IN THE SAME SERIES.
ON THE STUDY AND DIFFICULTIES OF MATHEMATICS. By Augustus
De Morgan. Entirely new edition, with portrait of the author,
index, and annotations, bibliographies of modern works on algebra,
the philosophy of mathematics, pan-geometry, etc. Pp., 288.
Cloth, $1.25 (5s.).
LECTURES ON ELEMENTARY MATHEMATICS. By Joseph Louis
Lagrange. Translated from the French by Thomas J. McCormack.
With photogravure portrait of Lagrange, notes, biography, marginal
analyses, etc. Only separate edition in French or English. Pages, 172.
Cloth, $1.00 (5s.).
HISTORY OF ELEMENTARY MATHEMATICS. By Dr. Karl Fink,
late Professor in T¨ubingen. Translated from the German by Prof.
Wooster Woodruff Beman and Prof. David Eugene Smith. (In preparation.)
THE OPEN COURT PUBLISHING CO.
324 dearborn st., chicago.
MATHEMATICAL ESSAYS
AND
RECREATIONS
BY
HERMANN SCHUBERT
PROFESSOR OF MATHEMATICS IN THE JOHANNEUM, HAMBURG, GERMANY
FROM THE GERMAN BY
THOMAS J. McCORMACK
Chicago, 1898
Produced by David Wilson
Transcriber’s notes
This e-text was created from scans of the book published at
Chicago in 1898 by the Open Court Publishing Company,
and at London by Kegan Paul, Trench, Truebner & Co.
The translator has occasionally chosen unusual forms of
words: these have been retained.
Some cross-references have been slightly reworded to take account
of changes in the relative position of text and floated figures.
Details are documented in the LATEX source, along with minor
typographical corrections.
TRANSLATOR’S NOTE.
The mathematical essays and recreations in this volume are by one of the most
successful teachers and text-book writers of Germany. The monistic construction
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-books, 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 arithmetic here
sketched, we may refer the reader to Professor Schubert’s volume Arithmetik und
Algebra, recently published in the G¨oschen-Sammlung (G¨oschen, 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 complete 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, Ill., December, 1898.

CONTENTS.
page
Notion and Definition of Number . . . . . . . . . . . . . 1
Monism in Arithmetic . . . . . . . . . . . . . . . . 8
On the Nature of Mathematical Knowledge . . . . . . . . . . 25
The Magic Square . . . . . . . . . . . . . . . . . 36
The Fourth Dimension . . . . . . . . . . . . . . . . 63
The Squaring of the Circle . . . . . . . . . . . . . . . 107


NOTION AND DEFINITION OF NUMBER.
Many essays have been written on the definition of number. But
most of them contain too many technical expressions, both philosophical
and mathematical, to suit the non-mathematician. 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 numeration
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 information
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 why 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 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
1
2 NOTION AND DEFINITION OF NUMBER.
element, which in some respects perhaps is not so absolutely essential;
namely, that the things to be counted shall be regarded 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. We may therefore lay down provisionally 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 ordinally, 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 spatial 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 example,
• • • • • • •• •• ••••• •• •• •• ••••• •• •• •• •• •• etc.
Now-a-days such natural number-pictures are rarely employed, and are
to be seen only on dominoes, dice, and sometimes, also, on playingcards.
It can be shown by archæological evidence that originally numeral
writing was made up wholly of natural number-pictures. For example,
the Romans in early times represented all numbers, which were
written at all, by assemblages of strokes. We have remnants of this
writing in the first three numerals of the modern Roman system. If we
needed additional evidence that the Romans originally employed natural
number-signs, we might cite the passage in Livy, VII. 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 Minerva, the “inventrix”
of counting, for the purpose of showing the number of years 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 notation.
* System der Arithmetik. (Potsdam: Aug. Stein. 1885.)
NOTION AND DEFINITION OF NUMBER. 3
Whosoever has carefully studied in any large ethnographical collection
the monuments of ancient Mexico, will surely have remarked 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 highly developed notation
of the latter, were not yet sufficiently advanced in civilisation
to feel any need of expressing numbers symbolically. But since the
most perfect of all systems of numeration, the Hindu system of “local
value,” was introduced and adopted in Europe in the twelfth century,
the Roman numeral system gradually disappeared, at least from practical
computation, and at present we are only reminded by the Roman
characters of inscriptions of the first and primitive stage of all numeral
notation. To-day we see natural number-pictures, except in the abovementioned
games, only very rarely, as where the tally-men of wharves
or warehouses 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 consequence
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 their utterance. We arrive thus
at natural number-words. For example, 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. Numberwords
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.
4 NOTION AND DEFINITION OF NUMBER.
Word-forms of the kind described are too inconvenient, however, 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 immemorial 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 construct their words for larger numbers
from words for smaller numbers. What number we shall begin with in
the formation of compound numeral words is quite indifferent, so far
as the idea of number 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 possessed ten
fingers.
Granting it were impossible to prove in any 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 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 thousand, tenty-ty-ty for ten thousand, etc. But in
the utterance of such words, the syllable ty would be so frequently repeated
that the same inconvenience would be produced as above in our
individual number-pictures. For this reason the genius which controls
the formation of speech took the first course.
NOTION AND DEFINITION OF NUMBER. 5
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 expressed by a one and
fifty-four appended naughts. In no other language 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 logical
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, quatre-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.
As we see, the languages of humanity now no longer possess natural
number-signs and number-words, but employ names and systems
of notation adopted subsequently to this first stage. Accordingly, 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 number, which we do by
simply saying that by number we understand the results of counting or
numeration. These are naturally composed of two elements. 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
6 NOTION AND DEFINITION OF NUMBER.
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 number-picture belonging to the whole. In this way we arrive
on the one hand at the idea of addition, and on the other, at the notion
of “unnamed” number. Since we have no means of telling 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 ultimately
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 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 necessary to ascend from the named to the unnamed number, an end
which we accomplish by not asking first what five plus three is, but
by 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 positive
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
NOTION AND DEFINITION OF NUMBER. 7
purpose of giving to the chief tool of the mathematician, 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, therefore, can scarcely be dispensed with,
yet could, if it were a question of principle, be avoided. Kronecker has
shown that any problem in which positive whole numbers are given,
and only such are sought, always admits of solution without the help
of other kinds of numbers, although the employment of the latter wonderfully
simplifies the solution.
How these derived species of numbers, by the logical application
of a single principle, flow naturally from the notion of number and
of addition above deduced, I shall show in the next article entitled
“Monism in Arithmetic.”
MONISM IN ARITHMETIC.
In 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 dominate 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 almost absolute monistic form, by the reduction
of all phenomena to the motions of molecules. For example, optical
and electrical phenomena, 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 arithmetic 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, evolution, and the finding
of logarithms, no other operations are definable 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, the boughs and branches of which may
still increase in size or even give forth fresh sprouts, but whose main
trunk has attained its fullest development.
8
MONISM IN ARITHMETIC. 9
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 arrive,
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. Schr¨oder, and Kronecker, the author of the present article, 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 essays, whose purpose will rather be the
study of the logical construction of the science from some single fundamental
principle, the following pages will simply give of the notions and
laws of arithmetic what is absolutely necessary for an understanding of
its development.
The starting-point of arithmetic is the idea of counting and of number
as the result of counting. On this subject, the reader is requested to
read the first essay of this collection. It is there shown that the idea of
addition springs immediately from the idea of counting. 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 + b = b + a,
is called the commutative law 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 distinguished 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 always something that
is to be increased, b the increase. Accordingly it has been proposed to
call the number which is regarded in addition 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 increment.
10 MONISM IN ARITHMETIC.
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; increment comes from increscere, to grow, and signifies
as in its ordinary meaning what is added.
Besides the commutative law one other follows from the idea of
counting—the associative law of addition. This law, which has reference
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 language of arithmetic this
law reads,
a + (b + c) = (a + b) + c.
To obtain now all the rules of addition we have only to apply the two
laws of commutation and association above stated, though frequently,
in the deduction of the same rule, each must be applied many times. I
may pass over here both the rules and their establishment.
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 reckoning, the solution of the two questions
is the same, since by the commutative 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 operations do differ, and it is accordingly desirable in a logical
investigation of the structure of arithmetic, to distinguish the two by
different names. As in all probability no terms have yet been suggested
for these two kinds of subtraction, I propose here for the first time the
following words for the two operations, namely, detraction to denote
the finding of the increment, and subtertraction to denote the finding
of the augend. We obtain these terms simply enough by thinking of
the augmentation of some object already existing. 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
MONISM IN ARITHMETIC. 11
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,
subtertrahere). For this reason, the finding of the increment is called
subtertraction. Owing to the commutative law, the two inverse operations,
as matters of computation, become one, which bears the name of
subtraction. The sign of this operation is the minus sign, a horizontal
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 different operations in
one single operation, subtraction, we employ for the number which before
was increment or augend, the term subtrahend, a word which on
account of its passive ending is not very good, and for which, accordingly,
E. Schr¨oder proposes to substitute 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 number which increased by b gives a, or 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 subtraction. These
rules we here omit.
From the foregoing it is plain that the minuend is necessarily larger
than the subtrahent. For in the process of addition the minuend was
the sum, and the sum grew out of the union of two natural numberpictures.*
Thus 5 minus 9, 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
* See page 2, supra.
12 MONISM IN ARITHMETIC.
done? Shall we banish entirely from arithmetic 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 result 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 convenient 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, therefore, to comprise all these symbols under
some one single symbol, and to construct this latter symbol so that
it will appear unequivocally 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 form 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 invention was 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 tsiphra, empty, whence is derived the English
cipher. The form of this sign has been different in different times and
with different peoples. But for the last two or three centuries, since the
symbolic language of arithmetic has become thoroughly established as
an international character, the form of the sign has been 0 (French z´ero,
German null ).
MONISM IN ARITHMETIC. 13
In calling this symbol and the symbols formed of a minus sign
followed by a result of counting, numbers, 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 11 is equal to 1 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 with real differences are we led to a statement of identity between
7 minus 11 and 1 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, numeri ficti (imaginary
numbers), as distinguished from numeri veri (real numbers). Not until
Descartes, in the first half of the seventeenth century, was any one bold
enough to substitute numeri ficti and numeri veri indiscriminately for
the same letter of algebraic expressions.
We have invested, thus, combinations of signs originally meaningless,
in which a smaller number stood before than after a minus sign,
with a meaning which enables us to reckon with such apparent 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 characterise, therefore,
as the foundation-principle of its whole construction. This principle,
which Hankel once called the principle of permanence, 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 two previously
defined numbers by a sign for a previously defined operation (plus,
minus, times, etc.) shall be invested with meaning, even where the original
definition of the operation used excludes such a combination; and
the meaning imparted is to be such that the combination considered shall
obey the same formula of definition as a combination having from the
outset a signification, 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 facility as
at the combinations above discussed, whose results are termed negative
14 MONISM IN ARITHMETIC.
numbers and zero. To think of such combinations as results and to call
the products reached also “numbers” is a misuse of language. It were
better if we used the phrase forms of numbers for all numbers that are
not the results of counting. But usus tyrannus!
It will now be my task to show how all numbers at which arithmetic
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,
a + (b + c + d) + (e + f) = (a + b + c) + (d + e) + f.
The necessary consequence of this is that we may neglect the consideration
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 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 b numbers each of which is called a. The number 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 counting,
or a number in the original sense of the word, but that the multiplicand
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 unnamed 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 c) = (a times b) times c.
MONISM IN ARITHMETIC. 15
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 commutative
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 (×), which is read
“times.” Joined with the fundamental formula above written are a
group of subsidiary formulæ 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 discussion of these rules
here.
As the combination of two numbers by a sign of multiplication has
no significance according to our definition of multiplication, when 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 symbols 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 multiplying by 5, and subtracting the differences obtained, and
because agreeably to the principle of no exception we must say that the
multiplication must be performed according to exactly the same rule
with a symbol which has the form of a difference whose minuend is less
by one than its subtrahent.
As from addition two inverse operations, detraction and subtertraction,
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 different 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
16 MONISM IN ARITHMETIC.
exception. Let us take this example,
4 yards × 3 = 12 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 yards,
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 measuring. 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 multiplier is sought we call the division measuring,
and if the multiplicand is sought, we call it parting. 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 horizontal
stroke, or a combination of both. Its definitional formula reads,
(a ÷ b)  b = a, or,
a
b  b = a.
Accordingly, dividing a by b means, to find the number which multiplied
by b gives a, or to find the number with which b must be multiplied
to produce a. From this formula, together with the formulæ relative
to multiplication, the well-known rules of division are 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 defined number.
For example, if the divisor is 5 the dividend can only be 5, 10,
15, and so forth, and 0, −5, −10 and so forth. Accordingly, 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, 3
5 or 12
5 are meaningless 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
MONISM IN ARITHMETIC. 17
of such a quotient form with its divisor equal to its dividend. In this
way we reach the definition of broken numbers or fractions, which by
the application of the principle of no exception spring from division exactly
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 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 fractions
and shall only direct the reader’s attention to the connexion which
exists between fractional and non-fractional or, as we usually say, whole
numbers. Since the number 12 lies between the numbers 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 < 12
5 < 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 12
5 . 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 1 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 of
the 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 fractional 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 0
0 , the expression is one which stands for any number
whatsoever, because any number whatsoever, 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 arithmetical rules to a ÷ 0 all such forms may
18 MONISM IN ARITHMETIC.
be equated to one another both when a is positive and also when a
is negative. We may therefore invent two new signs for such quotient
forms, namely +1 and −1. We find, further, that in transferring
the notions greater and less to these symbols, +1 is greater than any
positive number, however great, and −1 is smaller than any negative
number, however small. We read these new signs, accordingly, “plus
infinitely great” and “minus infinitely great.”
But even here arithmetic has not reached its completion, although
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 advance is possible in arithmetic.
For in virtue of the laws of commutation and association, which
also fortunately obtain in multiplication, 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 a+a+a+a we read
4 a, 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 arithmetic. But if after having introduced this operation we
repeatedly apply the monistic principle of arithmetic, the principle of
no exception, we reach new means of computation which have led to
undreamt 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 b factors of which each is called a, ab, which we read a to the
bth power. Here a new direct operation, that of involution, is defined,
and from now on we are justified in distinguishing operations which
are not inverses of others, as addition, multiplication, and involution,
by numbers of degree. Addition is the direct operation of the first degree,
multiplication that of the second degree, and involution that of
the third degree. In the expression ab the passive number a is called
the base, the active number b the exponent, the result, the power.
But whilst in the direct operations of the first and second degree,
the laws of commutation and association hold, here in involution, the
operation of third degree, the two laws are inapplicable, and the result
of their inapplicability is that operations of a still higher degree than the
MONISM IN ARITHMETIC. 19
third form no possible advancement of pure arithmetic. The product
of b factors a is not equal to the product of a factors b; that is, the
law of commutation does not hold. The only two different integers for
which a to the bth power is equal to b to the ath power are 2 and 4, for
2 to the 4th 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 (bc)th power of a or the cth power
of ab.
From the definition of involution follow the usual rules for reckoning
with powers, of which we shall only mention one, namely, that the
(b − c)th power of a is equal to the result of the division of a to the bth
power by a to the cth power. If we put here c equal to b, we are obliged,
by the principle of no exception, to put a to the 0th power equal to 1;
a new result not contained in the original notion 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. According to the latter, a to the (b−c)th power must
still remain equal to ab divided by ac even when b is smaller than c.
Whence follows that a to the minus dth power is equal to 1 divided by
a to the dth power, or to take specific numbers, that 3 to the minus 2nd
power is equal to 1
9 .
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 ab = c, we may ask two questions: first, what the base is which
raised to the bth power gives c; the second, what the exponent of the
power is to which a must be raised to produce c. In the first case we
seek the base, and term the operation which yields this result 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 pb c = a (which we read, the bth root of c is equal to a), and call c
the radicand, b the exponent of the root, and a the root. In the second
case, we write loga c = b (which we read, the logarithm of c to the base
a is equal to b), and call c the logarithmand or number, a the base of
the logarithm, and b the logarithm.
While, owing to the validity of the law of commutation in addition
and multiplication, the two inverse processes of those operations are
20 MONISM IN ARITHMETIC.
identical so far as computation is concerned, here in the case of involution
the two inverse operations are in this regard essentially different,
for in this case the law of commutation does not hold.
From the definitional formulæ for evolution and the finding of logarithms,
namely,
(pb c)b = c, and (a)loga c = 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 pb c has meaning only when c is the
bth power of some number already defined; and, secondly, that for the
present also loga c has 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, ab was meaningless
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 “a raised to the fractional power p
q .” The principle of no
exception compels us to give to the arithmetical combination “a to the
p
q
th power” a significance such that all the rules of computation will
hold with respect to it. Now, one rule that holds is, that the mth power
of the nth power of a is equal to the (m×n)th power of a. Consequently,
the qth power of a raised to the p
q
th power must be equal to a raised to
a power whose exponent is equal to p
q times q. But the last-mentioned
product gives, according to the definition of division, the number p.
Consequently the symbol a to the p
q
th power is so constituted that its
qth power is equal to a to the pth power; i. e., it is equal to the qth root
of ap. Similarly, we find that the symbol “a to the minus p
q
th power”
must be put equal to 1 divided by the qth root of a to the pth power, if
we are to reckon with this symbol as we do with real powers. Again,
just as a to the bth power is invested with meaning when b is a fractional
number, so some meaning harmonious with the principle of no
exception must be imparted to the bth root of c where b is a positive or
negative fractional number. For example, the three-fourthsth root of 8
MONISM IN ARITHMETIC. 21
is equal to 8 to the 4
3 power, that is, to the cube root of 8 to the 4th
power, or 16.
The principle underlying arithmetic now also compels us to give
to the symbol the “bth root of c” a meaning when c is not the bth
power of any number yet defined. First, let c be any positive integer
or fraction. Then always to be able to reckon with the bth root of c in
the same way that we do with extractible roots, we must agree always
to put the bth power of the bth root of c equal to c—for example, (p2 3)2
always exactly equal to 3. A careful inspection 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 numbers 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 quantity as we please. For example,
( 7
5 )3 = 343
125 = 2 93
125 < 3 < 33
8 = 27
8 = ( 3
2 )3.
The number 3, as we see, is here included between two limits which are
the third powers of two numbers 7
5 and 3
2 whose difference is 1
10 . We
could also have arranged it so that the difference should be equal to
1
100 , or to any specified number, however small. Now, instead of putting
the symbol “less than” between ( 7
5 )3 and 3, and between 3 and ( 3
2 )3,
let us put it between their third roots; for example, let us say:
7
5 < p3 3 < 3
2 , meaning by this that ( 7
5 )3 < 3 < ( 3
2 )3.
In this sense we may say that the new numbers always lie between
two old numbers whose difference may be made as small as we please.
Numbers possessing this property are called irrational numbers, 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 numbers 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.
22 MONISM IN ARITHMETIC.
The square root of minus 9 is neither equal to plus 3 nor to minus
3, since each multiplied by itself gives plus 9, 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 (p2 −9)2 shall always
be put equal to minus 9.* Keeping to this definition we see at once
that p−a, where a is any positive rational or irrational number, is a
symbol which can be put equal to the product of p+a by p−1. In
extending to these new numbers the rights of arithmetical 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 number some concise symbol. This symbol is the first
letter of the word imaginary, namely, i; so that we can always put for
such an expression as p−9, 3  i.
If we combine real and imaginary numbers by operations of the first
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+ib, where a and b stand for real numbers. 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 i we always substitute minus 1.
In the numerical combination a + ib, which we also call number,
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 numbers; if a is zero, it
stands for all purely imaginary numbers. This general number a + ib
* Henceforward we shall use the simpler sign p for p2 .
MONISM IN ARITHMETIC. 23
is called a complex number, so that the complex number includes in
itself as special cases all numbers heretofore defined. By the introduction
of irrational, purely imaginary, and the still more general complex
numbers, all combinations become invested with meaning which the
operations of the third degree 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 1 to the base 2 is a purely
imaginary number; the fourth root of minus 1 is a complex number.
Indeed, we may recognise, proceeding still further, that every combination
of two complex numbers, by means of any of the operations of the
first, second, or third degree will lead in turn to a complex number, that
is to say, never furnishes occasion, by application of the principle of no
exception, 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 investigation
of addition and its inverse operations, we reached no other numbers
except zero and positive and negative whole numbers, and every combination
of such numbers by operations of the first degree led to no
new numbers. After the investigation of multiplication and its inverse
operations, the positive or negative fractional 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 combination
does not necessarily always lead to a single number, but may lead to
many regularly arranged numbers. For example, the combination “logarithm
of minus one to a positive base” furnishes a countless number of
results which form an arithmetical series of purely imaginary numbers.
Still, 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 might furnish the germ of other new numbers which are
not reducible to a+ib. As a matter of fact, mathematicians have asked
themselves this question and investigated the direct operation of the
fourth degree, together with its inverse processes. The result of their
24 MONISM IN ARITHMETIC.
investigations was, that an operation which springs from involution as
involution sprang from multiplication is incapable of performing any
real mathematical service; the reason of which is, that in involution
the laws of commutation and association do not hold. It also further
appeared that the operations of the fourth degree could not give rise to
new numbers. 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 valuable
means of investigation in geometry and mechanics they are not numbers
of arithmetic, because the rules of arithmetic are not unconditionally
applicable to them.
The building up of arithmetic is thus completed. The extensions
of the domain of number are ended. It only remains to be asked why
the science of arithmetic appears in its structure so logical, 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 arithmetic, the principle of no
exception.
ON THE NATURE OF MATHEMATICAL
KNOWLEDGE.
“Mathematically certain and unequivocal” is a phrase 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 mathematical knowledge
really is more certain and unequivocal than other knowledge.
The intrinsic character of mathematical research and knowledge is
based essentially on three properties: first, on its conservative attitude
towards the old truths and discoveries of mathematics; secondly, on its
progressive mode of development, due to the incessant acquisition of
new knowledge 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 regarding 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 way to newer conceptions. Think of
the profound changes which the conceptions of theoretical medicine,
zo¨ology, botany, mineralogy, and geology have undergone. It is the
same, too, in the other sciences. In philology, comparative linguistics,
and history our ideas are quite different from what they formerly were.
25
26 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.
In no other science is it so indispensable a condition that whatever
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 proposition 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 of 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 different 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 decades
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 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 speculations 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 which their methods fail.
ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 27
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 necessarily
for a subsequent period, but with it the demonstration of those results
is more rigorously established. Despite all this, however, in no science
does cognition bear so unmistakably the imprint of truth as in pure
mathematics. And this fact bestows on mathematics its conservative
character.
This conservative character again is displayed in the objects of
mathematical research. The physician, the historian, the geographer,
and the philologist have to-day quite different fields of investigation
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 rˆole for hundreds, 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 problem 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 impossibility of solving this problem
exactly by the use of straight edge and compasses alone. (See pp. 111,
133–136.)
It is, of course, the insoluble problems that have the longest history;
partly because it is harder to show that a thing is impossible than that
it is possible, and, on the other hand, because problems that have
long defied solution are ever evoking anew the spirit of inquiry and the
ambition of mathematicians, and because the uncertainty of insolubility
lends to such problems a peculiar charm. Of the geometrical problems
that have occupied competent and incompetent 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
28 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.
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 trisection 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 discovering
new truths and developing new theories, all of which perhaps
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 mathematical inquiry,
there are of course other problems in mathematics 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 assumes 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 formula 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
research, and therefore also to mathematical knowledge, a conON
THE NATURE OF MATHEMATICAL KNOWLEDGE. 29
servative character, on the other hand, by the continuous outgrowth of
new truths and methods from the old, progressiveness is also one of its
characteristics. In marvellous profusion old knowledge is augmented by
new, which has the old as its necessary condition, and, therefore, could
not have arisen had not the old preceded 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 stories, which are the elementary propositions. But it is
necessary for this that all the stones should be properly fitted together;
and it 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
belonging in what is now the uppermost layer of the edifice, is Lindemann’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. Lindemann’s
demonstration could not have been produced in the preceding
century, because it rests necessarily on theories whose development falls
in the present century. It is true, 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 investigations
were necessary before the demonstration could be successfully
accomplished.
As we see, the majority of the mathematical truths now possessed
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 uninitiated 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.
30 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.
There is no royal road to the knowledge of mathematics, as Euclid once
said to the first Egyptian Ptolemy. This same inaccessibility of mathematics,
although it secures for it a lofty and aristocratic place among
the sciences, 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 uninitiated 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 investigations
on the mode of development of languages, which is more or 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 zo¨ology, botany, mineralogy, geology, and
chemistry work with given data. In order not to become involved in
futile speculations the last-mentioned sciences are constantly and inevitably
obliged to revert to observations by the senses. It is then their
task to link together these individual observations by 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 physics,
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 permanently
established, is able to pursue its course of development independently
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 practical art to which it owed its birth. Herodotus recounts that
geometry had its origin in Egypt where the inundations of the Nile
ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 31
obliterated 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 measurement,
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 mathematics
are not stimulated by the thought that their researches are
likely to be of practical value, yet that result is still frequently realised,
often after the lapse of centuries. The history of mathematics 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 planets 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 modern mathematical research, whether the same consist in the
extended development 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 endowed with intelligence
existed on other planets, the truths of mathematics 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 comfortless and wearisome
operation of heaping up demonstration on demonstration, of rivetting
* Cf. Felix Klein, “Remarks Given at the Opening of the Mathematical and Astronomical
Congress at Chicago.” The Monist (Vol. IV, No. 1, October, 1893).
32 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.
truth on truth, and of tormenting themselves 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 intelligible when we reflect that the progress
which has been made, particularly in the last few decades, and which is
uninfluenced from without, does not consist solely in the accumulation
of new truths and in the enunciation of new problems, nor solely in deductions
and solutions, but culminates rather in the discovery of new
methods and points of view in which the old disconnected and isolated
results appear suddenly in a new connexion or as different interpretations
of a common fundamental truth, or finally, as a single organic
whole.
Thus, for example, the idea of representing imaginary and complex
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 xn = 1, 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 functions and the theory of algebraic
curves, a relation which led to the discovery of the condition under
which two curves can be co-ordinated to each other, point for point,
and hence also adequately represented on each other. Of course such
combinations and extensions of view possess a much greater charm for
the mathematician than the mere accumulation of truths and solutions,
whose fascination consists entirely in their truth or correctness.
From these three cardinal characteristics, now, which distinguish
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 always 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
ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 33
which mathematics thus appears to have over the other sciences. In
order to show more distinctly the nature of these characteristics, which
we prefer to call negative, we shall select and confine our remarks to a
branch which is commonly taken to be synonymous with mathematics,
namely, to arithmetic in the broadest sense of the word.
The subject of inquiry in arithmetic is numbers and their combinations.
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 understanding 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 differences of number. Uninfluenced by
matter and force, independently, too, of the properties of geometrical
magnitudes, arithmetic could be conducted solely by its own intrinsic
potencies to its highest 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 extremely 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 arithmetical
34 ON THE NATURE OF MATHEMATICAL KNOWLEDGE.
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 exclusively to pure arithmetical
developments and leave it to others at 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 experience perceived by our senses. The geometry which Bolyai,
Lobach´evski, 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 demonstration
that geometry is a science that rests on the observation of nature,
and therefore in the correct sense of the word, is a natural science.
Yet what a difference there is, for instance, between geometry
and chemistry! Both derive their constructive materials from senseperception.
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 certainty of
the truth they contain not qualitatively but only quantitatively different.
In chemistry the probability of error is greater than in geometry,
because more numerous and more difficult observations have to be made
there than in geometry, where only the very first premises, which no
man with sound senses could ever impugn, 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-mathematical sciences,
where mechanics has the first claim to our attention. 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
ON THE NATURE OF MATHEMATICAL KNOWLEDGE. 35
of force, work, horse-power, atmospheric pressure, etc. The knowledge
of mechanics, 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 complexity
of the observational material which must here be put to the test.
Still, although mathematical knowledge does not lead to absolutely
certain results, it yet invests known results with incomparably greater
trustworthiness than does the knowledge of the other sciences. 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 sciences, so to shape their problems
and investigations that they can be made susceptible of mathematical
treatment. Then will mathematics gladly perform her duty. The moment
a science has advanced 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
advancement 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.
I.
INTRODUCTORY.
Among the philosophies of modern times there is none which has
emphasised so much the importance of form and formal thought
as the monism of The Monist. An expression of this philosophy 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 great results which engineering and other applied
sciences accomplish through 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 mankind 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.
36
THE MAGIC SQUARE. 37
We shall devote the following pages to a brief review of magic
squares, the consideration of which has made many a man believe in
mysticism. And yet there is no mysticism about them unless we either
consider everything mystical, even that twice two is four, or join the
sceptic in his exclamation that we can truly not know whether twice
two might not be five in other spheres of the universe.
The author of the short article on “Magic Squares” in the English
Cyclopædia (Vol. III, 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. Without 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 Nuremberg
painter, Albrecht D¨urer, is found among a number of other emblems,
* The term melancholy meant in D¨urer’s time, as it did also in Shakespeare’s and
Milton’s, “thought or thoughtfulness.” Says Milton in Il Penseroso:
“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, 12.
Thought that does not lead to action produces a gloomy state of mind.
Thoughtfulness 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.”
D¨urer’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 element of this melancholy, but on the whole the genius of thought appears
38 THE MAGIC SQUARE.
ALBERT D¨URER’S ENGRAVING
MELANCHOLY
OR THE
Genius of the Industrial Science of Mechanics.
THE MAGIC SQUARE. 39
which the reader will notice in our reproduction of the cut, the subjoined
square. This arrangement of the sixteen natural numbers from
1 14 15 4
12 7 6 9
8 11 10 5
13 2 3 16
Fig. 1.
1 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 arrangement of numbers is
termed a magic square, and the square which we have reproduced above
is the 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 D¨urer’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 quadratic
arrangement of the nine numbers from 1 to 9 in such a manner that
the sum of each horizontal, vertical, or diagonal row, always remains
bright, self-possessed, and strong.
D¨urer 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 simple tools used in the sixteenth century. A ladder leans against the house,
that assists 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 confidence.
40 THE MAGIC SQUARE.
the same, namely 15. This square is the adjoined.
2 7 6
9 5 1
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 1, or 4 and 3 and 8, or 2 and 9 and 4, or 7 and 5 and
3, or 6 and 1 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 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 D¨urer’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 1 to 16 is formed in the
simplest manner as follows. We inscribe the numbers from 1 to 16 in
their natural order in the squares, thus:
1 2 3 4
5 6 7 8
9 10 11 12
13 14 15 16
Fig. 3.
We then leave the numbers in the four corner-squares, viz. 1, 4, 13,
16, as well also as the numbers in the four middle-squares, viz. 6, 7,
10, 11, 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 instead of 8, 8
instead of 9, 5 instead of 12, 3 instead of 14, and 2 instead of 15. We
THE MAGIC SQUARE. 41
obtain thus the magic square
1 15 14 4
12 6 7 9
8 10 11 5
13 3 12 16
=34
=34
=34
=34
34 34 34 34
=34
34=
Fig. 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, 1, 4,
13, 16, or 6, 7, 10, 11, or 15, 14, 3, 2, or 12, 9, 5, 8, or 15, 8, 2, 9,
or 14, 12, 3, 5. We may easily convince ourselves that this square is
obtainable from the square of D¨urer by interchanging with one another
the two middle vertical rows.
II.
EARLY METHODS FOR THE CONSTRUCTION OF
ODD-NUMBERED SQUARES.
Since early times rules have also been known for the construction 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 1, 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 1 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 1
makes 26, and the half of 26 times 5 is 65. Analogously, for 6 times 6
squares the summation 111 is obtained, for 7 times 7 squares 175, for 8
times 8 squares 260, for 9 times 9 squares 369, 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
1 in the centre of the topmost row, then write 2 in the lowest space
of the vertical column next adjacent to the right, and then so inscribe
42 THE MAGIC SQUARE.
30 39 48 1 10 19 28
38 47 7 9 18 27 29
46 6 8 17 26 35 37
5 14 16 25 34 36 45
13 15 24 33 42 44 4
21 23 32 41 43 3 12
22 31 40 49 2 11 20 =175
=175
=175
=175
=175
=175
=175
175 175 175 175 175 175 175
=175
175=
Fig. 5.
the remaining numbers in their natural order in the squares diagonally
upwards towards the right, that on reaching the right-hand margin 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 preceding square of 7 times 7 cells is
formed, in which the reader is requested 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 D¨urer who lived about the year 1500, to the celebrated
arithmetician Adam Riese, and to the mathematician Michael Stifel,
which two last lived about 1550. In the seventeenth century Bachet
de M´eziriac, and Athanasius Kircher employed themselves on magic
squares. About 1700, finally, the French mathematicians De la Hire
and Sauveur made considerable contributions to the theory. 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.
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 following diagram (Fig. 6).
THE MAGIC SQUARE. 43
4 12 20 28
11 19 27
10 18 26 34
17 25 33
16 24 32 40
23 31 39
22 30 38 46
5 13 21
6 14
7
29 37 45
36 44
43
1
8
2
15
9
3
47
41
35
48
42
49
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 M´eziriac, we obtain the following magic square of the numbers from
1 to 49:
4 29 12 37 20 45 28
35 11 36 19 44 27 3
10 42 18 43 26 2 34
41 17 49 25 1 33 9
16 48 24 7 32 8 40
47 23 6 31 14 39 15
22 5 30 13 38 21 46
Fig. 7.
44 THE MAGIC SQUARE.
III.
MODERN MODES OF CONSTRUCTION OF ODD-NUMBERED
SQUARES.
The reader will justly ask whether there do not exist other correct magic
squares which are constructed after a different method from that just
given, and whether there do not exist modes of construction which will
lead to all the imaginable and possible magic squares of a definite number
of cells. A general mode of construction 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 1 to 5 five times; and in the second, five
times, the following multiples of five, viz.: 0, 5, 10, 15, 20. It is clear
now that by adding each of the numbers of the series from 1 to 5 to
each of the numbers 0, 5, 10, 15, 20, we shall get all the 25 numerals
from 1 to 25. All that additionally remains to be done therefore, is,
so to inscribe the numbers that by the addition of the two numbers in
any two corresponding cells each combination 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 1 to 5 added together make 15, and the numbers 0, 5,
10, 15, 20 make 50.
We effect the required method of inscription by imagining the numbers
1, 2, 3, 4, 5 (or 0, 5, 10, 15, 20) arranged in cyclical succession,
that is 1 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 1 2 is a cycle of the first order, 2 4 1 3 5 is
a cycle of the second order, 1 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 vertical 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 MAGIC SQUARE. 45
the other auxiliary square, that is lie in similarly situated cells. The
following auxiliary squares are, for example, thus possible:
3 4 5 1 2
5 1 2 3 4
2 3 4 5 1
4 5 1 2 3
1 2 3 4 5
Fig. 8.
and
0 10 20 5 15
5 15 0 10 20
10 20 5 15 0
15 0 10 20 5
20 5 15 0 10
Fig. 9.
Adding in pairs the numbers which occupy similarly situated cells,
we obtain the following correct magic square:
3 14 25 6 17
10 16 2 13 24
12 23 9 20 1
19 5 11 22 8
21 7 18 4 15
Fig. 10.
It will be seen that we are able thus to construct a very large
number of magic squares of 5 times 5 spaces by varying in every possible
manner the numbers in the two auxiliary squares. Furthermore, 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 constant sum of 65.
For example: 3 and 7, 11, 20, 24; or 10, 14 and 18, 22, 1. 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
46 THE MAGIC SQUARE.
will form a correct magic square, as the following figure will show:
2 13 24 10 16 2 13 24 10 16 2
9 20 1 12 23 9 20 1 12 23 9
11 22 8 19 5 11 22 8 19 5 11
18 4 15 21 7 18 4 15 21 7 18
25 6 17 3 14 25 6 17 3 14 25
2 13 24 10 16 2 13 24 10 16 2
9 20 1 12 23 9 20 1 12 23 9
11 22 8 19 5 11 22 8 19 5 11
18 4 15 21 7 18 4 15 21 7 18
25 6 17 3 14 25 6 17 3 14 25
2 13 24 10 16 2 13 24 10 16 2
9 20 1 12 23 9 20 1 12 23 9
11 22 8 19 5 11 22 8 19 5 11
Fig. 11.
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 two numbers
of every two similarly situated cells, the magic square, exhibited in
Diagram 14, in which each row gives the same sum 671.
IV.
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, different 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
THE MAGIC SQUARE. 47
1 2 3 4 5 6 7 8 9 10 11
3 4 5 6 7 8 9 10 11 1 2
5 6 7 8 9 10 11 1 2 3 4
7 8 9 10 11 1 2 3 4 5 6
9 10 11 1 2 3 4 5 6 7 8
11 1 2 3 4 5 6 7 8 9 10
2 3 4 5 6 7 8 9 10 11 1
4 5 6 7 8 9 10 11 1 2 3
6 7 8 9 10 11 1 2 3 4 5
8 9 10 11 1 2 3 4 5 6 7
10 11 1 2 3 4 5 6 7 8 9
Fig. 12.
0 11 22 33 44 55 66 77 88 99 110
33 44 55 66 77 88 99 110 0 11 22
66 77 88 99 110 0 11 22 33 44 55
99 110 0 11 22 33 44 55 66 77 88
11 22 33 44 55 66 77 88 99 110 0
44 55 66 77 88 99 110 0 11 22 33
77 88 99 110 0 11 22 33 44 55 66
110 0 11 22 33 44 55 66 77 88 99
22 33 44 55 66 77 88 99 110 0 11
55 66 77 88 99 110 0 11 22 33 44
88 99 110 0 11 22 33 44 55 66 77
Fig. 13.
1 13 25 37 49 61 73 85 97 109121
36 48 60 72 84 96 108120 11 12 24
71 83 95 107119 10 22 23 35 47 59
106118 9 21 33 34 46 58 70 82 94
20 32 44 45 57 69 81 93 105117 8
55 56 68 80 92 104116 7 19 31 43
79 91 103115 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 101113 4 16
63 75 87 99 100112 3 15 27 39 51
98 110111 2 14 26 38 50 62 74 86
Fig. 14.
of 4, we start with the natural sequence of the numbers and must then
find the complements of the numbers with respect to some other certain
number (as 17 in the square of 4) and also effect certain exchanges of
the numbers with one another. 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
48 THE MAGIC SQUARE.
with respect 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.
1 35 34 3 32 6
30 8 28 27 11 7
24 23 15 16 14 19
13 17 21 22 20 18
12 26 9 10 29 25
31 2 4 33 5 36
Fig. 15.
This square may also be constructed by the method of De la Hire,
from two auxiliary squares with the numbers 1, 2, 3, 4, 5, 6 and 0,
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 90 respectively. In this manner we get the magic
square last given above from the two following auxiliary squares:
1 5 4 3 2 6
6 2 4 3 5 1
6 5 3 4 2 1
1 5 3 4 2 6
6 2 3 4 5 1
1 2 4 3 5 6
Fig. 16.
and
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. 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 six
times the numbers from 1 to 6 that each number shall appear once and
only once in each horizontal, vertical, and diagonal row; for example,
THE MAGIC SQUARE. 49
in the following manner:
1 2 3 4 5 6
2 4 6 1 3 5
3 6 5 2 1 4
5 3 1 6 4 2
6 5 4 3 2 1
4 1 2 5 6 3
Fig. 18.
But if we attempt so to insert, in a like manner, the other set of numbers
0, 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 condition will result in failure.
It is therefore necessary to select auxiliary 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 square 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
1 2 3 4
4 3 2 1
2 1 4 3
3 4 1 2
Fig. 19.
and
0 4 8 12
8 12 0 4
12 8 4 0
4 0 12 8
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 1, 2, 3, 4 by the Ace, the King, the Queen, and the Knave,
and the numbers 0, 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 manner the four Aces,
the four Kings, the Four Queens, and the four Knaves, that in each
50 THE MAGIC SQUARE.
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 furnish 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 condition 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.
The magic squares of even numbers thus constructed are not the
only possible ones. On the contrary, there are very many others possible,
which obey different laws of formation. It has been calculated,
for example, that with the square of 4 it is possible to construct 880,
and with the square of 6, several million, different magic squares. The
number of odd-numbered magic squares constructible by the method
THE MAGIC SQUARE. 51
1 63 62 4 5 59 58 8
56 10 11 53 52 14 15 49
48 18 19 45 44 22 23 41
25 39 38 28 29 35 34 32
33 31 30 36 37 27 26 40
24 42 43 21 20 46 47 17
16 50 51 13 12 54 55 9
57 7 6 60 61 3 2 64
Fig. 22.
1 99 3 97 96 5 94 8 92 10
90 12 88 14 86 85 17 83 19 11
80 79 23 77 25 26 74 28 22 71
31 69 68 34 66 65 37 33 62 40
60 42 58 57 45 46 44 53 49 51
50 52 43 47 55 56 54 48 59 41
61 32 38 64 36 35 67 63 39 70
21 29 73 27 75 76 24 78 72 30
20 82 18 84 15 16 87 13 89 81
91 9 93 4 6 95 7 98 2 100
Fig. 23.
52 THE MAGIC SQUARE.
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.
V.
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 1 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 attention 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 number 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 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 divide 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 1 to 16 we have to insert
in the squares the numbers from 465 to 480. As the number of the
year 1892 is divisible by eleven, it must be possible to deduce from
the magic square constructed by us at the conclusion of Section III a
second magic square in which each row of 11 cells will give the number
of the year 1892. To do this, we subtract from 1892 the sum of the
original square, namely 671, and divide the remainder by 11, whereby
we get 111 and thus perceive that the numbers from 112 to 232 are to
be inscribed in the cells of the square required. We get in this way the
following square, from which one and the same sum, namely 1892, can
THE MAGIC SQUARE. 53
112 124 136 148 160 172 184 196 208 220 232
147 159 171 183 195 207 219 231 122 123 135
182 194 206 218 230 121 133 134 146 158 170
217 229 120 132 144 145 157 169 181 193 205
131 143 155 156 168 180 192 204 216 228 119
166 167 179 191 203 215 227 118 130 142 154
190 202 214 226 117 129 141 153 165 177 178
225 116 128 140 152 164 176 188 189 201 213
139 151 163 175 187 199 200 212 224 115 127
174 186 198 210 211 223 114 126 138 150 162
209 221 222 113 125 137 149 161 173 185 197 =1892
=1892
=1892
=1892
=1892
=1892
=1892
=1892
=1892
=1892
=1892
1892 1892 1892 1892 1892 1892 1892 1892 1892 1892 1892
Fig. 24.
be obtained 44 times, first from each of the 11 horizontal rows, secondly
from each of the 11 vertical rows, thirdly from each of the two diagonal
rows, and fourthly twenty additional times from each and every pair of
any two rows that lie parallel to a diagonal, have together 11 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 another be
taken away from each side, the smaller inner squares remaining 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. In the first of these two squares the internal square of 3 times
54 THE MAGIC SQUARE.
4 5 6 43 39 38 40
49 15 16 33 30 31 1
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 11 12 46
Fig. 25.
1 56 55 11 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. 26.
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 1 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 numbers from 25 to 40 in such a manner that each row
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 1 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 1 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 problem also has been
THE MAGIC SQUARE. 55
successfully solved, as the following diagram will show.
1 4 63 62 5 8 59 58
64 61 2 3 60 57 6 7
42 43 24 21 34 35 32 29
23 22 41 44 31 30 33 36
13 16 55 50 9 12 55 54
52 49 14 15 56 53 10 11
38 39 28 25 46 47 20 17
27 26 37 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 1 to 81. This square, which will be found below (Fig. 28), is divided
by parallel lines into 9 parts, of which each contains 9 consecutive
numbers that severally make up a magic square by themselves.
31 36 29 76 81 74 13 18 11
30 32 34 75 77 79 12 14 16
35 28 33 80 73 78 17 10 15
22 27 20 40 45 38 58 63 56
21 23 25 39 41 43 57 59 61
26 19 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 69 8 1 6 53 46 51
Fig. 28.
Wonderful as the properties of this square may appear, the law
by which the author constructed it is equally simple. We have simply
to regard the 9 parts as the 9 cells of a magic square of the numbers
from I to IX, and then to inscribe by the magic prescript in the square
56 THE MAGIC SQUARE.
designated as I the numbers from 1 to 9, in the square designated as
II 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 1 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 methodical 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 chessboard
the numbers from 1 to 64 that successive numbers, as well also
as the numbers 64 and 1, 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
THE MAGIC SQUARE. 57
gave the same sum 260. Ultimately he discovered several squares of this
description, which were published in the Berlin Chess Journal. One of
these is here appended:
47 10 23 64 49 2 59 6
22 63 48 9 60 5 50 3
11 46 61 24 1 52 7 58
62 21 12 45 8 57 4 51
19 36 25 40 13 44 53 30
26 39 20 33 56 29 14 43
35 18 37 28 41 16 31 54
38 27 34 17 32 55 42 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-problem of this kind
which will give 260 not only in the horizontal and vertical but also in
the two diagonal rows.
IX.
MAGICAL POLYGONS.
So far we have considered only such extensions of the idea underlying
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 consideration of the methods for
the construction of such figures, we will give here of magical polygons
simply a few examples, all supplied by Professor Scheffler:
1) The numbers from 1 to 32 admit of being written in a rectangle
of 4 × 8 in such a manner that the long horizontal rows give the sum
58 THE MAGIC SQUARE.
of 132 and the short vertical rows the sum of 66; thus:
1 10 11 29 28 19 18 16
9 2 30 12 20 27 7 25
24 31 3 21 13 6 26 8
32 23 22 4 5 14 15 17
Fig. 31.
2) The numbers from 1 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:
26 3 6 10 24 27
18 20 9 11 21 2
16 17
15 8
22 5
12
7 13
4 23
19
1 14
25
Fig. 32.
3) The numbers from 1 to 80 admit of being formed about a point
as common centre into 4 pentagons, such that each side of the first
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 line with the centre, is also the
same; namely, 92.
THE MAGIC SQUARE. 59
1
26 54
31 49
10 80
76 9
50 32
55 27
5 2
30 53
35 48
6 79
77 8
46 33
51 28
4 29 34 7 78 47 52 3
15
36 44
70 72
71 66
45 37
11 14
40 43
69 73
75 67
41 38
12 39 68 74 42 13
16
25 65
61 24
20 17
21 64
62 23
19 22 63 18
60
56 59
57 58
Fig. 33.
4) The numbers from 1 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 same 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 and along the
six that are constructed at right angles to its sides; this sum, for the
first hexagon from within, is 111, for the second 185, and for the third
259.
60 THE MAGIC SQUARE.
1 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
3 47 39 40 44 56
67 51 41 37 33 23 7
66 50 34 35 54 11
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 principle 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 description 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 belong in the compartments written on the upper surface of
the same and the numbers then taken off in layers of 16 from the top
downwards. We obtain thus 4 squares of 16 cells each, which together
THE MAGIC SQUARE. 61
make up the magic cube; as the following diagrams will show:
First Layer
from the Top.
1 48 32 49
60 21 37 12
56 25 41 8
13 36 20 61
Second Layer
from the Top.
63 18 34 15
6 43 27 54
10 39 23 58
51 30 46 3
Third Layer
from the Top.
62 19 35 14
7 42 26 55
1 38 22 59
50 31 47 2
Fourth Layer
from the Top.
4 45 29 52
57 24 40 9
53 28 44 5
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: 1, 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 any principal diagonal shall have one and the same
summation, 315.
Just as the magic squares of an odd number of cells could be formed
with the aid of two auxiliary squares, so also odd-numbered magic cubes
can be constructed with the help of three auxiliary cubes.
First Layer from Top.
121 27 83 14 70
11 6 117 48 79
44 100 1 57 113
53 109 40 91 22
87 18 74 105 31
Second Layer from Top.
2 58 114 45 96
36 92 23 54 110
75 101 32 88 19
84 15 66 122 28
118 49 80 6 62
Third Layer from Top.
33 89 20 71 102
67 123 29 85 11
76 7 63 119 50
115 41 97 3 59
24 55 106 37 93
Fourth Layer from Top.
64 120 46 77 8
98 4 60 111 42
107 38 94 25 51
16 72 103 34 90
30 81 12 68 124
Lowest Layer.
95 21 52 108 39
104 35 86 17 73
13 69 125 26 82
47 78 9 65 116
56 112 43 99 5
62 THE MAGIC SQUARE.
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 1 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 numbers of compartments.
CONCLUSION.
Musing on such problems as are the magic squares is fascinating to
thinkers of a mathematical turn of mind. We take delight 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 philosophical problem of the worldorder.
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 he heard
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 employing these definitions
as useful means of investigation. They had no idea of requiring
people to imagine four-dimensional 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 common geometry and to algebra operating with more
than three unknown quantities. At this stage, however, the spiritualists
came in, and coolly took possession of this private property of the
mathematicians. They were in great perplexity as to where they should
put the spirits of the dead. To give them a place in the world accessible
to our senses was not exactly practicable. They were compelled, therefore,
to look around after some terra incognita, which should oppose to
the spirit of research inborn in humanity an insuperable 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 possessed. With an intellectual perversity
63
64 THE FOURTH DIMENSION.
which science has no idea of, these spiritualists boldly asserted, first,
that the whole world was 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 ice-world.
Since spiritualism, under the leadership of a Leipsic Professor, Z¨ollner,
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
excellent 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 probably say, that the
definition in question referred to something different. A cubical box,
THE FOURTH DIMENSION. 65
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 unmistakably 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 determine
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 removed
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 decided 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 sufficient 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, one-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-dimensional. We, as
inhabitants of the earth, have naturally chosen as 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 establish and distinguish any definite point of time
66 THE FOURTH DIMENSION.
amid the infinite totality of all the points of time, is a single number.
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 denominator 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-dimensional
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-dimensional, 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 magnitude.
In conformity with the fundamental ideas by which we define the
notion of a one-dimensional manifoldness, it will be seen that the attribute
two-dimensional must be applied to all totalities of things in
which two numbers are necessary (and sufficient) to distinguish any
determinate individual thing amid the totality. The simplest twodimensioned
complex which we know of is the plane. To determine
accurately the position of a point in a plane, the simplest 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 Descartes the
fundamental idea of analytical geometry, a branch of mathematics in
which by the simple artifice of ascribing to every 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-dimensional. 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,
THE FOURTH DIMENSION. 67
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 perpendicular
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 follows 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 question might be asked, In what way and to what extent in
this case is the specification of two numbers requisite and sufficient to
determine amid all the rays which pass through the specified 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 azimuth, or the angular
distance between the circle on which the altitude is measured and the
meridian of the observer. It will be seen thus that the totality of all the
68 THE FOURTH DIMENSION.
luminous rays that an eye, conceived as a point, can receive from the
outer world is two-dimensional, and also that a luminous 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 following 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 bearings 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 perpendicular distances by which
the point in question is removed in a positive or negative sense from
these three planes. It is customary to denote the numbers which are
the measures of these three distances by x, y, and z, the positive x,
positive y, and positive z ordinarily 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 isolated from the three-dimensional manifoldness of all the
points of space a totality of less dimensions. If, for example, z is equal
to seven units or measures, this is equivalent to a statement that only
the two-dimensional totality of the points is meant, which constitute
the plane that can be laid at right angles to the upward-passing z-axis
at a distance of seven measures from the zero-point. Consequently,
every imaginable equation between x, y, and z isolates and defines a
two-dimensional aggregate of points. If two different equations obtain
THE FOURTH DIMENSION. 69
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 may 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 reference 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 numerous 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 ndimensional
totalities. All particles of air are four-dimensional in
magnitude when in addition to their position in space we also consider
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 fourdimensional
magnitudes, for their centres form a three-dimensional
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
assume 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 positions;
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 positions 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
70 THE FOURTH DIMENSION.
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
dimensions. Through the point obtained for the position of the second
corner of the square we must now conceive all the possible directions
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 increased by one, for only
one one-dimensional totality of perpendicular directions is possible to
one straight line in one of its points. Three corners of the square are
now fixed and therewith the position 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 dimensions.
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
points that differ only in respect of position, cannot in agreement with
our notions of things possess more than three dimensions. But the
question now arises, whether, if the progress of science tends in such a
direction, it is permissible to extend the notion 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 permissible,
although it leads to things which we cannot perceive by our
senses, I may call the reader’s attention to the fact that in arithmetic
we are accustomed from our youth upwards to extensions of ideas,
THE FOURTH DIMENSION. 71
which, accurately viewed, as little admit of graphic conception 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 counting of things, completely
from the individual and characteristic attributes of these things, that
is, we consider them as homogeneous. Second, we associate individually
with the things which we count 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
abbreviated 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 subtraction. But at this point
it clearly appears that not every problem 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 perfectly 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 solvable, the
science of arithmetic would never have received its full development,
and humanity would not have advanced as far in civilisation as it has.
* See the essay Notion and Definition of Number in this collection.
72 THE FOURTH DIMENSION.
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 pleasing opportunity of investing them
with one. The question is, then, what significance is the problem “5
minus 8” to be invested with?
The most natural and therefore the most advantageous solution
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 differences in which
the numbers that stand before the minus symbol are less by an equal
amount than those which follow it may be regarded as equal; so that the
simplest course seems to be to introduce 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 introduced. But in calling differential forms of this description
numbers, 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 FOURTH DIMENSION. 73
The point in all this, which the reader must carefully note, is, that
all the numbers of arithmetic, with the exception of the positive whole
numbers, are artificial products of human thought, invented to make
the language of arithmetic more flexible, and to accelerate the progress
of science. All these numbers lack the attributes 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 slightest 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 admissible,
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 advancement of
science is a different question. It must be admitted, therefore, that
the mathematician is justified in the extension of the notion of space
as a point-aggregate of three dimensions, and in the introduction of
space or point-aggregates of more than three dimensions, 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 gravitation, and of electricity would be a jumble of isolated
and unconnected mysteries. The assumption of an ether, however, comprises
in a systematic scheme all these isolated events, facilitates our
74 THE FOURTH DIMENSION.
mental control of the phenomena of nature, and enables us to produce
these phenomena at will. But it must not be forgotten in such reflexions
that the ether itself is even a greater problem for man, and that
the ether-hypothesis does not solve the difficulties of phenomena, but
only puts them in a unitary conceptual shape. Notwithstanding 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 introduction 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 “a3,” (which we read, a to the third power,)
and investigate in detail the operation of involution thus defined, so we
might also introduce some shorthand symbol for “a to the ath power to
the ath 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 ath , or “a to the ath to the ath 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 commutation
does not hold good. In addition, the numbers to be added may be
interchanged and the introduction of multiplication is therefore 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
THE FOURTH DIMENSION. 75
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 investigators 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 equations, derives much advantage from the notion of multidimensioned
spaces. If we have only three unknown quantities, 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 “moving,” “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 consideration 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 consideration
of more than three variables. But when we do this, we assume, 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-dimensioned manifoldness of numerical
quantities, which we may just as well regard as a four-dimensioned
aggregate of points. Two equations 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 determination of the contents of a square or a cube
by the involution 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
a2 stands for the contents of a square, and a3 for the content of a cube,
we naturally inquire after the contents of a structure which is produced
from the cube as the cube is produced from the square and which
also will have the contents a4. We cannot, it is true, clearly picture
to ourselves a structure of this description, but we can, nevertheless,
76 THE FOURTH DIMENSION.
establish its properties with mathematical exactness.* It is bounded
by 8 cubes just as the cube is bounded by 6 squares; it has 16 corners,
24 squares, and 32 edges, so that from every corner 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 multidimensioned
space, it must be conceded that the conception although
convenient is yet not indispensable. It is true, algebra is in need of the
idea of multiple dimensions, but it is not so absolutely in need of the
idea of point-aggregates of multiple dimensions.
This notion is, however, necessary and serviceable for a profound
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 present century the idea of elementary
geometry was indissolubly bound up with the name of Euclid, so that in
England where people adhered longest to the rigid deductive system of
the Grecian mathematician, 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 fundamental
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 possibly
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´evski, Riemann, and
Helmholtz, finally stated the decisive reasons why any attempt to prove
the axiom of the parallels must necessarily be futile. These reasons
* 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. Schlegel’s models are made by Brill of Darmstadt.
† Also called the twelfth axiom, also the fifth postulate.—Tr.
THE FOURTH DIMENSION. 77
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 demonstration of it. In view of
the intimate connection, which both in an historical and epistemological
point of view exists between the extension 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 geometry,
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 together less
than two right angles, these straight lines, being continually 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 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 interior
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 180°.
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 perception that the eleventh axiom
does not admit of demonstration without the employment of one of
78 THE FOURTH DIMENSION.
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 determination except those above mentioned.
a b
c
I
a b
c
II
b a
c
III
Fig. 35.
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 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, however, 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 position 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, whether
this is only possible in the plane, or whether it can also be done on
other surfaces.
THE FOURTH DIMENSION. 79
We find that there are certain surfaces in which this is possible, and
certain others in which it is not. For instance, it is impossible 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 everywhere the same curvature, but that the surface of an
egg at different 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 constant
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. There is no necessity
of our entering in any detail into the character and structure of the
last-mentioned surfaces.
Intimately related with the plane, however, are all those surfaces,
which, like the plane, have the curvature zero; in this category belong
especially cylindrical surfaces and conical surfaces. A sheet of paper
of the form of the sector of a circle may, for example, be readily bent
into the shape of a conical surface. If two congruent 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 bending 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 curvature. 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.
80 THE FOURTH DIMENSION.
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
other 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 angles 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 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 triangle it is more than two right angles,
the spherical triangle being 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 way 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-dimensional 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 remain 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 henceforth
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 without suffering in
THE FOURTH DIMENSION. 81
the transference any distension or any contraction. The space of experience,
therefore, has a constant measure of curvature. 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 possesses, 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 accurately expressed, in a shortest line, would
sometime, though perhaps 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 arrive 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
82 THE FOURTH DIMENSION.
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, “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 observation
whether the measure of curvature of experiential space is exactly 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 of 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 of observation at our command and
especially our telescopes are not competent to determine the amount of
the deviation. 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 institute
a comparison between the space of experience and other threedimensional
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 originally 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 service to science in the investigation of
THE FOURTH DIMENSION. 83
three-dimensional space, we shall give one a place here which is within
the comprehension of non-mathematicians.
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 intersection
12
13
14
25
35 45
15
34
23 24
Fig. 36.
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
of plane geometry only
with difficulty and great
circumstantiality. But by
resorting to the threedimensional
space of experience,
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
84 THE FOURTH DIMENSION.
meet some interposed plane in ten points whose relative positions are
exactly those of the ten points above described. 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 1, 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 conversely from every point also three straight
lines proceed.
Now, just as this figure is a section of a complete three-dimensional
pentagon, so another remarkable figure, of similar properties, may be
246 236
123
234
124
126
145
346
136
146
134
135
125
156
235
356
456
345
245
256
Fig. 37.
obtained from the section of a figure of four-dimensioned space. Imagine
six points, 1, 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
THE FOURTH DIMENSION. 85
points. (See Fig. 37.) Figures of this description, which are so composed
of points and rays that an equal number of rays proceed from
every point and an equal 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 service 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 chemistry, 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 structural formulæ by the different positions of
the atoms in space. (Compare La chimie dans l’espace 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 involve 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 cannot 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 will then be independent of one another.
86 THE FOURTH DIMENSION.
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 chemical research,
the science is really compelled, if it deals with molecules 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.
V.
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 multidimensioned
spaces has done, and bids fair to do, for geometrical 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 nonmathematicians
until the consequences which the spiritualists fancied
it was permissible to draw from this mathematical notion were publicly
known. But it is a tremendous step from the four-dimensioned space
of the 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 question of the
THE FOURTH DIMENSION. 87
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 existence of
a four-dimensional space containing the world, the least reprehensible
are those which are based on the existence of symmetry. We spoke
above of two triangles in the same plane which have all their sides and
A
C
B
D
F
E
Fig. 38.
angles congruent, but which cannot be made to coincide 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
ABF about the side AB, 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, F 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, 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
88 THE FOURTH DIMENSION.
equal to each other, and of which each may be viewed as the reflected
image of the other, are called symmetrical. For instance, the right and
the left hand are symmetrical; or, a right and a left glove. Now just
as in two 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, however, 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 anterior and
posterior sides are interchanged. If the structure which is to be converted
into its symmetrical counterpart is made of a flexible 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 angular velocity of the apparent motions
of the fixed stars is explicable 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 dependent on the sum of the experiences which we have before
THE FOURTH DIMENSION. 89
acquired, and partly on the necessity, due to the imperfection of reason,
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, Z¨ollner has also adduced cosmological
reasons to prove the existence of a four-dimensional space. To
these reasons belong especially the questions, whether the number 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 distance; 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 Z¨ollner
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
in a three-dimensional space, the phenomena therein which
are full of mystery to it. A four-dimensioned space would in such case
possess for the metaphysician a value similar to that which the ether
possesses for the physicist. Still more convincing than these cosmological
reasons to the majority of men is the physio-psychological reason
drawn from the phenomena of vision which Z¨ollner adduces. Into this
main argument we will enter in more detail.
When we “see” an object, as we all know, the light which proceeds
or is reflected therefrom produces an image on the retina of our eye;
this image is conducted to our consciousness by means of the optic
nerve, and our reason draws therefrom an inference respecting 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
90 THE FOURTH DIMENSION.
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 threedimensional
world must appear to the eye two-dimensional, because all
points which lie behind each other in the direction outwards from the
eye produce on the retina only one image. This is due to the fact that
the retinal images are themselves two-dimensional; for which reason,
Z¨ollner says, the world must appear to a child as two-dimensional, if
it be supposed to live in a primitive condition of unconscious mental
activity. To such a child two objects which are moving the one behind
the other, must appear as suffering displacement on a surface, 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
Z¨ollner further argues, in the first years of childhood forced to adopt a
hypothesis concerning the constitution of space and to assume that the
world is three-dimensional, although the eye can really perceive it as
only two-dimensional. Z¨ollner then further says, that in the explanation
of the effects of the external world, man constantly finds this hypothesis
of his childish 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 Z¨ollner’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 riddles 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. Z¨ollner also contends that our conception of the entire visible
world as possessed of three dimensions is a product of our reason,
which the mind was driven to form by the contradictions which would
be presented to it on the assumption of only two dimensions by the perspective
distortions, coincidences, and changes of magnitude of objects.
THE FOURTH DIMENSION. 91
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 surface
of the retina as the real object, and not the hand which lies beyond 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. Z¨ollner’s contention is, therefore, that man originally
had only a two-dimensional intuition of space, but was forced
by experience to represent to himself the objects 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 Z¨ollner’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 twentieth century as since Copernicus’s time the explanation of
the motions of the heavenly bodies by means of a three-dimensional
motion has been.
Z¨ollner’s arguments from the phenomena of vision may be refuted
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 photography 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
thickness. It is also wrong to say, as Z¨ollner says, that when we see
92 THE FOURTH DIMENSION.
the shadow of a hand which is cast upon a wall we see something twodimensional.
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-dimensional 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 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 reflected from every material particle strikes our eye. Now, by
thinking exclusively of those material particles which are reflected, in
contrariety to the empty space without and the hidden and therefore
non-reflected particles within, we form the notion of a surface.
It is evident from this, that all that we perceive is three-dimensional,
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 Z¨ollner. 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
Z¨ollner 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
THE FOURTH DIMENSION. 93
blind who have suddenly 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 irritations 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 possessed no peripheral nerves. According to
Z¨ollner’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 unintelligible
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-dimensioned 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 simply displace them to another 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-dimensioned space does exist, as the arguments adduced in sections
III and IV have rendered plain. If, namely, the space of four
94 THE FOURTH DIMENSION.
dimensions 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 actuality 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, therefore, as we can locally
fix the idea of a two-dimensional material world, as little can we ever
verify the notion of a four-dimensional 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, Z¨ollner 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 experiential space. To
invest this opinion with greater strength, Z¨ollner 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 Z¨ollner,
that this greatest philosopher 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 different from what Z¨ollner sees in them or pretends to
see. Z¨ollner 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,
represents it. Z¨ollner similarly deals with the Kantian “thing-in-itself,”
which is also regarded as an object of higher dimensions.
To show Kant in the light of a predecessor, Z¨ollner quotes the following
passage from the former’s “Tr¨aume eines Geistersehers, erl¨autert
THE FOURTH DIMENSION. 95
durch Tr¨aume der Metaphysik” (1766, Collected Works, Vol. VII. page
32 et seq.): “I confess that I am very much inclined 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 latter are connected, but
upon some internal principle of its own condition. 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 immaterial 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 spiritual 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 observations,—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. This contention,
which is that of very many thinkers, does not, however, entail the
consequence that the “transcendental subject of Kant” must be fourdimensional,
as Z¨ollner 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 therefore 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 Z¨ollner’s hypothesis. S. v. Waltershausen relates of Gauss in his
Gruss zum Ged¨achtniss, (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
96 THE FOURTH DIMENSION.
are above and outside our world may look down upon us; and there
were, he continued, in a jesting tone, 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 abstract 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
congruent 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 spirits.
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 gravitation. 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 explain the phenomenon of gravitation
as a force governing matter, is, however, essentially different
from the spiritual world of Z¨ollner, the function of which is to explain
supposed supersensuous phenomena which stand in the most glaring
contradiction with the established 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 spiritualism
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 excited the Chinese as
much as it has aroused, during the last few decades, the European and
THE FOURTH DIMENSION. 97
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 phenomena
into physical and intellectual. Of the first class he enumerates the
following: the moving of chairs and tables; the animation of walkingsticks,
slippers, and broomsticks; the miraculous throwing of objects;
spirit-rappings (Luther heard a sound in the Wartburg, “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
Z¨ollner); the double appearance of the same person; the penetration
of matter (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 exudations 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 phenomena are cited:
increased eloquence or suddenly acquired power of speaking in foreign
languages; spirit-effects at a distance; inability 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 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.
98 THE FOURTH DIMENSION.
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 dimension, that they cannot
be passed over.
First, with respect to clairvoyance, the American visionary Davis
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 adjoining 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 Hemisphere, 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´emoires d’un m´edicin, 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. Z¨ollner explains
clairvoyance by means of the fourth dimension thus:
A man who is accustomed to viewing things on a plain is supposed
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 distances. The plain,
that is, the two-dimensioned space, is accordingly viewed by him from
points outside of the plain as “open” in all directions. Exactly so, in
Z¨ollner’s theory, must three-dimensioned 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. Z¨ollner thus explains
clairvoyance as a condition in which the soul has displaced itself out
of its three-dimensioned space and reached a point which with respect
* Quoted by Cranz.
THE FOURTH DIMENSION. 99
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 balloon than
from places on the earth is simply this, that between the suspended
balloon and the objects seen at a distance nothing intervenes 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 explanation of Z¨ollner 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 Z¨ollner. Of these experiments
we will speak of the two most important, the experiment with
the glass sphere and the experiment with the knots. To explain the
connection of the glass sphere experiment with the fourth dimension,
we must first conceive of two-dimensional reasoning beings, or, let us
say, two-dimensional worms, living and moving in 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 transportation of the object from a position within the circle
to any position outside it could be effected by the three-dimensional
being simply taking the object out of the plane and placing it at the
desired point. This object, therefore, would, in an inexplicable manner,
100 THE FOURTH DIMENSION.
suddenly disappear before the eyes of the worms who were assembled
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 dimension, we shall derive
from this trick, which is wholly removed from the sense-perception
of the flattened worms, the following experiment, 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 Z¨ollner. 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, controlled
his four-dimensional space as we do our three-dimensional space. It
never occurred to Z¨ollner 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 that we cannot
explain a trick easily and naturally does not irrevocably 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 experiments
with knots. To explain this in connection with the fourth dimension,
we must resort again to the plane and the flat worm inhabiting 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, because 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
THE FOURTH DIMENSION. 101
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
breaking the seals at the two ends of the string. Everybody knows that
this problem is not soluble, but it may be calculated mathematically
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-dimensional
motion. Now as Slade untied the knot before Z¨ollner’s eyes without
apparently making any use of the ends fastened in the walls, Z¨ollner
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. (Compare
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 everything of this senseperceptible
world a real transcendental and four-dimensional “thing-initself”
whose projection in the space of experience 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
coryphæi of modern science offer, a lack of comprehension of real philosophy,”
the magazine The Proof of Faith expresses its delight at
the discovery of spiritualism in the following manner: “Every Christian
will surely rejoice at the deep and perhaps mortal wound which
102 THE FOURTH DIMENSION.
these new discoveries have in all probability administered 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 Z¨ollner, as the most prominent
representative of modern spiritualism, has put forward respecting its
relations to the doctrines of Christianity (Wissensch. 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 Z¨ollner’s view 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
disappeared 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 “Spiritualism 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: “(1) 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,
departed 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
THE FOURTH DIMENSION. 103
slate-writings belong throughout in the category of imbecility; they are
totally bereft of any contents. (3) The most favored, apparently, is the
moral condition of the soul. According to the testimony which we have,
its character 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 people 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 slow as it is, yet always
leads to a sure extension of knowledge, in the hope that some aimless,
foolhardy venture into the realm of uncertainty 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 problems of the world that are yet unsolved.
*
* *
Now 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 by our
senses is three-dimensional and that everything that possesses four or
more dimensions can only be regarded as abstractions or fictions which
our reason forms in its constant efforts after an extension 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-dimensional 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-dimensioned 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
104 THE FOURTH DIMENSION.
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 preceding
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 second fourdimensional
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 Z¨ollner and his adherents maintain, 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, generally, 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, Z¨ollner were
right, the plan of creation would possess the wonderful feature that
four-dimensional beings are capable of arbitrarily affecting the threedimensional
world, but that three-dimensional beings have no right in
their turn consciously to affect a two-dimensional world.
The supporters of Z¨ollner’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 consciously on
two-dimensional worlds, and that at any rate it is simply an inference
by analogy to conclude from the non-existence of a relation 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 Z¨ollner’s hypothesis, but only to
stamp it as very improbable. But despite this improbability Z¨ollner
would still be right if the phenomena of the material 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 mysterious phenomena, and there always will be. Yet,
as we have often seen that the progress of science has again and again
revealed as natural what former generations held to be supernatural, it
THE FOURTH DIMENSION. 105
is certainly wholly wrong to bring in for the explanation of phenomena
which now seem mysterious an hypothesis like that of Z¨ollner, by which
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 experiment, or
any botanical or zo¨ological culture. If the spirits are the authors of the
phenomena that are mysterious to us, why should they also not have
control of the phenomena which are not mysterious? The existence of
mysterious phenomena justifies in no manner or form the assumption
that spirits exist which produce them. Would it not be much simpler, if
we must have supernatural influences, to adopt the na¨ıve 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 supernatural
depend to a great extent on the stage of culture which humanity 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 be
absolutely master of the conditions under which the experiment is or is
not to succeed. Now, this criterion of scientific experimenting is totally
lacking in all spiritualistic experiments. We can never assign in their
case the conditions under which they will or will not succeed. When
all the preparations in a spiritualistic s´eance have been properly made,
but nothing takes place, the beautiful excuse is always forthcoming that
106 THE FOURTH DIMENSION.
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-psychological experiments in
question should only be performed by men who possess the critical eye
of inquiry, who are free from the dangers 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 arrive 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 employment
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
REMOTEST TIMES.
I.
UNIVERSAL INTEREST IN THE PROBLEM.
For two and a half thousand years, both competent and incompetent
minds have striven in vain to solve the problem known as
the squaring of the circle. Now that geometers have at last succeeded
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 justifiable 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 professional mathematicians.
The Proceedings of the French Academy for the year 1775 contain
at page 61 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 overwhelming 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 important 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,
107
108 THE SQUARING OF THE CIRCLE.
however, sees in this high-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
newspapers 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 examined
by the light? Almost always they will be found to be imperfectly
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 conceit 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 attempted 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 mathematical 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 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 considerable labor to extricate the
kernel of the demonstration from the turgid language and bombastic
* For the full psychogeny and psychiatry of the circle-squarer see A. De Morgan,
A Budget of Paradoxes (London, 1872).—Tr.
THE SQUARING OF THE CIRCLE. 109
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 circumscribed 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 elementary
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 enticed, but by the allurement which
everything exerts that is calculated to raise the individual above the
mass of ordinary humanity, 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. 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 attack
the problem, but which even in the hands of the most distinguished
mathematical strategists did not suffice to take the stronghold.
110 THE SQUARING OF THE CIRCLE.
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 mathematical questions? The
question is answered by the fact that the squaring 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 mathematical 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, tetragwnÐzein, 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 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 contrary, 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
THE SQUARING OF THE CIRCLE. 111
of modern mathematics, no one succeeded in supplying the wishedfor
demonstration of the problem’s impossibility. At last, Professor
Lindemann, of K¨onigsberg, in June, 1882, succeeded in furnishing a
demonstration,—and the first demonstration,—that it is impossible by
employing 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 accomplished 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 centuries 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 others have failed, and that just
they have been appointed by Providence to solve the famous puzzle.
But unfortunately the ineradicable mania for solving the quadrature of
the circle has also its serious side. Circle-squarers are not always so selfsatisfied
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 unable to
solve, darkens their soul and, lost to the world, they become 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
112 THE SQUARING OF THE CIRCLE.
circles both large and small. Now, since the time of Archimedes, all
civilised nations that have cultivated mathematics have denoted the
number that tells how many times larger the circumference of a circle
is than the diameter by the symbol ,—the Greek initial letter of
the word periphery.* To compute , therefore, means to calculate how
many times larger the circumference 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 diameter is
perhaps most important; that is, the computation of how great an area
that part of a plane which lies within a circle measures. 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 . For it is demonstrated in elementary geometry, that
the area of a circle is equal to the area of a triangle produced by drawing
in the circle a radius, erecting at the extremity of 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  amounts to.
The numerical rectification and numerical quadrature of the circle
based upon the computation of the number  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 rectification” or “constructive
quadrature.” Approximately, of course, by employing an approximate
value for , these problems are easily solvable. But to solve a problem
of construction in geometry, means to solve it with mathematical exactitude.
If the value  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 31
7 times greater than its diameter; then the diameter could
be divided into seven equal parts, which could easily be done by the
* The Greek symbol  was first employed by W. Jones in 1706 and did not come
into general use until about the middle of the eighteenth century through the
works of Euler.—Trans.
THE SQUARING OF THE CIRCLE. 113
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 matter 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 . Consequently, 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  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 circumference of a circle;
thus demonstrating at once the impossibility of solving the problem.
This question is to be answered in the negative. 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 constructed from the other by employing only straight edge
and compasses. This might be the case, too, with the rectification of
the circle; and consequently from the impossibility of representing  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 circumference. 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 of straight edge and compasses be
converted into a square exactly equal to it in area. So that, supposing
the rectification of the circumference of a circle to have been successfully
effected, a square could be constructed that would be exactly equal in
area to the circle.
114 THE SQUARING OF THE CIRCLE.
The dependence upon one another of the three problems of the
computation of the number , 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  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 construct a figure
there are so large a number of conditions annexed 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 only five original
problems. They are:
1. The construction of a straight line that shall pass through two
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 intersection)
exist.
THE SQUARING OF THE CIRCLE. 115
For the solution of the three last of these five problems the eye
alone is needed, while for the solution of the first two problems, besides
pencil, ink, chalk, or the like, additional special instruments 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 that it is no concern of geometry
what mechanical instruments are employed in the solution of the
five problems mentioned. Geometry 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 postulates.* 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 compasses,” 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 unsuccessful efforts at elementary solution, have upon the
* Usually geometers mention only two postulates (Nos. 1 and 2). But since to
geometry proper it is indifferent whether only the eye, or additional special
mechanical instruments are necessary, the author has regarded it more correct
in point of method to assume five postulates.
116 THE SQUARING OF THE CIRCLE.
whole advanced the science of geometry, and contributed much to the
clarification of geometrical ideas.
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 British 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 oneninth
of its length and upon the shortened line thus obtained a square
erected. Of course, the area of a square of this construction is only
approximately equal to the area of the circle. An idea may be obtained
of the degree of exactness of this original, primitive 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 Archimedes, yet much more correct than many
a one later employed. It is not known how Ahmes or his predecessors
arrived at this approximate quadrature; but it is certain that it was
handed down in Egypt from century to century, and in late Egyptian
times it appears repeatedly.
In addition to the effort of the Egyptians, we also find in pre-
Grecian antiquity an attempt at circle-computation among the Babylonians.
This is not a quadrature, but is intended as a rectification
of the circumference. The Babylonian mathematicians had discovered,
that if the radius of a circle be successively inscribed as a chord within
THE SQUARING OF THE CIRCLE. 117
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 computation may even be found in the Bible;
for in 1 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 quadrature, 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 Anaxagoras belongs the merit of having called attention
to a problem that was to bear rich fruit by inciting Grecian scholars to
busy themselves 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 purpose:
first, to trisect an angle, and second to square the circle. This curved
line is the tetragwnÐzousa so often mentioned by the later Greek mathematicians,
and by the Romans called “quadratrix.” Regarding the
nature of this curve we have exact knowledge from Pappus. But it
will be sufficient here to state that the quadratrix is not a circle nor
a portion of a circle, so that its construction 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
118 THE SQUARING OF THE CIRCLE.
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 insuperable.
Some time afterwards, about the year 350 B. C., the mathematician
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 rˆole in Grecian mathematics as the related problem of
quadrature.
As these problems gradually became known to the non-mathematicians
of Greece, attempts at solution at once sprang up that are worthy
of a place by the side of the solutions of modern amateur circlesquarers.
The Sophists especially believed themselves competent by
seductive dialectic to take the stronghold that had defied the intellectual
onslaughts of the greatest mathematicians. 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 itself, it was produced. The number 36, accordingly,
was, as they thought, the one that embodied the solution of the
famous problem.
Contrasted with this twisting of words the speculations of Bryson
and Antiphon, both contemporaries of Socrates, though inexact, appear
in a high degree intelligent. Antiphon divided the circle into four
equal arcs, and by joining the points of division obtained a square; he
then divided each arc again into two equal parts and thus obtained an
inscribed octagon; thence he constructed an inscribed 16-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
THE SQUARING OF THE CIRCLE. 119
area; and upon the basis of this fact Antiphon regarded the squaring
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 circumscribed 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. Notwithstanding this error, however, to
Bryson belongs the merit—first, of having introduced into mathematics
by his emphasis of the necessity 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 circumscribed polygons with a circle, of having indicated
to Archimedes the way by which an approximate value of  was to be
reached.
Not long after Antiphon and Bryson, Hippocrates of Chios treated
the problem, which had now become more and more famous, 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 rectilinearly bounded figure, and
which therefore could be converted by straight edge and compasses
into a square equal in area. First, Hippocrates found that the crescentshaped
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 untiring scholar were directed towards converting a
circle into a crescent. Naturally he was unable to attain this object, but
by his efforts 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
120 THE SQUARING OF THE CIRCLE.
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 Grecian
geometricians comes the systematist Euclid, whose rigid formulation of
geometrical principles has remained the standard presentation 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 computation 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 contributions. 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 Roman 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  unquestionably belongs. Like Bryson he started with regular
inscribed and circumscribed polygons. He showed how it was possible,
beginning with the perimeter of an inscribed hexagon, which is equal to
six radii, 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 : 20171/4, and on the other hand, that the corresponding ratio
with respect to the circumscribed 96-sided polygon was smaller than
14688 : 46731/2. He inferred from this, that the number , the ratio of
the circumference to the diameter, was greater than the fraction 6336
2017 1
4
and smaller than 14688
4673 1
2
. Reducing the two limits thus found for the
value of , Archimedes then showed that the first fraction was greater
than 310
71 , and that the second fraction was smaller than 31
7 , whence it
followed with certainty that the value sought for  lay between 31
7 and
310
71 . The larger of these two approximate values is the only one usually
learned and employed. That which fills us with most astonishment in
the case of Archimedes’s computation of , is, first, the great acumen
THE SQUARING OF THE CIRCLE. 121
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  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 Archimedes,
all refer to and employ the approximate value of 31
7 for , without,
however, contributing anything essentially new to the problems
of quadrature and of cyclometry. Thus Hero of Alexandria, the father
of surveying, who flourished about the year 100 B. C., employs for
purposes of practical measurement sometimes the value 31
7 for  and
sometimes even the rougher approximation  = 3. The astronomer
Ptolemy, who lived in Alexandria 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 8
60 and 30
3600 , or as we now say 3 degrees, 8 minutes (partes minutae
primae), and 30 seconds (partes minutae secundae). As a matter
of fact, the expression 3 + 8
60 + 30
3600 = 3 17
120 represents the number 
more exactly than 31
7 ; but on the other hand, is, by reason of the magnitude
of the numbers 17 and 120 as compared with the numbers 1 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 Romans
stood upon the shoulders of the Greeks. Indeed, with respect to cyclometry,
they not only did not add anything new to the Grecian discoveries,
but frequently even evinced that they either did not know
* For Archimedes’s actual researches, see Rudio, Archimedes, Huygens, Lambert,
Legendre, vier Abhand. ¨uber die Kreismessung (Leipsic, 1892), where translations
of the works of these four authors on cyclometry will be found.—Tr.
122 THE SQUARING OF THE CIRCLE.
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 diameter must measure 121
2
feet in circumference; in other words, he made  equal to 31
8 . And, similarly,
a treatise on surveying, preserved to us in the Gudian manuscript
of the library of Wolfenb¨uttel, 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  = 4.
The mathematical performances of the Hindus were not only
greater than those of the Romans, but in certain directions surpassed
even those of the Greeks. In the most ancient source of the mathematics
of India that we know of, the Culvasˆutras, 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 problem 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  would have to be if the construction were
exactly correct. We find out in this way that the value of  upon which
the Indian circling of the square is based, is about from five to six
hundredths smaller than the true value, whereas the approximate  of
Archimedes, 31
7 , 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 approximation that
in exactness surpasses even that of Ptolemy. The Hindu result gives
3.1416 for , while  really lies between 3.141592 and 3.141593. How
the Hindus obtained this excellent value is told by Gane¸ca, the commentator
of Bhˆaskara, an author of the twelfth century. Gane¸ca says
THE SQUARING OF THE CIRCLE. 123
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 circumferences of the inscribed and circumscribed
384-sided polygons they found that  was equal to 3927 : 1250. It will
be seen that the value given by Bhˆaskara is identical with the value
of Aryabhatta. It is further worthy of remark that the earlier of these
two Hindu mathematicians does not mention either the value 31
7 of
Archimedes or the value 3 17
120 of Ptolemy, but that the later one knows
of both values and especially recommends that of Archimedes as the
most useful for practical applications. Strange to say, the good approximate
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 10 when the radius
is unity. The value of  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  equal to the square root of 10, 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 respect
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, accomplished 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  several stages farther, and of having obtained in this way a
much more exact value for it—a circumstance that is explainable when
we consider that the Hindus are the inventors of our present system
of numeral notation, possessing 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 , or 3; but they possessed knowledge of
124 THE SQUARING OF THE CIRCLE.
the approximate value of Archimedes at least since the end of the sixth
century. Besides this, there appears in a number of Chinese mathematical
treatises an approximate value peculiarly their own, in which
 = 3 7
50 ; a value, however, which notwithstanding it is written in larger
figures, is no better than that of 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 preserved
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 Archimedean approximate value and
the two Hindu values, the square root of 10 and the ratio 62832 : 20000.
This distinction occurs also in Muhammed Ibn Musa Alchwarizmˆı, 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 Mohammedan world. The Arabians, however,
did not study the numerical quadrature of the circle only, but also the
constructive; for instance, 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
unfortunately has not yet been edited.
Christian civilisation, to which we are now about to pass, produced
up to the second half of the fifteenth century extremely insignificant
results in mathematics. Even with regard to our present problem we
have but a single important work to mention; the work, namely, of
Frankos von L¨uttich 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 inconsiderable 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 Nicolas de Cusa, a
man highly esteemed for his astronomical and calendarial studies. He
claimed to have discovered the quadrature of the circle by employing
only straight edge and compasses and thus attracted the attention of
THE SQUARING OF THE CIRCLE. 125
scholars to the historic problem. People believed the famous Cardinal,
and marvelled at his wisdom, until Regiomontanus, in letters written in
1464 and 1465 and published in 1533, rigorously demonstrated that the
Cardinal’s quadrature was incorrect. The construction of Cusa was as
follows. The radius of a circle is prolonged a distance equal to the side
of the inscribed 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
 = 31
7 . For by Cusa’s method  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 without 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 geometrical 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 quadrature, like most of his other professed
discoveries, was incorrect.
In the succeeding period the number of circle-squarers so increased
that we shall have to limit ourselves to those whom mathematicians
recognise. And particularly is Simon Van Eyck to be mentioned, who
towards the close of the sixteenth century published a quadrature which
was so approximate that the value of  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 31
7 . The
erroneous quadrature of Van Eyck was thus the occasion of Metius’s
discovery that the ratio 355 : 113, or 3 16
113 , varied from the true value of
 by less than one one-millionth, eclipsing accordingly all values hitherto
obtained. Moreover, it is demonstrable by the theory of continued
fractions, that, admitting figures to four places only, no two numbers
more exactly represent the value of  than 355 and 113.
126 THE SQUARING OF THE CIRCLE.
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 discovery 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 services to
astronomy, the Neapolitan John Porta, and Gregory of St. Vincent.
Longomontanus made  = 3 14185
100000 and was so convinced of the correctness
of his result as to thank God fervently, in the preface to his work
Inventio Quadraturae Circuli, 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  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 representing  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  if we cause the operations of extracting
the square root of 1
2 , and certain related additions and multiplications,
to succeed each other in a certain manner, and that  must come out
exactly, if this series of operations could be continued indefinitely. Vieta
thus found that to a diameter of 10000 million units a circumference
belongs of from 31415 million 926535 units to 31415 million 926536
units of the same length.
THE SQUARING OF THE CIRCLE. 127
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 circumference
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  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 declared to be correct. Ludolf was justly proud of his work,
and following the example of Archimedes, requested in his will that the
result of his most important mathematical performance, the computation
of  to 35 decimal places, be engraved upon his tombstone; a
request which is said to have been carried out. In honor of Ludolf,  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 sufficient
for any practical purpose that could ever arise, neither the problem
of constructive rectification nor that of constructive quadrature had
been in any respect theoretically advanced thereby. The investigations
conducted by the famous mathematicians and physicists Huygens and
Snell about the middle of the seventeenth century, were more important
from a mathematical point of view than the work of Ludolf. In his
book Cyclometricus Snell took the position 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 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 beautiful 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.; Theoremata De Circuli et Hyperbolae
Quadratura, 1651), as well as perfected in many ways by him. Snell and
128 THE SQUARING OF THE CIRCLE.
Huygens were fully aware that they had advanced 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 English 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 controversy, to which we owe many valuable tracts, was,
that Huygens finally demonstrated in an incontrovertible manner the
incorrectness 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 contemporary world, we shall omit
entirely to speak of the circle-squarers of our own time.
First to be mentioned is the celebrated English philosopher Hobbes.
In his book De Problematis Physicis, in which he proposes 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  = 31
5 . 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 special treatise, where to sustain at least the
appearance of being right, he disputed the fundamental principles of
geometry and the theorem of Pythagoras.
THE SQUARING OF THE CIRCLE. 129
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  = 3; Mathulon, who offered
in legal form a reward of a thousand dollars to the person who would
point out an error in his solution of 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 confidence 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
definite 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 quadrature
in which  equalled 31
8 , and promised fifty ducats to the person who
could prove that it was incorrect. Hesse of Berlin wrote an arithmetic
in 1776, in which a true quadrature was also “made known,”  being
exactly equal to 314
99 . About the same time Professor Bischoff of
Stettin defended a quadrature previously published by Captain Leistner,
Preacher Merkel, and Schoolmaster B¨ohm, which virtually made
 equal to the square of 62
35 , 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 construction can be easily made
with straight edge and compasses. Constructions of this kind, simple
in form and yet sufficiently exact for practical purposes, have been produced
for centuries in great numbers. 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
130 THE SQUARING OF THE CIRCLE.
rectification of the circle 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 intersection
with that point on the other perpendicular which is distant 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 difference between the true length
of the circumference and the line thus constructed is less than 3
100000 of
the diameter.
Although such constructions of approximation are very interesting
in themselves, they nevertheless play but a subordinate rˆole 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 thirtyfive
decimal places which Ludolf found, and on the other hand they
are not adapted to advance in any way the question 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 and
Leibnitz, and known as the differential and the integral calculus.
About the middle of the seventeenth century, before Newton and
Leibnitz represented  by series of powers, the English mathematicians
Wallis and Lord Brouncker, Newton’s predecessors in certain lines, succeeded
in representing  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  is represented
by the regularly formed product
2
3 × 4
3 × 4
5 × 6
5 × 6
7
8
7
8
9 × 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
THE SQUARING OF THE CIRCLE. 131
elegant result without proof, demonstrated the same in his Arithmetic
of Infinites.
The computation of  could scarcely have been pushed to a greater
degree of exactness by these results than that to which Ludolf 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  to hundreds of decimal
places.
Gregory, Newton, and Leibnitz found that the fourth part of  was
equal exactly to
1 − 1
3 + 1
5 − 1
7 + 1
9 − 1
11 + 1
13 − . . .
if we conceive this series, which is called the Leibnitz series, continued
indefinitely. This series is wonderfully simple but is not adapted to
the computation of , for the reason that entirely too many members
have to be taken into account to obtain  accurately to a few decimal
places only. The original formula, however, from which this series is
derived, gives other formulæ which are excellently adapted to the actual
computation. The original formula is the general series:
 = a − 1
3a3 + 1
5a5 − 1
7a7 + . . . ,
where  is the length of the arc belonging to any central angle in a
circle of radius 1, and a the tangent to this angle. From this we derive
the following:

4
= (a + b + c + . . . ) − 1
3 (a3 + b3 + c3 + . . . )
+ 1
5 (a5 + b5 + c5 + . . . ) − . . . ,
where a, b, c . . . are the tangents of angles whose sum is 45°. Determining,
therefore, the values of a, b, c . . . , which are equal to small and
convenient fractions and fulfil the conditions just mentioned, we obtain
series of powers which are adapted to the computation of .
The first to add by the aid of series of this description additional
decimal places to the old 35 in the number  was the English arithmetician
Abraham Sharp, who, following Halley’s instructions, in 1700
worked out  to 72 decimal places. A little later Machin, professor of
astronomy in London, computed  to 100 decimal places, by putting,
132 THE SQUARING OF THE CIRCLE.
in the series given above, a = b = c = d = 1
5 and e = − 1
239 ; that is, by
employing the following series:

4
= 4  1
5 −
1
3  52 +
1
5  55 −
1
7  57 + . . . 
−  1
239 −
1
3  2393 +
1
5  2395 − . . . 
In the year 1819, Lagny of Paris outdid the computation of Machin,
determining in two different ways the first 127 decimal places of .
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 = 1
2 , b = 1
5 , c = 1
8 . Finally,
at a recent date,  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 theoretical nor
a practical value. That the computation of  to say 15 decimal places
more than sufficiently satisfies the subtlest requirements of practice may
be gathered from a concrete example of the degree of exactness thus
obtainable. Imagine a circle to be described with Berlin as centre, and
the circumference to pass through Hamburg; then let the circumference
of the circle be computed by multiplying its diameter by the value of
 to 15 decimal places, and then conceive it to be actually measured.
The deviation from the true length in so large a circle as this even could
not be as great as the 18 millionth part of a millimetre.
An idea can hardly be obtained of the degree of exactness produced
by 100 decimal places. But the following example may possibly give us
some conception of it. Conceive a sphere constructed with the earth
as centre, and imagine its surface to pass through Sirius, which is 1341
2
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 these diminutive animalcula are
present. Now conceive these microbes to be all unpacked and so distributed
singly along a straight line, that every two microbes are as far
distant from each other as Sirius from us, that is 1341
2 million million
* In 1873 the approximation was carried by Shanks to 707 places of decimals.—
Trans.
THE SQUARING OF THE CIRCLE. 133
kilometres. Conceive the long line thus fixed by all the microbes, as the
diameter of a circle, and imagine the circumference of it to be calculated
by multiplying its diameter by  to 100 decimal places. Then, in
the case of a circle of this enormous magnitude even, the circumference
so calculated would not vary from the real circumference by a millionth
part of a millimetre.
This example will suffice to show that the calculation of  to 100
or 500 decimal places is wholly useless.
Before we close this chapter upon the evaluation of , we must
mention the method, less fruitful than curious, which Professor Wolff
of Zurich employed some decades ago to compute the value of  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 exactly 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  − 3. Consequently,
a sufficient number of casts of the needle according to the law of large
numbers must give the value of  approximately. As a matter of fact,
Professor Wolff, after 10000 trials, obtained the value of  correctly to
3 decimal places.
Fruitful as the calculus of Newton and Leibnitz was for the evaluation
of , 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 convinced 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 celebrated 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 
was neither a rational number nor even the square root of a rational
number; that is, that neither  nor the square of  could be exactly
* See also A. De Morgan, A Budget of Paradoxes, pp. 169–171.—Tr.
134 THE SQUARING OF THE CIRCLE.
represented by a fraction the denominator and numerator 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 particular 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.
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 postulates only. Slight
reflection showed, that a problem, to be elementarily 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 consequently its rectification were solvable elementarily, the number
, 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 numbers; that
is, there would have to exist an equation, made up entirely of whole
numbers, which would be correct if its unknown quantity were made
equal to .
Since the beginning of this century, consequently, the efforts of a
number of mathematicians have been bent upon proving that  generally
is not algebraical, that is, that it cannot be the root of an equation
having whole numbers for coefficients. But mathematics 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 preparatory assistance in his
treatise Sur la Fonction Exponentielle, published in the seventy-seventh
volume of the Comptes Rendus, Professor Lindemann, at that time of
Freiburg, now of Munich, finally succeeded, in June 1882, in rigorously
* Given in Legendre’s Geometry, in the Appendix to De Morgan, op. cit., p. 495,
and in Rudio, op. cit.—Tr.
THE SQUARING OF THE CIRCLE. 135
demonstrating that the number  is not algebraical,* and so supplied
the first proof that the problems of the rectification and squaring of
the circle with the help only of algebraical instruments like straight
edge and compasses 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 construct 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
* 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
e = 1 +
1
1
+
1
1  2
+
1
1  2  3
+
1
1  2  3  4
+ . . .
developed relations between certain definite integrals (Comptes Rendus of the
Paris Academy, Vol. 77, 1873). Proceeding from the relations thus established,
Professor Lindemann first demonstrates the following proposition: If the coefficients
of an equation of the nth degree are all real or complex whole numbers
and the n roots of this equation z1, z2, . . . , zn are different from zero and from
each other it is impossible for
ez1 + ez2 + ez3 · · · + ezn
to be equal to a
b , where a and b are real or complex whole numbers. It is then
shown that also between the functions
erz1 + erz2 + erz3 + . . . erzn,
where r denotes an integer, no linear equation can exist with rational coefficients
different from zero. Finally the beautiful theorem results: If z is the root
of an irreducible algebraic equation the coefficients of which are real or complex
whole numbers, then ez cannot be equal to a rational number. Now in reality
ep−1 is equal to a rational number, namely, −1. Consequently, p−1, and
therefore  itself, cannot be the root of an equation of the nth degree having
whole numbers for coefficients, and therefore also not of such an equation having
rational coefficients. The property last mentioned, however,  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.—Tr.]
136 THE SQUARING OF THE CIRCLE.
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.
INDEX.
Abstraction, 91
Academy, French, 107
Addition, 74; associative law of, 10;
commutative law of, 9
Ahmes, 116
Alchwarizmˆı, Muhammed Ibn Musa,
124
Algebraic curves, theory of, 32
Alhaitam, Ibn, 124
Analytical geometry, 66
Anaxagoras, 117
Antiphon, 118
Apollonius, 25, 31
Approximation, construction of, 129
Arabs, magic squares of, 39; squaring
of the circle among, 121 et seq.
Archimedes, 25, 108, 112, 120 et seq.
Arithmetic, 33; pure, 8 et seq.; monism
in, 8–24
Aryabhatta, 122
Association, laws of, 17
Associative law of addition, 10
Augend, 9
Axioms, Euclid’s, 76 et seq.
Aztecs, 5, 71
Babylonians, squaring of the circle
among, 116 et seq.
Bacteriology, 33
Basselin, 129
Beman, W. W., 135
Bhˆaskara, 122
Bible, speaks of four dimensions, 102;
squaring of the circle in the, 117
Bischoff, Professor, 129
B¨ohm, 129
Bolyai, 34, 76
Bovillius, 125
Brahmagupta, 123
Brouncker, Lord, 130
Bryson, 119
Cagliostro, 98
Calculus, Differential, 27
Cardan, 13
Cards, problem at, 49 et seq.
Chemistry, 34; application of the idea
of multi-dimensional space to
theoretical, 85
Chess-knight, magic squares that
involve the move of the, 56 et seq.
Chinese, 123; squaring of the circle
among, 121 et seq.
Christian nations, squaring of the circle
among, 121 et seq.
Cipher, 12
Circle-squarers, 108 et seq.
Circumversion, congruence by, 78
Clairvoyance, 98
Clavius, 126
Clebsch, Alfred, 32
Clerget, 129
Clocks, 3
Commutation, 14–17
Commutative law of addition, 9
Complex numbers, 23, 32
Concentric magic squares, 53 et seq.
Configurations, 85
Congruent figures, 78
Conic sections, 31
137
138 INDEX.
Construction, problems of, 112
Continuous functions, 26
Copernicus, 91
Corsonich, Lieutenant-Colonel, 129
Counting, 1 et seq., 9, 71 et seq.
Cranz, 64, 97
Culvasˆutras, 122
Cumberland, 106
Curvature, negative, 79 et seq.;
positive, 79 et seq.; zero, 79; of
space, 82
Cyclometry, 121 et seq.
Dase, Zacharias, 132
Davis, the American visionary, 98
De Cusa, Nicolas, 124
Definitional formulæ, 11, 20
Degrees, 121
Degree, numbers of, 18
De la Hire, 42, 44, 49, 52
De M´eziriac, Bachet, 42, 43
De Morgan, 37, 108, 133, 134
Depth, 90
Descartes, 66
De Serres, Oliver, 129
Detraction, 10
Difference, 11
Differential forms, 12 et seq., 72
Dimension, concept of, 64 et seq.
Dinostratus, 118
Displacement, congruence by, 78
Distance of objects, 93
Dumas, Alexander, 98
Duplication of the cube, 28
Du Prel, Carl, 101
D¨urer, Albert, 37, 39, 40, 42
Egyptians, the squaring of the circle
among, 116 et seq.
Eisenlohr, 27, 116
Equations between x, y and z, 68
Ether, 73 et seq.
Etruscans, 2
Euclid, 26, 34, 76, 120
Euclid’s axioms, 76–80
Euler, 112, 129
Evolution, 19, 20
Extension of notions in mathematics,
73 et seq.
Finaeus, Orontius, 125
Fingers, 1 et seq.; in reckoning,
4 et seq.
Five, as a numeral basis, 5
Foundation-principle, 13
Four-dimensional point-aggregates,
70 et seq.
Four-dimensioned space, refutation of
the existence of, 86 et seq.
Fourth degree, operation of, 74
Fourth dimension, the, 63–106
Four variable quantities, 75
Fractional numbers, negative, 17
Fractions, 17 et seq.
Functions, theory of, 32
Gane¸ca, 122
Gauss, 34, 76, 95
Geometry, 34; origin of, 30
Glaukon, 94
Grassmann, 9
Greeks, 3, 12; squaring of the circle
among, 110, 116 et seq.
Gregory of St. Vincent, 126
Gregory, James, 128, 131
Gudian manuscript, 122
Halley, 131
Hamlet, 37
Hankel, 9, 13
Harmony of the spheres, 62
Have, Mr., 97
Helmholtz, 34, 76
Hermite, Professor, 134
Herodotus, 30
Hesse, 129
Hindus, 12; mathematical performances
of the, 122 et seq.; squaring of the
circle among, 121 et seq.
Hippias, 117, 118
Hippocrates, 119
History, 30
Hobbes, 128
INDEX. 139
Homer, 81
Hugel, 60
Hume, 36
Huygens, 127, 128
Imaginary numbers, 22, 32
Impossibility of demonstration, 111
Increment, 9
Infinitely great, 18, 82
Insoluble problems, 27, 135
Involution, 18, 20, 74
Irrational numbers, 21
Jones, W., 112
Kant, 36, 94 et seq.
Kepler, 31
Kircher, Athanasius, 42
Klein, Felix, 29, 135
Knight, move of a, 50
Knight-problem, 56
Knots, experiment with the, 99, 100
Kochansky, 60, 130
Kronecker, 7, 9
Lagny, 132
Lambert, 29, 133
Law of association, 14
Legendre, 77, 134
Leibnitz, 26, 131
Leistner, 129
Lessing, 28
Liger, 129
Limits, 119
Lindemann, 29, 111, 134
Livy, 2
Lobach´evski, 34, 76
Local value, Hindu system of, 3
Logarithm, finding of the, 19
Logarithmand, 19
Longomontanus, 126
Ludolf, 130
Lunes of Hippocrates, 119
Luther, 97
Machin, 131
Magic cubes, 60 et seq.
Magic Squares, 36–62; problem and
origin of, 39 et seq.; concentric,
53 et seq.; even-numbered,
46 et seq.; odd-numbered,
41 et seq.; with magical parts,
54 et seq.; whose summation gives
the number of a year, 52 et seq.;
that involve the move of the
chess-knight, 56 et seq.
Mathematical knowledge, on the
nature of, 25–35; characteristics of,
32; intrinsic character of, 25
Mathematics, 36; most conservative of
all sciences, 25; self-sufficiency of,
30
Mathulon, 129
Measuring, 16
Mechanical instruments of geometry,
115 et seq.
Mechanics, 34
Mediums, 102
Melancholy, 37, 38
Merkel, 129
Methods, discovery of, 32
Metius, Peter, 125, 126
Mexico, ancient, 2
Mill, J. S., 36
Milton, 37
Minerva, 2
Minus sign, 11
Minutes, 121
Models of three-dimensional nets, 76
Monism in arithmetic, 8–24
Monist, The, 36
Moschopulus, 42
Multi-dimensional magnitudes and
spaces, 66 et seq., 75
Multiplication, 14, 74
Mysticism, 37
140 INDEX.
Named number, 6
N-dimensional totalities, 68 et seq.
Negative exponents, 19, 20
Negative numbers, 12 et seq., 15, 17, 72
Newton, 26, 31, 131
Nonius, Petrus, 125
Notation, systems of, 5
Number, notation and definition of,
1–7; denominate, 4; abstract, 6; as
the result of counting, 9; complex,
23, 32; forms of, 14
Numbers, as symbols, 7; imaginary, 22,
32; irrational, 21; real, 22
Numbering, 5
Number-pictures and signs, natural,
2 et seq.
Number-words, natural, 3
Numerals, the formation of, 4 et seq.
Numeral words, 4 et seq.; compound,
4; in the Indo-Germanic tongues, 4
Numeral writing, 2 et seq.
Numeration, 1 et seq.
Numeri ficti, 13
Numeri veri, 13
Pappus, 117
Parallels, theory of, 26, 76 et seq.
Parting, 16
Perception, 91
Physics, 8, 30, 34
, 112 et seq., 120, 130 et seq.
Plato, 36, 94
Polygons, magical, 57 et seq.
Porta, John, 126
Postulates, 115
Powers, 19
Prime numbers, problem of the
frequency of, 28
Principle of no exception, 13
Prizes offered circle-squarers, 110
Probabilities, 133
Problems, fundamental, 114; solvable
with straight edge and compasses,
134
Ptolemy, 30, 121, 122
Pure arithmetic, 8
Pythagoras, 25, 117, 128
Quadratrix, 117
Quadrature of the circle, in Egypt, 109;
stands and falls with the problem
of rectification, 113; numerical,
112 et seq.
Quantities, four variable, 75; negative,
12 et seq.
Quaternions, 24
Radicand, 19
Rays, 67
Real numbers, 22
Rectification of the circle, numerical,
112 et seq.
Regiomontanus, 125
Rhind Papyrus, 27, 116
Riemann, 34, 76, 96
Riese, Adam, 42
Romans, 3, 71; squaring of the circle
among, 121 et seq.
Romanus, Adrianus, 126, 127
Rudio, 121, 134
Sauveur, 42, 60
Scaliger, Joseph, 126
Scheffler, 42, 44, 57, 60
Schlegel, 64, 76
Schr¨oder, E., 9, 11
Science, 106
Seconds, 121
Series, infinite, 126 et seq.; Leibnitz’s,
131
Shakespeare, 37
Shanks, 132
Sharp, Abraham, 131
Shortest line, 79
Slade, 97
Smith, D. E., 135
Snell, 127
Socrates, 94
Solvability of problems, 115
Sophists, 118
Space, curvature of, 82;
four-dimensional, 63; intuition of,
91; of experience, 80; of multiple
dimensions, great service to
science, 82
INDEX. 141
Spheres in space, 69 et seq.
Spirits, existence of four-dimensional,
94 et seq.; of the dead, 64
Spirit-rappings, 97
Spirit-writing, 97
Spiritualistic, mediums, 86 et seq.;
s´eance, 105
Spiritualists, 63
Squaring of the circle, 27, 107–136
Squares in space, 69 et seq.
Stifel, Michael, 42
Straight edge and compasses, to
construct with, 113 et seq.
Subtertraction, 10
Subtraction, 10 et seq., 71 et seq.
Subtrahent, 11
Summands, 9
Supernatural influences, 105 et seq.
Sylvester II., 124
Symbols, 2
Symmetry, 88
Talmud, 117
Ten, 4
tetragwnÐzein, 110
Thales, 117
Thing-in-itself, 94
Third degree, operations of the, 18
Three Bodies, problem of, 28
Time, 33
Triangle, sum of the angles of a, 80
Trisection of the angle, 28
Twenty, as base, 5
Ulrici, 102
Ulysses, 81
Unlimited, 82
Unnamed number, 6
Van Ceulen, Ludolf, 127
Van Eyck, Simon, 125
Van’t Hoff, 85
Vicenary system, 5
Vieta, 126
Vision, 89 et seq.
Vitruvius, 122
Von L¨uttich, Frankos, 124
Wallis, 128, 130
Waltershausen, 95
Weierstrass, 29
Wenzelides, 56
Wislicenus, 85
Wolfenb¨uttel Library, 122
Wolff, Professor, 133
Words, numeral, 4 et seq.
Worms, two-dimensional, 99
Wundt, 102
Zero, 12, 15
Zero exponents, 20
Z¨ollner, 64, 89 et seq.

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perception, understanding, reflection, and ideation, each of these faculties being founded on one of the senses.
Intellectual errors are classified as fallacies of sensation, fallacies of perception, fallacies of understanding, fallacies
of reflection, and fallacies of ideation, and a war is waged against the metaphysic of the idealists in the interest of the
philosophy of science.
In the chapters on fallacies there is a careful discussion of the theory of ghosts, especially as treated in the publications
of the Society for Psychical Research, and by various other authors on the same subject.
No student of the sciences can afford to neglect this book. The discussion is clear and entertaining.
On the Study and Difficulties of Mathematics
By AUGUSTUS DE MORGAN.
Just Published—New corrected and annotated edition, with references to
date, of the work published in 1831 by the Society for the Diffusion of Useful
Knowledge. The original is now scarce.
With a fine Portrait of the great Mathematical Teacher, Complete Index, and
Bibliographies of Modern Works on Algebra, the Philosophy of Mathematics, Pangeometry,
etc.
Pp. viii + 288. Cloth, $1.25 (5s.).
“A Valuable Essay.”—Prof. Jevons, in the Encyclopædia Britannica.
“The mathematical writings of De Morgan can be commended unreservedly.”—Prof.
W. W. Beman, University of Michigan.
“It is a pleasure to see such a noble work appear as such a gem of the book-maker’s art.”—
Principal David Eugene Smith, Brockport Normal School, N. Y.
THE OPEN COURT PUBLISHING CO., CHICAGO.
324 Dearborn St.
London: Kegan Paul, Trench, Tr¨ubner & Co.
LICENSING. 145


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