last-named one is a one-dimensional totality.

Now that we have sufficiently discussed the attributes that are
characteristic of one and two-dimensional aggregates, we may,
without any further investigation of the subject, propose the follow-
ing definition, that, generally, an n- dimensional totality of infinitely
numerous things is such that the specification of n numbers is necessary
and sufficient to indicate definitely any individual amid all the infinitely
numerous individuals of that totality.

Accordingly, the point-aggregate made up of the world-space
which we inhabit, is a three-dimensional totality. To get our bear-
ings in this space and to define any determinate point in it, we have
simply to lay through any point which we take as our zero-point
three axes at right angles to each other, one running from right to
left, one backwards and forwards, and one upwards and downwards.
We then join each two of these axes by a plane and are enabled
thus to specify the position of every point in space by the three per-
pendicular distances by which the point in question is removed in a

70 ■ THE FOURTH DIMENSION.

positive or negative sense from these three planes. It is customary
to denote the numbers which are the measures of these three dis-
tances by X, y, and z, the positive x, positive j, and positive z ordi-
narily being reckoned in the right hand, the hitherward, and the
upward directions from the origin. If now, with direct reference to
this fundamental axial system, any particular specification of x, y,
and z be made, there will, by such an operation, be cut out and iso-
lated from the three-dimensional manifoldness of all the points of
space a totality of less dimensions. If, for example, z is equal to
Beven units or measures, this is equivalent to a statement that only
the two-dimensional totality of the points is meant, which consti-
tute the plane that can be laid at right angles to the upward-passing,
s-axis at a distance of seven measures from the zero-point. Conse-
quently, every imaginable equation between x,y^ and z isolates and
defines a two dimensional aggregate of points. If two different equa-
tions obtain between x, y, and z, two such two-dimensional totalities
will be isolated from among all the points of space. But as these
last must have some one-dimensional totality in common, we ma}'
say that the co-existence of two equations between x,y, and z defines
a one-dimensional totality of points, that is to say a straight line, a
line curved in a plane, or even, perhaps, one curved in space. It
is evident from this that the introduction of the three axes of refer-
ence forms a bridge between the theory of space and the theory of
equations involving three variable quantities, x, y, z. The reason
that the theory of space cannot thus be brought into connection
with algebra in general, that is, with the theory of indefinitely nu-
merous equations, but only with the algebra of three quantities, x,
y, z, is simply to be sought in the fact that space, as we picture it,
can have only three dimensions.

We have now only to supply a few additional examples of n-
dimensional totalities. All particles of air are four-dimensional in
magnitude when in addition to their position in space we also con
sider the variable densities which they assume, as they are expressed
by the different heights of the barometer in the different parts of the
atmosphere. Similarly, all conceivable spheres in space are four-
dimensional magnitudes, for their centres form a three-dimensional

THE FOURTH DIMENSION, 71

point-aggregate, and around each centre there may be additionally
conceived a one-dimensional totality of spheres, the radii of which
can be expressed by every numerical magnitude from zero to infinity.
Further, if we imagine a measuring-stick of invariable length to as-
sume every conceivable position in space, the positions so obtained
will constitute a five-dimensional aggregate. For, in the first place,
one of the extremities of the measuring stick may be conceived to
assume a position at every point of space, and this determines for
one extremity alone of the stick a three-dimensional totality of po-
sitions ; and secondly, as we have seen above, there proceeds from
every such position of this extremity a two-dimensional totality of
directions, and by conceiving the measuring-stick to be placed
lengthwise in every one of these directions we shall obtain all the
conceivable positions which the second extremity can assume, and
consequently, the dimensions must be 3 plus 2 or 5. Finally, to
find out how many dimensions the totality of all the possible posi-
tions of a square, invariable in magnitude, possesses, we first give
one of its corners all conceivable positions in space, and we thus
obtain three dimensions. One definite point in space now being
fixed for the position of one corner of the square, we imagine drawn
through this point all possible lines, and on each we lay off the
length of the side of the square and thus obtain two additional di-
mensions. Through the point obtained for the position of the sec-
ond corner of the square we must now conceive all the possible di-
rections drawn that are perpendicular to the line thus fixed, and
we must lay off once more on each of these directions the side of
the square. By this last determination the dimensions are only in-
creased by one, for only one one- dimensional totality of perpen-
dicular directions is possible to one straight line in one of its points.
Three corners of the square are now fixed and therewith the posi-
tion of the fourth also is uniquely determined. Accordingly, the
totality of all equal squares which differ from one another only by
their position in space, constitutes a manifoldness of six dimen-
sions.

72 THE FOURTH DIMENSION,

III.

THE INTRODUCTION OF THE NOTION OF FOUR-DIMENSIONAL
POINT-AGGREGATES, PERMISSIBLE.

In the preceding section it was shown that we can conceive not
only of manifoldnesses of one, two, and three dimensions, but also
of manifoldnesses of any number of dimensions. But it was at the
same time indicated that our world-space, that is, the totality of all
conceivable /<?/;//i- that differ only in respect of position, cannot in
agreement with our notions of things possess more than three di-
mensions. But the question now arises, whether, if the progress of
science tends in such a direction, it is permissible to extend the no
tion of space by the introduction of point-aggregates of more, than
three dimensions, and to engage in the study of the properties of
such creations, although we know that notwithstanding the fact
that we may conceptually establish and explore such aggregates of
points, yet we cannot picture to ourselves these creations as we do
the spatial magnitudes which surround us, that is, the regular three-
dimensional aggregates of points.

To show the reader clearly that this question must be answered
in the affirmative, that the extension of our notion of space is per-
missible, although it leads to things which we cannot perceive by
our senses, I may call the reader's attention to the fact that in arith-
metic we are accustomed from our youth upwards to extensions of
ideas, which, accurately viewed, as little admit of graphic concep-
tion as a four-dimensional space, that is, a point-aggregate of four
dimensions. By his senses man first reaches only the idea of whole
numbers — the results of counting. The observation of primitive
peoples* and of children clearly proves that the essential decisive
factors of counting are these three : First, we abstract, in the count-
ing of things, completely from the individual and characteristic at-
tributes of these things, that is, we consider them as homogeneous.
Second, we associate individually with the things which we count

* See the essay Notion and Definition of Number in this collection.

THE FOURTH DIMENSION. 73

other homogeneous things. These other things are even now, among
uncivilised peoples, the ten fingers of the two hands. They may,
however, be simple strokes, or, as in the case of dice and dominoes,
black points on a white background. Third, we substitute for the
result of this association some concise symbol or word ; for example,
the Romans substituted for three things counted, three strokes placed
side by side, namely: III ; but for greater numbers of things they
employed abbreviated signs. The Aztecs, the original inhabitants
of Mexico, had time enough, it seems, to express all the numbers
up to nineteen by equal circles placed side by side. They had ab-
breviated signs only for the numbers 20, 400, 8000, and so forth.
In speaking, some one same sound might be associated with the
things counted ; but this method of counting is nowadays employed
only by clocks: the languages of men since prehistoric times have
fashioned concise words for the results of the association in question.
From the notion of number, thus fixed as the result of counting, man
reached the notion of the addition of two numbers, and thence the
notion that is the inverse of the last process, the notion of subtrac-
tion. But at this point it clearly appears that not every problen^
which may be propounded is soluble ; for there is no number which
can express the result of the subtraction of a number from one
which is equally large or from one which is smaller than itself. The
primary school pupil who says that 8 from 5 "won't go" is per-
fectly right from his point of view. For there really does not exist
any result of counting which added to eight will give five.

If humanity had abided by this point of view and had rested
content with the opinion that the problem "5 minus 8" is not solv-
able, the science of arithmetic would never have received its full
development, and humanity would not have advanced as far in civ-
ilisation as it has. Fortunately, men said to themselves at this
crisis: "If 5 minus 8 won't go, we'll make it go; if 5 minus 8 does
not possess an intelligible meaning, we will simply give it one." As
a fact, things which have not a meaning always afford men a pleas-
ing opportunity of investing them with one. The question is, then,
what significance is the problem "5 minus 8" to be invested with?

74 THE FOURTH DIMENSION.

The most natural and therefore the most advantageous solu-
tion undoubtedly is to abide by the original notion of subtraction as
the inverse of addition, and to make the significance of 5 minus 8
such, that for 5 minus 8 plus 8 we shall get our original minuend 5.
By such a method all the rules of computation which apply to real
differences will also hold good for unreal differences, such as 5 minus
8. But it then clearly appears that all forms expressive of differ-
ences in which the numbers that stand before the minus symbol
are less by an equal amount than those which follow it may be re-
garded as equal ; so that the simplest course seems to be to intro-
duce as the common characteristic of all equal differential forms of
this description a common sign, which will indicate at the same
time the difference of the two numbers thus associated. Thus it
came about that for 5 minus 8, as well as for every differential form
which can be regarded as equal thereto, the sign '* — 3" was intro-
duced. But in calling differential forms of this description num-
bers, the notion of number was extended and a new domain was
opened up, namely the domain of negative numbers.

In the further development of the science of arithmetic, through
the operation of division viewed as the inverse of multiplication, a
second extension of the idea of number was reached, namely, the
notion of fractional numbers as the outcome of divisions that had
led to numbers hitherto undefined. We find, thus, that the science
of arithmetic throughout its whole development has strictly adhered
to the principle of conformity and consistency and has invested
every association of two numbers, which before had no significance,
by the introduction of new numbers, with a real significance, such
that similar operations in conformity with exactly the same rules
could be performed with the new numbers, viewed as the results of
this association, as with the numbers which were before known and
perfectly defined. Thus the science proceeded further on its way and
reached the notions of irrational, imaginary, and complex numbers.

The point in all this, which the reader must carefully note, is,
that all the numbers of arithmetic, with the exception of the posi-
tive whole numbers, are artificial products of human thought, in-
vented to make the language of arithmetic more flexible, and to

THE FOURTH DIMENSION. 75

accelerate the progress of science. All these numbers lack the at-
tributes of representability.

No man in the world can picture to himself "minus three
trees." It is possible, of course, to know that when three trees
of a garden have been cut down and carried away, three are
missing, and by substituting for "missing" the inverse notion of
"added," we may say, perhaps, that "minus three trees" are added.
But this is quite different from the feat of imagining a negative
number of trees. We can only picture to ourselves a number of
trees that results from actual counting, that is, a positive whole
number. Yet, notwithstanding all this, people had not the slight-
est hesitation in extending the notion of number. Exactly so must
it be permitted us in geometry to extend the notion of space, even
though such an extension can only be mentally defined and can never
be brought within the range of human powers of representation.

In mathematics, in fact, the extension of any notion is admis-
sible, provided such extension does not lead to contradictions with
itself or with results which are well established. Whether such
extensions are necessary, justifiable, or important for the advance-
ment of science is a different question. It must be admitted, there-
fore, that the mathematician is justified in the extension of the no-
tion of space as a point-aggregate of three dimensions, and in the
introduction of space or point-aggregates of more than three dimen-
sions, and in the employment of them as means of research. Other
sciences also operate with things which they do not know exist, and
which, though they are sufficiently defined, cannot be perceived by
our senses. For example, the physicist employs the ether as a
means of investigation, though he can have no sensory knowledge
of it. The ether is nothing more than a means which enables us to
comprehend mechanically the effects known as action at a distance
and to bring them within the range of a common point of view.
Without the assumption of a material which penetrates everything,
and by means of whose undulations impulses are transmitted to the
remotest parts of space, the phenomena of light, of heat, of gravi-
tation, and of electricity would be a jumble of isolated and uncon-
nected mysteries. The assumption of an ether, however, comprises

76 THE FOURTH DIMENSION.

in a systematic scheme all these isolated events, facilitates our men-
tal control of the phenomena of nature, and enables us to produce
these phenomena at will. But it must not be forgotten in such re-
flexions that the ether itself is even a greater problem for man, and
that the ether-hypothesis does not solve the difficulties of phenom-
ena, but only puts them in a unitary conceptual shape. Notwith-
standing all this, physicists have never had the least hesitation in
employing the ether as a means of investigation. And as little do
reasons exist why the mathematicians should hesitate to investigate
the properties of a four-dimensioned point-aggregate, with the view
of acquiring thus a convenient means of research.

IV.

THE INTRODUCTION OF THE IDEA OF FOUR-DIMENSIONED POINT-
AGGREGATES OF SERVICE TO RESEARCH.

From the concession that the mathematician has the right to
define and investigate the properties of point-aggregates of more
than three dimensions, it does not necessarily follow that the intro-
duction of an idea of this description is of value to science. Thus,
for example, in arithmetic, the introduction of operations which
spring from involution, as involution and its two inverse operations
proceed from multiplication, is undoubtedly permitted. Just as for
"a times a times a" we write the abbreviated symbol "ü:^," (which
we read, a to the third power,) and investigate in detail the opera-
tion of involution thus defined, so we might also introduce some
shorthand symbol for "a to the a* power to the a* power" and thus
reach an operation of the fourth degree, which would regard a as
a passive number and the number 3, or any higher number, as the
active number, that is, as the number which indicates how often a
is taken as the base of a power whose exponent may be a, or "a to
the a*," or "a to the a* to the a^ power."

But the introduction of such an operation of the fourth degree
has proved itself to be of no especial value to mathematics. And
the reason is that in the operation of involution the law of commuta-
tion does not hold good. In addition, the numbers to be added may
be interchanged and the introduction of multiplication is therefore

THE FOURTH DIMENSION. 77

of great value. So, also, in multiplication the numbers which are
combined, that is, the factors, may be changed about in any way,
and thus the introduction of involution is of value. But in involution
the base and the exponent cannot be interchanged, and consequently
the introduction of any higher operation is almost valueless.

But with the introduction of the idea of point-aggregates of
multiple dimensions the case is wholly different. The innovation
in question has proved itself to be not only of great importance to
research, but the progress of science has irresistibly forced investi-
gators to the introduction of this idea, as we shall now set forth in
detail.

In the first place, algebra, especially the algebraical theory of
systems of eqations, derives much advantage from the notion of
multi-dimensioned spaces. If we have only three unknown quanti-
ties, X, y, z, the algebraical questions which arise from the possible
problems of this class admit, as we have above seen, of geometrical
representation to the eye. Owing to this possibility of geometrical
representation, some certain simple geometrical ideas like "mov-
ing," "lying in," "intersecting," and so forth, may be translated
into algebraical events. Now, no reason exists why algebra should
stop at three variable quantities ; it must in fact take into considera-
tion any number of variable quantities.

For purposes of brevity and greater evidentness, therefore, it is
quite natural to employ geometrical forms of speech in the consid-
eration of more than three variables. But when we do this, we as-
sume, perhaps without really intending to do so, the idea of a space
of more than three dimensions. If we have four variable quantities,
X, y, z, u, we arrive, by conceiving attributed to each of these four
quantities every possible numerical magnitude, at a four-dimen-
sioned manifoldness of numerical quantities, which we may just as
well regard as a four-dimensioned aggregate of points. Two equa-
tions which exist on this supposition between x, y, z, and u, define
two three-dimensioned aggregates of points, which intersect, as we
may briefly say, in a two-dimensioned aggregate of points, that is,
in a surface ; and so on. In a somewhat different manner the de-
termination of the contents of a square or a cube by the involution

78 THE FOURTH DIMENSION.

of a number which stands for the length of its sides, leads to the
notion of four-dimensioned structures, and, consequently, to the
notion of a four dimensioned point-space. When we note that a'
stands for the contents of a square, and a^ for the content of a cube,
we naturally inquire after the contents of a structure which is pro-
duced from the cube as the cube is produced from the square and
which also will have the contents a*. We cannot, it is true, clearly
picture to ourselves a structure of this description, but we can,
nevertheless, establish its properties with mathematical exactness,*
It is bounded by 8 cubes just as the cube is bounded by 6 squares ;
it has 1 6 corners, 24 squares, and 32 edges, so that from every cor-
ner 4 edges, 6 squares, and 4 cubes proceed, and from every edge
3 squares and 3 cubes.

Yet despite the great service to algebra of this idea of multi-
dimensioned space, it must be conceded that the conception al-
though convenient is yet not indispensable. It is true, algebra is
in need of the idea of multiple dimensions, but it is not so abso-
lutely in need of the idea of /^/«/aggregates of multiple dimensions.

This notion is, however, necessary and serviceable for a pro-
found comprehension of geometry. The system of geometrical
knowledge which Euclid of Alexandria created about three hundred
years before Christ, supplied during a period of more than two
thousand years a brilliant example of a body of conclusions and
truths which were mutually consistent and logical. Up to the pres-
ent century the idea of elementary geometry was indissolubl}' bound
up with the name of Euclid, so that in England where people ad-
hered longest to the rigid deductive system of the Grecian mathe-
matician, the task of "learning geometry" and "reading Euclid"
were until a few years ago identical. Every proposition of this
Euclidean system rests on other propositions, as one building-stone
in a house rests upon another. Only the very lowest stones, the
foundations, were without supports. These are the axioms or fun-

* Victor Schlegel, indeed, has made models of the three-dimensional nets of all
the six structures which correspond in four-dimensioned space to the five regular
bodies of our space, in an analogous manner to that by which we draw in a plane
the nets of five regular bodies. Schlegels models are made by Brill of Darmstadt.

THE FOURTH DIMENSION. 79

damental propositions, truths on which all other truths are, directly
or indirectly, founded, but which themselves are assumed without
demonstration as self-evident.

But the spirit of mathematical research grew in time more and
more critical, and finally asked, whether these axioms might not pos-
sibly admit of demonstration. Especially was a rigid proof sought
for the eleventh* axiom of Euclid, which treats of parallels.

After centuries of fruitless attempts to prove Euclid's eleventh
axiom. Gauss, and with him Bolyai and Lobach^vski, Riemann,
and Helmholtz, finally stated the decisive reasons why any attempt
to prove the axiom of the parallels must necessarily be futile. These
reasons consist of the fact that though this axiom holds good enough
in the world-space such as we do and can conceive it, yet three-
dimensioned spaces are ideally conceivable though not capable of
mental representation, where the axiom does not hold good. The
axiom was thus shown to be a mere fact of observation, and from that
time on there could no longer be any thought of a deductive demon-
stration of it. In view of the intimate connection, which both in an
historical and epistemological point of view exists between the ex-
tension of the concept of space and the critical examination of the
axioms of Euclid, we must enter at somewhat greater length into
the discussion of the last mentioned propositions.

Of the axioms which Euclid lays at the foundation of his
geometr}^ only the following three are really geometrical axioms :

Eighth axiom: Magnitudes which coincide with one another
are equal to one another.

Eleventh axiom: If a straight line meet two straight lines so as
to make the two interior angles on the same side of it taken to-
gether less than two right angles, these straight lines, being con-
tinually produced, shall at length meet on that side on which are
the angles which are less than two right angles.

Twelfth axiom: Two straight lines cannot inclose a [finite]
space.

The numerous proofs which in the course of time were adduced

*Also called the twelfth axiom, also the fifth postulate. — Tr.

8o THE FOURTH DIMENSION.

in demonstration of these axioms, especially of the eleventh, all
turn out on close examination to be pseudo-proofs. Legendre drew
attention to the fact that either of the following axioms might be
substituted for the eleventh:

a) Given a straight line, there can be drawn through a point
in the same plane with that line, one and one line only which shall
not intersect the first (parallels) however far the two lines may be
produced ;

b) If two parallel lines are cut by a third straight line, the in-
terior alternate angles will be equal.

c) The sum of the angles of a triangle is equal to two right
angles, that is, to the angle of a straight line or to i8o°.

By the aid of any one of these three assertions, the eleventh
axiom of Euclid may be proved, and, vice versa, by the aid of the
latter each of the three assertions may be proved, of course with
the help of the other two axioms, eight and twelve. The percep-
tion that the eleventh axiom does not admit of demonstration with-
out the employment of one of the foregoing substitutes may best
be gained from the consideration of congruent figures. Every
reader will remember from his first instruction in geometry that the
congruence of two triangles is demonstrated by the superposition
of one triangle on the other and by then ascertaining whether the
two completely coincide, no assumptions being made in the deter-
mination except those above mentioned.

In the case of triangles which are congruent, as are I and II in
the preceding cut, this coincidence may be effected by the simple
displacement of one of the triangles; so that even a two-dimensional
being, supposed to be endowed with powers of reasoning, but only
capable of picturing to itself motions within a plane, also might
convince itself that the two triangles I and II could be made to
coincide. But a being of this description could not convince itself

•J HE FOURTH DIMENSION. ÖI

in like manner of the congruence of triangles I and III. It would
discover the equality of the three sides and the three angles, but it
could never succeed in so superposing the two triangles on each
other as to make them coincide. A three-dimensional being, how-
ever, can do this very easily. It has simply to turn triangle I about
one of its sides and to shove the triangle, thus brought into the po-
sition of its reflexion in a mirror, into the position of triangle III.
Similarly, triangles II and III may be made to coincide by moving
either out of the plane of the paper around one of its sides as axis
and turning it until it again falls in the plane of the paper. The
triangle thus turned over can then be brought into the position of
the other.

Later on we shall revert to these two kinds of congruence :
"congruence by displacement " and * 'congruence by circumversion. "
For the present we will start from the fact that it is always possible
within the limits of a plane to take a triangle out of one position
and bring it into another without altering its sides and angles. The
question is, wheth£r this is only possible in the plane, or whether
it can also be done on other surfaces.

We find that there are certain surfaces in which this is possi-
ble, and certain others in which it is not. For instance, it is im-
possible to move the triangle drawn on the surface of an egg into
some other position on the egg's surface without a distension or
contraction of some of the triangle's parts. On the other hand, it
is quite possible to move the triangle drawn on the surface of a
sphere into any other position on the sphere's surface without a
distension or contraction of its parts. The mathematical reason of
this fact is, that the surface of a sphere, like the plane, has every-
where the same curvature, but that the surface of an egg at differ-
ent places has different curvatures. Of a plane we say that it has
everywhere the curvature zero ; of the surface of a sphere we say it
has everywhere a positive curvature, which is greater in proportion
as the radius is smaller. There are surfaces also which have a con-
stant negative curvature; these surfaces exhibit at every point in
directions proceeding from the same side a partly concave and a
partly convex structure, somewhat like the centre of a saddle.

82 THE FOURTH DIMENSION.

There is no necessity of our entering in any detail into the charac-
ter and structure of the last-mentioned surfaces.

Intimately related with the plane, however, are all those sur-
faces, which, like the plane, have the curvature zero ; in this cate-
gory belong especially cylindrical surfaces and conical surfaces. A
sheet of paper of the form of the sector of a circle may, for exam-
ple, be readily bent into the shape of a conical surface. If two con-
gruent triangles, now, be drawn on the sheet of paper, which may
by displacement be translated the one into the other, these triangles
will, it is plain, also remain congruent on the conical surface; that
is, on the conical surface also we may displace the one into the
other; for though a bendiog of the figures will take place, there
will be no distension or contraction. Similarly, there are surfaces
which, like the sphere, have everywhere a constant positive curva-
ture. On such surfaces also every figure can be transferred into
some other position without distension or contraction of its parts.
Accordingly, on all surfaces thus related to the plane or sphere,
the assumption which underlies the eighth axiom of Euclid, that it
is possible to transfer into any new position any figure drawn on
such surfaces without distortion, holds good.

The eleventh axiom in its turn also holds good on all surfaces
of constant curvature, whether the curvature be zero or positive ;
only in such instances instead of "straight" line we must say
"shortest" line. On the surface of a sphere, namely, two shortest
lines, that is, arcs of two great circles, always intersect, no matter
whether they are produced in the direction of the side at which the
third arc of a great circle makes with them angles less than two
right angles, or, in the direction of the oth^r side, where this arc
makes with them angles of more than two right angles. On the
plane, however, two straight lines intersect only on the side where
a third straight line that meets them makes with them interior an-
gles less than two right angles.

The twelfth axiom of Euclid, finally, only holds good on the
plane and on the surfaces related to it, but not on the sphere or other
surfaces which, like the sphere, have a constant positive curvature.
This also accounts for the fact that one of the three postulates

THE FOURTH DIMENSION. 83

which we regarded as substitutes for the eleventh axiom, though
valid for the plane, is not true for the surface of a sphere; namely,
the postulate that defines the sum of the angles of a triangle. This
sum in a plane triangle is two right angles ; in a spherical tri-
angle it is more than two right angles, the spherical triangle be-
ing greater, the greater the excess the sum of its angles is above
two right angles. It will be seen, from these considerations, that
in geometries in which curved surfaces and not fixed planes are
studied, the axioms of Euclid are either all or partially false.

The axioms of geometry thus having been revealed as facts of
experience, the question suggested itself whether in the same wa}'
in which it was shown that different two-dimensional geometries
were possible, also different three-dimensional systems of geometry
might not be developed ; and consequently what the relations were
in which these might stand to the geometry of the space given by
our senses and representable to our mind. As a fact, a three-dimen-
sional geometry can be developed, which like the geometry of the
surface of an egg will exclude the axiom that a figure or body can
be transferred from any one part of space to any other and yet re-
main congruent to itself. Of a three-dimensional space in which
such a geometry can be developed we say, that it has no constant
measure of curvature.

The space which is representable to us, and which we shall
lienceforth call the space of experience, possesses, as our experiences
without exception confirm, the especial property that every bodily
thing can be transferred from any one part of it to any other with-
out suffering in the transference any distension or any contraction.
The space of experience, therefore, has a constant measure of cur-
vature. The question, however, whether this measure of curvature
is zero or positive, that is, whether the space of experience possesses
the properties which in two-dimensional structures a plane pos-
sesses, or whether it is the three dimensional analogue of the surface
of a sphere is one which future experience alone can answer. If the
space of experience has a constant positive measure of curvature
which is different from zero, be the difference ever so slight, a point
which should move forever onward in a straight line, or, more ac-

84 THE FOURTH DIMENSION.

curately expressed, in a shortest line, would sometime, though per-
haps after having traversed a distance which to us is inconceivable,
ultimately have to arrive from the opposite direction at the place
from which it set out, just as a point which moves forever onward
in the same direction on the surface of a sphere must ultimately ar-
rive at its starting point, the distance it traverses being longer the
greater the radius of the sphere or the smaller its curvature.

It will seem, at first blush, almost incredible, that the space of
experience possibly could have this property. But an example,
which is the historical analogue of this modern transformation of our
conceptions, will render the idea less marvellous. Let us transport
ourselves to the age of Homer. At that time people believed that
the earth was a great disc surrounded on all sides by oceans which
were conceived to be in all directions infinitely great. Indeed, for
the primitive man, who has never journeyed far from the place of
his birth, this is the most natural conception. But imagine now that
some scholar had come, and had informed the Homeric hero Ulysses
that if he would travel forever on the earth in the same direction he
would ultimately come back to the point from which he started ;
surely Ulysses would have gazed with as much astonishment upon
this scholar as we now look upon the mathematician who tells us
that it is possible that a point which moves forever onward in space
in the same direction may ultimately arrive at the place from which
it started. But despite the fact that Ulysses would have regarded
the assertion of the scholar as false because contradictory to his
familiar conceptions, that scholar, nevertheless, would have been
right ; for the earth is not a plane but a spherical surface. So also
the mathematician may be right who bases this more recent strange
view on the possible fact that the space of experience may have a
measure of curvature which is not exactly zero but slightly greater
than zero. If this were really the case, the volume of the space of
experience, though very large, would, nevertheless, be finite ; just
as the real spherical surface of the earth as contrasted with the
Homeric plane surface is finite, having so and so many square miles.
When the objection is here made that a finiteness of space is totally
at variance with our modes of thought and conceptions, two ideas,

THE FOURTH DIMENSION. 85

"infinitely great" and "unlimited," are confounded. All that is at
variance with our practical conceptions is that space can anywhere
have a boundary ; not that it may possibly be of tremendous but
finite magnitude.

It will now be asked if we cannot determine by actual observa-
tion whether the measure of curvature of experiential space is ex-
actly zero or slightly different therefrom. The theorem of the sum
of the angles of a triangle and the conclusions which follow from
this theorem do indeed supply us with a means of ascertaining this
fact. And the results of observation have been, that the measure oj
curvature of space is in all probability exactly equal to zero or if it is
slightly different from zero it is so little so that the technical means oJ
observation at our command and especially our telescopes are not compe-
tent to determine the amount of the deviatio?i. More, we cannot with
any certainty say.

All these reflections, to which the criticism of the hypotheses
that underlie geometry long ago led investigators, compel us to in-
stitute a comparison between the space of experience and other
three-dimensional aggregates of points (spaces), which we cannot
mentally represent but can in thought and word accurately define
and investigate. As soon, however, as we are fully implicated in the
task of accurately investigating the properties of three-dimensional
aggregates of points, we find ourselves similarly forced to regard
such aggregates as the component elements of a manifoldness of
more than three dimensions. In this way the exact criticism of
even ordinary geometry leads us to the abstract assumption of a
space of more than three dimensions. And as the extension of every
idea gives a clearer and more translucent form to the idea as it orig-
inally stood, here too the idea of multi-dimensioned aggregates
of points and the investigation of their properties has thrown a new
light on the truths of ordinary geometry and placed its properties
in clearer relief. Among the numerous examples which show how
the notion of a space of multiple dimensions has been of great ser-
vice to science in the investigation of three-dimensional space, we
shall give one a place here which is within the comprehension of
non -mathematicians.

86

THE FOURTH DIMENSION.

Imagine in a plane two triangles whose angles are denoted
by pairs of numbers — namely, by 1-2, 1-3, 1-4, and 2-5, 3-5, 4-5.
(See Fig. 36.) Let the two triangles so lie that the three lines which
join the angles 1-2 and 2-5, 1-3 and 3-5, and 1-4 and 4-5 intersect at
a point, which we will call 1-5. If now we cause the sides of the
triangles which are opposite to these angles to intersect, it will be
found that the points of intersection so obtained possess the peculiar
property of lying all in one and the same straight line. The point
of intersection of the connection 1-3 and 1-4 with the connection 4-5
and 3-5 may appropriately be called 3-4. Similarly, the point of in-
tersection 2-4 is produced
by the meeting of 4-5, 2-5
and 1-2, 1-4; and the point
of intersection 2-3, by the
meeting of 1-3, 1-2 and
3-5, 2-5. The statement,
that the three points of
intersection, 3-4, 2-4, 2-3,
thus obtained, lie in one
straight line, can be
proved by the principles
1^ of plane geometry only
with difficulty and great
circumstantiality. But by
resorting to the three-
dimensional space of ex-
perience, in which the plane of the drawing lies, the proposition
can be rendered almost self-evident.

To begin with, imagine any five points in space which may be
denoted by the numbers 1, 2, 3, 4, 5 ; then imagine all the possible
ten straight lines of junction drawn between each two of these points,
namely, 1-2, 1-3 .... 4-5 ; and finally, also, all the ten planes of
junction of every three points described, namely, the plane 1-2-3,
1-2-4, • • • • 3-4-5- A spatial figure will thus be obtained, whose ten
straight lines will meet some interposed plane in ten points whose
relative positions are exactly those of the ten points above described.

Fig. 36.

IHE FOURTH DIMENSION. 87

Thus, for example, on this plane the points 1-2, 1-3, and 2-3 will lie in
a straight line, for through the three spatial points i, 2, 3, a plane can
be drawn which will cut the plane of a drawing in a straight line.
The reason, therefore, that the three points 3-4, 2-4, 2-3, also must
ultimately lie in a straight line, consists in the simple fact that the
plane of the three points 2, 3, 4, must cut the plane of the drawing
in a straight line. The figure here considered consists of ten points
of which sets of three so lie ten times in a straight line that con-
versely from every point also three straight lines proceed.

Now, just as this figure is a section of a complete three-dimen-
sional pentagon, so another remarkable figure, of similar proper-

Fig. 37-

ties, may be obtained from the section of a figure of four-dimen-
sioned space. Imagine six points, i, 2, 3, 4, 5, 6, situated in
this four-dimensioned space, and every three of them connected by
a plane, and every four of them by a three-dimensioned space. We
shall obtain thus twenty planes and fifteen three-dimensioned spaces
which will cut the plane in which the figure is to be produced in
twenty points and fifteen rays which so lie that each point sends out
three rays and every ray contains four points. (See Fig. 37.) Fig-
ures of this description, which are so composed of points and rays
that an equal number of rays proceed from every point and an equal

88 THE FOURTH DIMENSION.

number of points lie in every ray, are called configurations. Other
configurations may, of course, be produced, by taking a different
number of points and by assuming that the points taken lie in a
space of different or even higher dimensions. The author of this
article was the first to draw attention to configurations derived from
spaces of higher dimensions. As we see, then, the notion of a space
of more than three dimensions has performed an important ser-
vice in the investigations of common plane geometry.

In conclusion, I should like to add a remark which Cranz makes
regarding the application of the idea of multi-dimensioned space
to theoretical chemistry. (See the treatise before cited.) In chem-
istry, the molecules of a compound body are said to consist of the
atoms of the elements which are contained in the body, and these are
supposed to be situated at certain distances from one another, and
to be held in their relative positions by certain forces. At first, the
centres of the atoms were conceived to lie in one and the same
plane- But Wislicenus was led by researches in paralactic acid
to explain the differences of isomeric molecules of the same struc-
tural formulae by the different positions of the atoms in space. (Com-
pare La chimie dans Vespace by van't Hoff, 1875, preface by J.
Wislicenus). In fact four points can always be so arranged in space
that every two of them may have any distance from each other ;
and the change of one of the six distances does not necessarily in-
volve the alteration of any other.

But suppose our molecule consists of five atoms? Four of these
may be so placed that the distance between any two of them can be
made what we please. But it is no longer possible to give the fifth
atom a position such that each of the four distances by which.it is
separated from the other atoms may be what we please. On the
contrary, the fourth distance is dependent on the three remaining
distances ; for the space of experience has only three dimensions.
If, therefore, I have a molecule which consists of five atoms I can-
not alter the distance between two of them without at least altering
some second distance. But if we imagine the centres of the atoms
placed in a four-dimensioned space, this can be done ; all the ten
distances which may be conceived to exist between the five points

THE FOURTH DIMENSION. bg

will then be independent of one another. To reach the same result
in the case of six atoms we must assume a five-dimensional space ;
and so on.

Now, if the independence of all the possible distances between
the atoms of a molecule is absolutely required by theoretical chem-
ical research, the science is really compelled, if it deals with mole-
cules of more than four atoms, to make use of the idea of a space of
more than three dimensions. This idea is, in this case, simply an
instrument of research, just as are, also, the ideas of molecules and
atoms — means designed to embrace in a perspicuous and systematic
form the phenomena of chemistry and to discover the conditions
under which new phenomena can be evoked. Whether a four-
dimensioned space really exists is a question whose insolubility
cannot prevent research from making use of the idea, exactly as
chemistry has not been prevented from making use of the notion
of atom, although no one really knows whether the things we call
atoms exist or not.

REFUTATION OF THE ARGUMENTS ADDUCED TO PROVE THE

EXISTENCE OF A FOUR DIMENSIONED SPACE INCLUSIVE

OF THE VISIBLE WORLD.

The considerations of the preceding section will have convinced
the cultured non-mathematician of the service which the theory of
multi-dimensioned spaces has done, and bids fair to do, for geo-
metrical research. In addition thereto is the consideration that
every extension of one branch of mathematical science is a constant
source of beneficial and helpful influence to the other branches. The
knowledge, however, that mathematicians can employ the notion of
four-dimensioned space with good results in their researches, would
never have been sufficient to procure it its present popularity; for
every man of intelligence has now heard of it, and, in jest or in
earnest, often speaks of it. The knowledge of a four-dimensioned
space did not reach the ears of cultured non-mathematicians until
the consequences which the spiritualists fancied it v/as permissible
to draw from this mathematical notion were publicly known. But
it is a tremendous step from the four-dimensioned space of th^

go THE FOURTH DIMENSION.

mathematicians to the space from which the spirit-friends of the
spiritualistic mediums entertain us with rappings, knockings, and
bad English. Before taking this step we will first discuss the ques-
tion of the real existence of a four-dimensional space, not deciding
the question whether this space, if it really does exist, is inhabited
by reasonable beings who consciously act upon the world in which
we exist.

Among the reasons which are put forward to prove the exist-
ence of a four-dimensional space containing the world, the least rep-
rehensible are those which are based on the existence of symmetry.
We spoke above of two triangles in the same plane which have all

IS

Fig. 38.

their sides and angles congruent, but which cannot be made to co-
incide by simple displacement within the plane; but we saw that
this coincidence could be effected by holding fast one side of one
triangle and moving it out of its plane until it had been so far turned
round that it fell back into its plane. Now something similar to
this exists in space. Cut two figures, exactly like that of Fig. 38, out
of a piece of paper, and turn the triangle ABJ^ about the side A£,
ACE about the side AC, BCD about the side BC, and in one figure
above and in the other below; then in both cases the points D,E, P
will meet at a point, because AE is equal to AF, BF is equal to
BD, CD is equal to CE. In this manner we obtain two pyramids
which in all lengths and all angles are congruent, yet which cannot,

THE FOURTH DIMENSION. Ql

no matter how we try, be made to coincide, that is, be so fitted the
one into the other that they shall both stand as one pyramid. But
the reflected image of the one could be brought into coincidence
with the other. Two spatial structures whose sides and angles are
thus equal to each other, and of which each may be viewed as the
reflected image of the other, are called synnnetrical. For instance,
the right and the left hand are symmetrical ; or, a right and a left
glove. Now just as in tv/o dimensions it is impossible by simple
displacement to bring into congruence triangles which like those
above mentioned can only be made to coincide by circumversion,
so also in three dimensions it is impossible to bring into congruence
two symmetrical pyramids. Careful mathematical reflection, how-
ever, declares that this could be effected, if it were possible, while
holding one of the surfaces, to move the pyramid out of the space
of experience, and to turn it round through a four-dimensioned
space until it reached a point at which it would return again into
our experiential space. This process "would simply be the four-
dimensional analogue of the three-dimensional circumversion in
the above-mentioned case of the two triangles. Further, the interior
surfaces in this process would be converted into exterior surfaces,
and vice versa, exactly as in the circumversion of a triangle the an-
terior and posterior sides are interchanged. If the structure which
is to be converted into its symmetrical counterpart is made of a flex-
ible material, the interchange mentioned of the interior and exterior
surfaces may be effected by simply turning the structure inside out ;
for example, a right glove may thus be converted into a left glove.
Now from this truth, that every structure can be converted, by
means of a four-dimensional space inclusive of the world, into a
structure symmetrical with it, it has been sought to establish the
probability of the real existence of a four-dimensioned space. Yet
it will be evident, from the discussion of the preceding section, that
the only inference which we can here make is, that the idea of a
four- dimensioned space is competent, from a mathematical point of
view, to throw some light upon the phenomena of symmetry. To
conclude from these facts that a space of this kind really exists,
would be as daring as to conclude from the fact that the uniform

92 THE FOURTH DIMENSION.

angular velocity of the apparent motions of the fixed stars is expli-
cable from the assumption of an axial motion of the firmament, that
the fixed stars are really rigidly placed in a celestial sphere rotating
about its axis. It must not be forgotten that our comprehension of
the phenomena of the real world consists of two elements : first, of
that which the things really are ; and, second, of that by which we
rationally apprehend the things. This latter element is partly de-
pendent on the sum of the experiences which we have before ac-
quired, and partly on the necessity, due to the imperfection of rea-
son, of our classifying the multitudinous isolated phenomena of the
world into categories which we ourselves have formed, and which,
therefore, are not wholly derived from the phenomena themselves,
but to a great extent are dependent on us.

Besides geometrical reasons, Zöllner has also adduced cosmo-
logical reasons to prove the existence of a four-dimensional space.
To these reasons belong especially the questions, whether the num-
ber of the fixed stars is infinitely great, whether the world is finite
or infinite in extension, whether the world had a beginning or will
have an end, whether the world is not hastening towards a condition
of equilibrium or dead level by the universal distribution of its matter
and energy ; the problems, also, of gravitation and action at a dis-
tance ; and finally, the questions concerning the relations between
the phenomena in the world of sense-perception to the unknown
things-in-themselves. All these questions which can be decided in
no definite sense, led Zöllner and his followers to the assumption
that a four-dimensioned space inclusive of the space of experience
must really exist. But more careful reflection will show that this
assumption does not dispose of the difficulties but simply displaces
them into another realm. Furthermore, even if four-dimensioned
space did unravel and make clear all the cosmological problems
which have bothered the human mind, still, its existence would not
be proved thereby; it would yet remain a mere hypothesis, designed
to render more intelligible to a being who can only make experiences
irt a three-dimensional space, the phenomena therein which are full
of mystery to it. A four-dimensioned space would in such case pos-
sess for the metaphysician a value similar to that which the ethpr

THE FOURTH DIMENSION. 93

possesses for the physicist. Still more convincing than these cos-
mological reasons to the majority of men is the physio-psychological
reason drawn from the phenomena of vision which Zöllner adduces.
Into this main argument we will enter in more detail.

When we " see " an object, as we all know, the light which pro-
ceeds or is reflected therefrom produces an image on the retina of
pur eye ; this image is conducted to our consciousness by means of
the optic nerve, and our reason draws therefrom an inference re-
specting the object. When, now, we look at a square whose sides
are a decimetre in length, and whose centre is situated at the distance
of a metre from the pupil of our eye, an image is produced on the
retina. . But exactly the same image will be produced there if we
look at a square whose sides are parallel to the sides of the first
square but two decimetres in length, and whose centre is situated
at a distance of two metres from the pupil of the eye. Proceeding
thus further, we readily discover that an eye can perceive in any
length or line only the ratio of its magnitude to the distance at which
it is situated from it, and that generally a three-dimensional world
must appear to the eye two-dimensional, because all points which
lie behind each other in the direction outwards from the eye pro-
duce on the retina only one image. This is due to the fact that the
retinal images are themselves two-dimensional ; for which reason,
Zöllner says, the world must appear to a child as two-dimensional,
if it be supposed to live in a primitive condition of unconscious men-
tal activity. To such a child two objects which are moving the one
behind the other, must appear as suffering displacement on a sur
face, which we conceive behind the objects, and on which the latter
are projected. In all these apparent displacements, coincidences
and changes of form also are effected. All these things must appear
puzzling to a human being in the first stages of its development,
and the mind thus finds itself, as Zöllner further argues, in the first
years of childhood forced to adopt a hypothesis concerning the con-
stitution of space and to assume that the world is three-dimensional,
although the eye can really perceive it as oniy two-dimensional.
Zöllner then further says, that in the explanation of the effects oi
the external world, man constantly finds this hypothesis of his chi^d-

94 1HE FOURTH DIMENSION.

ish years confirmed, and that in this way it has become in his mind
so profound a conviction that it is no longer possible for him to
think it away. Consonant with this argumentation, also, is ZoU-
ner's remark, that the same phenomenon has presented itself in
astronomical methods of knowledge. To explain the movements of
the planets, which appear to describe regular paths on the surface
of a celestial sphere, we were compelled in the solution of the rid-
dles which these motions presented, to assume in the structure of the
heavens a dimension of "depth," and the complicated motions in
the two-dimensioned firmament were converted into very simple
motions in three-dimensioned space. Zöllner also contends that our
conception of the entire visible world as possessed of three dimen-
sions is a product of our reason, which the mind was driven to form
by the contradictions which would be presented to it on the assump-
tion of only two dimensions by the perspective distortions, coinci-
dences, and changes of magnitude of objects. When a child moves
its hand before its eyes, turns it, brings it nearer,, or pushes it farther
away, this child successively receives the most various impressions
on the surface of its retina of one and the same object of whose
identity and constancy its feelings offer it a perfect assurance. If
the child regarded the changeable projection of the hand on the sur-
face of the retina as the real object, and not the hand which lies be-
yond it, the child would constantly be met with contradictions in its
experience, and to avoid this it makes the hypothesis that the space
of experience is three-dimensional. Zöllner's contention is, there-
fore, that man originally had only a two-dimensional intuition of
space, but was forced by experience to represent to himself the ob-
jects which on the retinal surface appeared two-dimensional, as
three-dimensional, and thus to transform his two-dimensional space-
intuition into a three-dimensional one. Now, in exactly the same
way, according to Zöllner's notion, will man, by the advancement
and increasing exactness of his knowledge of the phenomena of the
outer world, also be compelled to conceive of the material world as
a "shadow cast by a more real four-dimensional world," so that
these conceptions will be just as trivial for the people of the twen-
tieth century as since Copernicus's time the explanation of the mo-

THE FOURTH DIMENSION. 95

tions of the heavenly bodies by means of a three-dimensional mo-
tion has been.

Zöllner's arguments from the phenomena of vision may be re-
futed as follows : In the first place it is incorrect to say that we see
the things of the external world by means of two-dimensional retinal
images. The light which penetrates the eye causes an irritation
of the optic nerves, and any such effect which, though it be not
powerful, is, nevertheless, a mechanical one, can only take place on
things which are material. But material things are always three-
dimensional. The effect of light on the sensitive plates of photog-
raphy can with just as little justice be regarded as two-dimensional.
Our senses can have perception of nothing but three-dimensional
things, and this perception is effected by forces which in their turn
act on three-dimensional things, namely our sensory nerves. It is
wrong to call an image two-dimensional, for it is only by abstraction
that we can conceive of a thickness so growing constantly smaller
and smaller as to admit of our regarding a three-dimensional picture
as two-dimensional, by giving it in mind a vanishingly small thick-
ness. It is also wrong to say, as Zöllner says, that when we see the
shadow of a hand which is cast upon a wall we see something two-
dimensional. What we really perceive is that no light falls upon
our eye from the region included by the shadow, while from the
entire surrounding region light does fall on our eye. But this light
is reflected from the material particles which form the surface of the
wall, that is, from three-dimensional particles of matter. We must
always remember that our eye communicates to us only three-dimen-
sional knowledge, and that for the comprehension of anything which
has two, one, or no dimensions, a purely intellectual act of abstraction
must be added to the act of perception. When we imagine we have
made a lead-pencil mark on paper, we have, exactly viewed, simply
heaped alongside of each other little particles of graphite in such a
manner that there are by far fewer graphite particles in the lateral
and upward directions than there are in the longitudinal direction,
and thus our reason arrives by abstraction at the notion of a straight
line. When we look at an object, say a cube of wood, we recognise
the object as three-dimensional, and it is only by abstraction that

96 THE FOURTH DIMENSION.

we can conceive of its two-dimensional surfaces, of its twelve one-
dimensional edges, and of its eight no-dimensional corners. For we
reach the perception of its surface, for example, solely by reason of
the fact that the material particles which form the cube prevent the
transmission of light, and reflect it, whereby a part of the light re-
flected from every material particle strikes our eye. Now, by think-
ing exclusively of those material particles which are reflected, in
contrariety to the empty space without and the hidden and there-
fore non-reflected particles within, we form the notion of a surface.
It is evident from this, that all that we perceive is three-dimen-
sional, that we cannot reach anything two-dimensional without
an intellectual abstraction, and that, therefore, we cannot conceive
of anything two-dimensional exerting effects upon material things.
But this fact is a refutation of the retinal argument of Zöllner. If
vision consisted wholly and exclusively in the creation of a two-
dimensional image, the things which take place in the world could
never come into our consciousness. The child, therefore, does not
originally apprehend the world, as Zöllner says, as two-dimensional;
on the contrary, it apprehends it either not at all, or it apprehends
it as three-dimensional. Of course the child must first "learn how"
to see. It is found from the observation of children during the first
months of their lives, and of the congenitally blind who have sud-
denly acquired the power of vision by some successful operation,
that seeing does not consist alone in the irritations which arise in
the optic nerves, but also in the correct interpretation of these irri-
tations by reason. This correct interpretation, however, can be
accomplished only by the accumulation of a considerable stock of
experience. Especially must the recognition of the distance of the
object seen be gradually learned. In this, two things are especially
helpful ; first, the fact that we have two eyes and, consequently,
that we must feel two irritations of the optic nerves which are not
wholly alike ; and, secondly, the fact that we are enabled by our
power of motion and our sense of touch to convince ourselves of
the distance and form of the bodies seen. The question now arises,
what sort of an intuition of space would a creature have that had
only one eye, that could neither move itself nor its eye, and also

THE FOURIH DIMENSION. 97

possessed no peripheral nerves. According to Zöllner's view, this
creature could, owing to its two-dimensional retinal images, have
only a two-dimensional intuition of space. The author's opinion,
however, is, that such a creature could not see at all, as it has no
possibility of collecting experiences which are adapted in any way
to interpreting the effects of things on its retina. The light which
proceeded from the objects roundabout and fell on the retina could
produce no other effect on the being than that of a wholly unintel-
ligible irritation, or perhaps even pain.

The reflections presented sufficiently show that neither the
phenomena of symmetry nor the retinal images of the objects of
vision necessarily force upon us the assumption of a four-dimen-
sioned space. If the material world should ever present problems
which could not in the progress of knowledge be solved in a natural
way, the assumption that a four-dimensional space containing the
world exists would also be incompetent to resolve the difficulties
presented ; it would simply convert these difficulties into others,
and not dispose of the problems but siniply displace them to an-
other world. Yet the question might be asked, is the existence of
a four-dimensional space really impossible? To answer this question
we must first clearly know what we mean by "exist." If existence
means that the intellectual idea of a thing can be formed and that
this idea shall not lead to contradictions with other well-established
ideas and with experience, we have only to say that four-dimen
sioned space does exist, as the arguments adduced in sections III
and IV have rendered plain. If, namely, the space of four dimen-
sions did not exist as a clear idea in the minds of mathematicians,
mathematicians could certainly not have been led by this idea to
results which are recognised by the senses as true, and which really
take place in our own representable space. But if existence means
"material actuality," we must say that we neither now nor in the
future can know anything about it. For we know material actual-
ity only as three-dimensional, our senses can only make three-
dimensional experiences, and the inferences of our reason, although
they can well abstract from material things, can never ascend to
.the point of explaining a four-dimensional materiality. Just as little.

98 THE FOURTH DIMENSION.

therefore, as we can locally fix the idea of a two-dimensional mate-
rial world, as little can we ever verify the notion of a four-dimen-
sional material existence.

VI.

EXAMINATION OF THE HYPOTHESIS CONCERNING THE EXISTENCE
OF FOUR-DIMENSIONAL SPIRITS.

In connection with the belief that the visible world is contained
in a four-dimensioned space, Zöllner and his adherents further hold
that this higher space is inhabited by intelligent beings who can
act consciously and at will on the human beings who live in expe-
riential space. To invest this opinion with greater strength, Zöllner
appealed to the fact that the greatest thinkers of antiquity and of
modern times were either wholly of this opinion or at least held
views from which his contentions might be immediately derived.
Plato's dialogue between Socrates and Glaukon in the seventh book
of the Republic, is evidence, says Zöllner, that this greatest philos-
opher of antiquity possessed some presentiment of this extension of
the notion of space. Yet any one who has connectedly studied and
understood Plato's system of philosophy must concede that the so-
called "ideas" of the Platonic system denote something wholly dif-
ferent from what Zöllner sees in them or pretends to see. Zöllner
says that these Platonic ideas are spatial objects of more than three
dimensions and represent "real existence" in the same sense that
the material world, as contrasted with the images on the retina, rep-
resents it. Zöllner similarly deals with the Kantian "thing-in-it-
self," which is also regarded as an object of higher dimensions.

To show Kant in the light of a predecessor, Zöllner quotes the
following passage from the former's "Träume eines Geistersehers,
erläutert durch Träume der Metaphysik " (1766, Collected Works^
Vol. VII. page 32 et seq.): "I confess that I am very much in-
"clined to assert the existence of immaterial beings in the world,
"and to rank my own soul as one of such a class. It appears, there
*'is a spiritual essence existent which is intimately bound up with
"matter but which does not act on those forces of the elements by
"which the latterare connected, but upon some internal principle

THE FOURTH DIMKKSION. gg

•'of its own coudition. It will, in the time to come — I know not
"when or where — be proved, that the human soul, even in this life,
"exists in a state of uninterrupted connection with all the imma-
"terial natures of the spiritual world ; that it alternately acts on
"them and receives impressions from them, of which, as a human
" soul, it is not, in the normal state of things, conscious. It would
" be a great thing, if some such systematic constitution of the spirit-
" ual world, as we conceive it, could be deduced, not exclusively
"from our general notion of spiritual nature, which is altogether too
"hypothetical, but from some real and universally admitted ob-
"servations, — or, for that matter, if it could even be shown to be
"probable."

What Kant really asserts here is, first, the partly independent
and partly dependent existence of the soul, and of spiritual beings
generally, on matter, and, second, that spiritual beings have some
common connection with and mutually influence one another. Th s
contention, which is that of very many thinkers, does not, how-
ever, entail the consequence that the "transcendental subject of
Kant" must be four-dimensional, as Zöllner asserts it does. Kant
never even hinted at the theory that the psychical features of the
world owe their connection with the material features to the fact
that they are four-dimensional and, therefore, include the three-
dimensional. Is it a necessary conclusion that if a thing exists and
is not three-dimensional, as is the case with the soul, it is there-
fore four-dimensional? Can it not in fact be so constituted that it
is wholly meaningless to speak of dimensions at all in connection
with it ?

Yet still more strangely than the words of Plato and Kant do
certain utterances of the mathematicians Gauss and Riemann speak
in favor of Zöllner's hypothesis. S. v. Waltershausen relates of
Gauss in his Gruss zum Gedäclitniss, (Leipsic, 1856), that Gauss
had often remarked that the three dimensions of space were only
a specific peculiarity of the human mind. We can think ourselves,
he said, into beings who are conscious of only two dimensions ;
similarly, perhaps, beings who are above and outside our world may
look down upon us; and there were, he continued, in a jesting tone.

lOO THE FOURTH DIMENSION,

a number of problems which he had here indefinitely laid aside, but
hoped to treat in a superior state by superior geometrical methods.
Leaving aside this jest, which quite naturally suggested itself, the
remarks of Gauss are quite correct. We possess the power to ab-
stract and can think, therefore, what kind of geometry a being that
is only acquainted with a two-dimensional world would have ; for
instance, we can imagine that such a being could not conceive of
the possibility of making two triangles coincide which were con-
gruent in the sense above explained, and so on. So, also, we can
understand that a being who has control of four dimensions can only
conceive of a geometry of four-dimensional space, yet may have the
capacity of thinking itself into spaces of other dimensions. But it
does not follow from this that a four-dimensional space exists, let
alone that it is inhabited by reasonable beings.

Riemann, on the other hand, speaks directly of a world of spir-
its. In his Neue mathematische Principien der Naturphilosophie he
puts forth the hypothesis that the space of the world is filled with
a material that is constantly pouring into the ponderable atoms,
there to disappear from the phenomenal world. In every ponderable
atom, he says, at every moment of time, there enters and appears a
determinate amount of matter, proportional to the force of gravita-
tion. The ponderable bodies, according to this theory, are the
place at which the spiritual world enters and acts on the material
world. Riemann's world of spirits, the sole office of which is to ex-
plain the phenomenon of gravitation as a force governing matter,
is, however, essentially different from the spiritual world of Zöllner,
the function of which is to explain supposed supersensuous phe-
nomena which stand in the most glaring contradiction with the es-
tablished known laws of the material world.

Besides this appeal to the testimony of eminent men like Plato,
Kant, Gauss, and Riemann, the scientific prophet of modern spirit-
ualism also bases his theory on the belief, which has obtained at all
times and appeared in various forms among all peoples, that there
exist in the world forces which at times are competent to evoke
phenomena that are exempt from the ordinary laws of nature. We
have but to think of the phenomena of table-turning which once ex-

THE FOURTH DIMENSION. lOI

cited the Chinese as much as it has aroused, during the last few
decades, the European and American worlds ; or of the divining-
rod, by whose help our forefathers sought for water, in fact, as we
do now in parts of Europe and America.

Cranz, in his essay on the subject, divides spiritualistic phe-
nomena into physical and intellectual. Of the first class he enume-
rates the following : the moving of chairs and tables ; the animation
of walking-sticks, slippers, and broomsticks; the miraculous throw-
ing of objects; spirit-rappings (Luther heard a sound in the Wart-
burg, *< as if three score casks were hurled down the stairs") ; the
ecstatic suspension of persons above the floor ; the diminution of
the forces of gravity ; the ordeals of witches ; the fetching of wished-
for objects ; the declination of the magnetic needle by persons at a
distance ; the untying of knots in a closed string ; insensibility to
injury and exemption therefrom when tortured, as in handling red-
hot coals, carrying hot irons, etc.; the music of invisible spirits ;
the materialisations of spirits or of individual parts of spirits (the
footprints in the experiments of Slade, photographed by Zöllner) ;
the double appearance of the same person ; the penetration of mat-
ter (of closed doors, windows, and so forth). As numerous also is
the selection presented by Cranz of intellectual phenomena, namely:
spirit-writing (Have's instrument for the facilitation of intercourse
with spirits), the clairvoyance and divination of somnambulists, of
visionary, ecstatic, and hypnotised persons, prompted or controlled
by narcotic medicines, by sleeping in temples, by music and dancing,
by ascetic modes of life and residence in barren localities, by the ex-
udations of the soil and of water, by the contemplation of jewels,
mirrors, and crystal-pure water, and by anointing the finger-nails
with consecrated oil. Also the following additional intellectual phe-
nomena are cited : increased eloquence or suddenly acquired power
of speaking in foreign languages; spirit-effects at a distance; in-
ability to move, transferences of the will, and so forth.

All these phenomena, presented with the aspect of truth, and
associated more or less with trickery, self-deception, and humbug,
are adduced by the spiritualists to substantiate the belief in a world
of spirits which intentionally and consciously take part in the events

I02 THE FOURTH DIMENSION.

of the material world, and to prove that these phenomena may be
sufficiently and consistently explained by the effects of the activity
of such a world. It is impossible for us to discuss and put to the
test here the explanations of all these supersensuous phenomena.
Anything and everything can be explained by spirits who act at will
upon the world. There are only a few of these phenomena, namely,
clairvoyance and Slade's experiments, whose explanations are so
intimately connected with our main theme, the so-called fourth di-
mension, that they cannot be passed over.

First, with respect to clairvoyance, the American visionary Da-
vis describes the experiences which he claims to have made in this
condition, induced by " magnetic sleep," as follows :* "The sphere
of my vision now began to expand. At first, I could only clearly
discern the walls of the house. At the start they seemed to me dark
and gloomy; but they soon became brighter and finally transparent.
I could now see the objects, the utensils, and the persons in the ad-
joining house as easily as those in the room in which I sat. But my
perceptions extended further still ; before my wandering glance,
which seemed to control a great semi-circle, the broad surface of
the earth, for hundreds of miles about me, grew as transparent as
water, and I saw the brains, the entrails, and the entire anatomy of
the beasts that wandered about in the forests of the Eastern Hemi-
sphere, hundreds and thousands of miles from the room in which I
sat." The belief in the possibility of such states of clairvoyance is
by no means new. Alexander Dumas made use of it, for example,
in his M^moires d^un mddicin, in which Count Balsamo, afterwards
called Cagliostro, is said to possess the power to throw suitable
persons into this wonderful condition and thus to find out what
other persons at distant places are doing. Zöllner explains clair-
voyance by means of the fourth dimension thus :

A man who is accustomed to viewing things on a plain is sup-
posed to ascend to a considerable height in a balloon. He will
there enjoy a much more extended prospect than if he had remained
on the plain below, and will also be able to signal to greater dis-

* Quoted by Cranz.

THE FOURTH DIMENSION. IO3

tances. The plain, that is, the two-dimensioned space, is accord-
ingly viewed by him from points outside of the plain as "open" in
all directions. Exactly so, in Zöllner's theory, must three-dimen-
sioned spaces appear, when viewed from points in four-dimensioned
space, namely, as "open"; and the more so in proportion as the
point in question is situated at a greater distance from the place of
our body, or in proportion as the soul ascends to a greater height in
this fourth dimension. Zöllner thus explains clairvoyance as a con-
dition in which the soul has displaced itself out of its three-dimen-
sioned space and reached a point which with respect to this space
is four-dimensionally situated and whence it is able to contemplate
the three dimensional world without the interference of intervening
obstacles.

The following remark is to be made to this explanation. The
reason why we have a better and more extended view from a bal-
loon than from places on the earth is simply this, that between the
suspended balloon and the objects seen at a distance nothing inter-
venes but the air, and air allows the transmission of light, whereas,
at the places below on the earth there are all kinds of material
things about the observer which prevent the transmission of light
and either render difficult or absolutely impossible the sight of
things which lie far away. In the same way, also, from a point in
four-dimensioned space, a three-dimensional object will be visible
only provided there are no obstacles intervening. If, therefore, this
awareness of a distant object is a real, actual vision by means of a
luminous ray which strikes the eye, there is contained in the ex-
planation of Zöllner the tacit assumption that the medium with
which the four-dimensional world is filled is also pervious to light
exactly as the atmosphere is.

The theory that there are four-dimensional spirits who produce
the phenomena cited by the spiritualists received special support
from the experiments which the prestidigitateur Slade, who claimed
he was a spiritualistic medium, performed before Zöllner. Of these
experiments we will speak of the two most important, the experi-
ment with the glass sphere and the experiment with the knots. To
explain the connection of the glass sphere experiment with the fourth

104 THE FOURTH DIMENSION.

dimension, we must first conceive of two-dimensional reasoning be-
ings, or, let us say, two-dimensional worms, living and moving ip a
plane. For a creature of this kind it will be self-evident that there
are no other paths between two points of its plane than such as lie
within the plane. It must, accordingly, be beyond the range of
conception of this worm, how any two-dimensional object which lies
within a circle in its space can be brought to any other position in
its space outside the circle without the object passing through the
barriers formed by the circle's circumference. But if this worm
could procure the services of a three-dimensional being, the trans-
portation of the object from a position within the circle to any posi-
tion outside it could be effected by the three-dimensional being sim-
ply taking the object out ofxho. plane and placing it at the desired
point. This object, therefore, would, in an inexplicable manner,
suddenly disappear before the eyes of the worms who were assem-
bled as spectators, and after the lapse of an interval of time would
again appear outside the circle without having passed at any point
through the circle's circumference. If now we add another dimen-
sion, we shall derive from this trick, which is wholly removed from
the sense-perception of the flattened worms, the following experi-
ment, which is wholly beyond the perception of us human beings.
Inside a glass sphere, which is closed all around, a grain of corn is
placed ; the problem is to transport the corn to some place outside
the sphere without passing through the glass surface. Now we
should be able to perform this trick if some four-dimensional being
would render us the same aid that we before rendered the two-
dimensional worm. For the four-dimensional being could take the
grain of corn into his four-dimensional space and then replace it in
our space in the desired spot outside of the glass sphere. Slade
performed this trick before Zöllner. Its mere performance sufficed
to convince this latter investigator that Slade had here made use of
a four-dimensional agent, who, in respect of power of motion, con-
trolled his four-dimensional space as we do our three-dimensional
space. It never occurred to Zöllner that this experiment was the
cleverly executed trick of a prestidigitateur, or, as it would at once
occur to us, that the whole thing was a sensory illusion. The fact

THE FOURTH DIMErSION. IO5

that we cannot explain a trick easily and naturally does not irrevo-
cably prove that it is accomplished by other means than those
which the world of matter presents.

Still better known than this last performance is Slade's experi-
ments with knots. To explain this in connection with the fourth
dimension, we must resort again to the plane and the flat worm in-
habiting it. To two parallel lines in a plane let the two ends of a
third line, which has a double point, that is, intersects itself once,
be attached. Our flat worm would not be able to untie the loop
formed by the doubled third line, which we will call a string, be-
cause it cannot execute motions in three dimensions. If, therefore,
a two dimensional prestidigitateur should appear and accomplish
the trick of untying this loop without removing the two ends of the
string from the parallel lines, he will have accomplished for our flat
worm a supersensuous experiment. A human being engaged in the
service of the prestidigitateur could execute for him the experiment
by simply lifting the string a little out of the plane, pulling it taut,
and placing it back again. This suggests the following analogous
experiment for three-dimensional beings. The two ends of a string
in which a common knot has been made are sealed to the opposite
walls of a room. The problem is to untie this knot without break-
ing the seals at the two ends of the string. Everybody knows that
this problem is not soluble, but it may be calculated mathemati-
cally that the knot in the string can be untied as easily by motions
in a fourth dimension of space as in the experiment above described
the knot in the two-dimensional string was untied by a three-dimen-
sional motion. Now as Slade untied the knot before Zöllner's eyes
without apparently making any use of the ends fastened in the
walls, Zöllner was still more firmly confirmed in the view that Slade
had power over spirits who performed the experiments for him.

Still more far-reaching is the theory of Carl du Prel concerning
the relations of the material and the four-dimensional world. (Com
pare his numerous essays in the spiritualistic magazine Sphinx.)
Just as the shadows of three-dimensional objects cast on a wall are
controlled in their movements by the things whose projections they
are, in the same way it is claimed does there exist back of every-

1C6 THE FOURTH DIMENSION.

thing of this sense-perceptible world a real transcendental and four-
dimensional "thing-in itself " whose projection in the space of ex-
perience is what we falsely regard as the independent thing. Thus
every man besides existing in his terrestrial self also exists in a
spiritual or astral self which constantly accompanies him in his
walks through life and whose existence is especially proclaimed in
states of profound sleep, of somnambulism, and in the conditions
of mediums. In this way Du Prel explains the wonderful feats of
somnambulists, and the aerial journeys of sorcerers and witches.
Whereas, ordinarily the separation of the material body from the
astral body is only effected at the moment of death; in the case of
somnambulists this separation may take place at any time, or, as Du
Prel says, "the threshold of feeling may be permanently displaced."

In view of the natural relations of such theories to the dogmas
of Christianity it is explainable that theologians also have raised
their voices for or against spiritualism. While the Protestant Church
Times beheld in the ''repulsive thaumaturgic performances which
these coryphaei of modern science offer, a lack of comprehension of
real philosophy," the magazine The Proof of Faith expresses its de-
light at the discovery of spiritualism in the following manner :
"Every Christian will surely rejoice at the deep and perhaps mor-
tal wound which these new discoveries have in all probability ad-
ministered to modern materialism."

We shall pass by the childish opinion that the Bible also speaks
of four dimensions, as both in Job (xi. 8-9) and in the Epistle to
the Ephesians (iii. 18) only breadth, length, depth, and height,
that is, four directions of extension, are mentioned. Yet we will
still add, as Cranz has done, the reflections which Zöllner, as the
most prominent representative of modern spiritualism, has put for-
ward respecting its relations to the doctrines of Christianity ( Wis-
sensch. AbhandL, Vol. III). By the foundation of transcendental
physics on the basis of spiritualistic phenomena, the "new light"
has arisen which is spoken of in the New Testament. The rending of
the veil of the Temple on the crucifixion of Christ, the resurrection,
the ascension, the transfiguration, the speaking with many tongues
on the giving out of the Holy Ghost, all these are in Zöllner's view

THE FOURTH DIMENSION. IO7

spiritualistic phenomena. Similarly, he sees a reference to the
four-dimensional world of spirits in all those sayings of Christ in
which the latter speaks to his Apostles of the impossibility of their
having any image or notion of the place to which when he dis-
appeared he would go and whence he would return. (Gospel of
St. John, xii. 33, 36; xiv. 2, 3, 28; xvi. 5, 13.)

Ulrici, however, goes farthest in the mingling of spiritualistic
and Christian beliefs ; for he sees in the doctrine of spiritualism a
means of strengthening belief in a supreme moral world-order and
in the immortality of the soul. In answer to Ulrici's tract "Spirit-
ualism So-called, a Question of Science" (1889) Wundt wrote an
annihilating reply bearing the title "Spiritualism, a Question of
Science So-called." Wundt criticises the future condition of our
soul according to spiritualistic hypotheses in the following sarcastic
yet pertinent words, which Cranz also quotes: "(i) Physically, the
"souls of the dead come into the thraldom of certain living beings
"who are called mediums. These mediums are, for the present at
"least, a not widely diffused class, and they appear to be almost
"exclusively Americans. At the command of these mediums, de-
" parted souls perform mechanical feats which possess throughout
"the character of absolute aimlessness. They rap, they lift tables
"and chairs, they move beds, they play on the harmonica, and do
"other similar things. (2) Intellectually, the souls of the dead
"enter a condition which, if we are to judge from the productions
"which they deposit on the slates of the mediums, must be termed
"a very lamentable one. These slate- writings belong throughout
"in the category of imbecility ; they are totally bereft of any con-
" tents. (3) The most favored, apparently, is the moral condition of
"the soul. According to the testimony which we have, its charac-
"ter cannot be said to be anything else than that of harmlessness.
"From brutal performances, such, for instance, as the destruction
"of bed-canopies, the spirits most politely refrain." Wundt then
laments the demoralising effect which spiritualism exercises on peo-
ple who have hitherto devoted their powers to some serious pursuit
or even to the service of science. In fact it is a presumptuous and
flagrant procedure to forsake the path of exact research, which

[08 THE FOURTH DIMENSION.

slow as it is, yet always leads to a sure extension of knowledge, in
the hop"? that some aimless, foolhardy venture into the realm of
uncerta'nty will carry us farther than the path of slow toil, and that
we can ever thus easily lift the veil which hides from man the prob-
lems r* the world that are yet unsolved.

*

•low that we have presented the opinions of others respecting
the existence of a four dimensional world of spirits, the author would
like to develop one or two ideas of his own on the subject. In the
preceding section it was stated that everything that we perceive b}'
our senses is three-dimensional and that everything that possesses
'our or more dimensions can only be regarded as abstractions or fic-
tions which our reason forms in its constant efforts after an exten-
sion and generalisation of knowledge. To speak of two-dimensional
matter is as self-contradictory as the notion of four-dimensional
matter. But a two or a four-dimensional world might exist in some
other manner than a material manner, and for all we know in one
which to us does not admit of representation. But in such a case,
if it were without the power of affecting the material world, we should
never be able to acquire any knowledge concerning its existence,
and it would be totally indifferent to the people of the three-dimen-
sional world, whether such a world existed or not. Just as an artist
during his lifetime produces a number of different works of art, so
also the Creator might have created a number of different-dimen-
sioned worlds which in no wise interfere with one another. In such
a case, any one world would not be able to know anything of any
other, and we must consequently regard the question whether a
four-dimensional world exists which is incapable of affecting ours,
as insoluble. We have only to examine, therefore, the question
whether the phenomenal world perhaps is a single individual in a
great layer of worlds of which every successive one has one more
dimension than the preceding and which are so connected with one
another that each successive world contains and includes the pre-
ceding world, and, therefore, can produce effects in it. For our
reason, which draws its inferences from the phenomena of this world,
tells us, that if outside the three-dimensional world there exists a

THE FOURTH DIMENSION. lOQ

second four-dimensional world, containing ours, there is no reason
why worlds of more or less dimensions should not, with the same
right, also exist. But if now, as Zöllner and his adherents main-
tain, four-dimensional spirits exist which can act by the mere efforts
of their own wills on our world, there is surely no reason why we
three-dimensional beings should not also be able to produce effects
on some two-dimensional animated world. Whether we have, gen-
erally, any such influence we do not know, but we certainly do know
that we do not act purposely and consciously on a two-dimensional
world. If, therefore, Zöllner were right, the plan of creation would
possess the wonderful feature that four-dimensional beings are cap-
able of arbitrarily affecting the three-dimensional world, but that
three-dimensional beings have no right in their turn consciously to
affect a two-dimensional world.

The supporters of Zöllner's hypothesis will perhaps reply to
the objection just made, that the plan of creation might, after all,
possibly possess this wonderful peculiarity, that we human beings
perhaps, in some higher condition of culture, will be able to act con-
sciously on two-dimensional worlds, and that at any rate it is simply
an inference by analogy to conclude from the non-existence of a rela-
tion between three and two dimensions that the same relation is also
wanting in the case of four and three dimensions. As a matter of fact,
the objection above made is not intended to refute Zöllner's hypoth-
esis, but only to stamp it as very improbable. But despite this im-
probability Zöllner would still be right if the phenomena of the ma-
terial world actually made his hypothesis necessary. That, however,
is by no means the case. Although most of the phenomena to which
the spiritualists appeal are probably founded on sense-illusions,
humbug, and self-deception, it cannot be denied that there possibly
do exist phenomena which cannot be brought into harmony with
the natural laws now known. There always have been mysteriou«
phenomena, and there always will be. Yet, as we have often seen
that the progress of science has again and again revealed as natura'
what former generations held to be supernatural, it is certainly
wholly wrong to bring in for the explanation of phenomena whicb
now seem mysterious an hypothesis like that of Zöllner, by which

no THE- FOURTH DIMENSION.

everything in the world can be explained. If we adopt a point of
view which regards it as natural for spirits arbitrarily to interfere in
the workings of the world, all scientific investigation will cease, for
we could never more trust or rely upon any chemical or physical ex-
periment, or any botanical or zoological culture. If the spirits are
\he authors of the phenomena that are mysterious to us, why should
they also not have control of the phenomena which are not myste-
rious? The existence of mysterious phenomena justifies in no man-
ner or form the assumption that spirits exist which produce them.
Would it not be much simpler, if we musf have supernatural in-
fluences, to adopt the naive religious point of view, according to
which everything that happens is traceable to the direct, actual, and
personal interference of a single being which we call God? Things
which formerly stood beyond the sphere of our knowledge and were
regarded as marvellous events, as a storm, for example, now stand
in the most intimate connection with known natural laws. Things
that formerly were mysterious are so no longer. If one hundred and
fifty years ago some scientists were in the possession of our present
knowledge of inductional electricity and had connected Paris and
Berlin with a wire by whose aid one could clearly interpret in
Berlin what another person had at that very moment said in Paris,
people would have regarded this phenomenon as supernatural and
assumed that the originator of this long-distance speaking was in
league with spirits.

We recognise, thus, that the things which are termed super-
natural depend to a great extent on the stage of culture which hu-
manity has reached. Things which now appear to us mysterious,
may, in a very few decades, be recognised as quite natural. This
knowledge, however, is not to be obtained by the lazy assumption
of bands of spirits as the authors of mysterious phenomena, but by
performing what in physics and chemistry is called experiment.
But the first and essential condition of all scientific experimenting
is that the experimenter shall 1 e absolutely master of the conditions
under which the experiment is or is not to succeed. Now, this cri-
terion of scientific experimenting is totally lacking in all spiritualistic
experiments. We can never assign in their case the conditions un-

THE FOURTH DIMENSION. Ill

der which they will or will not succeed. When all the preparations
in a spiritualistic sdancehaye been properly made, but nothing takes
place, the beautiful excuse is always forthcoming , that the "spirits
were not willing," that there were "too many incredulous persons
present," and so forth. Fortunately, in physical experiments these
pretexts are not necessary. By the path of experiment, and not by
that of transcendental speculation, physics has thus made incredible
progress and has piled new knowledge strata on strata upon the
old. Accordingly, the prospect is left that the mysteries which the
conditions and properties of the human soul still present can be
solved more and more by the methods of scientific experiment. To
this end, however, it is especially necessary that the physio-psycho-
logical experiments in question should only be performed by men
who possess the critical eye of inquiry, who are free from the dan-
gers of self-illusion, and who are competent to keep apart from their
experiments all superstition and deception. The history of natural
science clearly teaches that it is only by this road that man can ar-
rive at certain and well-established knowledge. If, therefore, there
really is behind such phenomena as mind-reading, telepathy, and
similar psychical phenomena, something besides humbug and self-
illusion, what we have to do is to study privately and carefully by
serious experiments the success or non-success of such phenomena,
and not allow ourselves to be influenced by the public and dramatic
performances of psychical artists, like Cumberland and his ilk.

The high eminence on which the knowledge and civilisation of
humanity now stands was not reached by the thoughtless employ-
ment of fanciful ideas, nor by recourse to four-dimensional worlds,
but by hard, serious labor, and slow, unceasing research. Let all men
of science, therefore, band themselves together and oppose a solid
front to methods that explain everything that is now mysterious to
us by the interference of independent spirits. For these methods,
owing to the fact that they can explain everything, explain nothing,
and thus oppose dangerous obstacles to the progress of real research,
to which we owe the beautiful temple of modern knowledge.

THE SQUARING OF THE CIRCLE.

AN HISTORICAL SKETCH OF THE PROBLEM FROM THE RE-
MOTEST TIMES.
I.
UNIVERSAL INTEREST IN THE PROBLEM.

T7OR two and a half thousand years, both competent and incom-
-^ petent minds have striven in vain to solve the problem kno\Vn
as the squaring of the circle. Now that geometers have at last suc-
ceeded in giving a rigorous demonstration of the impossibility of
solving the problem with straight edge and compasses, it seems
fitting and opportune to cast a glance into the nature and history
of this very ancient problem. And this will be found the more jus-
tifiable in view of the fact that the squaring of the circle, at least
in name, is very widely known outside of the narrow circle of pro-
fessional mathematicians.

The Proceedings of the French Academy for the year 1775 con-
tain at page 6i a resolution of the Academy not to examine from
that time on, any so-called solutions of the quadrature of the circle.
The Academy was driven to this determination by the overwhelm-
ing multitude of professed solutions of the famous problem which
were sent to it every month, in the year, — solutions which of course
were an invariable attestation of the ignorance and self-conceit of
their authors, but which suffered collectively from the very impor-
tant drawback in mathematics of being wrong. Since that time all
professed solutions of the problem received by the Academy find a
sure and permanent resting-place in the waste-basket, and remain
unanswered for all time. The circle-squarer, however, sees in this

THE SQUARING OF THE CIRCLE. II3

liigh-handed manner of rejection only the envy of the great and
powerful at his grand intellectual discovery. He is determined to
secure recognition, and appeals therefore to the public. The news-
papers must obtain for him the appreciation that scientific societies
have denied. And every year the old mathematical sea-serpent
more than once disports itself in the columns of our newspapers in
the shape of an announcement that Mr. N. N., of P. P., has at last
solved the problem of the quadrature of the circle.

But what manner of people are these circle-squarers, when ex-
amined by the light? Almost always they will be found to be im-
perfectly educated persons, whose mathematical knowledge does
not exceed that of a modern high-school student. It is seldom that
they know accurately what the requirements of the problem are and
what its nature ; they are totally ignorant of the two and a half
thousand years' history of the problem ; and they have no idea
whatever of the important investigations which have been made
with regard to it by great and real mathematicians in every century
down to our own time.*

Yet great as is the quantum of ignorance that circle-squarers
intermix with their intellectual products, the lavish supply of con-
ceit and egotism with which they season their performances is still
greater. I have not far to go to furnish a verification of this. A
book printed in Hamburg in the year 1840 lies before me, in which
the author thanks Almighty God at every second page that He has
selected him and no one else to solve the " problem phenomenal"
of mathematics, "so long sought for, so fervently desired, and at-
tempted by millions." After this modest author has proclaimed
himself the unmasker of Archimedes's deceit, he says : "And thus
it hath pleased our mother Nature to withhold this precious math-
ematical jewel from the eye of human investigation, until she
thought it fitting to reveal truth to simplicity."

This will suffice to show the great fatuity of the author. But
it does not suffice to prove his ignorance. He has no conception

* For the full psychogeny and psychiatry of the circle-squarer see A. De Mor-
g in, A Budget of Paradoxes (London, 1872). — TV

114 1"HE SQUARING OF THE CIRCLE.

of mathematical demonstration ; he takes it for granted that things
are so because they seem so to him. Errors of logic, also, abound
in his book. But, minor fallacies apart, wherein does the real error
of this "unmasker" of Archimedes consist? It requires consider-
able labor to extricate the kernel of the demonstration from the
turgid language and bombastic style in which the author has buried
his conclusions. But it is this. The author inscribes a square in a
circle, circumscribes another about it, then points out that the inside
square is made up of four congruent triangles, whereas the circum-
scribed square is made up of eight such triangles ; from which fact,
seeing that the circle is larger than the one square and smaller than
the other, he draws the bold conclusion that the circle is equal in
area to six such triangles. It is hardly conceivable that a rational
being could infer that something which is greater than 4 and less
than 8 must necessarily be 6. But with a man that attempts the
squaring of the circle this kind of ratiocination is possible.

It is the same with all the other attempted solutions of the
problem ; in all of them either logical fallacies or violations of ele-
mentary arithmetical or geometrical truths can be pointed out
Only they are not always of such a trivial nature as in the book
just mentioned.

Let us now inquire into the origin of this propensity which
leads people to occupy themselves with the quadrature of the circle.

Attention must first be called to the antiquity of the problem.
A quadrature was attempted in Egypt 500 years before the exodus
of the Israelites. Among the Greeks the problem never ceased to
play a part that greatly influenced the progress of mathematics.
And in the middle ages also the squaring of the circle sporadically
appears as the philosopher's stone of mathematics. The problem
has thus never ceased to be dealt with and considered. But it
is not by the antiquity of the problem that circle-squarers are en-
ticed, but by the allurement which everything exerts that is cal-
culated to raise the individual above the mass of ordinary human-
ity, and to bind about his temples the laurel crown of celebrity
Ambition spurred men on in ancient Greece and still spurs them
on in modern times to crack this primeval mathematical nut.

THE SQUARING OF THE CIRCLE. II5

Whether they are competent thereto is a secondary consideration.
They look upon the squaring of the circle as the grand prize of a
lottery that can just as well fall to their lot as that of any other
man. They do not remember that —

"Toil before honor is placed by sagacious decrees of Immortals."

and that it requires years of consecutive study to gain possession of
the mathematical weapons that are indispensably necessary to at-
tack the problem, but which even in the hands of the most distin-
guished mathematical strategists did not suffice to take the strong-
hold.

But why is it, we must further ask, that it happens to be the
squaring of the circle and not some other unsolved mathematical
problem upon which the efforts of people are bestowed who have
no knowledge of mathematics yet busy themselves with mathemati"
cal questions? The question is answered by the fact that the squar-
ing of the circle is about the only mathematical problem that is
known to the unprofessional world, — at least by name. Even among
the Greeks the problem was very widely known outside of mathe-
matical circles. In the eyes of the Grecian layman, as at present
among many of his modern brethren, occupation with this problem
was regarded as the most important and essential business of
mathematicians. In fact, they had a special word to designate this
species of activity, namely, Terpayojvi'^eiv, which means to busy one's
self with the quadrature. In modern times, also, every educated
person, though he be not a mathematician, knows the problem by
name, and knows that it is insoluble, or at least, that despite the
efforts of the most famous mathematicians it has not yet been
solved. For this reason the phrase "to square the circle," is now
generally used in the sense of attempting the impossible.

But in addition to the antiquity of the problem, and the fact
also that it is known to the lay world, there is an important third
factor that induces people to occupy themselves with it. This is the
report that has been current for more than a century now, that the
Academies, the Queen of England, or some other influential person
has offered a large prize to be given to the one that first solves the

Il6 THE SQUARING OF THE CIRCLE.

problem. As a matter of fact, the hope of obtaining this large
prize of money is with many circle-squarers the principal incitement
to action. And the author of the book above referred to begs his
readers to lend him their assistance in obtaining the prizes offered.

Although the opinion is widely current in the unprofessional
world, that professional mathematicians are still busied with the
solution of the problem, this is by no means the case. On the con-
trary, for some two hundred years, the endeavors of many great
mathematicians have been directed solely towards demonstrating
with exactness that the problem is insoluble. It is, as a rule, —
and naturally, — more difficult to prove that a thing is impossible
than to prove that it is possible. And thus it has happened, that
up to within a few years ago, despite the employment of the most
varied and the most comprehensive methods of modern mathemat-
ics, no one succeeded in supplying the wished-for demonstration of
the problem's impossibility. At last, Professor Lindemann, now
of Munich, in June, 1882, succeeded in furnishing a demonstra-
tion,— and the first demonstration, — that it is impossible by em-
ploying only straight edge and compasses to construct a square
that is mathematically exactly equal in area to a given circle. The
demonstration, naturally, was not effected with the help of the old
elementary methods ; for if it were, it would have been accom-
plished centuries ago ; but methods were requisite that were first
furnished by the theory of definite integrals and departments of
higher algebra developed in the last few decades ; in other words,
it required the direct and indirect preparatory labor of many cen-
turies to make finally possible a demonstration of the insolubility
of this historic problem.

Of course, this demonstration will have no more effect than
the resolution of the Paris Academy of 1775, in causing the fecund
race of circle-squarers to vanish from the face of the earth. In the
future as in the past, there will be people who know nothing of
this demonstration and will not care to know anything, and who
believe that they cannot help succeeding in a matter in which oth-
ers have failed, and that just they have been appointed by Provi-
dence to solve the famous puzzle. But unfortunately the inerad-

THE SQUARING OF THE CIRCLE. I I 7

icable mania for solving the quadrature of the circle has also its
serious side. Circle-squarers are not always so self-satisfied as
the author of the book above mentioned. They often see, or at
least divine, the insuperable difficulties that tower up before them,
and the conflict between their aspirations and their performances,
the consciousness that the problem they long to solve they are un-
able to solve, darkens their soul and, lost to the world, they be-
come interesting subjects for the science of psychiatry.

II.

NATURE OF THE PROBLEM.

It is easy to determine the length of the radius of a circle, or
the length of its diameter, which must be double that of the radius;
and the question next arises, what is the number that tells how many
times larger the circumference of the circle, that is the length of the
circular line, is than its radius or its diameter. From the fact that
all circles have the same shape it follows that this proportion will
be the same for all circles both large and small. Now, since the
time of Archimedes, all civilised nations that have cultivated math-
ematics have denoted the number that tells how many times larger
the circumference of a circle is than the diameter by the symbol n,
— the Greek initial letter of the word periphery.* To compute n,
therefore, means to calculate how many times larger the circumfer-
ence of a circle is than its diameter. This calculation is called
"the numerical rectification of the circle."

Next to the calculation of the circumference, the calculation of
the superficial contents of a circle by means of its radius or diam-
eter is perhaps most important ; that is, the computation of how
great an area that part of a plane which lies within a circle meas-
ures. This calculation is called the "numerical quadrature." It
depends, however, upon the problem of numerical rectification ;
that is, upon the calculation of the magnitude of n. For it is de-
monstrated in elementary geometry, that the area of a circle is

*The Greek symbol »r was first employed by W. Jones in 1706 and did not
:ome into general use until about the middle of the eighteenth century through
(Iv works of Euler. — Trovs

Il8 THE SQUARING OF THE CIRCLE.

equal to the area of a triangle produced by drawing in the circle a
radius, erecting at the extremity o£ the same a tangent, — that is, in
this case, a perpendicular, — cutting off upon the latter the length
of the circumference, measuring from the extremity, and joining
the point thus obtained with the centre of the circle. It follows
from this that the area of a circle is as many times larger than the
square upon its radius as the number n amounts to.

The numerical rectification and numerical quadrature of the
circle based upon the computation of the number n are to be clearly
distinguished from problems that require a straight line equal in
length to the circumference of a circle, or a square equal in area to
a circle, to be constructively produced from its radius or its diameter;
problems which might properly be called "constructive rectifica-
tion" or "constructive quadrature." Approximately, of course, by
^jmploying an approximate value for 7t, these problems are easily
iolvable. But to solve a problem of construction in geometry,
means to solve it with mathematical exactitude. If the value n
were exactly equal to the ratio of two whole numbers to each
other, the constructive rectification would present no difficulties.
For example, suppose the circumference of a circle were exactly
3^ times greater than its diameter ; then the diameter could be di-
vided into seven equal parts, which could easily be done by the
principles of planimetry with straight edge and compasses ; then
by prolonging to the amount of such a part a straight line exactly
three times as long as the diameter, we should obtain a straight
line exactly equal to the circumference of the circle. But as a mat-
ter of fact, — and this has actually been demonstrated, — there do
not exist two whole numbers, be they ever so great, that exactly
represent by their proportion to each other the number n. Con-
sequently, a rectification of the kind just described does not attain
the object desired.

It might be asked here, whether from the demonstrated fact
that the number n is not equal to the ratio of two whole numbers
however great, it does not immediately follow that it is impossible
to construct a straight line exactly equal in length to the circum-
ference of a circle ; thus demonstrating at once the impossibility of

THE SQUARING OF THE CIRCLE. II9

solving the problem. This question is to be answered in the nega-
tive. For in geometry there can easily exist pairs of lines of which
the one can be readily constructed from the other, notwithstanding
the fact that no two whole numbers can be found to represent the
ratio of the two lines. The side and the diagonal of a square, for
instance, are so constituted. It is true the ratio of the latter two
magnitudes is nearly that of 5 to 7. But this proportion is not
exact, and there are in fact no two numbers that represent the ratio
exactly. Nevertheless, either of these two lines can be readily con-
structed from the other by employing only straight edge and com-
passes. This might be the case, too, with the rectification of the
circle ; and consequently from the impossibility of representing n
by the ratio between two whole numbers the impossibility of the
problem of rectification is not inferable.

The quadrature of the circle stands and falls with the problem
of rectification. This rests upon the truth above mentioned, that
a circle is equal in area to a right-angled triangle, in which one
side is equal to the radius of the circle and the other to the circum-
ference. Supposing, accordingly, that the circumference of the circle
had been rectified, then we could construct this triangle. But every
triangle, as we know from plane geometry, can, with the help ol
straight edge and compasses be converted into a square exactly
equal to it in area. So that, supposing the rectification of the cir-
cumference of a circle to have been successfully effected, a square
could be constructed that would be exactly equal in area to the
circle.

The dependence upon one another of the three problems of the
computation of the number n, the quadrature of the circle, and its
rectification, thus obliges us, in dealing with the history of the
quadrature, to regard investigations with respect to the value of n
and attempts to rectify the circle as of equal importance, and to
consider them accordingly.

We have used repeatedly in the course of this discussion the
expression " to construct with straight edge and compasses." It
will be necessary to explain what is meant by the specification of
these two instruments. When to a requirement in geometry to

l^u THE SQUARING OF THE CIRCLE.

construct a figure there are so large a number of conditions an-
nexed that the construction of only one figure or a limited number
of figures is possible in accordance with those conditions ; such a
full and stated requirement is called a problem of construction, or
briefly a problem. When a problem of this kind is presented for
solution it is necessary to reduce it to simpler problems, already
recognised as solvable ; and since these latter depend in their turn
upon other, still simpler problems, we are finally brought back to
certain fundamental problems upon which the rest are based but
which are not themselves reducible to problems less simple. These
fundamental problems are, so to speak, the lowermost stones of the
edifice of geometrical construction. The question next arises as to
what problems may be properly regarded as fundamental; and it
has been found, that the solution of a great part of the problems
that arise in elementary plane geometry rests upon the solution of
■)nly five original problems. They are :

1. The construction of a straight line that shall pass through
'wo given points.

2. The construction of a circle the centre of which is a given
point and the radius of which has a given length.

3. The determination of the point lying coincidently on two
given straight lines prolonged as far as necessary, — in case such a
point (point of intersection) exists.

4. The determination of the two points that lie coincidently
on a given straight line and a given circle, — in case such common
points (points of intersection) exist.

5. The determination of the two points that lie coincidently on
two given circles, — in case such common points (points of inter-
section) exist.

For the solution of the three last of these five problems the
eye alone is needed, while for the solution of the first two prob-
lems, besides pencil, ink, chalk, or the like, additional special in-
struments are required : for the solution of the first problem a
straight edge or ruler is most generally used, and for the solution
of the second a pair of compasses. But it must be remembered
tliat it is no concern of geometry what mechanical instruments are

THE SQUARING OF THE CIRCLE, 121

employed in the solution of the five problems mentioned. Geom-
etry simply limits itself to the presupposition that these problems
are solvable, and regards a complicated problem as solved if, upon
a specification of the constructions of which the solution consists,
no other requirements are demanded than the five above mentioned.
Since, accordingly, geometry does not itself furnish the solution of
these five problems, but rather exacts them, they are termed postu-
lates * All problems of plane geometry are not reducible to these
five problems alone. There are problems that can be solved only
by assuming other problems as solvable which are not included in
the five given ; for example, the construction of an ellipse, having
given its centre and its major and minor axes. Many problems,
however, possess the property of being solvable with the assistance
of the above-formulated five postulates alone, and where this is the
case they are said to be "constructible with straight edge and com-
passes," or "elementarily" constructible.

After these general remarks upon the solvability of problems
of geometrical construction, which an understanding of the history
of the squaring of the circle makes indispensable, the significance
of the question whether the quadrature of the circle is or is not
solvable, that is elementarily solvable, will become intelligible.
But the conception of elementary solvability only gradually took
clear form, and we therefore find among the Greeks as well as
among the Arabs endeavors, successful in some respects, that aimed
at solving the quadrature of the circle with other expedients than
the five postulates. We have also to take these endeavors into
consideration, and especially so as they, no less than the unsuccess-
ful efforts at elementary solution, have upon the whole advanced
the science of geometry, and contributed much to the clarification
of geometrical ideas.

♦Usually geometers mention only two postulates (Nos. i and 2). But since to
geometry proper it is indifferent whether only the eye, or additional special mechan-
ical instruments are necessary, the author has regarded it more correct in point of
method to assume hve postulates.

122 THE SQUARING OF THE CIRCLE.

III.
THE EGYPTIANS, BABYLONIANS, AND GREEKS.
In the oldest mathematical work that we possess we. find a rule
telling us how to construct a square which is equal in area to a
given circle. This celebrated book, the Rhind Papyrus of the Brit-
ish Museum, translated and explained by Eisenlohr (Leipsic, 1877),
was written, as stated in the work itself, in the thirty-third year of
the reign of King Ra-a-us, by a scribe of that monarch, named
Ahmes. The composition of the work falls accordingly in the period
of the two Hyksos dynasties, that is, in the period between 2000
and 1700 B. C. But there is another important circumstance to be
noted. Ahmes mentions in his introduction that he composed his
work after the model of old treatises, written in the time of King
Raenmat; whence it appears that the originals of the mathematical
expositions of Ahmes are half a thousand years older still than the
Rhind Papyrus.

The rule given in this papyrus for obtaining a square equal to
a circle specifies that the diameter of the circle shall be shortened
one-ninth of its length and upon the shortened line thus obtained
a square erected. Of course, the area of a square of this construc-
tion is only approximately equal to the area of the circle. An idea
may be obtained of the degree of exactness of this original, primi-
tive quadrature by remarking, that if the diameter of the circle in
question is one metre in length, the square that is supposed to be
equal to the circle is a little less than half a square decimetre too
large ; an approximation not so accurate as that made by Archi-
medes, yet much more correct than many a one later employed. It
is not known how Ahmes or his predecessors arrived at this ap-
proximate quadrature ; but it is certain that it was handed down in
Egypt from century to century, and in late Egyptian times it ap-
pears repeatedly.

In addition to the effort of the Egyptians, we also find in pre-
Grecian antiquity an attempt at circle-computation among the Ba-
bylonians. This is not a quadrature, but is intended as a rectifica-

THE SQUARING OF THE CIRCLE. 1 23

tion of the circumference. The Babylonian mathematicians had
discovered, that if the radius of a circle be successively inscribed
as a chord within its circumference, after the sixth inscription we
arrive at the point from which we set out, and they concluded from
this that the circumference of a circle must be a little larger than a
line which is six times as long as the radius, that is three times as
long as the diameter. A trace of this Babylonian method of com-
putation may even be found in the Bible; for in i Kings vii. 23,
and 2 Chron. iv. 2, the great laver is described, which under the
name of the "molten sea" constituted an ornament of the temple
of Solomon ; and it is said of this vessel that it measured ten cubits
from brim to brim, and thirty cubits round about. The number 3
as the ratio of the circumference to the diameter is still more plainly
given in the Talmud, where we read that "that which measures
three lengths in circumference is one length across."

With regard to the earlier Greek mathematicians — as Thales
and Pythagoras— we know that they acquired their elementary
mathematical knowledge in Egypt. But nothing has been handed
down to us which shows that they knew of the old Egyptian quad
rature, or that they dealt with the problem at all. But ^tradition
says, that, subsequently, the teacher of Euripides and Pericles, the
great philosopher and mathematician Anaxagoras, whom Plato so
highly praised, "drew the quadrature of the circle" in prison, in the
year 434 B. C. This is the account of Plutarch in the seventeenth
chapter of his work De Exilio. The method is not told us in which
Anaxagoras is supposed to have solved the problem, and it is not
said whether, knowingly or unknowingly, he gave an approximate
solution after the manner of Ahmes. But at any rate, to Anaxago-
ras belongs the merit of having called attention to a problem that
was to bear rich fruit by inciting Grecian scholars to busy them-
selves with geometry, and thus more and more to advance that
science.

Again, it is reported that the mathematician Hippias of Elis
invented a curved line that could be made to serve a double pur-
pose : first, to trisect an angle, and second to square the circle.
This curved line is the TCTpaywvt^ouo-a so often mentioned by the

1 24 THE SQUARING OF THE CIRCLE.

later Greek mathematicians, and by the Romans called "quadrat
rix." Regarding the nature of this curve we have exact knowledge
from Pappus. But it will be sufficient here to state that the quad-
ratrix is not a circle nor a portion of a circle, so that its construc-
tion is not possible by means of the postulates enumerated in the
preceding section. And therefore the solution of the quadrature
of the circle founded on the construction of the quadratrix is not
an elementary solution in the sense discussed in the last section.
We can, it is true, conceive a mechanism that will draw this curve
as well as compasses draw a circle ; and with the assistance of a
mechanism of this description the squaring of the circle is solvable
with exactitude. But if it be allowed to employ in a solution an
apparatus especially adapted thereto, every problem may be said to
be solvable. Strictly taken, the invention of the curve of Hippias
substitutes for one insuperable difficulty another equally insuper-
able. Some time afterwards, about the year 350 B. C, the mathe-
matician Dinostratus showed that the quadratrix could also be used
to solve the problem of rectification, and from that time on this
problem plays almost the same role in Grecian mathematics as the
related problem of quadrature.

As these problems gradually became known to the non-math-
ematicians of Greece, attempts at solution at once sprang up
that are worthy of a place by the side of the solutions of modern
amateur circle-squarers. The Sophists especially believed them-
selves competent by seductive dialectic to take the stronghold that
had defied the intellectual onslaughts of the greatest mathemati-
cians. With verbal nicety, amounting to puerility, it was said that
the squaring of the circle depended upon the finding of a number
which represented in itself both a square and a circle ; a square by
being a square number, a circle in that it ended with the same
number as the root number from which, by multiplication with it-
self, it was produced. The number 36, accordingly, was, as they
thought, the one that embodied the solution of the famous prob-
lem.

Contrasted with this twisting of words the speculations of Bry-
son and Antiphon, both contemporaries of Socrates, though inex-

THE SQUARING OF THE CIRCLE. 1 25

act, appear in a high degree intelligent. Ant .fion divided the
circle into four equal arcs, and by joining the points of division ob-
tained a square ; he then divided each arc again into two equal
parts and thus obtained an inscribed octagon ; thence he constructed
an inscribed i6-gon, and perceived that the figure so inscribed
more and more approached the shape of a circle. In this way, he
said, one should proceed, until there was inscribed in the circle a
polygon whose sides by reason of their smallness should coincide
with the circle. Now this polygon could, by methods already
taught by the Pythagoreans, be converted into a square of equal
area ; and upon the basis of this fact Antiphon regarded the squar-
ing of the circle as solved.

Nothing can be said against this method except that, however
far the bisection of the arcs is carried, the result still remains an
approximate one.

The attempt of Bryson of Heraclea was better still ; for this
scholar did not rest content with finding a square that was very
little smaller than the circle, but obtained by means of circum-
scribed polygons another square that was very little larger than the
circle. Only Bryson committed the error of believing that the area
of the circle was the arithmetical mean between an inscribed and a
circumscribed polygon of an equal number of sides. Notwith-
standing this error, however, to Bryson belongs the merit — first, of
having introduced into mathematics by his emphasis of the neces-
sity of a square which was too large and one which was too small,
the conception of upper and lower "limits" in approximations;
and secondly, by his comparison of the regular inscribed and cir-
cumscribed polygons with a circle, of having indicated to Archime-
des the way by which an approximate value of n was to be reached.

Not long after Antiphon and Bryson, Hippocrates of Chios
treated the problem, which had now become more and more fa-
mous, from a new point of view. Hippocrates was not satisfied
with approximate equalities, and searched for curvilinearly bounded
plane figures which should be mathematically equal to a recti-
linearly bounded figure, and which therefore could be converted by
straight edge and compasses into a square equal in area. First,

126 THE SQUARING OF THE CIRCLE.

Hippocrates found that the crescent-shaped plane figure produced
by drawing two perpendicular radii in a circle and describing upon
the line joining their extremities a semicircle, is exactly equal in
area to the triangle that is formed by this line of junction and the
two radii ; and upon the basis of this fact the endeavors of this un-
tiring scholar were directed towards converting a circle into a cres-
cent. Naturally he was unable to attain this object, but by his ef-
forts he discovered many new geometrical truths ; among others
being the generalised form of the theorem mentioned, which bears
to the present day the name of lunulae Hippocratis, the lunes of
Hippocrates. Thus, in the case of Hippocrates, it appears in the
plainest light, how precisely the insolvable problems of science are
qualified to advance science ; in that they incite investigators to
devote themselves with persistence to its study and thus to fathom
its utmost depths.

Following Hippocrates in the historical line of the great Gre-
cian geometricians comes the systematist Euclid, whose rigid form-
ulation of geometrical principles has remained the standard presen-
tation down to the present century. The Elements of Euclid,
however, contain nothing relating to the quadrature of the circle
or to circle-computation. Comparisons of surfaces which relate to
the circle are indeed found in the work, but nowhere a computa-
tion of the circumference of a circle or of the area of a circle. This
palpable gap in Euclid's system was filled by Archimedes, the
greatest mathematician of antiquity.

Archimedes was born in Syracuse in the year 287 B. C, and
devoted his life, which was spent in that city, to the mathematical
and the physical sciences, which he enriched with invaluable con-
tributions. He lived in Syracuse till the taking of the town by
Marcellus, in the year 212 B. C, when he fell by the hand of a Ro-
man soldier whom he had forbidden to destroy the figures he had
drawn in the sand. To the greatest performances of Archimedes
the successful computation of the number n unquestionably be-
longs. Like Bryson he started with regular inscribed and circum-
scribed polygons. He showed how it was possible, beginning with
the perimeter of an inscribed hexagon, which is equal to six radii,

THE SQUARING OF THE CIRCLE. 1 27

to obtain by calculation the perimeter of a regular dodecagon, and

then the perimeter of a figure having double the number of sides of

that, and so on. Treating, then, the circumscribed polygons in a

similar manner, and proceeding with both series of polygons up to

a regular 96-sided polygon, he discovered on the one hand that the

ratio of the perimeter of the inscribed 96-sided polygon to the

diameter was greater than 6336 : 2017^, and on the other hand, that

the corresponding ratio with respect to the circumscribed 96-sided

polygon was smaller than 14688:4673^. He inferred from this,

that the number n, the ratio of the circumference to the diameter,

was greater than the fraction ^'„, and smaller than 77^1-rr. Reducing
"^ 2017^ 4673^ °

the two limits thus found for the value of n, Archimedes then
showed that the first fraction was greater than 3^-^, and that the
second fraction was smaller than 3^, whence it followed with cer-
tainty that the value sought for n lay between 3^ and 3^^. The
larger of these two approximate values is the only one usually
learned and employed. That which fills us with most astonish-
ment in the case of Archimedes's computation of n, is, first, the
great acumen and accuracy displayed by him in all the details of
the computation, and secondly the unwearied perseverance which
he exercised in calculating the limits of n without the help of the
Arabian system of numerals and the decimal notation. For it must
be considered that at many stages of the computation what we call
the extraction of roots was necessary, and that Archimedes could
only by extremely tedious calculations obtain ratios that expressed
approximately the roots of given numbers and fractions.*

With regard to the mathematicians of Greece that follow Archi-
medes, all refer to and employ the approximate value of 3^ for ;r,
without, however, contributing anything essentially new to the
problems of quadrature and of cyclometry. Thus Hero of Alex-
andria, the father of surveying, who flourished about the year
100 B. C, employs for purposes of practical measurement some-

* For Archimedes's actual researches, see Rudio, Archimedes, Huygens, Lam-
bert, Legendre, vier Abhand. über die Kreismessung (Leipsic, 1892), where
translations of the works of these four authors on cyclometry will be found.- -7r.

128 THE SQUARING OF THE CIRCLE.

times the value 3^ for n and sometimes even the rougher ap-
proximation ;r^3. The astronomer Ptolemy, who lived in Alex-
andria about the year 150 A. D. , and who was famous as being the
author of the planetary system universally recognised as correct
down to the time of Copernicus, was the only one who furnished a
more exact value ; this he designated, in the sexagesimal system of
fractional notation which he employed, by 3, 8, 30, — that is 3 and
g\ and jf fo' o"^ ^s we now say 3 degrees, 8 minutes (^partes minutat
primae), and 30 seconds {partes niinutae secundae). As a matter of
fact, the expression 3 + A + ?f ¥7 = 3tW represents the number n
more exactly than 3^ ; but on the other hand, is, by reason of the
magnitude of the numbers 17 and 120 as compared with the num-
bers I and 7, more cumbersome.

IV.

THE ROMANS, HINDUS, CHINESE, ARABS, AND THE CHRISTIAN
NATIONS TO THE TIME OF NEWTON.

In the mathematical sciences, more than in any other, the Ro-
mans stood upon the shoulders of the Greeks. Indeed, with re-
spect to cyclometry, they not only did not add anything new to the
Grecian discoveries, but frequently even evinced that they either
did not know of the beautiful result obtained by Archimedes, or at
least could not appreciate it. For instance, Vitruvius, who lived
during the time of Augustus, computed that a wheel 4 feet in diam-
eter must measure i2|^ feet in circumference; in other words, he
made n equal to 3^. And, similarly, a treatise on surveying, pre-
served to us in the Gudian manuscript of the library of Wolfen-
büttel, contains the following instructions for squaring the circle :
Divide the circumference of a circle into four parts and make one
part the side of a square ; this square will be equal in area to the
circle. Apart from the fact that the rectification of the arc of a
circle is requisite to the construction of a square of this kind, the
Roman quadrature, viewed as a calculation, is more inexact even
than any other known computation ; for its result is that n=.^.

The mathematical performances of the Hindus were not only
greater than those of the Romans, but in certain directions sui-

THE SQUARING OK THE CIRCLE, I 29

passed even those of the Greeks. In the most ancient source of
the mathematics of India that we know of, the Culvasutras, which
date back to a little before our chronological era, we do not find, it
is true, the squaring of the circle treated of, but the opposite prob-
lem is dealt with, which might fittingly be termed the circling of
the square. The half of the side of a given square is prolonged in
length one third of the excess of half the diagonal over half the
side, and the line thus obtained is taken as the radius of the circle
equal in area to the square. The simplest way to obtain an idea
of the exactness of this construction is to compute how great n
would have to be if the construction were exactly correct. We
find out in this way that the value of n upon which the Indian cir-
cling of the square is based, is about from five to six hundredths
smaller than the true value, whereas the approximate n of Archi-
medes, 34^, is only from one to two thousandths too large, and that
the old Egyptian value exceeds the true value by from one to two
hundredths.

Cyclometry very probably made great advances among the
Hindus in the first four or five centuries of our era ; for Aryabhatta,
who lived about the year 500 after Christ, states, that the ratio of
the circumference to the diameter is 62832:20000, an approxima-
tion that in exactness surpasses even that of Ptolemy. The Hindu
result gives 3-1416 for n, while n really lies between 3 141592 and
3-141593. How the Hindus obtained this excellent value is told by
Ganefa, the commentator of Bhaskara, an author of the twelfth
century. Gane9a says that the method of Archimedes was carried
still farther by the Hindu mathematicians ; that by continually
doubling -the number of sides they proceeded from the hexagon to
a polygon of 384 sides, and that by the comparison of the circum-
ferences of the inscribed and circumscribed 384-sided polygons they
found that n was equal to 3927:1250. It will be seen that the value
given by Bhaskara is identical with the value of Aryabhatta. It is
further worthy of remark that the earlier of these two Hindu math-
ematicians does not mention either the value 3| of Archimedes or
the value i^-^^ of Ptolemy, but that the later one knows of both
values and especially recommends that of Archimedes as the most

IßO THE SQUARING OF THE CIRCLE.

useful for practical applications. Strange to say, the good ap-
proximate value of Aryabhatta does not occur in Brahmagupta, the
great Hindu mathematician who flourished in the beginning of the
seventh century ; but we find the curious information in this author
that the area of a circle is exactly equal to the square root of lo
when the radius is unity. The value of n as derivable from this
formula, — a value from two to three hundredths too large, — has
unquestionably arisen on Hindu soil. For it occurs in no Grecian
mathematician ; and Arabian authors, who were in a better position
than we to know Greek and Hindu mathematical literature, declare
that the approximation which makes it equal to the square root of
lo, is of Hindu origin. It is possible that the Hindu people, who
were addicted more than any other to numeral mysticism, sought
to find in this approximation some connection with the fact that
man has ten fingers, and that accordingly ten is the basis of their
numeral system.

Reviewing the achievements of the Hindus generally with re-
spect to the problem of quadrature, we are brought to recognise
that this people, whose talents lay more in the line of arithmetical
computation than in the perception of spatial relations, accom-
plished as good as nothing on the purely geometrical side of the
problem, but that the merit belongs to them of having carried the
Archimedean method of computing n several stages farther, and of
having obtained in this way a much more exact value for it — a cir-
cumstance that is explainable when we consider that the Hindus
are the inventors of our present system of numeral notation, pos-
sessing which they easily outdid Archimedes, who employed the
awkward Greek system.

With regard to the Chinese, this people operated in ancient
times with the Babylonian value for tt, or 3 ; but they possessed
knowledge of the approximate value of Archimedes at least since
the end of the sixth century. Besides this, there appears in a num-
ber of Chinese mathematical treatises an approximate value pe-
culiarly their own, in which 7t^=i-^\ a value, however, which not-
withstanding it is written in larger figures, is no better than that of

THE SQUARING OF THE CIRCLE. I3I

Archimedes. Attempts at the constructive quadrature of the circle
are not found among the Chinese.

Greater were the merits of the Arabians in the advancement of
mathematics; and especially in virtue of the fact that they pre-
served from oblivion the results of both Greek and Hindu research
and handed them down to the Christian countries of the West. The
Arabians expressly distinguished between the Archimedian approx-
imate value and the two Hindu values, the square root of lo and
the ratio 62832:20000. This distinction occurs also in Muhammed
Ibn Musa Alchwarizmi, the same scholar who in the beginning of
the ninth century brought the principles of our present system of
numerical notation from India and introduced it into the Moham-
medan world. The Arabians, however, did not study the numerical
quadrature of the circle only, but also the constructive ; for in-
stance, an attempt of this kind was made by Ibn Alhaitam, who
lived in Egypt about the year 1000 and whose treatise upon the
squaring of the circle is preserved in a Vatican codex, which un-
fortunately has not yet been edited.

Christian civilisation, to which we are now about to pass, pro-
duced up to the second half of the fifteenth century extremely in
significant results in mathematics. Even with regard to our pres-
ent problem we have but a single important work to mention; the
work, namely, of Frankos von Lüttich on the squaring of the circle,
published in six books, but preserved only in fragments. The
author, who lived in the first half of the eleventh century, was
probably a pupil of Pope Sylvester II., who was himself a not in-
considerable mathematician for his time and the author of the most
celebrated geometrical treatise of the period.

Greater interest came to be bestowed upon mathematics, and
especially on the problem of the quadrature of the circle, in the
second half of the fifteenth century, when the sciences again began
to revive. This interest was principally aroused by Cardinal Nico-
las de Cusa, a man highly esteemed for his astronomical and calen-
darial studies. He claimed to have discovered the quadrature of
the circle by employing only straight edge and compasses and thus
attracted the attention of scholars to the historic problem. People

132 THE SQUARING OF THE CIRCLE.

believed the famous Cardinal, and marvelled at his wisdom, until
Regiomontanus, in letters written in 1464 and 1465 and published
ii^ ^533> rigorously demonstrated that the Cardinal's quadrature
was incorrect. The construction of Cusa was as follows. The ra-
dius of a circle is prolonged a distance equal to the side of the in-
scribed square; the line so obtained is taken as the diameter of a
second circle, and in the latter an equilateral triangle is described ;
then the perimeter of the latter is equal to the circumference of the
original circle. If this construction, which its inventor regarded as
exact, be considered as a construction of approximation, it will be
found to be more inexact even than the construction resulting from
the value 7t = 2,\- For by Cusa's method n would be from five to
six thousandths smaller than it really is.

In the beginning of the sixteenth century a certain Bovillius
appears, who also gave the construction of Cusa, — this time with-
out notice. But about the middle of the sixteenth century a book
was published which the scholars of the time at first received with
interest. It bore the proud title De Rebus Mathematicis Hactenus
Desideratis. Its author, Orontius Finaeus, represented that he had
overcome all the difficulties that had ever stood in the way of geo-
metrical investigators ; and incidentally he also communicated to
the world the "true quadrature" of the circle. His fame was short-
lived. For soon afterwards, in a book entitled De Erratis Orontii,
the Portuguese Petrus Nonius demonstrated that Orontius's quad-
rature, like most of his other professed discoveries, was incorrect.

In the succeeding period the number of circle-squarers so in-
creased that we shall have to limit ourselves to those whom mathe-
maticians recognise. And particularly is Simon Van Eyck to be
mentioned, who towards the close of the sixteenth century pub-
lished a quadrature which was so approximate that the value of n
derived from it was more exact even than that of Archimedes ; and
to disprove it the mathematician Peter Metius was obliged to seek
a still more accurate value than 3^. The erroneous quadrature of
Van Eyck was thus the occasion of Metius's discovery that the ra-
tio 355:113» or 3^y*j, varied from the true value of n by less than
one one-millionth, eclipsing accordingly all values hitherto ob-

THE SQUARING OF IHE CIRCLE. 1 33

tained. Moreover, it is demonstrable by the theory of continued
fractions, that, admitting figures to four places only, no two num-
bers more exactly represent the value of n than 355 and 113.

In the same way the quadrature of the great philologist Joseph
Scaliger led to refutations. Like most circle-squarers who believe
in their discovery. Scaliger also was little versed in the elements of
geometry. He solved the famous problem, however, — at least in
his own opinion, — and published in 1592 a book upon it, which
bore the pretentious title Nova Cyclometria, and in which the name
of Archimedes was derided. The baselessness of his supposed dis-
covery was demonstrated to him by the greatest mathematicians of
his time ; namely, Vieta, Adrianus Romanus, and Clavius.

Of the erring circle-squarers that flourished before the middle
of the seventeenth century three others deserve particular mention,
— Longomontanus of Copenhagen, who rendered such great ser-
vices to astronomy, the Neapolitan John Porta, and Gregory of St.
Vincent. Longomontanus made 7r = 3^öVöW' ^^^ was so convinced
of the correctness of his result as to thank God fervently, in the
preface to his work Inventio Quadraturae Circuit, that He had
granted him in his old age the strength to conquer the celebrated
difficulty. John Porta followed the example of Hippocrates and
endeavored to solve the problem by a comparison of lunes. Gregory
of St. Vincent published a quadrature, the error of which was very
hard to detect but was finally discovered by Descartes.

Of the famous mathematicians who dealt with our problem in
the period between the close of the fifteenth century and the time
of Newton, we first meet with Peter Metius, before mentioned, who
succeeded in finding in the fraction 355:113 the best approximate
value for n involving small numbers only. The problem received
a different advancement at the hands of the famous mathematician
Vieta. Vieta was the first to whom the idea occurred of represent-
ing 7t with mathematical exactness by an infinite series of definitely
prescribed operations. By comparing inscribed and circumscribed
polygons, Vieta found that we approach nearer and nearer to tc if
we cause the operations of extracting the square root of \, and
certain related additions and multiplications, to succeed each other

134 THE SQUARING OF THE CIRCLE.

in a certain manner, and that n must come out exactly, if this series
of operations could be continued indefinitely. Vieta thus found
that to a diameter of loooo million units a circumference belongs of
from 31415 million 926535 units to 31415 million 926536 units of
the same length.

But Vieta was outdone by the Netherlander Adrianus Romanus,
who added five additional decimal places to the ten of Vieta. To
accomplish this he computed with unspeakable labor the circum-
ference of a regular circumscribed polygon of 1073741824 sides.
This number is the thirtieth power of 2. Yet great as the labor of
Adrianus Romanus was, that of Ludolf Van Ceulen was still greater;
for the latter calculator succeeded in carrying the Archimedean
process of approximation for the value of n to 35 decimal places ;
that is, the deviation from the true value was smaller than one one-
thousand quintillionth, a degree of exactness that we can have
scarcely any conception of. Ludolf published the figures of the
tremendous computation that led to his result. His calculation
was carefully examined by the mathematician Griemberger and de-
clared to be correct. Ludolf was justly proud of his work, and fol-
lowing the example of Archimedes, requested in his will that the
result of his most important mathematical performance, the com-
putation of 7t to 35 decimal places, be engraved upon his tomb-
stone ; a request which is said to have been carried out. In honor
of Ludolf, 71 is called to-day in Germany the Ludolfian number.

Although through the labor of Ludolf a degree of exactness for
cyclometrical operations was now obtained that was more than suf-
ficient for any practical purpose that could ever arise, neither the
problem of constructive rectification nor that of constructive quad-
rature had been in any respect theoretically advanced thereby. The
investigations conducted by the famous mathematicians and phys-
icists Huygens and Snell about the middle of the seventeenth cen-
tury, were more important from a mathematical point of view than
the work of Ludolf. In his book Cyclometricus Snell took the posi-
tion that the method of comparison of polygons, which originated
with Archimedes and was employed by Ludolf, was not necessarily
the best method of attaining the end sought ; and he succeeded by

THE SQUARING OF THE CIRCLE, I35

employing propositions which state that certain arcs of a circle are
greater or smaller than certain straight lines connected with the
circle, in obtaining methods that make it possible to reach results
like the Ludolfian with much less labor of calculation. The beau-
tiful theorems of Snell were proved a second time, and better
proved, by the celebrated Dutch promoter of the science of optics,
Huygens {^Opera Varia, p. 365 et seq. ; Theoremaia De Ctrcuii ei
Hyperbolae Quadratur a, 165 1), as well as perfected in many ways
by him. Snell and Huygens were fully aware that they had ad-
vanced the problem of numerical quadrature only, and not that of
the constructive quadrature. This plainly appeared in Huygens's
case from the vehement dispute which he conducted with the Eng-
lish mathematician James Gregory. This controversy is significant
for the history of our problem, from the fact that Gregory made
the first attempt to prove that the squaring of the circle with straight
edge and compasses was impossible. The result of the contro-
versy, to which we owe many valuable tracts, was, that Huygens
finally demonstrated in an incontrovertible manner the incorrect-
ness of Gregory's proof of impossibility, adding that he also was of
opinion that the solution of the problem with straight edge and
compasses was impossible, but nevertheless was not himself able
to demonstrate this fact. And Newton later expressed himself to
the same effect. As a matter of fact a period of over 200 years
elapsed before higher mathematics was far enough advanced to
furnish a rigorous demonstration of impossibility.

V.

FROM NEWTON TO THE PRESENT.

Before we proceed to consider the promotive influence which
the invention of the differential and the integral calculus exercised
upon our problem, we shall enumerate a few at least of that never-
ending succession of erring quadrators who delighted the world
with the products of their ingenuity from the time of Newton to
the present ; and out of a pious and sincere regard for the contem-
porary world, we shall omit entirely to speak of the circle-squarers
of our own time.

136 iHK SQUARING OF THE CIRCLE.

First to be mentioned is the celebrated English philosopher
Hobbes. In his book De Problematis Physicis, in which he pro-
poses to explain the phenomena of gravity and of ocean tides, he
also takes up the quadrature of the circle and gives a very trivial
construction, which in his opinion definitively solved the problem.
It made n-='>^\. In view of Hobbes's importance as a philosopher,
two mathematicians, Huygens and Wallis, thought it proper to
refute him at length. But Hobbes defended his position in a spe-
cial treatise, where to sustain at least the appearance of being right,
he disputed the fundamental principles of geometry and the the-
orem of Pythagoras,

In the last century France especially was rich in circle-squarers.
We will mention : Oliver de Serres, who by means of a pair of
scales determined that a circle weighed as much as the square upon
the side of the equilateral triangle inscribed in it, that therefore
they must have the same area, an experiment in which 7r=3;
Mathulon, who offered in legal form a reward of a thousand dol-
lars to the person who would point out an error in his solution ot
the problem, and who was actually compelled by the courts to pay
the money ; Basselin, who believed that his quadrature must be
right because it agreed with the approximate value of Archimedes,
and who anathematised his ungrateful contemporaries, in the con-
fidence that he would be recognised by posterity ; Liger, who
proved that a part is greater than the whole and to whom therefore
the quadrature of the circle was child's play ; Clerget, who based
his solution upon the principle that a circle is a polygon of a defi-
nite number of sides, and who calculated, also, among other things,
how large the point is at which two circles touch.

Germany and Poland also furnish their contingent to the army
of circle-squarers. Lieutenant-Colonel Corsonich produced a quad-
rature in which it equalled 3^, and promised fifty ducats to the per-
son who could prove that it was incorrect. Hesse of Berlin wrote
an arithmetic in 1776, in which a true quadrature was also "made
known," n being exactly equal to 3^^. About the same time Pro-
fessor Bischoff of Stettin defended a quadrature previously pub-
lished by Captain Leistner, Preacher Merkel, and Schoolmaster

THE SQUARING OF THE CIRCLE, I 37

Böhm, which virtually made n equal to the square of ||, not even
attaining the approximation of Archimedes.

From attempts of this character are to be clearly distinguished
constructions of approximation in which the inventor is aware that
he has not found a mathematically exact construction, but only an
approximate one. The value of such a construction will depend
upon two things — first, upon the degree of exactness with which it
is numerically expressed, and secondly on whether the construc-
tion can be easily made with straight edge and compasses. Con
structions of this kind, simple in form and yet sufficiently exact for
practical purposes, have been produced for centuries in great num-
bers. The great mathematician Euler, who died in 1783, did not
think it out of place to attempt an approximate construction of this
kind. A very simple construction for the rectification of the circlt-
and one which has passed into many geometrical text-books is that
published by Kochansky in 1685 in the Leipziger Berichte. It is as
follows: "Erect upon the diameter of a circle at its extremities
perpendiculars ; with the centre as vertex and the diameter as side
construct an angle of 30°; find the point of intersection of the
line last drawn with the perpendicular, and join this point of inter-
section with that point on the other perpendicular which is dis-
tant three radii from the base of the perpendicular. The line of
junction so obtained is very approximately equal to one-half of the
circumference of the given circle." Calculation shows that the dif-
ference between the true length of the circumference and the line
thus constructed is less than y^/^^^^ of the diameter.

Although such constructions of approximation are very inter-
esting in themselves, they nevertheless play but a subordinate role
in the history of the squaring of the circle; for on the one hand
they can never furnish greater exactness for circle-computation
than the thirty-five decimal places which Ludolf found, and on the
other hand they are not adapted to advance in any way the ques-
tion whether the exact quadrature of the circle with straight-edge
■ and compasses is possible.

The numerical side of the problem, however, was considerably
advanced by the new mathematical methods perfected by Newton

138 THE SQUARING OF THE CIRCLE.

and Leibnitz, and known as the differential and the integral cal-
culus.

About the middle of the seventeenth century, before Newton
and Leibnitz represented tt by series of powers, the English mathe-
maticians Wallis and Lord Brouncker, Newton's predecessors in
certain lines, succeeded in representing tt by an infinite series of
figures combined according to the first four rules of arithmetic. A
new method of computation was thus opened. Wallis found that
the fourth part of tt is represented by the regularly formed product

- f Xf XiXf XfXfXfX etc.
more and more exactly the farther the multiplication is continued,
and that the result always comes out too small if we stop at a proper
fraction but too large if we stop at an improper fraction. Lord
Brouncker, on the other hand, represents the value in question by
a continued fraction in which the denominators are all 2 and the
numerators are the squares of the odd numbers. Wallis, to whom
Brouncker had communicated his elegant result without proof, de-
monstrated the same in his Arithmetic of Infinites.

The computation of n could scarcely have been pushed to a
greater degree of exactness by these results than that to which Lu-
dolf and others had carried it by the older and more laborious
methods. But the series of powers derived from the differential
calculus of Newton and Leibnitz furnished a means of computing
7t to hundreds of decimal places.

Gregory, Newton, and Leibnitz found that the fourth part of
7t was equal exactly to

i-i + i-i + l-yV + T»^

if we conceive this series, which is called the Leibnitz series, con-
tinued indefinitely. This series is wonderfully simple but is not
adapted to the computation of n, for the reason that entirely too
many members have to be taken into account to obtain n accurately
to a few decimal places only. The original formula, however, from
which this series is derived, gives other formulae which are excel-
lently adapted to the actual computation. The original formula is
the general series :

THE SQUARING OF THE CIRCLE. I39

where a is the length of the arc belonging to any central angle in a
circle of radius i, and a the tangent to this angle. From this we
derive the following :

-fK«' + '^' + ^'+- ••) — •••.

where a, b, c . . . are the tangents of angles whose sum is 45*. De-
termining, therefore, the values of a, b, c . . . , which are equal to
small and convenient fractions and fulfil the conditions just men-
tioned, we obtain series of powers which are adapted to the com-
putation of 7t.

The first to add by the aid of series of this description addi-
tional decimal places to the old 35 in the number n was the Eng-
lish arithmetician Abraham Sharp, who, following Halley's instruc-
tions, in 1700 worked out n to 72 decimal places. A little later
Machin, professor of astronomy in London, computed n to 100
decimal places, by putting, in the series given above, a^b^c==^d
= \ and (?= — ifi^j that is, by employing the following series:

7C

5 S-S' 5-5' 7-5'

' ' +'

L239 3.2393 • 5.239"

In the year 1819, Lagny of Paris outdid the computation of
Machin, determining in two different ways the first 127 decimal
places of 7t. Vega then obtained as many as 140 places, and the
Hamburg arithmetician Zacharias Dase went as far as 200 places.
The latter did not use Machin's series in his calculation, but the
series produced by putting in the general series above given a = ^,
b = \, c = \. Finally, at a recent date, tc has been computed to
500 places.*

The computation to so many decimal places may serve as an
illustration of the excellence of the modern methods as contrasted
with those anciently employed, but it has otherwise neither a theo-

*In 1873 the approximation was carried by Shanks to 707 places of decimals.
— Tratis.

140 THE SQUARING OF THE CIRCLE.

retical nor a practical '/alue. That the computation of n to say 15
decimal places more than sufficiently satisfies the subtlest require-
ments of practice may be gathered from a concrete example of the
degree of exactner.s thus obtainable. Imagine a circle to be de-
scribed with Berli\i as centre, and the circumference to pass through
Hamburg ; then 'et the circumference of the circle be computed by
multiplying its ''liameter by the value of ;r to 15 deciinal places,
nnd then conce /e it to be actually measured. The deviation from
the true length /n so large a circle as this even could not be as great
as the 18 mill- )nth part of a millimetre.

An idea an hardly be obtained of the degree of exactness pro-
ihiced by ir i decimal places. But the following example may pos-
sibly give ' s some conception of it. Conceive a sphere constructed
with the '' arth as centre, and imagine its surface to pass tlirough
Sirius, vhich is 1342- millions of millions of kilometres distant from
the earth. Then imagine this enormous sphere to be so packed
with microbes that in every cubic millimetre millions of millions of
ihesc diminutive animalcula are present. Now conceive these mi-
:.rcbes to be all unpacked and so distributed singly along a straight
ire, that every two microbes are as far distant from each other as
/'irius from us, that is 134^ million million kilometres. Conceive
ihe long line thus fixed by all the microbes, as the diameter of a
circle, and imagine the circumference of it to be calculated by mul-
tiplying its diameter by n to 100 decimal places. Then, in the
case of a circle of this enormou-s magnitude even, the circumference
so calculated would not vary from the real circumference by a mil-
lionth part of a millimetre.

This example will suffice to show that the calculation of n to
100 or 500 decimal places is wholly useless.

Before we close this chapter upon the evaluation of n, we
must mention the method, less fruitful than curious, which Profes-
sor Wolff of Zurich employed some decades ago to compute the
value of TT to 3 places.* The floor of a room is divided up into equal
squares, so as to resemble a huge chess-board, and a needle ex-

•See also A. De Morgan, A Budget of Paradoxes, pp. 169-171. — Tr.

THE SQUARING OF THE CIRCLE. I4I

actly equal in length to the side of each of these squares, is cast
haphazard upon the floor. If we calculate, now, the probabilities
of the needle so falling as to lie wholly within one of the squares,
that is so that it does not cross any of the parallel lines forming the
squares, the result of the calculation for this probability will be
found to be exactly equal to n — 3. Consequently, a sufi&cient
number of casts of the needle according to the law of large num-
bers must give the value of n approximately. As a matter of fact,
Professor Wolff, after loooo trials, obtained the value of n correctly
to 3 decimal places.

Fruitful as the calculus of Newton and Leibnitz was for the
evaluation of n, the problem of converting a circle into a square
having exactly the same area was in no wise advanced thereby.
Wallis, Newton, Leibnitz, and their immediate followers distinctly
recognised this. The quadrature of the circle could not be solved ;
but it also could not be proved that the problem was insolvable
with straight edge and compasses, although everybody was con-
vinced of its insolvability. In mathematics, however, a conviction
is only justified when supported by incontrovertible proof ; and in
the place of endeavors to solve the quadrature there accordingly
now come endeavors to prove the impossibility of solving the cele-
brated problem.

The first step in this direction, small as it was, was made by
the French mathematician Lambert, who proved in the year 1761
that n was neither a rational number nor even the square root of a
rational number ; that is, that neither n nor the square of n could
be exactly represented by a fraction the denominator and numera-
tor of which are whole numbers, however great the numbers be
taken. Lambert's proof* showed, indeed, that the rectification and
the quadrature of the circle could not be accomplished in one par-
ticular simple way, but it still did not exclude the possibility of the
problem being solvable in some other more complicated way, and
without requiring further aids than straight edge and compasses.

♦Given in Legendre's Geometry, in the Appendix to De Morgan, of. cit., p
495, and in Rudio, op. cit. — TV.

142 THE SQUARING OF THE CIRCLE.

Proceeding slowly but surely it was next sought to discover
the essential properties which distinguish problems solvable with
straight edge and compasses from problems the construction of
which is elementarily impossible, that is by employing the postu-
lates only. Slight reflection showed, that a problem, to be elemen-
tarily solvable, must always be such that the unknown lines of its
figure are connected with the known lines by an equation for the
solution of which equations of the first and second degree only are
requisite, and which can be so arranged that the measures of the
known lines will appear as integers only. The conclusion to be
drawn from this was that if the quadrature of the circle and conse-
quently its rectification were solvable elementarily, the number n,
which represents the ratio of the unknown circumference to the
known diameter, must be the root of a certain equation, of a very
high degree perhaps, but in which all the numbers are whole num-
bers ; that is, there would have to exist an equation, made up en-
tirely of whole numbers, which would be correct if its unknown
quantity were made equal to n.

Since the beginning of this century, consequently, the efforts
of a number of mathematicians have been bent upon proving that
n generally is not algebraical, that is, that it cannot be the root of
an equation having whole numbers for coefficients. But mathe-
matics had to make tremendous strides forward before the means
were at hand to accomplish this demonstration. After the French
Academician, Professor Hermite, had furnished important prepara-
tory assistance in his treatise Sur la Fonction Exponentielle, pub-
lished in the seventy-seventh volume of the Comptes Rendus, Pro-
fessor Lindemann, at that time of Freiburg, now of Munich, finally
succeeded, in June 1882, in rigorously demonstrating that the num-
ber TT is not algebraical,* -and so supplied the first proof that the

* For the benefit of my mathematical readers I shall present here the most
important steps of Lindemann's demonstration. M. Hermite in order to prove the
transcendental character of

developed relations between certain definite integrals (Comptes Rendus o£ the
Paris Academy, Vol. 77, 1873). Proceeding from the relations thus established,

THE SQUARING OF THK CIRCLE. 143

problems of the rectification and squaring of the circle with the
help only of algebraical instruments like straight edge and com-
passes are insolvable. Lindemann's proof appeared successively
in the Reports of the Berlin Academy (June, 1882), in the Comptes
Rendus of the French Academy (Vol. 115, pp. 72 to 74), and in the
Mathematische Annalen (Vol. 20, pp. 213 to 225).

"It is impossible with straight edge and compasses to con-
struct a square equal in area to a given circle." These are the
words of the final determination of a controversy which is as old as
the history of the human mind. But the race of circle-squarers,
unmindful of the verdict of mathematics, the most infallible of
arbiters, will never die out as long as ignorance and the thirst for
glory remain united.

Professor Lindemann first demonstrates the following proposition : If the coeffi
cients of an equation of the wth degree are all real or complex whole numbers and
the n roots of this equation ^j, ?j, . . ., Zn are different from zero and from each
other it is impossible for

<.'l-|_^**-|-g»S. . . .-f ^'•

to be equal to y , where a and b are real or complex whole numbers. It is then
shown that also between the functions

where r denotes an integer, no linear equation can exist with rational coefficients
different from zero. Finally the beautiful theorem results : If ^ is the root of an
irreducible algebraic equation the coefiBcients of which are real or complex whole
numbers, then e' cannot be equal to a rational number. Now in reality e'^~^ is
equal to a rational number, namely, — i. Consequently, irv — i, and therefore tt
itself, cannot be the root of an equation of the nth degree having whole numbers
for coefiBcients, and therefore also not of such an equation having rational coefiB-
cients. The property last mentioned, however, n would have if the squaring of the
circle with straight edge and compasses were possible. [The questions involved
in the discussions of the last three pages have been excellently treated by Klein in
Famous Problems of Elementary Geometry recently translated by Beman and
Smith (Ginn & Co., Boston). Lindemann's proof is here presented in a simplified
form, and so brought within the comprehension of students conversant only with
algebra. — 7V.]

INDEX.

Abstraction, 95.

Academy, French, 112.

Addition, 76 ; associative law of, 10 ;

commutative law of, 9.
Ahmes, 122.

Alch warizmi, Muhammed Ibn Musa, 131.
Algebraic curves, theory of, 35.
Alhaitam, Ibn, 131.
Analytical geometry, 67.
Anaxagoras, 123.
Antiphon, 125.
ApoUonius, 27, 34.

Approximation, constructions of, 137.
Arabs, magic squares of, 42 ; squaring

of the circle among, 128 et seq.
Archimedes, 27, 113, 117, 126 et seq.
Arithmetic, 36; pure, 8 et seq.; monism

in, 8-26.
Aryabhatta, 129.
Association, laws of, 18.
Associative law of addition, 10.
Augend, 10.

Axioms, Euclid's, 79 et seq.
Aztecs, 5, 73.

Babylonians, squaring of the circle

among, 122 et seq.
Bacteriology, 36.
Basselin, 136.
Beman, W. W., 143.
Bhaskara, 129.
Bible, speaks of four dimensions, 106 ;

squaring of the circle in the, 123.
Bischofif, Professor, 136.
Böhm, 137.
Bolyai, 37, 79.

Bovillius, 132.
Brahmagupta, 130.
Brouncker, Lord, 13S.
Bryson, 125.

Cagliostro, 102.

Calculus, Differential, 29.

Cardan, 13.

Cards, problem at, 51 et seq.

Chemistry, 37; application of the idea of
multi-dimensioned space to theoret-
ical, 88.

Chess-knight, magic squares that m-
volve the move of the, 57 et seq.

Chinese, 130; squaring of the circle
among, 128 et seq.

Christian nations, squaring of the circle
among, 128 et seq.

Cipher, 13.

Circle-squarers, 113 et seq.

Circumversion, congruence by, 81.

Clairvoyance, 103.

Clavius, 133.

Clebsch, Alfred, 35.

Clerget, 136.

Clocks, 4.

Commutation, 15-18.

Commutative law of addition. 9.

Complex numbers, 24, 35.

Concentric magic squares, 55 et seq.

Configurations, 88.

Congruent figures, 80.

Conic sections, 34.

Construction, problems of, 118.

Continuous functions, 28.

Copernicus, 94.

146

Corsonich, Lieutenant Colonel, 136.
Counting, i et seq , 9, 72 et seq.
Cranz, 65, 101.
Culvasütras, 129.
Cumberland, 11 1.

Curvature, negative, 81 at seq.; posi-
tive, 81 et seq. ; zero, 82 ; of space, 85.
Cyclometry, 127 et seq.

Dase, Zacharias, 139.

Davis, the American visionary, x02.

De Cusa, Nicolas, 131.

Definitional formulae, 11, 21.

Degrees, 128.

Degree, numbers of, ig.

De la Hire, 45, 46, 51, 53.

De Meziriac, Bachet, 45, 46.

De Morgan, 41, 113, 140, 141.

Depth, 94.

Descartes, 67.

De Series, Oliver, 136.

Detraction, 11.

Difference, 11.

Differential forms, 13 et seq., 74.

Dimension, concept of, 65 et seq.

Dinostratus, 124.

Displacement, congruence by, 81.

Distance of objects, 96.

Dumas, Alexander, 102.

Duplication of the cube, 30.

Du Prel, Carl, 105.

Dürer, Albert, 40, 43, 45.

Egyptians, the squaring of the circle

among, 122 et seq.
Eisenlohr, 29, 122.
Equations between x, y, and z, 70.
Ether, 75 et seq.
Etruscans, 3.
Euclid, 28, 37, 78, 126.
Euclid's axioms, 79-82.
Euler, 117, 137.
Evolution, 20, 21.
Extension of notions in mathematics, 75

et seq.

Finaeus, Orontius, 132.

Fingers, x et seq. ; in reckoning, 4 et

seq.
Five as a numeral basis, 5.

Foundation-principle, 14.
Four-dimensional point-aggregates, "72

et seq.
Four dimensioned space, refutation of

the existence of, 89 et seq.
Fourth degree, operation of, 76.
Fourth dimension, the, 64-1 11.
Four variable quantities, 77.
Fractional numbers, negative, 18.
Fractions, 17 et seq.
Functions, theory of, 35.

Gane§a, 129.

Gauss, 37, 79, 99.

Geometry, 37; origin of, 33.

Glaukon, 98.

Grassmann, 9.

Greeks, 3, 13; squaring of the circle

among, 115, 122 et seq.
Gregory of St. Vincent, 133.
Gregory, James, 135, 138.
Gudian manuscript, 128.

Halley, 139.

Hamlet, 41.

Hankel, 9, 14.

Harmony of the spheres, 63.

Have, Mr., loi.

Helmholtz, 37, 79.

Hermite, Professor, 142.

Herodotus, 33.

Hesse, 136.

Hindus, 13; mathematical performances

of the, 128 et seq.; squaring of the

circle among, 128 et seq.
Hippias, 123, 124.
Hippocrates, 125.
History, 33.
Hobbes, 136.
Homer, 84.
Hugel, 61.
Hume, 39.
Huygens, 135, 136.

Imaginary numbers, 23, 35.
Impossibility of demonstration, 116.
Increment, 10.
Infinitely great, 19, 85.
Insoluble problems, 30, 143.
Involution, 19, 21, 76.
Irrational numbers, 23.

INDEX.

147

Jones, W., 117.

Kant, 39, 98 et seq.

Kepler, 34.

Kircher, Athanasius, 45.

Klein, Felix, 3r, 143.

Knight, move of a, 52.

Knight-problem, 58.

Knots, experiment with the, 103, 105.

Kochansky, 61, 137.

Kronecker, 7, g.

Lagny, 139.

Lambert, 32, 141.

Law of association, 15.

Legendre, 80, 141.

Leibnitz, 28, 138.

Leistner, 136.

Lessing, 30.

Liger, 136.

Limits, 125.

Lindemann, 31, 116, 142.

Livy, 2.

Lobachevski, 37, 79.

Local value, Hindu system of, 3.

Logarithm, finding of the, 20.

Logarithmand, 21.

Longomontanus, 133.

Ludolf, 137.

Lunes of Hippocrates, 126.

Luther, loi.

Machin, 139.

Magic cubes, 61 et seq.

Magic Squares, 39-63 ; problem and
origin of, 42 ; concentric, 55 et seq ;
even-numbered, 49 et seq. ; odd-num-
bered, 44 et seq, ; with magical parts,

56 et seq.; whose summation gives
the number of a year, 53 et seq. ; that
involve the move of the chess-knight,

57 et secj.

Mathematical knowledge, on the nature
of, 27-38; characteristics of, 35; in-
trinsic character of, 27.

Mathematics, 39 ; most conservative of
all sciences, 27; self-sufficiency of, 32.

Mathulon, 136.

Measuring, 17.

Mechanical instruments ot geometry,

120 et seq.
Mechanics, 38.
Mediums, 107.
Melancholy, 40, 41.
Merkel, 136.

Methods, discovery of, 35.
Metius, Peter, 132, 133.
Mexico, ancient, 3.
Mill, J. S., 39.
Milton, 41.
Minerva, 3.
Minus sign, 11.
\iinutes, 128.

Models of three-dimensional nets, 78.
Alonism in arithmetic, 8-26.
A/om'st, The, 39.
Moschopulus, 45.
Multi-dimensional magnitudes and

spaces, 67 et seq., 77.
Multiplication, 15, 77.
Mysticism, 39.

Named number, 6.

N -dimensional totalities, 69 et seq.

Negative exponents, 20. 21.

Negative numbers, 12 et &eq., 16, iS. 74.

Newton, 28, 34, 138.

Nonius, Petrus, 132.

Notation, systems of, 6.

Number, notation and definition of, 1-7;

denominate, 4 ; abstract, 7 ; as the

result of counting, 9; complex, 24,35;

forms of, 14.
Numbers, as symbols, 7; imaginary, 23,

35; irrational, 23; real, 23.
Numbering, 6.
Number-pictures and signs, natural, 2

at seq.
Number-words, natural, 4.
Numerals, the formation of, 5 et seq.
Numeral words, 4 et seq. ; compound, 4;

in the Indo-Germanic tongues, 5.
Numeral writing, 2 et seq.
Numeration, i et seq.
Numeri ficti, 13.
Numeri veri, 13.

Pappus, 124.

Parallels, theory of, 28, 79 et seq.

148

INDEX.

Parting, 17.

Perception, 95.

Physics, 8, 33, 38.

TT, 117 et seq., 127, 138 et seq.

Plato, 39, 98.

Polygons, magical, 59 et seq.

Porta, John, 133.

Postulates, 121.

Powers, 20.

Prime numbers, problem of the fre-
quency of, 31.

Principle of no exception, 14.

Prizes ofifered circle-squarers, 116.

Probabilities, iai.

Problems, fundamental, 120; solvable
with straight edge and compasses, 142.

Ptolemy, 32, 128, 129.

Pure arithmetic, 8 et seq.

Pythagoras, 27, 123, 136.

Quadratrix, 124.

Quadrature of the circle, in Egypt, 114;

D stands and falls with the problem of

rectification, 119; numerical, 117 et

seq.
Quantities, four variable, 77; negative,

12 et seq.
Quaternions, 25.

Radicand, 20.

Rays, 69.

Real numbers, 23.

Rectification of the circle, numerical,

117 et seq.
Regiomontanus, 132.
Rhind Papyrus, 29, 122.
Riemann, 37, 79, 100.
Riese, Adam, 45.
Romans, 3, 73; squaring of the circle

among, 128 et seq.
Romanus, Adrianus, 133, 134.
Rudio, 127, 141.

Sauveur, 45, 61.
Scaliger, Joseph, 133.
Scheffler, 45, 46, 59, 61.
Schlegel, 65, 78.
Schröder, E. , 9, 11.
Science, iii.

Seconds, 128.

Series, infinite, 133 et seq. ; Leibnitz's,
138.

Shakespeare, 41.

Shanks, 139.

Sharp, Abraham, 139.

Shortest line, 82.

Slade, loi.

Smith, D E., 143.

Snell, 134.

Socrates, 98.

Solvability of problems. 121.

Sophists, 124.

Space, curvature of, 85 ; four-dimen-
sional, 64 ; intuition of, 94 ; of expe-
rience, 83 ; of multiple dimensions,
great service to science, 85.

Spheres in space, 70 et seq.

Spirits, existence of four-dimensional, 98
et seq.; of the dead, 65.

Spirit-rappings, loi.

Spirit-writing, loi.

Spiritualistic, mediums, 90 et seq.;
seance, iii.

Spiritualists, 64.

Squaring of the circle, 30, 1 12-143.

Squares in space, 71.

Stifel, Michael, 45.

Straight edge and compasses, to con-
struct with, iig et seq.

Subtertraction, 11.

Subtraction, 10 et seq., 73 et seq.

Subtrahent, 11.

Summands, 9.

Supernatural influences, iio et seq.

Sylvester II., 131.

Symbols, 2.

Symmetry, 91.

Talmud, 123.

Ten, 4.

T£Tpayuvi(,eiv, 115.

Thaies, 123.

Thing-in-itself, 98.

Third degree, operations of the, 19.

Three Bodies, problem of, 31.

Time, 36.

Triangle, sum of the angles of a, 83.

Trisection of the argle, 30.

Tweniy, as base, 5.

INDKX.

149

Ulrici, 107.
Ulysses, 84.
Unlimited, 85.
Unnamed number, 6.

Van Ceulen, Ludolf, 134.
Van Eyck, Simon, 132.
Van't Hoff, 88.
Vicenary system, 5,
Vieta, 133.
Vision, 93 et seq.
Vitruvius, 128
Von Lüttich, Frankos, 131.

Wallis, 136, 138.
Waltershausen, 99.
Weierstrass, 31.
Wenzelides, 58.
Wislicenus, 88.
Wolfenbüttel Library, 128.
Wolff, Professor, 140.
Words, numeral, 4 et seq.
Worms, two-dimensional, 104.
Wundt, 107.

Zero, 13, 16.

Zero exponents. 21.

Zöllner, 65, 92, 93 et seo

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