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THE NATURE OF 
MATHEMATICS 

P -E B JOTIRDAIHM.A 


THE PEOPLE'S -BOOKS 

THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA 

LOS ANGELES 

M C AINSH&CO 

14 COLLEGE ST TORONTO 

THE PEOPLE'S BOOKS 

THE NATURE OF MATHEMATICS 

THE NATURE OF 
MATHEMATICS 

Bv PHILIP E. B. JOURDAIN, M.A. 

LONDON: T. C. & E. C. JACK 
67 LONG ACRE, W.C., AND EDINBURGH 
NEW YORK: DODGE PUBLISHING CO. 

Engineering & Mathematical ( Sciences Library o>ax 

PREFACE 

THE aim of the following pages is fully stated in the Intro- 
ductory Chapter. Here I need only mention that some of the 
reflections in this book are taken from articles of mine in the 
Monist of 1908 and in Nature of 1909. To the Editors of 
these periodicals I wish to express my thanks for allowing me 
again to say some things which their kindness allowed me to 
say before. I must also thank those of my friends who have 
kindly read and helpfully criticised parts of this book. 



872007 

CONTENTS 



PAGE 
INTRODUCTION 5 

CHAP. 

I. THE GROWTH OF MATHEMATICAL SCIENCE IN 

ANCIENT TIMES ...... 10 

II. THE RISE AND PROGRESS OP MODERN MATHE- 
MATICS ALGEBRA 23 

III. THE RISE AND PROGRESS OP MODERN MATHE- 

MATICS ANALYTICAL GEOMETRY AND THE 
METHOD OP INDIVISIBLES .... 39 

IV. THE BEGINNINGS OF THE APPLICATION OF MATHE- 

MATICS TO NATURAL SCIENCE THE SCIENCE 

OF DYNAMICS 53 

V. THE RISE OF MODERN MATHEMATICS THE IN- 
FINITESIMAL CALCULUS 65 

VI. MODERN VIEWS OF LIMITS AND NUMBERS . . 77 

VII. THE NATURE OF MATHEMATICS .... 83 

BIBLIOGRAPHY ....... 89 

INDEX 90 

THE NATURE OF MATHEMATICS 



INTRODUCTION 

AN eminent mathematician once remarked that he was never 
satisfied with his knowledge of a mathematical theory until he 
could explain it to the next man he met in the street. That 
is hardly exaggerated; however, we must remember that a 
satisfactory explanation entails duties on both sides. Any one 
of us has the right to ask of a mathematician, " What is the 
use of mathematics ? " Any one may, I think and will try to 
show, rightly suppose that a satisfactory answer, if such an 
answer is anyhow possible, can be given in quite simple terms. 
Even men of a most abstract science, such as mathematics or 
philosophy, are chiefly adapted for the ends of ordinary life ; 
when they think, they think, at the bottom, like other men. 
They are often more highly trained, and have a technical facility 
for thinking that comes partly from practice and partly from 
the use of the contrivances for correct and rapid thought given 
by the signs and rules for dealing with them that mathematics 
and modern logic provide. But there is no real reason why, 
with patience, an ordinary person should not understand, 
speaking broadly, what mathematicians do, why they do it, 
and what, so far as we know at present, mathematics is. 

Patience, then, is what may rightly be demanded of the 
inquirer. And this really implies that the question is not 
merely a rhetorical one an expression of irritation or scepti- 
cism put in the form of a question for the sake of some fancied 
effect. If Mr. A. dislikes the higher mathematics because he 
rightly perceives that they will not help him in the grocery 
business, he asks disgustedly, " What's the use of mathematics ? " 
and does not wait for an answer, but turns his attention to 

5 

6 THE NATURE OF MATHEMATICS 

grumbling at the lateness of his dinner. Now, we will admit 
at once that higher mathematics is of no more use in the 
grocery trade than the grocery trade is in the navigation of a 
ship ; but that is no reason why we should condemn mathe- 
matics as entirely useless. I remember reading a speech made 
by an eminent surgeon, who wished, laudably enough, to spread 
the cause of elementary surgical instruction. "The higher 
mathematics," said he with great satisfaction to himself, " do 
not help you to bind up a broken leg ! " Obviously they do 
not ; but it is equally obvious that surgery does not help us 
to add up accounts ; ... or even to think logically, or to 
accomplish the closely allied feat of seeing a joke. 

To the question about the use of mathematics we may reply 
by pointing out two obvious consequences of one of the applica- 
tions of mathematics : mathematics prevents much loss of life 
at sea, and increases the commercial prosperity of nations. 
Only a few men a few intelligent philosophers and more 
amateur philosophers who are not highly intelligent would 
doubt if these two things were indeed benefits. Still, probably, 
all of us act as if we thought that they were. Now, I do not 
mean that mathematicians go about with life-belts or serve 
behind counters ; they do not usually do so. What I mean I 
will now try to explain. 

Natural science is occupied very largely with the prevention 
of waste of the labour of thought and muscle when we want to 
call up, for some purpose or other, certain facts of experience. 
Facts are sometimes quite useful. For instance, it is useful for 
a sailor to know the positions of the stars and sun on the nights 
and days when he is out of sight of land. Otherwise, he cannot 
find his whereabouts. Now, some people connected with a 
national institution publish periodically a Nautical Almanac 
which contains the positions of stars and other celestial things 
you see through telescopes, for every day and night years and 
years ahead. This Almanac, then, obviously increases the 
possibilities of trade beyond coasting-trade, and makes travel 
by ship, when land cannot be sighted, much safer ; and there 
would be no Nautical Almanac if it were not for the science of 
astronomy ; and there would be no practicable science of 
astronomy if we could not organise the observations we make 
of sun and moon and stars, and put hundreds of observations 

INTRODUCTION 7 

in a convenient form and in a little space in short, if we could 
not economise our mental or bodily activity by remembering or 
carrying about two or three little formulse instead of fat books 
full of details ; and, lastly, we could not economise this activity 
if it were not for mathematics. 

Just as it is with astronomy, so it is with all other sciences 
both that of Nature and mathematical science : the very 
essence of them is the prevention of waste of the energies of 
muscle and memory. There are plenty of things in the un- 
known parts of science to work our brains at, and we can only 
do so efficiently if we organise our thinking properly, and con- 
sequently do not waste our energies. 

The purpose of this little volume is not to give like a text- 
book a collection of mathematical methods and examples, but 
to do, firstly, what text-books do not do : to show how and 
why these methods grew up. All these methods are simply 
means, contrived with the conscious or unconscious end of 
economy of thought-labour, for the convenient handling of long 
"and complicated chains of reasoning. This reasoning, when 
applied to foretell natural events, on the basis of the applica- 
tions of mathematics, as sketched in the fourth chapter, often 
gives striking results. But the methods of mathematics, 
though often suggested by natural events, are purely logical. 
Here the word "logical" means something more than the 
traditional doctrine consisting of a series of extracts from the 
science of reasoning, made by the genius of Aristotle and frozen 
into a hard body of doctrine by the lack of genius of his school. 
Modern logic is a science which has grown up with mathematics, 
and, after a period in which it moulded itself on the model of 
mathematics, has shown not only that the reasonings but also 
conceptions of mathematics are logical in their nature. 

In this book I shall not pay very much attention to the 
details of the elementary arithmetic, geometry, and algebra of 
the many text-books, but shall be concerned with the dis- 
cussion of those conceptions such as that of negative number 
which are used and not sufficiently discussed in these 
books. Then, too, I shall give a somewhat full account of the 
development of analytical methods and certain examinations of 
principles. 

8 THE NATURE OF MATHEMATICS 

I hope that I shall succeed in showing that the process of 
mathematical discovery is a living and a growing thing. Some 
mathematicians have lived long lives full of calm and unwavering 
faith for faith in mathematics, as I will show, has always been 
needed some have lived short lives full of burning zeal, and 
so on ; and in most of the faith of mathematicians there has 
been much error. 

Now we come to the second object of this book. In the 
historical part we shall see that the actual reasonings made by 
mathematicians in building up their methods have often not 
been in accordance with logical rules. How, then, can we say 
that the reasonings of mathematics are logical in their nature ? 
The answer is that the one word " mathematics " is habitually 
used in two senses, and so, as explained in the last chapter, I 
have distinguished "mathematics," the methods used to dis- 
cover certain truths, and " Mathematics " the truths discovered. 
When we have passed through the stage of finding out, by 
external evidence or conjecture, how mathematics grew up with 
problems suggested by natural events, like the falling of a 
jtone, and then how something very abstract and intangible 
but very real separated out of these problems, we can turn our 
attention to the problem of the nature of Mathematics without 
troubling ourselves any more as to how, historically, it gradu- 
ally appeared to us quite clearly that there is such a thing at 
all as Mathematics something which exists apart from its 
application to natural science. History has an immense value 
in being suggestive to the investigator, but it is, logically 
speaking, irrelevant. Suppose that you are a mathematician ; 
what you eat will have an important influence on your dis- 
coveries, but you would at once see how absurd it would be to 
make, say, the momentous discovery that 2 added to 3 makes 
5 depend on an orgy of mutton cutlets or bread and jam. 
The methods of work and daily life of mathematicians, the 
connecting threads of suggestion that run through their work, 
and the influence on their work of the allied work of others, 
all interest the investigator because these things give him 
examples of research and suggest new ideas to him ; but these 
reasons are psychological and not logical. 

But it is as true as it is natural that we should find that the 

INTRODUCTION 9 

way to become acquainted with new ideas is to study the 
way in which knowledge about them grew up. This, then, is 
what we will do in the first place, and it is here that I must 
bring my own views forward. Briefly stated, they are these. 
Every great advance in mathematics with which we shall be 
concerned here has arisen out of the needs shown in natural 
science or out of the need felt to connect together, in one 
methodically arranged whole, analogous mathematical processes 
used to describe different natural phenomena. The application 
of logic to our system of descriptions, which we may make 
either from the motive of satisfying an intellectual need (often 
as strong, in its way, as hunger) or with the practical end in 
view of satisfying ourselves that there are no hidden sources of 
error that may ultimately lead us astray in calculating future 
or past natural events, leads at once to those modern refine- 
ments of method that are regarded with disfavour by the 
old-fashioned mathematicians. 

In modern times appeared clearly what had only been 
vaguely suspected before the true nature of Mathematics. 
Of this I will try to give some account, and show that, since 
mathematics is logical and not psychological in its nature, all 
those petty questions sometimes amusing and often tedious 
of history, persons, and nations are irrelevant to Mathematics 
in itself. Mathematics has required centuries of excavation, 
and the process of excavation is not, of course, and never will 
be, complete. But we see enough now of what has been 
excavated clearly to distinguish between it and the tools which 
have been or are used for excavation. This confusion, it should 
be noticed, was never made by the excavators themselves, but 
only by some of the philosophical onlookers who reflected on 
what was being done. I hope and expect that our reflections 
will not lead to this confusion. 

10 THE NATURE OF MATHEMATICS 
CHAPTER I 

THE GROWTH OP MATHEMATICAL SCIENCE IN ANCIENT TIMES 

IN the history of the human race, inventions like those of the 
wheel, the lever, and the wedge were made very early judging 
from the pictures on ancient Egyptian and Assyrian monuments. 
These inventions were made on the basis of an instinctive and 
unreflecting knowledge of the processes of nature, and with the 
sole end of satisfaction of bodily needs. Primitive men had to 
build huts in order to protect themselves against the weather, and, 
for this purpose, had to lift and transport heavy weights, and so 
on. Later, by reflection on such inventions themselves, possibly 
for the purposes of instruction of the younger members of a 
tribe or the newly-joined members of a guild, these isolated 
inventions were classified according to some analogy. Thus we 
see the same elements occurring in the relation of a wheel to its 
axle and the relation of the arm of a lever to its fulcrum the 
same weights at the same distance from the axle or fulcrum, as 
the case may be, exert the same power, and we can thus class 
both instruments together in virtue of an analogy. Here what 
we call " scientific " classification begins. We can well imagine 
that this pursuit of science is attractive in itself; besides 
helping us to communicate facts in a comprehensive, compact, 
and reasonably connected way, it arouses a purely intellectual 
interest. It would be foolish to deny the obvious importance 
to us of our bodily needs; but we must clearly realise two 
things : (1) The intellectual need is very strong, andjs as much 
a fact as hunger or thirst ; sometimes it is even stronger than 
bodily needs Newton, for instance, often forgot to take food 
when he was engaged with his discoveries; (2) Practical 
results of value often follow from the satisfaction of intellectual 
needs. It was the satisfaction of certain intellectual needs in 
the cases of Maxwell and Hertz that ultimately led to wireless 
telegraphy ; it was the satisfaction of some of Faraday's intel- 
lectual needs that made the dynamo and the electric telegraph 
possible. But many of the results of strivings after intellectual 
satisfaction have as yet no obvious bearing on the satisfaction 
of our bodily needs. However, it is impossible to tell whether 
or no they will always be barren in this way. This gives us a 

GROWTH OF MATHEMATICAL SCIENCE 11 

new point of view from which to consider the question, " What 
is the use of mathematics ? " To condemn even those branches 
of mathematics because their results cannot obviously be applied 
to some practical purpose is short-sighted. 

The formation of science is peculiar to human beings among 
animals. The lower animals sometimes, but rarely, make 
isolated discoveries, but never seem to reflect on these in- 
ventions in themselves with a view to rational classification 
in the interests either of the intellect, or of the indirect 
furtherance of practical ends. Perhaps the greatest difference 
between man and the lower animals is that men are capable of 
taking circuitous paths for the attainment of their ends, while 
the lower animals have their minds so filled up with their 
needs that they try to seize the object they want or remove 
that which annoys them in a direct way. Thus, monkeys 
often vainly snatch at things they want, while even savage 
men use catapults or snares or the consciously observed pro- 
perties of flung stones. 

The communication of knowledge is the first occasion that 
compels distinct reflection, as everybody can still observe in 
himself. Further, that which the old members of a guild 
mechanically pursue strikes a new member as strange, and 
thus an impulse is given to fresh reflection and investigation. 

When we wish to bring to the knowledge of a person any 
phenomena or processes of nature, we have the choice of two 
methods : we may allow the person to observe matters for 
himself, when instruction comes to an end; or, we may 
describe to him the phenomena in some way, so as to save him 
the trouble of personally making anew each experiment. To 
describe an event like the falling of a stone to the earth 
in the most comprehensive and compact manner requires that 
we should discover what is constant and what is variable in 
the processes of nature ; that we should discover the same law 
in the moulding of a tear and in the motions of the planets. 
This is the very essence of nearly all science, and we will 
return again to this point later on. 

We have thus some idea of what is known as " the econo- 
mical function of science." This sounds as if science were 
governed by the same laws as the management of a business ; 

12 THE NATURE OF MATHEMATICS 

and so, in a way, it is. But whereas the aims of a business 
are not, at least directly, concerned with the satisfaction of 
intellectual needs, science including natural science, logic, 
and mathematics uses business methods consciously for such 
ends. The methods are far wider in range, more reasonably 
thought out, and more intelligently applied than ordinary 
business methods, but the principle is the same. And this 
may strike some people as strange, but it is nevertheless true : 
there appears more and more as time goes on a great and 
compelling beauty in these business methods of science. 

The economical function appears most plainly in very ancient 
and modern science. In the beginning-, all economy had in 
immediate view the satisfaction simply of bodily wants. 
With the artisan, and still more so with the investigator, 
the most concise and simplest possible knowledge of a given 
province of natural phenomena a knowledge that is attained 
with the least intellectual expenditure naturally becomes in 
itself an aim ; but though knowledge was at first a means to 
an end, yet, when the mental motives connected therewith are 
once developed and demand their satisfaction, all thought of 
its original purpose disappears. It is one great object of science 
to replace, or save the trouble of making experiments, by the 
reproduction and anticipation of facts in thought. Memory is 
handier than experience, and often answers the same purpose. 
Science is communicated by instruction, in order that one man 
may profit by the experience of another and be spared the 
trouble of accumulating it for himself; and thus, to spare the 
efforts of posterity, the experiences of whole generations are 
stored up in libraries. And further, yet another function of 
this economy is the preparation for fresh investigation. 1 

The economical character of ancient Greek geometry is not 
so apparent as that of the modern algebraical sciences. We 
shall be able to appreciate this fact when we have gained 
some ideas on the historical development of ancient and 
modern mathematical studies. 

The generally accepted account of the origin and early 
development of geometry is that the ancient Egyptians were 
obliged to invent it in order to restore the landmarks which 
had been destroyed by the periodical inundations of the Nile. 

1 Cf. pp. 7, 16, 20, E3, 89. 

GROWTH OF MATHEMATICAL SCIENCE 13 

These inundations swept away the landmarks in the valley 
of the river, and, by altering the course of the river, increased 
or decreased the taxable value of the adjoining lands, rendered 
a tolerably accurate system of surveying indispensable, and 
thus led to a systematic study of the subject by the priests. 
Proclus (412-485 A.D.), who wrote a summary of the early 
history of geometry, tells this story, which is also told by 
Herodotus, and observes that it is by no means strange that 
the invention of the sciences should have originated in practical 
needs, and that, further, the transition from perception with 
the senses to reflection, and from reflection to knowledge, is to 
be expected. Indeed, the very name " geometry " which is 
derived from two Greek words meaning measurement of the 
enrth seems to indicate that geometry was not indigenous to 
Greece, and that it arose from the necessity of surveying. 
For the Greek geometricians, as we shall see, seem always to 
have dealt with geometry as an abstract science to have 
considered lines and circles and spheres and so on, and not 
the rough pictures of these abstract ideas that we see in the 
world around us and to have sought for propositions which 
should be absolutely true, and not mere approximations. The 
name does not therefore refer to this practice. 

However, the history of mathematics cannot with certainty 
be traced back to any school or period before that of the 
Ionian Greeks. It seems that the Egyptians' geometrical 
knowledge was of a wholly practical nature. For example, 
the Egyptians were very particular about the exact orientation 
of their temples ; and they had therefore to obtain with 
accuracy a north and south line, as also an east and west line. 
By observing the points on the horizon where a star rose and 
set, and taking a plane midway between them, they could 
obtain a north and south line. To get an east and west line, 
which had to be drawn at right angles to this, certain people 
were employed who used a rope ABCD, divided by knots or 
marks at B and 0, so that the lengths AB, BO, CD were in 
the proportion 3:4:5. The length BO was placed along the 
north and south line, and pegs P and Q inserted at the knots 
B and 0. The piece BA (keeping it stretched all the time) 
was then rotated round the peg P, and similarly the piece CD 
vas rotated round the peg Q, until the ends A and D coin- 

14 THE NATURE OF MATHEMATICS 

cided ; the point thus indicated was marked by a peg R. 
The result was to form a triangle PQR whose angle at P was 
a right angle, and the line PR would give an east and west 
line. A similar method is constantly used at the present time 
by practical engineers, and by gardeners in marking tennis 
courts, for measuring a right angle. This method seems also 
to have been known to the Chinese nearly three thousand 
years ago, but the Chinese made no serious attempt to classify 
or extend the few rules of arithmetic or geometry with which 
they were acquainted, or to explain the causes of the phenomena 
which they observed. 

The geometrical theorem of which a particular case is in- 
volved in the method just described is well known to readers 
of the first book of Euclid's Elements. The Egyptians must 
probably have known that this theorem is true for a right- 
angled triangle when the sides containing the right angle are 
equal, for this is obvious if a floor be paved with tiles of that 
shape. But these facts cannot be said to show that geometry 
was then studied as a science. Our real knowledge of the nature 
of Egyptian geometry depends mainly on the Rhind papyrus. 

The ancient Egyptian papyrus of Rhind, which was written 
by an Egyptian priest named Ahmes considerably more than 
a thousand years before Christ, and which is now in the 
British Museum, contains a fairly complete applied mathe- 
matics, in which the measurement of figures and solids plays 
the principal part ; there are no theorems properly so called ; 
everything is stated in the form of problems, not in general 
terms but in distinct numbers. For example : to measure a 
rectangle the sides of which contain two and ten units of 
length ; to find the surface of a circular area whose diameter is 
six units. We find also in it indications for the measurement 
of solids, particularly of pyramids, whole and truncated. The 
arithmetical problems dealt with in this papyrus which, by 
the way, is headed " Directions for knowing all dark things "- 
contain some very interesting things. In modern language, 
we should say that the first part deals with the reduction of 
fractions whose numerators are 2 to a sum of fractions each 
of whose numerators is 1. Thus ^ is stated to be the sum 
of -Jf, -^g-, yfj-, and -j^-. Probably Ahmes had no rule 
for forming the component fractions, and the answers given 

GROWTH OF MATHEMATICAL SCIENCE 15 

represent the accumulated experiences of previous writers. In 
one solitary case, however, he has indicated his method, for, 
after having asserted that is the sum of \ and , he 
added that therefore two-thirds of one-fifth is equal to the sum 
of a half of a fifth and a sixth of a fifth, that is, to -^ + $ 

That so much attention should have been paid to fractions 
may be explained by the fact that in early times their treat- 
ment presented considerable difficulty. The Egyptians and 
Greeks simplified the problem by reducing a fraction to the sum 
of several fractions, in each of which the numerator was unity, 
so that they had to consider only the various denominators : 
the sole exception to this rule being the fraction f. This 
remained the Greek practice until the sixth century of our era. 
The Romans, on the other hand, generally kept the denominator 
equal to twelve, expressing the fraction (approximately) as so 
many twelfths. 

In Ahmes' treatment of multiplication, he seems to have 
relied on repeated additions. Thus, to multiply a certain 
number, which we will denote by the letter a, by 13, he first 
multiplied by 2 and got 2a, then he doubled the results and 
got 4a, then he again doubled the result and got 8a, and lastly 
he added together a, 4a, and 8a. 

Now we have used the sign " a " to stand for any number : 
not a particular number like 3, but any one. This is what 
Ahmes did, and what we learn to do in what we call " algebra." 
When Ahmes wished to find a number such that it, added to 
its seventh, makes 19, he symbolised the number by the sign 
we translate " heap." He had also signs for our " + ," " ," 
and " = ." l Nowadays we can write Ahmes' problem as : Find 

the number x such that x +-^- = 19. Ahmes gave the 

answer in the form 16 + \ + |-. 

1 In this book, I shall take great care in distinguishing signs for 
what they signify. Thus : 2 is to be distinguished from " 2 " : by 





16 THE NATURE OF MATHEMATICS 

We shall fiud that algebra was hardly touched by those 
Greeks who made of geometry such an important science, partly, 
perhaps, because the almost universal use of the abacus l 
rendered it easy for them to add and subtract without any 
knowledge of theoretical arithmetic. And here we must 
remember that the principal reason why Ahmes' arithmetical 
problems seem so easy to us, is because of our use from childhood 
of the system of notation introduced into Europe by the Arabs, 
who originally obtained it from the Hindoos. In this system 
an integral number is denoted by a succession of digits, each 
digit representing the product of that digit and a power of ten, 
and the number being equal to the sum of these products. 
Thus, by means of the local value attached to nine symbols and 
a symbol for zero, any number in the decimal scale of notation 
can be expressed. It is important to realise that the long and 
strenuous work of the most gifted minds was necessary to 
provide us with simple and expressive notation which, in nearly 
all parts of mathematics, enables even the less gifted of us to 
reproduce theorems which needed the greatest genius to dis- 
cover. Each improvement in notation seems, to the uninitiated, 
but a small thing ; and yet, in a calculation, the pen sometimes 
seems to be more intelligent than the user. Our notation is an 
instance of that great spirit of economy which spares waste of 
labour on what is already systematised, so that all our strength 
can be concentrated either upon what is known but unsys- 
tematised, or upon what is unknown. 

Let us now consider the transformation of Egyptian geometry 
in Greek hands. Thales of Miletus (about 640-546 B.C.), 
who, during the early part of his life, was engaged partly in 
commerce and partly in public affairs, visited Egypt and first 
brought this knowledge into Greece. He discovered many 
things himself, and communicated the beginnings of many to 
his successors. We cannot form any exact idea as to how 

1 The principle of the abacus is that a number is represented by 
counters in a series of grooves, or beads strung on parallel wires ; 
as many counters being put on the first groove as there are units, as 
many on the second as there are tens, and so on. The rules to be 
followed in addition, subtraction, multiplication, and division are 
given in various old works on arithmetic. 






GROWTH OF MATHEMATICAL SCIENCE 17 

Thales presented his geometrical teaching. We infer, how- 
ever, from Proclus that it consisted of a number of isolated 
propositions which were not arranged in a logical sequence, but 
that the proofs were deductive, so that the theorems were not 
a mere statement of an induction from a large number of special 
instances, as probably was the case with the Egyptian geo- 
metricians. The deductive character which he thus gave to 
the science is his chief claim to distinction. Pythagoras (born 
about 580 B.C.) changed geometry into the form of an abstract 
science, regarding its principles in a purely abstract manner, 
and investigated its theorems from the immaterial and intel- 
lectual point of view. Among the successors of these men, the 
best known are Archytas of Tarentum (428-347 B.C.), Plato 
(429-348 B.C.), Hippocrates of Chios (born about 470 B.C.), 
Meuaechmus (about 375-325 B.C.), Euclid (about 330-275 B.C.), 
Archimedes (287-212 B.C.), and Apollonius (260-200 B.C.). 

The only geometry known to the Egyptian priests was that 
of surfaces, together with a sketch of that of solids, a geometry 
consisting of the knowledge of the areas contained by some 
simple plane and solid figures, which they had obtained by 
actual trial. Thales introduced the ideal of establishing by 
exact reasoning the relations between the different parts of a 
figure, so that some of them could be found by means of others 
in a manner strictly rigorous. This was a phenomenon quite 
new in the world, and due, in fact, to the abstract spirit of 
the Greeks. In connection with the new impulse given to 
geometry, there arose with Thales, moreover, scientific astro- 
nomy, also an abstract science, and undoubtedly a Greek 
creation. The astronomy of the Greeks differs from that of 
the Orientals in this respect : the astronomy of the latter, 
which is altogether concrete and empirical, consisted merely in 
determining the duration of some periods or in indicating, by 
means of a mechanical process, the motions of the sun and 
planets, whilst the astronomy of the Greeks aimed at the 
discovery of the geometrical laws of the motions of the heavenly 
bodies. 

Let us consider a simple case. The area of a right-angled 
field of length 80 yards and breadth 50 yards is 4000 square 
yards. Other fields which are not rectangular can be approxi- 
mately measured by mentally dissecting them a process which 






18 THE NATURE OF MATHEMATICS 

often requires great ingenuity and is a familiar problem to land- 
surveyors. Now, let us suppose that we have a circular field 
to measure. Imagine from the centre of the circle a large 
number of radii drawn, and let each radius make equal angles 
with the naxt ones on each side of it. By joining the points 
in succession where the radii meet the circumference of the 
circle, we get a large number of triangles of equal area, and 
the sum of the areas of all these triangles gives an approxima- 
tion to the area of the circle. It is particularly instructive 
repeatedly to go over this and the following examples mentally, 
noticing how helpful the abstract ideas we call " straight line," 
" circle," " radius," " angle," and so on, are. We all of us 
know them, recognise them, and can easily feel that they are 
trustworthy and accurate ideas. We feel at home, so to speak, 
with the idea of a square, say, and can at once give details 
about it which are exactly true for it, and very nearly true for 
a field which we know is very nearly a square. This replace- 
ment in thought by an abstract geometrical object economises 
labour of thinking and imagining by leading us to concentrate 
our thoughts on that alone which is essential for our purpose. 

Thales seems to have discovered and it is a good thing to 
follow these discoveries on figures made with the help of com- 
passes and ruler the proof of what may be regarded as the 
obvious fact that the circle is divided into halves by its 
diameter, that the angles at the base of a triangle with two 
equal sides an isosceles triangle are equal, that all the 
triangles described in a semi-circle with two of their angular 
points at the ends of the diameter and the third anywhere on 
the circumference contain a right angle ; and he measured the 
distance of vessels from the shore, presumably by causing two 
observers at a known distance apart to measure the two angles 
formed by themselves and the ship. This last discovery is an 
application of the fact that a triangle is determined if its base 
and base angles are given. 

When Archytas and Meuaechmus employed mechanical 
instruments for solving certain geometrical problems, " Plato," 
says Plutarch, " inveighed against them with great indignation 
and persistence as destroying and perverting all the good there 
is in geometry ; for the method absconds from incorporeal and 
intellectual to sensible things, and besides employs again such 






GROWTH OF MATHEMATICAL SCIENCE 19 

bodies as require much vulgar handicraft : in this way mechanics 
was dissimilated and expelled from geometry, and, being for a 
long time looked down upon by philosophy, became one of the 
arts of war." In fact, manual labour was looked down upon by 
the Greeks, and a sharp distinction was drawn between the 
slaves, who performed bodily work and really observed nature, 
and the leisured upper classes who speculated and often only 
knew Nature by hearsay. This explains much of the naive, 
hazy, and dreamy character of ancient natural science. Only 
seldom did the impulse to make experiments for oneself break 
through ; but when it did, a great progress resulted, as was 
the case with Archytas and Archimedes. Archimedes, like 
Plato, held that it was undesirable for a philosopher to seek to 
apply the results of science to any practical use ; but, whatever 
might have been his view of what ought to be the case, he did 
actually introduce a large number of new inventions. 

We will not consider further here the development of 
mathematics with other ancient nations, nor the chief problems 
investigated by the Greeks ; such details may be found in some 
of the books mentioned in the Bibliography at the end. The 
object of this chapter is to indicate the nature of the science of 
geometry, and how certain practical needs gave rise to investi- 
gations in which appears an abstract science which was worthy 
of being cultivated for its own sake, and which incidentally 
gave rise to advantages of a practical nature. 

There are two branches of mathematics which began to be 
cultivated by the Greeks, and which allow a connection to be 
formed between the spirits of ancient and modern mathematics. 

The first is the method of geometrical analysis to which 
Plato seems to have directed attention. The analytical method 
of proof begins by assuming that the theorem or problem is 
solved, and thence deducing some result. If the result be 
false, the theorem is not true or the problem is incapable of 
solution : if the result be true, if the steps be reversible, we 
get (by reversing them) a synthetic proof ; but if the steps be 
not reversible, no conclusion can be drawn. "We notice that 
the leading thought in analysis is that which is fundamental in 
algebra, and which we have noticed in the case of Ahmes : 
the calculation or reasoning with an unknown entity, which is 






20 THE NATURE OF MATHEMATICS 

denoted by a conventional sign, as if it were known, and the 
deduction at last, of some relation which determines what the 
entity must be. 

And this brings us to the second branch spoken of: algebra 
with the later Greeks. Diophantus of Alexandria, who pro- 
bably lived in the early half of the fourth century after Christ, 
and probably was the original inventor of an algebra, used 
letters for unknown quantities in arithmetic and treated 
arithmetical problems analytically. Juxtaposition of symbols 
represented what we now write as " + ," ar >d " " and " = " 
were also represented by symbols. All these symbols are 
mere abbreviations for words, and perhaps the most important 
advantage of symbolism the power it gives of carrying out 
a complicated chain of reasoning almost mechanically was not 
made much of by Diophantus. Here again we come across 
the economical value of symbolism : it prevents the wearisome 
expenditure of mental and bodily energy on those processes 
which can be carried out mechanically. We must remember 
that this economy both emphasises the unsubjugated that is 
to say, unsystematised problems of science, and has a charm 
an aesthetic charm, it would seem of its own. 

Lastly, we must mention "incommensurables," "loci," and 
the beginnings of " trigonometry." 

Pythagoras was, according to Eudemus and Proclus, the 
discoverer of "incommensurable quantities." Thus, he is said 
to have found that the diagonal and the side of a square are 
" incommensurable." Suppose, for example, that the side of 
the square is one unit in length ; the diagonal is longer than 
this, but is not two units in length. The excess of the length 
of the diagonal over one unit is not an integral submultiple of 
the unit. And we can proceed in this way without end. Ex- 
pressing the matter arithmetically, the remainder that is left 
over after each division of a remainder into the preceding 
divisor is not an integral submultiple of the remainder used as 
divisor. That is to say, the rule given in text-books on arith- 
metic and algebra for finding the greatest common measure 
does not come to an end. This rale, when applied to integer 
numbers, always comes to an end ; but, when applied to certain 
lengths, it does not. Pythagoras proved, then, that if we 






GROWTH OF MATHEMATICAL SCIENCE 21 

start with a line of any length, there are other lines whose 
lengths do not bear to the first length the ratio of one integer 
to another, no matter if we have all the integers to choose 
from. Of course, any two fractions have the ratio of two 
integers to one another. In the above case of the diagonal, if 
the diagonal were in length some number x of units, we should 
have a; 2 = 2, and it can be proved that no fraction, when 
" multiplied " in the sense to be given in the next chapter 
by itself gives 2 exactly, though there are fractions which give 
this result more and more approximately. 

On this account, the Greeks drew a sharp distinction be- 
tween "numbers," and "magnitudes'' or "quantities" or 
measures of lengths. This distinction was gradually blotted 
out as people saw more and more the advantages of identifying 
numbers with the measures of lengths. The invention of 
analytical geometry, described in the third chapter, did most 
of this blotting out. It is in comparatively modern times 
that mathematicians have adequately realised the importance 
of this logically valid distinction made by the Greeks. It is 
a curious fact that the abandonment of strictly logical thinking 
should have led to results which transgressed what was then 
known of logic, but which are now known to be readily in- 
terpretable in the terms of what we now know of Logic. This 
subject will occupy us again in the sixth chapter. 

The question of loci is connected with geometrical analysis, 
and is difficult to dissociate from a mental picture of a point 
in motion. Think of a point under certain restrictions, so that 
it cannot move in more than two directions at any instant, and 
so can only move in some curve. Thus, a point may move so 
that its distance from a fixed point is constant ; the peak of 
an angle may move so that the arms of the angle pass 
slipping through two fixed points, and the angle is always a 
right angle. In both cases the moving point keeps on the 
circumference of a certain circle. This curve is a " locus." It 
is evident how thinking of the locus a point can describe may 
help us to solve problems. 

We have seen that Thales discovered that a triangle is 
determined if its base and base angles are given. When we 
have to make a survey of either an earthly country or part of 






22 THE NATURE OF MATHEMATICS 

the heavens, for the purpose of map-making, we have to 
measure angles for example, by turning a sight, like those 
used on guns, through an angle measured on a circular arc of 
metal to fix the relative directions of the stars or points on 
the earth. Now, for terrestrial measurements, a piece of 
country is approximately a flat surface, while the heavens are 
surveyed as if the stars were, as they seem to be, scattered on 
the inside of a sphere at whose centre we are. Secondly, it is 
a network of triangles plane or spherical of which we 
measure the angles and sometimes the sides : for, if the angles 
of a triangle are known, the proportionality of the sides is 
known ; and this proportionality cannot be concluded from 
a knowledge of the angles of a rectangle, say. Hipparchus 
(born about 160 B.C.) seems to have invented this practical 
science of the complete measurement of triangles from certain 
data, or, as it is called, " trigonometry," and the principles laid 
down by him were worked out by Ptolemy of Alexandria (died 
168 A.D.) and also by the Hindoos and Arabians. Usually, 
only angles can be measured with accuracy, and so the question 
arises : given the magnitude of the angles, what can be con- 
cluded as to the kind of proportionality of the sides. Think 
of a circle described round the centre O, and let AP be the 
arc of this circle which measures the angle AOP. Notice that 
the ratio of AP to the radius is the same for the angle AOP 
whatever value the radius may have. Draw PM perpendicular 
to OA. Then the figure OPMAP reminds one of a stretched 
bow, and hence are derived the names " sine of the arc AP " 
for the line PM, and " cosine " for OM. Tables of sines and 
cosines of arcs (or of angles, since the arc fixes the angle if the 
radius is known) were drawn up, and thus the sides PM and 
OM could be found in terms of the radius, when the arc was 
known. It is evident that this contains the essentials for the 
finding of the proportions of the sides of plane triangles. Spheri- 
cal trigonometry contains more complicated relations which are 
directly relevant to the position of an astronomer and his 
measurements. 

Mathematics did not progress in the hands of the Romans : 
perhaps the genius of this people was too practical. Still, it 
was through Rome that mathematics came into medieval 






MODERN MATHEMATICS ALGEBRA 23 

Europe. The Arab mathematical text-books and the Greek 
books from Arab translation were introduced into Western 
Europe by the Moors in the period 1150-1450, and by the 
end of the thirteenth century the Arabic arithmetic had been 
fairly introduced into Europe, and was practised by the side 
of the older arithmetic which was founded on the work of 
Bjethius (about 475-526). Then came the Renascence. 
Mathematicians had barely assimilated the knowledge obtained 
from the Arabs, including their translations of Greek writers, 
when the refugees who escaped from Constantinople after the 
fall of the Eastern Empire (1453) brought the original works 
and the traditions of Greek science into Italy. Thus by the 
middle of the fifteenth century the chief results of Greek and 
Arabian mathematics were accessible to European students. 

The invention of printing about that time rendered the 
dissemination of discoveries comparatively easy. 



CHAPTER II 

THE RISE AND PROGRESS OP MODERN MATHEMATICS 
ALGEBRA 

MODERN mathematics may be considered to have begun 
approximately with the seventeenth century. It is well 
known that the first 1500 years of the Christian era produced, 
in Western Europe at least, very little knowledge of value in 
science. The spirit of the Western Europeans showed itself to 
be different from that of the ancient Greeks, and only slightly 
less so from that of the more Easterly nations ; and, when 
Western mathematics began to grow, we can trace clearly the 
historical beginnings of the use, in a not quite accurate form, 
of those conceptions variable and function which are 
characteristic of modern mathematics. We may say, in 
anticipation, that these conceptions, thoroughly analysed by 
reasoning as they are now, make up the difference of our 
modern views of Mathematics from, and have caused the 
likeness of them to, those of the ancient Greeks. The Greeks, 
seem, in short, to have taken up a very similar position 
towards the mathematics of their day to that which logic forces 
us to take up towards the far more general mathematics of to-day. 







24 THE NATURE OF MATHEMATICS 

The generality of character has been attained by the effort to 
put mathematics more into touch with natural sciences in par- 
ticular the science of motion. The main difficulty was that, 
to reach this end, the way in which mathematicians expressed 
themselves was illegitimate. Hence philosophers, who lacked 
the real sympathy that must inspire all criticism that hopes 
to be relevant, never could discover any reason for thinking 
that what the mathematicians said was true, and the world 
had to wait until the mathematicians began logically to 
analyse their own conceptions. No body of men ever needed 
this sympathy more than the mathematicians from the revival 
of letters down to the middle of the nineteenth century, for 
110 science was less logical than mathematics. 

The ancient Greeks never used the conception of motion 
in their systematic works. The idea of a locus seems to imply 
that some curves could be thought of as generated by moving 
points ; the Greeks discovered some things by helping their 
imaginations with imaginary moving points, but they never 
introduced the use of motion into their final proofs. This was 
because the Eleatic school, of which the principal representative 
was Zeno (495-435 B.C.), invented some exceedingly subtle 
puzzles to emphasize the difficulty there is in the conception of 
motion. We shall return in some detail to these puzzles, 
which have not been appreciated in all the ages from the time 
of the Greeks till quite modern times. Owing to this lack of 
subtlety, the conception of variability was freely introduced 
into mathematics. It was the conceptions of constant, variable, 
and function, of which we shall, from now on, often have 
occasion to speak, which were generated by ideas of motion, 
and which, when they were logically purified, have made both 
modern mathematics and modern logic, to which they were 
transferred by mathematical logicians Leibniz, Lambert, 
Boole, De Morgan, and the numerous successors of Boole and 
De Morgan from about 1850 onwards into a science much 
more general than, but bearing some close analogies with, the 
ideal of Greek mathematical science. Later on will be found a 
discussion of what can be meant by a " moving point." 

Let us now consider more closely the history of modern 
mathematics. Modern mathematics, like modern philosophy 






MODERN MATHEMATICS ALGEBRA 25 

and like one part the speculative and not the experimental part 
of modern physical science, may be considered to begin with 
Rene* Descartes (1596-1650). Of course, as \ve should expect, 
Descartes had many and worthy predecessors. Perhaps the 
greatest of them was the French mathematician Fran9ois Viete 
(1540-1603), better known by his Latinized name of " Vieta." 
But it is simpler and shorter to confine our attention to Descartes. 

Descartes always plumed himself on the independence of his 
ideas, the breach he made with the old ideas of the Aristotelians, 
and the great clearness and simplicity with which he described 
his ideas. But we must not uiider-estimate the part that 
"ideas in the air" play; and, further, we know now that 
Descartes' breach with the old order of things was not as great 
as he thought. 

Descartes, when describing the effect which his youthful 
studies had upon him when he came to reflect upon them, said : 

" I was especially delighted with the mathematics, on account 
of the certitude and evidence of their reasonings : but I had 
not as yet a precise knowledge of their true use ; and, thinking 
that they but contributed to the advancement of the mechanical 
arts, I was astonished that foundations so strong and solid 
should have had no loftier superstructure reared on them." 

And again : 

"Among the branches of philosophy, I had, at an earlier 
period, given some attention to logic, and, among those of the 
mathematics, to geometrical analysis and algebra three arts 
or sciences which ought, as I conceived, to contribute something 
to my design. But, on examination, I found that, as for logic, 
its syllogisms and the majority of its other precepts are of avail 
rather in the communication of what we already know, or even 
in speaking without judgment of things of which we are ignorant, 
than in the investigation of the unknown : and although this 
science contains indeed a number of correct and very excellent 
precepts, there are, nevertheless, so many others, and these 
either injurious or superfluous, mingled with the former, that it 
is almost quite as difficult to effect a severance of the true from 
the false as it is to extract a Diana or a Minerva from a rough 
block of marble. Then as to the analysis of the ancients and 
the algebra of the moderns ; besides that they embrace only 
matters highly abstract, and, to appearance, of no use, the 






26 THE NATURE OF MATHEMATICS 

former is BO exclusively restricted to the consideration of figures 
that it can exercise the understanding only on condition of 
greatly fatiguing the imagination ; and, in the latter, there is 
so complete a subjection to certain rules and formulas, that 
there results an art full of confusion and obscurity calcukted to 
embarrass, instead of a science fitted to cultivate the mind. 
By these considerations I was induced to seek some other 
method which would comprise the advantages of the three and 
be exempt from their defects. . . . 

" The long chains of simple and easy reasonings by means of 
which geometers are accustomed to reach the conclusions of 
their most difficult demonstrations had led me to imagine that 
all things to the knowledge of which man is competent are 
mutually connected in the same way, and that there is nothing 
so far removed from us as to be beyond our reach, or so hidden 
that we cannot discover it, provided only that we abstain from 
accepting the false for the true, and always preserve in our 
thoughts the order necessary for the deduction of one truth 
from another. And I had little difficulty in determining the 
objects with which it was necessary to begin, for I was already 
persuaded that it must be with the simplest and easiest to 
know, and, considering that, of all those who have hitherto 
sought truth in the sciences, the mathematicians alone have 
been able to find any demonstrations, that is, any certain and 
evident reasons, I did not doubt but that such must have been 
the rule of their investigations. I resolved to begin, therefore, 
with the examination of the simplest objects, not anticipating, 
however, from this any other advantage than that to be found 
in accustoming my mind to the love and nourishment of truth, 
and to a distaste for all such reasonings as were unsound. But 
I had no intention on that account of attempting to master all 
the particular sciences commonly denominated ' mathematics ' ; 
but observing that, however different their objects, they all 
agree in considering only the various relations or proportions 
subsisting among those objects, I thought it best for my 
purpose to consider these proportions in the most general form 
possible, without referring them to any objects in particular, 
except such as would most facilitate the knowledge of them, 
and without by any means restricting them to these, that 
afterwards I might thus be the better able to apply them to 






MODERN MATHEMATICS ALGEBRA 27 

every other class of objects to which they are legitimately 
applicable. Perceiving farther that, in order to understand 
these relations, I should sometimes have to consider them one 
by one, and sometimes only to bear in mind, or embrace them 
in the aggregate, I thought that, in order the better to consider 
them individually, I should view them as subsisting between 
straight lines, than which I could find no objects more simple, 
or capable of being more distinctly represented to my imagina- 
tion and senses ; and on the other hand, that in order to retain 
them in the memory, or embrace an aggregate of many, I should 
express them by certain characters the briefest possible. In 
this way I believed that I could borrow all that was best 
both in geometrical analysis and in algebra, and correct all the 
defects of the one by help of the other." 

Let us, then, consider the characteristics of algebra and 
geometry. 

We have seen, when giving an account, in the first chapter, 
of the works of Ahmes and Diophantus, that mathematicians 
early saw the advantage of representing an unknown number 
by a letter or some other sign that may denote various numbers 
ambiguously, writing down much as in geometrical analysis 
the relations which they bear, by the conditions of the problem, 
to other numbers, and then considering these relations. If the 
problem is determinate that is to say, if there are one or 
more definite solutions which can be proved to involve only 
numbers already fixed upon this consideration leads, by the 
use of certain rules of calculation, to the determination actual 
or approximate of this solution or solutions. Under certain 
circumstances, even if there is a solution, depending on a vari- 
able, we can find it and express it in a quite general way, by 
rules, but that need not occupy us here. Thus, suppose that 
you know my age, but that I do not know yours but wish to. 
You might say to me : "I was eight years old when you 
were born." Then I should think like this. Let x be the 
(unknown) number of years in your age at this moment and, 
say, 33 the number of years in my age at this moment ; then 
in essentials your statement can be translated by the equation 
"a:-8 = 33." The meaning of the signs "-,"" = ," and " + " 
are supposed to be known ; as indeed they are by most people 






28 THE NATURE OF MATHEMATICS 

nowadays quite sufficiently for our present purpose. Now, 
one of the rules of algebra is that any term can be taken from 
one side of the sign " = " to the other if only the " + " or " " 
belonging to it is changed into " " or " + ," as the case may 
l)e. Thus, in the present case, we have: "x = 33 + 8 = 41." 
This absurdly simple case is chosen intentionally. It is essential 
in mathematics to remember that even apparently insignificant 
economies of thought add up to make a long and complicated 
calculation readily performed. This is the case, for example, 
with the convention introduced by Descartes of using the last 
letters of the alphabet to denote unknown numbers, and the 
first letters to denote known ones. This convention is adopted, 
with a few exceptional cases, by algebraists to-day, and saves 
much trouble in explaining and in looking for unknown and 
known quantities in an equation. Then, again, the signs 
" +," " ," " = " have great merits, which those unused to 
long calculations cannot so readily understand. Even the 
saving of space made by writing " xy " for " x x y " (" x multi- 
plied by y ") is important, because we can obtain by it a shorter 
and more readily surveyed formula. Then, too, Descartes made 
a general practice of writing " powers " or " exponents " as we 
do now ; thus "x 5 " stands for "xxx" and "x 5 " for some less 
suggestive symbol representing the continued multiplication of 
five x's. 

One great advantage of this notation is that it makes the 
explanation of logarithms, which were the great and laborious 
discovery of John Napier (1550-1617), quite easy. We start 
from the equation "x m x n = x m+n ." Now, if yf y, and we cull 
p the " logarithm of y to the base x " ; in signs : "p = log z y " ; 
the equation from which we started gives, if we denote x m by " u " 
and x n by "v," so that ra = log x u and n = log x v, that log.,, (uv) = 
\og x u + log x v. Thus, if the logarithms of numbers to a given 
base (say x = 10) are tabulated ; calculations with large numbers 
are made less arduous, for addition replaces multiplication, 
when logarithms are found. Also subtraction of logarithms 
gives the logarithm of the quotient of two numbers. 

Let us now shortly consider the history of algebra from 
Diophantus to Descartes. 

The word "algebra" is the European corruption of an 






MODERN MATHEMATICS ALGEBRA 29 

Arabic phrase which means restoration and reduction the 
first word referring to the fact that the same magnitude may 
be added to or subtracted from both sides of an equation, and 
the last word meaning the process of simplification. The 
science of algebra was brought among the Arabs by Mohammed 
ben Musa (Mahomet the son of Moses), better known as 
Alkarismi, in a work written about 830 A.D., and was certainly 
derived by him from the Hindoos. The algebra of Alkarismi 
holds a most important place in the history of mathematics, 
for we may say that the subsequent Arab and the early 
medieval works on algebra were founded on it, and also that 
through it the Arabic or Indian system of decimal numera- 
tion was introduced into the West. It seems that 
the Arabs were quick to appreciate the work of others 
notably of the Greek masters and of the Hindoo mathe- 
maticians but, like the ancient Chinese and Egyptians, 
they did not systematically develop a subject to any con- 
siderable extent. 

Algebra was introduced into Italy in 1202 by Leonardo of 
Pisa (about 1175-1230) in a work based on Alkarismi's 
treatise, and into England by Robert Record (about 1510- 
1558) in a book called the Whetstone of Witte published in 
1557. Improvements in the method or notations of algebra 
were made by Record, Albert Girard (1595-1632), Thomas 
Harriot (1560-1621), Descartes, and many others. 

In arithmetic we use symbols of number. A symbol is any 
sign for a quantity which is not the quantity itself. If a man 
counted his sheep by pebbles, the pebbles would be symbols of 
the sheep. At the present day, when most of us can read and 
write, we have acquired the convenient habit of using marks on 
paper, 1, 2, 3, 4, and so on, instead of such things as pebbles. 
Our 1 + 1 is abbreviated into 2, 2 + 1 is abbreviated into 3, 
3 + 1 into 4, and so on. When " 1," " 2," " 3," &c., are used 
to abbreviate, rather improperly, " 1 mile," "2 miles," "3 miles," 
&c., for instance, they are called signs for concrete numbers. 
But when we shake off all idea of " 1," " 2," &c., meaning one, 
two, &c., of anything in particular, as when we say, " six and 
four make ten," then the numbers are called abstract numbers. 
To the latter the learner is first introduced in treatises on 






30 THE NATURE OF MATHEMATICS 

arithmetic, and does not always learn to distinguish rightly 
between the two. Of the operations of arithmetic only addition 
and subtraction can be performed with concrete numbers, and 
without speaking of more than one sort of 1. Miles can be 
added to miles, or taken from miles. Multiplication involves 
a new sort of 1, 2, 3, &c., standing for repetitions (or times, 
as they are called). Take 6 miles 5 times. Here are two 
kinds of units, 1 mile and 1 time. In multiplication, one of 
the units must be a number of repetitions or times, and to talk 
of multiplying 6 feet by 3 feet would be absurd. What notion 
can be formed of 6 feet taken " 3 feet " times ? In solving the 
following question, " If 1 yard cost 5 shillings how much will 
12 yards cost?" we do not multiply the 12 yards by the 5 
shillings ; the process we go through is the following : Since 
each yard costs 5 shillings, the buyer must put down 5 
shillings as often (as many times) as the seller uses a one-yard 
measure; that is, 5 shillings is taken 12 times. In division 
we must have the idea either of repetition or of partition, that 
is, of cutting a quantity into a number of equal parts. " Divide 
18 miles by 3 miles," means, find out how many times 3 miles 
must be repeated to give 18 miles: but "divide 18 miles by 
3 " means, cut 18 miles into 3 equal parts, and find how many 
miles are in each part. 

The symbols of arithmetic have a determinate connection; 
for instance, 4 is always 2 + 2, whatever the things mentioned 
may be, miles, feet, acres, &c. In algebra we take symbols 
for numbers which have no determinate connection. As in 
arithmetic we draw conclusions about 1, 2, 3, &c., which are 
equally true of 1 foot, 2 feet, &c., 1 minute, 2 minutes, &c. ; 
so in algebra we reason upon numbers in general, and draw 
conclusions which are equally true of all numbers. It is true 
that we also use, in kinds of algebra which have been developed 
within the last century, letters to represent things other than 
numbers for example, classes of individuals with a certain 
property, such as "horned animals," for logical purposes; or 
certain geometrical or physical things with directions in space, 
such as "forces" and signs like " + " and "-" to represent 
ways of combination of the things, which are analogous to, 
but not identical with, addition and subtraction. If " a " 
denotes " the class of horned animals " and " 6 " denotes " the 






MODERN MATHEMATICS ALGEBRA 31 

class of beasts of burden," the sign " ab " has been used to 
denote " the class of horned beasts of burden." We see that 
here ab =6a, just as in the multiplication of numbers, and the 
above operation has been called, partly for this reason, " logical 
multiplication," and denoted in the above way. Here we meet 
the practice of mathematicians and of all scientific men of 
using words in a wider sense for the sake of some analogy. 
This habit is all the more puzzling to many people because 
mathematicians are often not conscious that they do it, or even 
talk sometimes zs if they thought that they were generalising 
conceptions instead of words. But, when we talk of a "family 
tree," we do not indicate a widening of our conception of trees 
of the roadside. 

We shall not need to consider these modern algebras, but 
we shall be constantly meeting what are called the " generalisa- 
tions of number" and transference of methods to analogous 
cases. Indeed, it is hardly too much to say that in this lies 
the very spirit of discovery. An example of this is given by 
the extension of the word " numbers " to include the names of 
fractions as well. The occasion for this extension was given 
by the use of arithmetic to express such quantities as distances. 
This had been done by Archimedes and many others, and had 
become the usual method of procedure in the works of the 
mathematicians of the sixteenth century, and plays a great 
part in Descartes' work. 

Mathematicians, ever since they began to apply arithmetic 
to geometry, became alive to the fact that it was convenient 
to represent points on a straight line by numbers, and numbers 
by points on a straight line. What is meant by this may be 
described as follows. If we choose a unit of length, we can 
mark off points on a straight line corresponding to units 
which means that we select a point, called "the origin," to 
start from, 1 unit, 2 units, 3 units, and so on, so that " the 
point m," as we will call it for short, is at a distance of m 
units from the origin. Then we can divide up the line and 
mark points corresponding to the fractions , , f, y 1 ^, , 
or the point between 1 and 2 which is the same distance 
from 1 as f is from 0, and so on. Now, there is nothing 
here to distinguish fractions from numbers. Both are treated 
exactly in the same way ; the results of addition, subtraction, 

c 






32 THE NATURE OF MATHEMATICS 

multiplication, and division 1 are interpretable in much the 
same way as new points whether the "a" and "6" in "a + b," 
"a b," "ab," and so on, stand for numbers or factors, and 
we have, for example, 

a + b = b + a, ab = ba, a(b + c)=ab+ac, 

always. Because of this very strong analogy, mathema- 
ticians have called the fractions "numbers" too, and they 
often speak and write of " generalisations of numbers," of 
which this is the first example, as if the conception of 
number were generalised, and not merely the name "num- 
ber," in virtue of a great and close and important 
analogy. 

When once the points of a line were made to represent 
numbers, there seemed to be no further difficulty in admitting 
certain " irrational numbers " to correspond to the end-points 
of the incommensurable lines which had been discovered by 
the Greeks. This question will come up again at a later 
stage : there are necessary discussions of principle involved, 
but mathematicians did not go at all deeply into questions 
of principle until fairly modern times. Thus it has happened 
that, until the last sixty years or so, mathematicians were 
nearly all bad reasoners, as Swift remarked of the mathe- 
maticians of Laputa in Ghtlliver's Travels, and were unpar- 
donably hazy about first principles. Often they appealed to 
a sort of faith. To an intelligent and therefore doubting 
beginner, an eminent French mathematician of the eighteenth 
century said : " Go on, and faith will come to you." It is a 
curious fact that mathematicians have so often arrived at truth 
by a sort of instinct. 

Let us now return to our numerical algebra. Take, say, 
the number 8, and the fraction, which we will now call a 
" number " also ^. Add 1 to both ; the greater contains 

1 The operation of what is called, for the sake of analogy, 
" multiplication " of fractions is denned in the manner indicated 
in the following example. If of a yard costs lOd, how much 

10 x 4 x 7 
does of a yard cost ? The answer is - pence, and we define 

O X o 

4x7 1 

5 5 as " multiplied by " f , by analogy with what would happen 

o x o J- 

if | were 1 and % were, say, 3. 






MODERN MATHEMATICS ALGEBRA S3 

the less exactly 8 times. Now this property is possessed by 
any number, and not 8 alone. In fact, if we denote the 
number we start with by "a," we have, by the rules of algebra, 

a+ 1 

y- y = a This is an instance of a general property of 

numbers proved by algebra. 

Algebra contains many rules by which a complicated alge- 
braical expression can be reduced to its simplest terms. Owing 
to the suggestive and compact notation, we can easily acquire 
an almost mechanical dexterity in dealing with algebraical 
symbols. This is what Descartes means when he speaks of 
algebra as not being a science fitted to cultivate the mind. On 
the other hand, this art is due to the principle of the economy 
of thought, and the mechanical aspect becomes, as Descartes 
foresaw, very valuable if we could use it to solve geometrical 
problems without the necessity of fatiguing our imaginations 
by long reasonings on geometrical figures. 

I have already mentioned that the valuable notation " of 1 " 
was due to Descartes. This was published, along with all his 
other improvements in algebra, in the third part of his Geometry 
of 1637. I shall speak in the next chapter of the great dis- 
covery contained in the first two parts of this work ; here I 
will resume the improvements in notation and method made 
by Descartes and his predecessors, which make the algebraical 
part of the Geometry very like a modern book on algebra. 

It is still the custom in arithmetic to indicate addition by 
juxtaposition : thus " 2 " means " 2 -f ." In algebra, we 
always, nowadays, indicate addition by the sign " + " and 
multiplication by juxtaposition or, more rarely, by putting a 
dot or the sign " x " between the signs of the numbers to be 
multiplied. Subtraction is indicated by " ". 

Here we must digress to point out what is often, owing to 
confusion of thought, denied in text-books that, where "a" 
and "b" denote numbers, "a b" can only denote a number 
if a is equal to or greater than b. If a is equal to b, the 
number denoted is zero ; there is really no good reason for 
denying, say, that the numbers of Charles II. 's foolish sayings 
and wise deeds are equal, if a well-known epitaph be true. 
Here again we meet the strange way in which mathematics 






34 THE NATURE OF MATHEMATICS 

has developed. For centuries mathematicians used " negative " 
and "positive" numbers, and identified "positive" numbers 
with signless numbers like 1, 2, and 3, without any scruple, 
just as they used fractionary and irrational " numbers." And 
when logically-minded men objected to these wrong statements, 
mathematicians simply ignored them or said : " Go on ; faith 
will come to you." And the mathematicians were right, and 
merely could not give correct reasons or at least always gave 
wrong ones for what they did. We have, over again, the 
fact that criticism of the mathematicians' procedure, if it 
wishes to be relevant, must be based on thorough sympathy 
and understanding. It must try to account for the Tightness 
of mathematical views, and bring them into conformity with 
logic. Mathematicians themselves never found a competent 
philosophical interpreter, and so nearly all the interesting part 
of mathematics was left in obscurity until, in the latter half of 
the nineteenth century, mathematicians themselves began to 
cultivate philosophy or rather logic. 

Thus we must go out of the historical order to explain what 
"negative numbers" means. First, we must premise that 
when an algebraical expression is enclosed in brackets, it sig- 
nifies that the whole result of that expression stands in the 
same relation to surrounding symbols as if it were one letter 
only. Thus, "a (b c) " means that from a we are to take 
6 c, or what is left after taking c from 6. It is not, there- 
fore, the same as a 6 c. In fact we easily find that a (b 
c) is the same as 06 + c. Note also that " (a + 6) (c+d) " 
means (a + b) multiplied by (c + d). 

Now, suppose a and 6 are numbers, and a is greater than 6. 
Let a 6 be c. To get c from a, we carry out the operation 
of taking away 6. This operation, which is the fulfilment of 
the order: "Subtract 6," is a "negative number." Mathe- 
maticians call it a "number" and denote it by " 6" simply 
because of analogy : the same rules for calculation hold for 
" negative numbers " and " positive numbers " like " -f 6," 
whose meaning is now clear too, as do for our signless numbers ; 
when "addition," "subtraction," &c., are redefined for these 
operations. The way in which this redefinition must take 
place is evident when we represent integers, fractions, and 
positive and negative numbers by points on a straight line. 






MODERN MATHEMATICS ALGEBRA 35 

To the right of are the integers and fractions, to the left of 
are the negative numbers, and to the right of stretch the 
series of positive numbers, +a coinciding with a and being 
symmetrically placed with a as regards 0. Also we deter- 
mine that the operations of what we call " addition," &c., of 
these new " numbers " must lead to the same results as the 
former operations of the same name. Thus the same symbol 
is used in different senses, and we write 

a + 6-6 = a + = ( + o) + ( + 6) + (-6)= + a=a. 
This is a remarkable sequence of quick changes. 

We have used the sign of equality, " = ". It means origi- 
nally : " is the same as." Thus 3+1 = 4. But we write, by 
the above convention, "a= +a," and so we sacrifice exactness, 
which sometimes looks rather pedantic, for the sake of keeping 
our analogy in view, and for brevity. 

Let us bear this, at first sight, puzzling but, at second sight, 
justifiable peculiarity of mathematicians in mind. It has 
always puzzled intelligent beginners and philosophers. The 
laws of calculation and convenient symbolism are the things 
a mathematician thinks of and aims at. He seems to identify 
different things if they both satisfy the same laws which are 
important to him, just as a magistrate may think that there 
is not much difference between Mr. A., who is red-haired and 
a tinker and goes to chapel, and Mr. B., who is a brown-haired 
horse-dealer and goes to church, if both have been found out 
committing petty larceny. But their respective ministers of 
religion or wives may still be able to distinguish them. 

Any two expressions connected by the signs of equality form 
an " equation." Here we must notice that the words : " Solve 
the equation a; 2 + ace = 6," means, find the value or values of 
x such that, a and 6 being given numbers, cc 2 +ax becomes 6. 
Thus, if a = 2 and 6 = 1, the solution is x = 1. 

As we saw above, Descartes fixed the custom of employing 
the letters at the beginning of the alphabet to denote known 
quantities, and those at the end of the alphabet to denote 
unknown quantities. Thus, in the above example, a and 6 are 
some numbers supposed to be given, while x is sought. The 
question is solved when x is found in terms of a and 6 and fixed 
numbers (like 1, 2, 3) ; and so, when to a and b are attributed 






36 THE NATURE OF MATHEMATICS 

any fixed values, x becomes fixed. The signs "a" and "6" 
denote ambiguously, not uniquely like "2" does; and "x" 
does not always denote ambiguously when a and 6 are fixed. 
Thus, in the above case, when a = 2, 6= 1,- "x" denotes 
the one negative number 1. What is meant is this: In 
each member of the class of problems got by giving a and 6 
fixed values independently of one another, there is an un- 
known x, which may or may not denote different numbers, 
which only becomes known when the equation is solved. Con- 
sider now the equation ax + by = c, where a, 6, andc are known 
quantities and x and y are unknown. We can find x in terms 
of a, 6, c, and y, or y in terms of a, b, c, and x ; but x is only 
fixed when y is fixed, or y when x is fixed. Here in each case 
of fixedness of a, 6, and c, x is undetermined and " variable," 
that is to say, it may take any of a whole class of values. 
Corresponding to each x, one y belongs ; and y also is a " vari- 
able " depending on the " independent variable " x. The idea 
of " variability " will be further illustrated in the next chapter ; 
here we will only point out how the notion of what is called 
by mathematicians the "functional dependence" of y on x 
comes in. The variable y is said to be a " function " of the 
variable x if to every value of x corresponds one or more values 
of y. This use has, to some extent, been adopted in ordinary 
language. We should be understood if we were to say that 
the amount of work performed by a horse is a function of the 
food that he eats. 

Descartes also adopted the custom if he did not arrive at 
it independently advocated by Harriot of transferring all the 
terms of an equation to the same side of the sign of equality. 
Thus, instead of "x=l," "ax + b = c," and "Sx^ + g^hx." 
we write respectively " x 1 = 0," " ax + (b c) = 0," and 
"3a; 2 hx j rg = 0." The point of this is that all equations of 
the same degree in the unknown we shall have to consider cases 
of more unknowns than one in the next chapter that is to say, 
equations in which the highest power of x (x or a; 2 or x 3 . . . .) 
is the same, are easily recognisable. Further, it is convenient 
to be able to speak of the expression which is equated to 0, 
as well as of the equation. The equations in which a; 2 , and no 
higher power of a;, appears are called "quadratic" equations 
the result of equating a " quadratic " function to ; those in 








MODERN MATHEMATICS ALGEBRA 37 

which x 3 , and no higher power, appears are called " cubic " ; 
and so on for equations " of the fourth, fifth, ..." degrees. 
Now the quadratic equations: 3a; 2 + 




38 THE NATURE OF MATHEMATICS 

For the case of imaginaries, let a, 6, c, and d be any 
numbers, then 



_ 
= [(ac-M) + J-l) (ad + bc)] [(ac-bd) 



We get, then, an interesting and easily verifiable theorem on 
numbers by calculation with imaginaries, and imaginaries 
disappear from the conclusion. Mathematicians thought, then, 
that imaginaries, though apparently uninterpretable and even 
self-contradictory, must have a logic. So they were used with 
a faith that was almost firm and was only justified much later. 
Mathematicians indicated their growing security in the use of 
tj 1 by writing " * " instead of " J 1 " and calling it 
" the complex unity," thus denying, by implication, that there 
is anything really imaginary or impossible or absurd about it. 

The truth is that " * " is not uninterpretable. It represents 
an operation just as the negative numbers do, but is of a 
different kind. It is geometrically interpretable also, though 
not in a straight line, but in a plane. For this we must 
refer to the Bibliography ; but here we must point out that, in 
this "generalisation of number" again, the words "addition," 
" multiplication," and so on, do not have exactly the same, but 
an analogous, meaning to those which they had before, and that 
" complex numbers " form a domain like a plane in which a line 
representing the integers, fractions, and irrationals is contained. 
But we must leave the further development of these questions. 

It must be realised that the essence of algebra is its 
generality. In the most general case, every symbol and every 
statement of a proposition in algebra is interpretable in terms 
of certain operations to be undertaken with abstract things 
such as numbers or classes or propositions. These operations 
merely express the relations , of these things to one another. 
If the results at any stage of an algebraical process can be 
interpreted and this interpretation is often suggested by the 






ANALYTICAL GEOMETRY 39 

symbolism say, not as operations with operations with 
integers, but as other operations with integers, they express 
true propositions. Thus (a + 6) 2 = a 2 + 2a6 + 6 2 expresses, for 
example, a relation holding between those operations with 
integers that we call "fractionary numbers," or an analogous 
relation between integers. The language of algebra is a 
wonderful instrument for expressing shortly, perspicuously, and 
suggestively, the exceedingly complicated relations in which 
abstract things stand to one another. The motive for studying 
such relations was originally, and is still in many cases, the 
close analogy of relations between certain abstract things to 
relations between certain things we see, hear, and touch in the 
world of actuality round us, and our minds are helped in 
discovering such analogies by the beautiful picture of alge- 
braical processes made in space of two or of three dimensions 
made by the " analytical geometry " of Descartes, described in 
the next chapter. 



CHAPTER III 

THE RISE AND PROGRESS OF MODERN MATHEMATICS ANA- 
LYTICAL GEOMETRY AND THE METHOD OF INDIVISIBLES 

WE will now return to the consideration of the first two 
sections of Descartes' book Geometry of 1637. 

In Descartes' book we have to glean here and there what we 
now recognise as the essential points in his new method of 
treating geometrical questions. These points were not ex- 
pressly stated by him. I shall, however, try to state them in 
a small compass. 

Imagine a curve drawn on a plane surface. This curve may 
be considered as a picture of an algebraical equation involving 
x and y in the following way. Choose any point on the 
curve : and call " x " and " y " the numbers that express the 
perpendicular distances of this point, in terms of a unit of 
length, from two straight lines (called " axes ") drawn at right 
angles to one another in the plane mentioned. Now, as we 
move from point to point of the curve, x and y both vary, 
bid there is an unvarying relation which connects x and y, 
and this relation can be expressed by an algebraical equation 








40 THE NATURE OF MATHEMATICS 

called "the equation of the curve," and which contains, in 
germ as it were, all the properties of the curve considered. 
This constant relation between x and y is a relation like y z = 
4o#. We must distinguish carefully between a constant 
relation between variables and a relation between constants. 
We are always coming across the former kind of relation in 
mathematics ; we call such a relation a " function " of x and 
y the word was first used about fifty years after Descartes' 
Geometry was published, by Leibniz and write a function of 
x and y in general as "f(x, y)." In this notation, no hint is 
given as to any particular relation x and y may bear to each 
other, and, in such a particular function as y 1 4ax, we say 
that "the /OTTO of the function is constant," and this is only 
another way of saying that the relation between x and y is 
fixed. This may be also explained as follows. If x is fixed, 
there is fixed one or more values of y, and if y is fixed, there 
is fixed one or more values x. Thus the equation ax + by+c 
= gives one y for each x and one x for each y ; the equation 
y z 4aa; = gives two y's for each x and one x for each y. 1 

Consider the equation ax + by + c = 0, or, say, the more 
definite instance x + 2y 2 = 0. Draw axes and mark off 
points : having fixed on a unit of length, find the point je=l 
on the o;-axis, on the perpendicular to this axis measure where 
the corresponding y, got by substituting x = 1 in the above 
equation, brings us. We find y = . Take x = J, then y = % ; 
and so on. We find that all the points on the parallels to the 
7/-axis lie on one straight line. This straight line is determined 
by the equation x + 2y 2 = ; every point off that straight 
line is such that its x and y are not connected by the relation 
x + 2y 2 = 0, and every point of it is such that its x and y 
are connected by the relation x + 2^ 2 = 0. Similarly we 
can satisfy ourselves that every point on the circumference of 
a circle of radius c units of length, described round the point 
where the axes cross, is such that ic 2 + / 2 = c 2 , and every point 

1 We also denote a function of x by "f(x) " or " F(x) " or 
"0(0:)", fie. Here "/" is a sign for "function of," not for a 
number, just as later we shall find "sin" and "A" and "d" 
standing for functions and not numbers. This may be regarded as 
an extension of the language of early algebra. The equation 
y=f(x) is in a good form for graphical representation in the 
manner explained below. 






ANALYTICAL GEOMETRY 41 

not on this circumference does not have an x and y such that 
the constant relation o; 2 + 7/ 2 = c 2 is satisfied for it. 

There are two points to be noticed in the above general state- 
ment. Firstly, I have said that the curve " may be expressed," 
and so on. By this I mean that it is possible and not neces- 
sarily always true that the curve may be so considered. We 
can imagine curves that cannot be represented by a finite 
algebraical equation. Secondly, about the fundamental lines 
of reference the "axes" as they are called. One of these 
axes we have called the "#-axis," and the distance measured 
by the number x is sometimes called " the abscissa " ; while 
the line of length y units which is perpendicular to the end of 
the abscissa farthest from the origin, and therefore parallel to 
the other axis ("the y-axis ") is called " the ordinate." The 
name " ordinate " was used by the ancient Roman surveyors. 
The lines measured by the numbers x and y are called the 
" co-ordinates " of the point determining and determined by 
them. Sometimes the numbers x and y themselves are called 
" co-ordinates," and we will adopt that practice here. 

Sometimes the axes are not chosen at right angles to one 
another, but it is nearly always far simpler to do so, and in 
this book we always assume that the axes are rectangular. 
The whole plane is divided by the axes into four partitions, 
the co-ordinates are measured from the point called " the 
origin" where the axes cross. Here the interpretation in 
geometry of the " negative quantities " of algebra which so 
often seems so puzzling to intelligent beginners gives us a means 
of avoiding the ambiguity arising from the fact that there 
would be a point with the same co-ordinates in each quadrant 
into which the plane is divided. 

Consider the cc-axis. Measure lengths on it from the origin, 
so that to the origin (0) corresponds the number 0. Let OA, 
measured from left to right along the axis, be the unit of 
length; then to the point A corresponds the number 1. Then 
let lengths AB, BC, and so on, all measured from left to right, 
be equal to OA in length ; to the points B, C, and so on, 
correspond the numbers 2, 3, and so on. Further to the point 
that bisects OA, let the fraction ^ correspond ; and so on for 
the other fractions. In this way half of the x-axis is nearly 
filled up with points. But there are points, such as the point 






42 THE NATURE OF MATHEMATICS 

P, such that OP is the length of the circumference of a circle, 
say of unit diameter. For picturesqueness, we may imagine 
this point P got by rolling the circle along the re-axis from O 
through one revolution. The point P will fall a little to the 
left of the point 3^ and a little to the right of the point 3/^, 
and so on ; the point P is not one of the points to which 
names of fractions have been assigned by the process sketched 
above. This can be proved rigidly. If it were not true, it 
would be very easy to " square the circle." 

There are many other points like this. There is no fraction 
which, multiplied by itself, gives 2 ; but there is a length 
the diagonal of a square of unit side which is such that, if we 
were to assume that a number corresponded to every point on 
OX, it would be a number a such that a 2 = 2. We will return 
to this important question of the correspondence of points and 
lines to numbers, and will now briefly recall that " negative 
numbers'' are represented, in Descartes' analytical geometry, 
on the x-axis, by the points to the left of the origin, and, on 
the t/-axis, by the points below the origin. This was explained 
in the second chapter. 

Algebraical geometry gave us a means of classifying curves, 
All straight lines determine equations of the first degree be- 
tween x and y, and all such equations determine straight lines ; 
all equations of the second degree between x and y, that is to 
say, of the form 

ax 2 - + bxy + cy* + dx + ey +f = 0, 

determine curves which the ancient Greeks had studied and 
which result from cutting a solid circular cone, or two equal 
cones with the same axis, whose only point of contact is formed 
by the vertices. It is somewhat of a mystery why the Greek 
geometricians should have pitched upon these particular curves 
to study, and we can only say that it seems, from the present 
standpoint, an exceedingly lucky chance. For these conic 
sections of which, of course, the circle is a particular case 
are all the curves, and those only, which are represented by 
the above equation of the second degree. The three great 
types of curves the " parabola," the " ellipse," and the 
"hyperbola" all result from the above equation when the 
coefficients a, 6, c, d, e, f satisfy certain special conditions. 
Thus, the equation of a circle which is a particular kind of 






ANALYTICAL GEOMETRY 43 

lipse is always of the form got from the above equation by 
putting 6 = and c = a. 

It may be mentioned that, long after these curves were in- 
troduced as sections of a cone, Pappus discovered that they 
could all be defined in a plane as loci of a point P which 
moves so that the proportion that the distance of P from a 
fixed point (S) bears to the perpendicular distance of P(PN) 
to a fixed straight line is constant. As this proportion is less 
than equal to, or greater than 1, the curve is an ellipse, 
parabola, or hyperbola, respectively. 

It will not be expected that a detailed account should here 
be given of the curves which result from the development of 
equations of the second or higher degrees between x and y. I 
will merely again emphasize some points which are, in part, 
usually neglected or not clearly stated in text-books. The 
letters "a, 6, . . . x, y," here stand for "numbers" in the 
extended sense. We have seen in what sense we may, with 
the mathematicians, speak of fractionary, positive, and negative 
" numbers," and identify, say, the positive number + 2 and 
the fraction -| with the signless integer 2. Well, then, the 
above letters stand for numbers of that class which includes in 
this sense the fractionary, irrational, positive and negative 
numbers, but excludes the imaginary numbers. We call the 
numbers of this class " real " numbers. The question of 
irrational numbers will be discussed at greater length in the 
sixth chapter, but enough has been said to show how they were 
introduced. In mathematics it has, I think, always happened 
that conceptions have been used long before they were formally 
introduced, and used long before this use could be logically 
justified or whose nature clearly explained. The history of 
mathematics is the history of a faith whose justification has 
been long delayed, and perhaps is not accomplished even now. 

These numbers are the measurements of length, in terms of 
a definite unit, like the inch, of the abscissae and ordinates of 
certain points. We speak of such points simply by naming 
their co-ordinates, and say, for example, that " the distance of 
the point (x, y) from the point (a, 6) is the positive square root 
of (x-a) 2 + (x-b) 2 . 

Notice that or, for example, is the length of a line. It is 
natural to make, as algebraists before Descartes did,jc 2 8tandprww- 






44 THE NATURE OF MATHEMATICS 

arily for the number of square units in a square whose sides are x 
units in length, but there is no necessity in this. We shall often use 
the latter kind of measurement in the fourth and fifth chapters. 
The equation of a straight line can be made to satisfy two 
given conditions. We can write the equation in the form 



and thus have two ratios, and -, that we can determine 

a a 

according to the conditions. The equation ax + by + c = has 
apparently three "arbitrary constants," as they are called, but 
we see that this greater generality is only apparent. Now we 
can so fix these constants that two conditions are fulfilled by 
the straight line in question. Thus, suppose that one of these 
conditions is that the straight line should pass through the 
origin the point (0, 0). This means simply that when x = 0, 
then 2/ = 0. Putting, then, x = Q and y = in the above 

M 

equation, we get - = 0, and thus one of the constants is 
a 

determined. The other is determined by a new condition that, 
say, the line also passes through the point (, 2). Substituting, 

then, in the above equation, we have, as - = 0, as we know 

a 

already, + = 0, whence - = . Hence the equation of 
a a 



the line passing through (0, 0) and (, 2) is x \y = 0, or y Qx. 
Instead of having to pass through a certain point, a condition 
may be, for example, that the perpendicular from the origin on 
the straight line should be of a certain length, or that the line 
should make a certain angle with the awixis, and so on. 

Similarly, the circle whose equation is written in the form 



is of radius c and centre (a, 6). It can be determined to pass 
through any three points, or, say, to have a determined length of 
radius and position of centre. Fixation of centre is equivalent 
to two conditions. Thus, suppose the radius is to be of unit 
length: the above equation is (a; a) z + (y 6) 2 = 1. Then, 
if the centre is to be the origin, both a and 6 are determined to 
be 0, and this may be also effected by determining that the circle 
is to pass through the points (, 0) and ( - , 0), for example. 









ANALYTICAL GEOMETRY 45 



Now, if we are to find the points of intersection of the 
straight line 2x+2y=l and the circle x 2 + y z =l, we seek 
those points which are common to both curves, that is to say, 
all the pairs of values of x and y which satisfy both the above 
equations. Thus we need not trouble about the geometrical 
picture, but we only have to apply the rules of algebra for 
finding the values of x and y which satisfy two " simultaneous " 
equations in x and y. In the above case, if (X, Y) is a point 

1 2X 

of intersection, we have F = -^ and therefore, by substi- 

a 

f\ - 2Y\ 2 
tution in the other equation, X z + ( - J = 1. This gives 

a quadratic equation 

8X 2 - 4.X -3 = 

for X, and, by rules, we find that X must be either J (1 + >/7) 
or j(l- /s/7). Hence there are two values of the abscissa 
which are given when we ask what are the co-ordinates of the 
points of intersection ; and the value of y which corresponds to 
each of these re's is given by substitution in the equation 



Thus we find again the fact, obvious from a figure, that a 
straight line cuts a circle at two points at most. We can 
determine the points of intersection of any two curves .whose 
equations can be expressed algebraically, but of course the 
process is much more complicated in more general cases. Here 
we will consider an important case of intersection of a straight 
line. 

Think of a straight line cutting a circle at two points. 
Imagine one point fixed and the other point moved up towards 
the first. The intersecting line approaches more and more to 
the position of the tangent to the circle at the first point, and, 
by making the movable point approach the other closely enough, 
the secant will approach the tangent in position as nearly as 
we wish. Now, a tangent to a curve at a certain point was 
defined by the Greeks as a straight line through the point such 
that between it and the curve no other straight line could be 
drawn. Note that other curves might be drawn : thus various 
circles may have the same tangent at a common point on their 
circumference, but no circle and no curve met with in ele- 
mentary mathematics has more than one tangent at a point. 






46 THE NATURE OF MATHEMATICS 

Descartes and many of his followers adopted different forms of 
definition which really involve the idea of a limit, an idea 
which appears boldly in the infinitesimal calculus. A tangent 
is the limit of a secant as the points of intersection approach 
infinitely near to one another; it is a produced side of the 
polygon with infinitesimal sides that the curve is supposed to 
be ; it is the direction of motion at an instant of a point moving 
in the curve considered. The equation got from that of the 
curve by substituting for y from the equation of the intersecting 
straight line has, if this straight line is a tangent, two equal 
roots. In the above case, this equation was quadratic. In the 
case of a circle, we can easily deduce the well-known property 
of a tangent of being perpendicular to the radius ; and see that 
this property has no analogue in the case of other curves. 

We must remember that, just as -plane, curves determine 
and are determined by equations with two independent variables 
x and y, so surfaces spheres for instance in three-dimensional 
space determine and are determined by equations with three 
independent variables, x, y, and z. Here x, t/, and z are the 
co-ordinates of a point in space ; that is to say, the numerical 
measures of the distances of this point from three fixed planes 
at right angles to each other. Thus, the equation of a sphere 
of radius d and centre at (a, 6, c) is (a; a) 2 + (y 6) 2 
+ (z-c) 2 = d 2 . 

We may look at analytical geometry from another point of 
view which we shall find afterwards to be important, and 
which even now will suggest to us some interesting thoughts. 
The essence of Descartes' method also appears when we 
represent loci by the method. Consider a circle ; it is the 
locus of a point (P) which moves in a plane so as to preserve 
a constant distance from a fixed point (0). Here we may 
think of P as varying in position, and make up a very striking 
picture of what we call a variable in mathematics. We must, 
however, remember that, by what we call a " variable " for 
the sake of picturesqueness, we do not necessarily mean some- 
thing which varies. Think of the point of a pen as it moves 
over a sheet of writing paper ; it occupies different positions 
with respect to the paper at different times, and we under- 
standably say that the pen's point moves. But now think of 
a point in space. A geometrical point which is not the bit 






ANALYTICAL GEOMETRY 47 

of space occupied by the end of a pen or even an " atom " of 
matter is merely a mark of position. We cannot, then, 
speak of a point moving ; the very essence of point is to be 
position. The motion of a point of space, as distinguished 
from a point of matter, is a fiction, and is the supposition that 
a given point can be now one point and now another. Motion, 
in the ordinary sense, is only possible to matter and not to 
space. Thus, when we speak of a " variable position," we are 
speaking absurdly if we wish our words to be taken literally. 
But we do not really wish so when we come to think about it ; 
what we are doing is this : we are using a picturesque phrase 
for the purpose of calling up an easily imagined thought which 
helps us to visualize roughly a mathematical proposition which 
can only be described accurately by a prolix process. The 
ancient Greeks allowed prolixity, and it was only objected to 
by the uninitiated. Modern mathematics up to about sixty 
years ago successfully warred against prolixity ; hence the 
obscurity of its fundamental notions and processes and its great 
conquests. The great conquests were made by sacrificing very 
much to analogy : thus, entities like the integer 2, the ratio 2/1, 
and the real number which is denoted by " 2 " were identified, 
as we have seen, because of certain close analogies that they 
have. This seems to have been the chief reason why the 
procedure of the mathematicians has been so often condemned 
by logicians and even by philosophers. In fact, when mathe- 
maticians began to try to find out the nature of mathematics, 
they had to examine their entities and the methods which they 
used to deal with them, with the minutest care, and hence to 
look out for the points when the analogies referred to break 
down, and distinguish between what mathematicians had 
usually failed to distinguish. Then the people who do not 
mind a bit what mathematics is, and are only interested in 
what it does, called these earnest inquirers "pedants" when 
they should have said "philosophers," and "logic-choppers" 
whatever they may be when they should have said " logicians." 
We have tried to show why ratios or fractions, and so on, are 
called " numbers," and apparently said to be something which 
they are not ; we must now try to get at the meaning of the 
words " constant " and " variable." 

By means of algebraic formulae, rules for the reconstruction 

D 






48 THE NATURE OF MATHEMATICS 

of great numbers sometimes an infinity of facts of nature 
may be expressed very concisely or even embodied in a single 
expression. The essence of the formula is that it is an expres- 
sion of a constant rule among variable quantities. These ex- 
pressions "constant" and "variable" have come down into 
ordinary language. We say that the number of miles which a 
certain man walks per day is a "variable quantity"; and we 
do not mean that, on a particular day, the number was not 
fixed and definite, but that on different days he walked, 
generally speaking, different numbers of miles. When, in 
mathematics, we speak of a " variable," what we mean is that 
we are considering a class of definite objects for instance, the 
class of men alive at the present moment and want to say 
something about any one of them indefinitely. Suppose that 
we say : " If it rains, Mr. A will take his umbrella out with 
him " ; the letter " A " here is what we call the sign of the 
" variable." We do not mean that the above proposition is 
about a variable man. There is no such thing ; we say that 
a man varies in health and so in time, but, whether or not 
such a phrase is strictly correct, the meaning we would have to 
give the phrase " a variable " in the above sentence is not one 
and the same man at different periods of his own existence, but 
one and the same man who is different men in turn. What we 
mean is that if "A" denotes any man, and not Smith or Jones 
or Robinson alone, then he takes out his umbrella on certain 
occasions. The statement is not always true ; it depends on A. 
If " A " stands for a bank manager, the statement may be true ; 
if for a tramp or a savage, it probably is not. Instead of "A," 
we may put " B " or " C " or " X " ; the kind of mark on 
paper does not really matter in the least. But we attach, by 
convention, certain meanings to certain signs ; and so, if we 
wrote down a mark of exclamation for the sign of a variable, 
we might be misunderstood and even suspected of trying to be 
funny. We shall see, in the seventh chapter, the importance 
of the variable in logic and mathematics. 

" Laws of nature " express the dependence upon one another 
of two or more variables. This idea of dependence of variables 
is fundamental in all scientific thought, and reaches its most 
thorough examination in mathematics and logic under the name 
of " functionality." On this point, we must refer back to the 






ANALYTICAL GEOMETRY 49 

second chapter. The ideas of function and variable were not 
prominent until the time of Descartes, and names for these 
ideas were not introduced until much later. 

The conventions of analytical geometry as to the signs of 
co-ordinates in different quadrants of the plane had an important 
influence in the transformation of trigonometry from being a 
mere adjunct to a practical science. In the same notation as 
that used at the end of the first chapter, we may conveniently 

AP 

call the number - -, which is the same for all lengths of OP, 

by the name "w," for short, and define and as the 

"sine of u," and the "cosine of u" respectively. Thus 
" sin u " and " cos u," as we write them for short, stand for 
numerical functions of u. Considering as the origin of a 
system of rectangular co-ordinates of which OA is the ce-axis, so 

that u measures the angle POA and - and - are cos u and sin u 

r r 

respectively. Now, even if u becomes so great that POA is 
successively obtuse, more than two right angles . . ., these 
definitions can be preserved, if we pay attention to the signs 
of x and y in the various quadrants. Thus sin u and cos u 
become separated from geometry, and appear as numerical 
functions of the variable u, whose values, as we see on re- 
flection, repeat themselves at regular intervals as u becomes 
larger and larger. Thus, suppose that OP turns about O in a 
direction opposite to that in which the hands of a clock move. 

In the first quadrant, sin u and cos u are ^ and - : in the 

r r 

second they are - and ; in the third they are -^ and ? : 

r r r r 

Al SYl 

in the fourth they are - and - ; in the fifth they are 

At fV 

- and - again ; and so on. Trigonometry was separated from 

geometry mainly by John Bernoulli and Euler, whom we shall 
mention later. 

We will now turn to a different development of mathematics. 



The ancient Greeks seem to have had something approaching 
a general method for finding areas of curvilinear figures. In- 
deed, infinitesimal methods, which allow indefinitely close 
approximation, naturally suggest themselves. The determina- 
tion of the area of any rectilinear figure can be reduced to that 
of a rectangle, and can thus be completely effected. But this 
process of finding areas this " method of quadratures "- 
failed for areas or volumes bounded by curved lines or surfaces, 
respectively. Then the following considerations were applied. 
"When it is impossible to find the exact solution of a question, 
it is natural to endeavour to approach to it as nearly as possible 
by neglecting quantities which embarrass the combinations, if 
it be foreseen that these quantities which have been neglected 
cannot, by reason of their small value, produce more than a 
trifling error in the result of the calculation. For example, as 
the properties of curves are with difficulty discovered, it is 
natural to consider them as polygons of a great number of 
sides. If a regular polygon be supposed to be inscribed in a 
circle, it is evident that these two figures, although always 
different, are nevertheless more and more alike according as 
the number of the sides of the polygon increases. Their peri- 
meters, their areas, the solids formed by their revolving round 
a given axis, the angles formed by these lines, and so on, are, 
if not respectively equal, at any rate so much the nearer ap- 
proaching to equality as the number of sides becomes increased. 
Whence, by supposing the number of these sides very great, it 
will be possible, without any perceptible error, to assign to the 
circumscribed circle the properties that have been found be- 
longing to the inscribed polygon. Thus, if it is proposed to 
find the area of a given circle, let us suppose this curve to be 
a regular polygon of a great number of sides : the area of any 
regular polygon whatever is equal to the product of its peri- 
meter into the half of the perpendicular drawn from the centre 
upon one of its sides ; hence, the circle being considered as a 
polygon of a great number of sides, its area ought to equal the 
product of the circumference into half the radius. Now, this 
result is exactly true. However, the Greeks, with their taste 
for strictly correct reasoning, could not allow themselves to 
consider curves as polygons of an "infinity" of sides. They 
were also influenced by the arguments of Zeno, and thus re- 
garded the use of " infinitesimals " with suspiciou. 






ANALYTICAL GEOMETRY 51 

Zeno showed that we meet difficulties if we hold that time 
and space are infinitely divisible. Of the arguments which 
he invented to show this, the best known is the puzzle of 
Achilles and the Tortoise. Zeno argued that, if Achilles ran 
ten times as fast as a tortoise, yet, if the tortoise has (say) 
1000 yards start, it could never be overtaken. For, when 
Achilles had gone the 1000 yards, the tortoise would still be 
100 yards in front of him ; by the time he had covered these 
100 yards, it would still be 10 yards in front of him ; and 
so on for ever : thus Achilles would get nearer and nearer to 
the tortoise, but never overtake it. Zeno invented some other 
subtle puzzles for much the same purpose, and they could only 
be discussed really satisfactorily by quite modern mathematics. 

To avoid the use of infinitesimals, Eudoxus (408-355 B.C.) 
devised a method, exposed by Euclid in the Twelfth Book of 
his Elements and used by Archimedes to demonstrate many of 
his great discoveries, of verifying results found by the doubtful 
infinitesimal considerations. When the Greeks wished to dis- 
cover the properties of a curve, they regarded it as the fixed 
boundary to which the inscribed and circumscribed polygons 
approach continually, and as much as they pleased, according 
as they increased the number of their sides. Thus they ex- 
hausted in some measure the space comprised between these 
polygons and the curve, and doubtless this gave to this operation 
the name of "the method of exhaustion." As these polygons 
terminated by straight lines were known figures, their con- 
tinual approach to the curve gave an idea of it more and more 
precise, and, the law of continuity serving as a guide, the 
Greeks could eventually arrive at the exact knowledge of its 
properties. But it was not sufficient for geometricians to have 
observed, and, as it were, guessed at these properties ; it was 
necessary to verify them in an unexceptionable way ; and this 
they did by proving that every supposition contrary to the 
existence of these properties would necessarily lead to some 
contradiction : thus, after, by infinitesimal considerations, they 
had found the area (say) of a curvilinear figure to be a, they 
verified it by proving that, if it is not a, it would yet be 
greater than the area of some polygon inscribed in the curvi- 
linear figure whose area is palpably greater than that of the 
polygon. 

In the seventeenth century we have a complete contrast 






52 THE NATURE OF MATHEMATICS 

with the Grecian spirit. The method of discovery seemed 
much more important than correctness of demonstration. About 
the same time as the invention of analytical geometry by 
Descartes came the invention of a method for finding the 
areas of surfaces, the positions of the centres of gravity of 
variously shaped surfaces, and so on. In a book published 
in 1635, and in certain later works, Bonaventura Cavalieri 
(1598-1647) gave his "method of indivisibles," in which the 
cruder ideas of his predecessors, notably of Kepler (1571-1630) 
were developed. According to Cavalieri, a line is made up of 
an infinite number of points, each without magnitude, a surface 
of an infinite number of lines, each without breadth, and a 
volume of an infinite number of surfaces, each without thick- 
ness. The use of this idea may be illustrated by a simple 
example. Suppose it is required to find the area of a right- 
angled triangle. Let the base be made up of n points (or 
indivisibles), and similarly let the side not perpendicular to the 
base be made of na points, then the ordinates at the successive 
points of the base will contain a, 2a . . ., na points. There- 
fore the number of points in the area is a + 2a + . . . -f na ; 
the sum of which is ^(n 2 a+na). Since n is very large, we 
may neglect %na, for it is inconsiderable compared with |n 2 a. 
Hence the area is composed of a number |(no)n of points, 
and thus the area is measured in square units by multiplying 
half the linear measure of the altitude by that of the base. 
The conclusion, we know from other facts, is exactly true. 

Cavalieri found by this method many areas and volumes and 
the centres of gravity of many curvilinear figures. It is to 
be noticed that both Cavalieri and his successors held quite 
clearly that such a supposition that lines were composed of 
points was literally absurd, but could be used as a basis for a 
direct and concise method of abbreviation which replaced with 
advantage the indirect, tedious, and rigorous methods of the 
ancient Greeks. The logical difficulties in the principles of 
this and allied methods were strongly felt and commented on 
by philosophers sometimes with intelligence ; felt and boldly 
overcome by mathematicians in their strong and not unreason- 
able faith ; and only satisfactorily solved by mathematicians 
not the philosophers in comparatively modern times. 

The method of indivisibles whose use will be shown in 






ANALYTICAL GEOMETRY 53 

the next chapter in an important question of mechanics is 
the same in principle as "the integral calculus." The 
integral calculus grew out of the work of Cavalieri and his 
successors, among whom the greatest are Eoberval (1602- 
1675), Blaise Pascal (1623-1662), and John Wallis (1616- 
1703), and mainly consists in the provision of a convenient 
and suggestive notation for this method. The discovery of the 
infinitesimal calculus was completed by the discovery that the 
inverse of the problem of finding the areas of figures enclosed 
by curves was the problem of drawing tangents to these curves, 
and the provision of a convenient and suggestive notation 
for this inverse and simpler method, which was, for certain 
historical reasons, called " the differential calculus." 

Both analytical geometry and the infinitesimal calculus are 
enormously powerful instruments for solving geometrical and 
physical problems. The secret of their power is that long and 
complicated reasonings can be written down and used to solve 
problems almost mechanically. It is the merest superficiality 
to despise mathematicians for busying themselves, sometimes 
even consciously, with the problem of economising thought. 
The powers of even the most god-like intelligences amongst us 
are extremely limited, and none of us could get very far in 
discovering any part whatever of the Truth if we could not 
make trains of reasoning which we have thought through and 
verified, very ready for and easy in future application by 
being made as nearly mechanical as possible. In both 
analytical geometry and the infinitesimal calculus, all the 
essential properties of very many of the objects dealt with in 
mathematics, and the essential features of very many of the 
methods which had previously been devised for dealing with 
them are, so to speak, packed away in a well-arranged (and 
therefore readily got at) form, and in an easily usable way. 

CHAPTER IV 

THE BEGINNINGS OF THE APPLICATION OF MATHEMATICS 
TO NATURAL SCIENCE THE SCIENCE OF DYNAMICS 

THE end of very much mathematics and of the work of 
many eminent men is the simple and, as far as may be, 






54 THE NATURE OF MATHEMATICS 

accurate description of things in the world around us, of 
which we become conscious through our senses. 

Among these things, let us consider, say, a particular 
person's face, and a billiard ball. The appearance to the eye 
of the ball is obviously much easier to describe than that of 
the face. We can call up the image a very accurate one 
of a billiard ball in the mind of a person who has never seen 
it by merely giving the colour and radius. And, unless we 
are engaged in microscopical investigations, this description 
is usually enough. The description of a face is a harder 
matter: unless we are skilful modellers, we cannot do this 
even approximately ; and even a good picture does not attempt 
literal accuracy but only conveys a correct impression often 
better than a model, say in wax, does. 

Our ideal in natural science is to build up a working model 
of the universe out of the sort of ideas that all people carry 
about with them everywhere "in their heads, 1 ' as we say, and 
to which ideas we appeal when we try to teach mathematics. 
These ideas are those of number, order, the numerical measures 
of times and distances, and so on. One reason why we strive 
after this ideal is a very practical one. If we have a working 
model of, say, the solar system, we can tell, in a few minutes, 
what our position with respect to the other planets will be at 
all sorts of far future times, and can thus predict certain 
/uture events. Everybody can see how useful this is ; perhaps 
those persons who see it most clearly are those sailors who 
use the Nautical Almanac. We cannot make the earth 
tarry in its revolution round its axis in order to give us a 
longer day for finishing some important piece of work ; but, 
by finding out the unchanging laws concealed in the phenomena 
of the motions of earth, sun, and stars, the mathematician can 
construct the model just spoken of. And the mathematician 
is completely master of his model; he can repeat the occur- 
rences in his universe as often as he likes; something like 
Joshua, he can make his " sun " stand still, or hasten, in 
order that he may publish the Nautical Almanac several 
years ahead of time. Indeed, the "world" with which we 
have to deal in theoretical or mathematical mechanics is but 
a mathematical scheme, the function of which it is to imitate, 
by logical consequences of the properties assigned to it by 






THE SCIENCE OF DYNAMICS 55 

definition, certain processes of nature as closely as possible. 
Thus our "dynamical world" may be called a model of 
reality, and must not be confused with the reality itself. 

That this model of reality is constructed solely out of logical 
conceptions will result from our conclusion that mathematics 
is based on logic, and on logic alone ; that such a model is 
possible is really surprising on reflection. The need for 
completing facts of nature in thought was, no doubt, first felt 
as a practical need the need that arises because we feel it 
convenient to be able to predict certain kinds of future events. 
Thus, with a purely mathematical model of the solar system, 
we can tell, with an approximation which depends upon the 
completeness of the model, the relative positions of the sun, 
stars, and planets several years ahead of time ; this it is that 
enables us to publish the Nautical Almanac, and makes up to 
us, in some degree, for our inability "to grasp this sorry 
scheme of things entire . . . and re-mould it nearer to the 
heart's desire." 

Now, what is called "mechanics" deals with a very 
important part of the structure of this modeL We spoke of a 
billiard ball just now. Everybody gets into the way, at an 
early age, of abstracting from the colour, roughness, and so on, 
of the ball, and forming for himself the conception of a sphere. 
A sphere can be exactly described ; and so can what we call a 
" square," a " circle " and an " ellipse," in terms of certain 
conceptions such as those called " point," " distance," " straight 
line," and so on. Not so easily describable are certain other 
things, like a person or an emotion. In the world of moving 
and what we roughly class as inanimate objects that is to 
say, objects whose behaviour is not perceptibly complicated by 
the phenomena of what we call " life " and " will," people have 
sought from very ancient times, and with increasing success, to 
discover rules for the motions and rest of given systems of 
objects (such as a lever or a wedge) under given circumstances 
(pulls, pressures, and so on), Now, this discovery means : 
The discovery of an ideal, exactly describable motion which 
should approximate as nearly as possible to a natural motion 
or class of motions. Thus Galileo (15641642) discovered 
the approximate law of bodies falling freely, or on an inclined 
plane, near the earth's surface; and Newton (1642-1727) the 






56 THE NATURE OF MATHEMATICS 

still more accurate law of the motions of any number of bodies 
under any forces. 

Let us now try to think clearly of what we mean by such a 
rule, or, as it is usually called, a " scientific " or " natural law," 
and why it plays an important part in the arrangement of our 
knowledge in such a convenient way that we can at once, so to 
speak, lay our hand on any particular fact the need of which is 
shown by practical or theoretical circumstances. 

For this purpose, we will see how Galileo, in a work 
published in 1638, attacked the problem of a falling body. 
Consider a body falling freely to the earth : Galileo tried to 
find out, not why it fell but how it fell, that is to say, in what 
mathematical form the distance fallen through and the velocity 
attained depends on the time taken in falling and the space 
fallen through. Freely falling bodies are followed with more 
difficulty by the eye the farther they have fallen ; their impact 
on the hand receiving them is, in like measure, sharper; the 
sound of their striking louder. The velocity accordingly 
increases with the time elapsed and the space traversed. Thus, 
the modern inquirer would ask : What function is the number 
(v) representing the velocity of those (s and t} representing the 
distance fallen through and the time of falling 1 Galileo asked, 
in his primitive way : Is v proportional to s, is v proportional 
to 1 1 Tims he made assumptions, and then ascertained by 
actual trial the correctness or otherwise of these assumptions. 

One of Galileo's assumptions was, thus, that the velocity 
acquired in the descent is proportional to the time of the 
descent. That is to say, if a body falls once, and then falls 
again during twice as long an interval of time as it first fell, it 
will attain in the second instance double the velocity it acquired 
in the first. To find by experiment whether or not this 
assumption accorded with observed facts ; as it was difficult to 
prove by any direct means that the velocity acquired was 
proportional to the time of descent, but easier to investigate 
by what law the distance increased with the time, Galileo 
deduced from his assumption the relation that obtained between 
the distance and the time. This very important deduction he 
effected as follows. 

On the straight line OA, let the abscissae OE, OC, OQ, and 
so on, represent in length various lengths of time elapsed from 






THE SCIENCE OF DYNAMICS 57 

a certain instant represented by 0, and let the ordinates EF, 
CD, GH, and so on, corresponding to these abscissse represent 
in length the magnitude of the velocities acquired at the time 
represented by the respective abscissse. 

We observe, now, that, by our assumption, 0, F, D, H, lie 
in a straight line OB, and so : (1) At the instant C, at which 
one-half OC of the time of descent OA has elapsed, the velocity 
CD is also one-half of the final velocity AB ; (2) If E and Q 
are equally distant in opposite directions on OA from (7, the 
velocity GH exceeds the mean velocity CD by the same amount 
that the velocity EF falls short of it ; and for every instant 
antecedent to C there exists a corresponding one subsequent to 
C and equally distant from it. Whatever loss, therefore, as com- 
pared with uniform motion with half the final velocity, is suffered 
in the first half of the motion, such loss is made up in the second 
half. The distance fallen through we may consequently regard 
as having been uniformly described with half the final velocity. 

In symbols, if we call the number of units of velocity 
acquired in t units of time by the name v, and suppose that v 
is proportional to t, the number s of units of space descended 
through is proportional to |f 2 . In fact, s is given by ^vt, 
and, as v is proportional to t, s is proportional to ^ 2 . 

Now, Galileo verified this relation between s and t experi- 
mentally. The motion of free falling was too quick for 
Galileo to observe accurately with the very imperfect means 
such as water-clocks at his disposal. There were no 
mechanical clocks at the beginning of the seventeenth century ; 
they were first made possible by the dynamical knowledge of 
which Galileo laid the foundations. Galileo, then, made the 
motion slower, so that s and t were big enough to be measured, 
by rather primitive apparatus in which the moving balls ran 
down grooves in inclined planes. That the spaces traversed 
by the ball are proportional to the squares of the measures of 
the times in free descent as well as in motion on an inclined 
plane, Galileo verified by experimentally proving that a ball 
which falls through the height of an inclined plane attains the 
same final velocity as a ball which falls through its length. 
This experiment was an ingenious one with a pendulum whose 
string, when half the swing had been accomplished, caught on 
a fixed nail so placed that the remaining half of the swing was 






58 THE NATURE OF MATHEMATICS 

with a shorter string than the other half. This experiment 
showed that the bob of the pendulum rose, in virtue of the 
velocity acquired in its descent, just as high as it had fallen. 
This fact is in agreement with our instinctive knowledge of 
natural events ; for if a ball which falls down the length of an 
inclined plane could attain a greater velocity than one which 
falls through its height, we should only have to let the body 
pass with the acquired velocity to another more inclined plane 
to make it rise to a greater vertical height than that from 
which it had fallen. Hence we can deduce, from the accelera- 
tion on an inclined plane, the acceleration of free descent, for, 
since the final velocities are the same and s = %vt, the lengths 
of the sides of the inclined plane are simply proportional to 
the times taken by the ball to pass over them. 

The motion of falling that Galileo found actually to exist is, 
accordingly, a motion of which the velocity increases pro- 
portionally to the time. 

Like Galileo, we have started with the notions familiar to 
us (through the practical arts, for example), such as that of 
velocity. Let us consider this motion more closely. 

If a motion is uniform and c feet are travelled over in every 
second, at the end of t seconds it will have travelled ct feet. 
Put ct s for short. Then we call the " velocity " of the moving 

body the distance traversed in unit of time so that it is - units of 



length per second, the number which is the measure of the 
distance divided by the number which is the measure of the 
time elapsed. Galileo, now, attained to the conception of a 
motion in which the velocity increases proportionally to the 
time. If we draw a diagram and set off, from the origin O 
along the a;-axis OA, a series of abscissae which represent the 
times in length, and erect the corresponding ordinates to 
represent the velocities, the ends of these ordinates will lie on 
a line OB, which, in the case of the "uniformly accelerated 
motion" to which Galileo attained, is straight, as we have 
already seen. But if the ordinates represent spaces instead of 
velocities, the straight line OB becomes a curve. We see 
the distinction between the " curve of spaces " and " the curve 
of velocities," with times as abscissae in both cases. If the 
velocity is uniform, the curve of spaces is a straight line OB 






THE SCIENCE OF DYNAMICS 59 

drawn from the origin O, and the curve of velocities is a 
straight line parallel to the jr-axis. If the velocity is variable, 
the curve of spaces is never a straight line ; but if the motion 
is uniformly accelerated, the curve of velocities is a straight 
line like OB. The relations between the curve of spaces, the 
curve of velocities, and the areas of such curves AOB are, as 
we shall see, relations which are at once expressible by the 
"differential and integral calculus," indeed, it is mainly be- 
cause of this important illustration of the calculus that the 
elementary problems of dynamics have been treated here. 
And the measurement of velocity in the case where the velocity 
varies from time to time is an illustration of the formation of 
the fundamental conception of the differential calculus. 

It may be remarked that the finding of the velocity of a 
particle at a given instant and the finding of a tangent to a 
curve at a given point are both of them the same kind of 
problem the finding of the " differential quotient " of a func- 
tion. We will now enter into the matter more in detail. 

Consider a curve of spaces. If the motion is uniform, the 
number measuring any increment of the distance divided by 
the number measuring the corresponding increment of the time 
gives the same value for the measure of the velocity. But if 
we were to proceed like this where the velocity is variable, we 
should obtain widely differing values for the velocity. How- 
ever, the smaller the increment of the time, the more nearly 
does the bit of the curve of spaces which corresponds to this 
increment approach straightness, and hence uniformity of in- 
crease (or decrease) of s. Thus, if we denote the increment of 
t by " Az," where " A " does not stand for a number but for the 
phrase "the increment of," and the corresponding increment 
(or decrement) of s by " As," we may define the measure of 

As 
average velocity in this element of the motion as -r;. But, 

however small At is, the line represented by As is not, usually 
at least, quite straight, and the velocity at the instant t, which, 
in the language of Leibniz's differential calculus, is defined as 
the quotient of " infinitely small " increments and symbolised 

ds 

by , the A's being replaced by d's when we consider 
at 

"infinitesimals," appears to be only defined approximately. 






60 THE NATURE OF MATHEMATICS 

We have met this difficulty when considering the method of 
indivisibles, and will meet it again when considering the in- 
finitesimal calculus, and will only see how it is overcome when 
we have become familiar with the conception of a " limit." 

This new notion of velocity includes that of uniform velocity 
as a particular case. In fact, the rules of the infinitesimal 

calculus allow us to conclude, from the equation =a, where 

dt 



a is some constant, the equation 8 = at + b, where b is another 
constant. We must remember that all this was not expressly 
formulated until about fifty years after Galileo had published 
his investigations on the motion of falling. 

If we consider the curve of velocities, uniformly accelerated 
motion occupies in it exactly the same place as uniform velocity 
does in the curve of spaces. If we denote by v the numerical 
measure of the velocity at the end of t units of time, the 
acceleration, in the notation of the differential calculus, is 

measured by , and the equation = ft, where h is some 
dt dt 

constant, is the equation of uniformly accelerated motion. In 
Newtonian dynamics, we have to consider variably accelerated 
motions, and this is where the infinitesimal calculus or some 
practically equivalent calculus such as Newton's " method of 
fluxions " becomes so necessary in theoretical mechanics. 

We will now consider the curve of spaces for uniformly 
accelerated motion. On this diagram the arcs being t and s 
we will draw the curve 



where g denotes a constant. Of course, this is the same thing 

CJOC 

as drawing the curve y = y in a plane divided up by the 

m 

a;-axis and the y-axis of Descartes. This curve is a parabola 
passing through the origin. An interesting thing about this 
curve is that it is the curve that would be described by a body 
projected obliquely near the surface of the earth if the air did 
not resist, and is very nearly the path of such a projectile in the 
resisting atmosphere. A free body, according to Galileo's view, 
always falls towards the earth with a uniform vertical accelera- 
tion measured by the above number g. If we project a body 
vertically upwards with the initial velocity of c units, its velocity 






THE SCIENCE OF DYNAMICS 61 

at the end of t units of time is c + gt units, for if the 
direction downwards (of g) is reckoned positive, the direction 
upwards (of c) must be reckoned negative. If \ve project a 
body horizontally with the velocity of a units, and neglect the 
resistance of the air, Galileo recognised that it would describe, 
in the horizontal direction, a distance of at units in t units of 

at 2 

time, while simultaneously it would fall a distance of ? 

units. The two motions are to be considered as going on in- 
dependently of each other. Thus also, oblique projection may 
be considered as compounded of a horizontal and a vertical 
projection. In all these cases the path of the projectile is a 
parabola ; in the case of the horizontal projection, its equation 
in x and y co-ordinates is got from the two equations x at and 

at z 
y = ^-, and is thus 

2 v== 0^ 

y 2a 2 

Now, suppose that the velocity is neither uniform nor 
increases uniformly, but is different and increases at a different 
rate at different points of time. Then in the curve of 
velocities, the line OB is no longer straight. In the former 
case, the number s was the number of square units in the area 
of the triangle AOB. In this case the figure AOB is not a 
triangle, though we shall find that its area is the s units we 
seek, although v does not increase uniformity from O to A. 

Notice again that if, on OA, we take points C and E very 
close together, the little arc DF is very nearly straight, and 
the figure DGF very nearly a rectilinear triangle. Note that 
we are only trying, in this, to get a first approximation to the 
value of s, and so that, instead of the continuously changing 
velocities we know or think we know from our daily ex- 
perience, we are considering a fictitious motion in which the 
velocity increases (or decreases) so as to be the same as that of 
the motion thought of at a large number of points at minute 
and equal distances, and between successive points increases 
(or decreases) uniformly. 

Note also that we are assuming (what usually happens with 
the curves with which we shall have to do) that the arc DF 
which corresponds to CE becomes as straight as we wish if we 
take C and E close enough together. 






62 THE NATURE OF MATHEMATICS 

And now let us calculate s approximately. Starting from O, 
in the first small interval OH the rectilinear triangle OHK, 
where HK is the ordinate at H, represents approximately the 
space described. In the next small interval HL, where the 
length of HL is equal to that of OH, the space described is 
represented by the rectilinear figure KHLM. The rectangle 
KL is the space passed over the uniform velocity HK in time 
KL; and the triangle KNM is the space passed over by a 
motion in which the velocity increases from zero to MN. 
And so on for other intervals beyond HL. Thus s is ultimately 
given (approximately) as the number of square units in a 
polygon which closely approximates to the figure AOB 

We must now say a few words about the meaning of the 
letters in geometrical and mechanical equations which, following 
Descartes, we use instead of the proportions used by Galileo 
and many of his contemporaries and followers. It seems 
better, when beginning mechanics, to think in proportions, but 
afterwards, for convenience in dealing with the symbolism of 
mathematical data, it is better to think in equations. 

A typical proportion is : Final velocities are to one another 
as the times ; or, in symbols, " V : V : : T : T'." Here "V" 
(for example) is just short for " the velocity attained at the 
end of the period of time " (reckoned from some fixed instant) 
denoted by " T" and V : V, and T : T', are just numbers 
(real numbers) ; and the proportion states the equality of 
these numbers. Hence the proportion is sometimes written 
" V : V' = T : T'" If, now, v is the numerical measure, 

v t 

merely, of V, v' that of V', and so on, we have = ~, or 

v t 

vt' = v't. 

In the last equation, the letters v and t have a mnemonic 
significance, as reminding us that we started from velocities and 
times, but we must carefully avoid the idea that we are 
" multiplying " (or can do so) velocities by times ; what we are 
doing is multiplying the numerical measures of them. People 
who write on geometry and mechanics often say inaccurately, 
simply for shortness, " let s denote the distance, t the time," 
and so on; whereas, by a tacit convention, small italics are 
usually employed to denote numbers. However, in future, 






THE SCIENCE OF DYNAMICS 63 

for the sake of shortness, I shall do as the writers referred to, 
and speak of v as " the velocity." Equations in mechanics, 

such as "s = ;?L" are only possible if the left-hand side is of 
a 

the same kind as the right-hand side : we cannot equate spaces 
and times, for example. 

Suppose that we have fixed on the unit of length as one 
inch and the unit of time as one second. As unit of velocity 
we might choose the velocity with which, say, a inches are 
described uniformly in one second. If we did this, we should 
express the relation between the s units of space passed over 
by a body with a given velocity (v units) in a given time 
(t units) as s = avt ; whereas, if we defined the unit of velocity 
as the velocity with which the unit of length is travelled over 
in the unit of time, we should write s = vt. 

Among the units derived from the fundamental units such 
as those of length and time the simplest possible relations are 
made to hold. Thus, as the unit of area and the unit of 
volume, the square and the cube of unit sides are respectively 
used, the unit of velocity is the uniform rate at which unit of 
length is travelled over in the unit of time, the unit of 
acceleration is the gain of unit velocity in unit time, and so on. 

The derived units depend on the fundamental units, and the 
function which a given derived unit is of its fundamental units 
is called its "dimensions." Thus the velocity v is got by 
dividing the length s by the time t. The dimensions of a 
velocity are written 



and those of an acceleration denoted F 

w.m.ja. 

IT"'*! iT^H 2 

These equations are merely mnemonic ; the letters do not 
mean numbers. The mnemonic character comes out when we 
wish to pass from one set of units to another. Thus, if we 
pass to a unit of length 6 times greater and one of time c times 
greater, the acceleration / with the old units is related to that 
(/') with the new units by the equation 






64 THE NATURE OF MATHEMATICS 

As the units become greater, /' becomes less ; and, since the 
dimensions of F are jL-J, , the factor is obviously suggested 

to us the symbol " [2 1 ] 2 " suggesting a squaring of the number 
measuring the time. 

From Galileo's work resulted the conclusion that, where 
there is no change of velocity in a straight line, there is no force. 
The state of a body unacted upon by force is uniform rectili- 
near motion ; and rest in a special case of this motion, where 
the velocity is and remains zero. This " law of inertia " was 
exactly opposite to the philosophical opinion, derived from 
Aristotle, that force is requisite to keep up a uniform motion ; 
and may be roughly verified by noticing the behaviour of a 
body projected with a given velocity and moving under little 
resistance as a stone on a sheet of ice. Newton and his 
contemporaries saw how important this law was in the 
explanation of the motion of a planet say, about the sun. 
Think of a simple case, and imagine the orbit to be a circle. 
The planet tends to move along the tangent with uniform 
velocity, but the attraction of the sun simultaneously draws 
the planet towards itself, and the result of this continual 
combination of two motions is the circular orbit. Newton 
succeeded in calculating the shapes of the orbits for different 
laws of attraction, and found that, when attraction varies 
inversely as the square of the distance, the shapes are conic 
sections, as had been observed in the case of our solar system. 

The problem of the solar system appeared, then, in a 
mathematical dress ; various things move about in space, and 
this motion is completely described if we know the geometrical 
relations distances, positions, and angular distances between 
these things at some moment, the velocities at this moment, and 
the accelerations at every moment. Of course, if we knew all 
the positions of all the things at all the instants, our descrip- 
tion would be complete ; it happens that the accelerations are 
usually simpler to find directly than the positions : thus, in 
Galileo's case the acceleration was simply constant. Thus, 
we are given functional relations between these positions and 
their rates of change. We have to determine the positions 
from these relations. 








THE INFINITESIMAL CALCULUS 65 

It is the business of the " method of fluxions " or the 
"infinitesimal calculus" to give methods for finding the 
relations between variables from relations between their 
rates of change or between them and these rates. This 
shows the importance of the calculus in such physical 
questions. 

Mathematical physics grew up perhaps too much so on 
the model of theoretical astronomy, its first really extensive 
conquest. There are signs that mathematical physics is 
freeing itself from its traditions, but we need not go further 
into the subject in this place. 

Roberval devised a method of tangents which is based on 
Galileo's conception of the composition of motions. The 
tangent is the direction of the resultant motion of a point 
describing the curve. Newton's method, which is to be dealt 
with in the fifth chapter, is analogous to this, and the idea 
of velocity is fundamental in his "method of fluxions." 



CHAPTER V 

THE RISE OF MODERN MATHEMATICS THE 
INFINITESIMAL CALCULUS 

IN the third chapter we have seen that the ancient Greeks 
were sometimes occupied with the theoretically exact deter- 
mination of the areas enclosed by curvilinear figures, and that 
they used the " method of exhaustion," and, to demonstrate 
the results which they got, an indirect method. We have 
seen, too, a "method of indivisibles," which was direct 
and seemed to gain in brevity and efficiency from a certain 
lack of correctness in expression and perhaps even a small 
inexactness in thought. We shall find the same merits and 
demerits both, especially the merits, intensified in the 
" infinitesimal calculus." 

By the side of researches on quadratures and the finding of 
volumes and centres of gravity developed the methods of 
drawing tangents to curves. We have begun to deal with this 







66 THE NATURE OF MATHEMATICS 

subject in the third chapter: here we shall illustrate the 
considerations of Fermat (1601-1665) and Barrow (1630- 
1677) the intellectual descendants of Kepler by a simple 
example. 

Let it be proposed to draw a tangent at a given point P in 
the circumference of a circle of centre and equation x 2 + y z = 1. 
Let us take the circle to be a polygon of a great number of 
sides ; let PQ be one of these sides, and produce it to meet 
the -axis at T. Then PT will be the tangent in question. 
Let the co-ordinates of P be X and Y ; those of Q will be 
X + e and Y -f a, where e and a are infinitely small increments, 
positive or negative. From a figure in which the ordinates 
and abscissae of P and Q are drawn, so that the ordinate of P 
is PR, we can see, by a well-known property of triangles, that 
TR is to RP (or F) as e is to a. Now, X and Y are related 
by the equation X 2 + F 2 = l, and, since Q is also on the locus 
aj2 + 2/ 2 = l, we have (X + e) 2 +(F + a) 2 = 1. From the 
two equations in which X and Y occur, we conclude that 

0, and hence 
o 



e TR - 

But - = : hence TR = - * '. Now, a and e may 
a Y X + \ 

be neglected in comparison with X and F, and thus we can 

F 2 
say that, at any rate very nearly, we have TR = - But 

-A. 

this is exactly right, for, since TP is at right angles to OP, we 
know that OR is to RP as PR is to RT. Here X and F are 
constant, but we can say that the abscissa of the point where 
the tangent at any point (say y) of the circle cuts the a;-axis 

v 2 

is given by adding - - to x. 
x 

Thus, we can find tangents by considering the ratios of 
infinitesimals to one another. The method obviously applies 
to other curves besides circles ; and Barrow's method and 
nomenclature leads us straight to the notation and nomenclature 
of Leibniz. Barrow called the triangle PQS, where S is 
where a parallel to the #-axis through Q meets PR, the 
" differential triangle," and Leibniz denoted Barrow's a and e 
by dy and dx (short for "the differential ofy" and "the 












THE INFINITESIMAL CALCULUS 67 

differential of a;," BO that "d" does not denote a number but 
"dx" altogether stands for an " infinitesimal p> ) respectively, 
and called the collection of rides for working with his signs the 
"differential calculus." 

But before the notation of the differential calculus and the 
rules of it were discovered by Gottfried Wilhelm von Leibniz 
(1646-1716), the celebrated German philosopher, statesman, 
and mathematician, he had invented the notation and found 
some of the rules of the " integral calculus " : thus, he had 
used the now well-known sign "f" or long "s" as short for 
"the sum of," when considering the sum of an infinity of 
infinitesimal elements as we do in the method of indivisibles. 
Suppose that we propose to determine the area included 
between a certain curve y=f(x), the #-axis, and two fixed 
ordinates whose equations are x = a and x = 6 ; then, if we 
make use of the idea and notation of differentials, we notice 
that the area in question can be written as 

"fy.dx," 

the summation extending from x a to x = 6. We will not 
here further concern ourselves about these boundaries. Notice 
that in the above expression we have put a dot between the 
" y " and the " dx " : this is to indicate that y is to multiply 
dx. Hitherto we have used juxtaposition to denote multiplica- 
tion, but here d is written close to x with another end in 
view ; and it is desirable to emphasize the difference between 
" d " used in the sense of an adjective and " d " used in the 
sense of a multiplying number, at least until the student 
can easily tell the difference by the context. If, then, we 
imagine the abscissa divided into equal infinitesimal parts, 
each of length dx, corresponding to the constituents called 
" points " in the method of indivisibles, y . dx is the area 
of the little rectangle of sides dx and y which stand at 
the end of the abscissa x. If, now, instead of extending 
to x = 6, the summation extends to the ordinate at the 
indeterminate or " variable " point x, y .dx becomes a func- 
tion of x. 

Now, if we think what must be the differential of this sum, 
that the infinitesimal increment that it gets when the abscissa 
of length x, which is one of the boundaries, is increased by dx, 







68 THE NATURE OF MATHEMATICS 

we see that it must be y . dx Hence 



and hence the sign " d " destroys, so to speak, the effect of the 
sign " / ". We also have fdx = x, and find that this summation 
is the inverse process to differentiation. Thus the problems of 
tangents and quadratures are inverses of one another. The 
quantity which by its differentiation produces a proposed dif- 
ferential, is called the " integral " of this differential ; since we 
consider it as having been formed by infinitely small continual 
additions : each of these additions is what we have named the 
differential of the increasing quantity, it is a fraction of it : 
and the sum of all these fractions is the entire quantity which 
we are in search of. For the same reason we call "integrat- 
ing" or "taking the sum of" a differential the finding the 
integral of the sum of all the infinitely small successive 
additions which form the series, the differential of which, 
properly speaking, is the general term. 

It is evident that two variables which constantly remain 
equal increase the one as much as the other during the same 
time, and that consequently their differences are equal : and 
the same holds good even if these two quantities had differed 
by any quantity whatever when they began to vary ; provided 
that this primitive difference be always the same, their differ- 
entials will always be equal. 

Eeciprocally, it is clear that two variables which receive at 
each instant infinitely small equal additions must also either 
remain constantly equal to one another, or always differ by the 
same quantity : that is, the integrals of two differentials which 
are equal can only differ from each other by a constant quantity. 
For the same reason, if any two quantities whatever differ in 
an infinitely small degree from each other, their differentials 
will also differ from one another infinitely little : and recipro- 
cally, if two differential quantities differ infinitely little from 
one another, their integrals, putting aside the constant, can 
also differ but infinitely little one from the other. 

Now, some of the rules for differentiation are as follows. 
If y=f(x), dy=f(x + dx)f(x\ in which higher powers of 
differentials added to lower ones may be neglected. Thus, 
if y = x z , then dy = (x + dx) 2 -x 2 = '2x.dx+(dx)'* = l 2x .dx. 
Here it is well to refer back to the treatment of the problem 







THE INFINITESIMAL CALCULUS 69 

of tangents at the beginning of this chapter. Again, if y = 
a . x, where a is constant ; dy = a.dx. If y = x . z, then dy = 

X 

(x + dx}(z + dz)x . z = x . dz + z . dx. If y = , x = y . z, so 

Z 

dx = y.dz + z.dy; hence dy y ' Z . Since the integral 

m 

calculus is the inverse of the differential calculus, we have at once 
f2x . dx = x z , \a . dx = afdx, 

fx . dz + / z . dx = xz, 

and so on. More fully, from d(x 3 ) = 3cc 2 . dx, we conclude, not 
that fx 2 . dx = $x 3 , but that fx 2 . dx = $x 3 + c, where <: c " de- 
notes some constant depending on the fixed value for x from 
which the integration starts. 

Consideraparabola?/ 2 = a#; then 2y.dy a. dx, or dx = ' . 

a 

tf) O j 

Thus the area from the origin to the point x is j^L-l ? + c ; 

8 

2ty 3 ^i/ 2 cfo/ 2V 3 

but d- = y ' y ; thus the area is - + c, or, since y 2 ax, 
3ct a on 

%x .y + c. To determine c when we measure the area from to x, 
we have the area zero when x = ; hence the above equation 
gives c = 0. This whole result, now quite simple to us, is one 
of the greatest discoveries of Archimedes. 

Let us now make a few short reflections on the infinitesimal 
calculus. First, the extraordinary power of it in dealing with 
complicated questions lies in that the question is split up into 
an infinity of simpler ones which can all be dealt with at once, 
thanks to the wonderfully economical fashion in which the 
calculus, like analytical geometry, deals with variables. Thus, 
a curvilinear area is split up into rectangular elements, all the 
rectangles are added together at once when it is observed that 
integral is the inverse of the easily acquired practice of differ- 
entiation. We must never lose sight of the fact that, when we 
differentiate y or integrate y . dx, we are considering, not a par- 
ticular x or y, but any one of an infinity of them. Secondly, 
we have seen that what in the first place had been regarded 
but as a simple method of approximation, leads at any rate in 
certain cases to results perfectly exact. The fact is that the 







70 THE NATURE OF MATHEMATICS 

exact results are due to a compensation of errors : the error 
resulting from the false supposition made, for example, by re- 
garding a curve as a polygon with an infinite number of sides 
each infinitely small and which when produced is a tangent 
of the curve, is corrected or compensated for by that which 
springs from the very processes of the calculus, according to 
which we retain in differentiation infinitely small quantities of 
the same order alone. In fact, after having introduced these 
quantities into the calculation to facilitate the expression of 
the conditions of the problem and after having regarded them 
as absolutely zero in comparison with the proposed quantities, 
with a view to simplify these equations, in order to banish the 
errors that they had occasioned and to obtain a result perfectly 
exact, there remains but to eliminate these same quantities 
from the equations where they may still be. 

But all this cannot be regarded as a strict proof. There 
are great difficulties in trying to determine what infinitesimals 
are : at one time they are treated like finite numbers and at 
another like zeros or as "ghosts of departed quantities," as 
Bishop Berkeley, the philosopher, called them. 

Another difficulty is given by differentials " of higher orders 
than the first." Let us take up again the considerations of 

ds 

the fourth chapter. We saw that v = , and found that s was 

at 

got by integration : s = (v . dt. This is now an immediate 

ds di} 

inference, since dt = ds. Now, let us substitute for v in . 
dt dt 

Here t is the independent variable, and all of the older mathe- 
maticians treated the elements dt as constant the interval of 
the independent variable was split up into atoms, so to speak, 
which themselves were regarded as known, and in terms of 
which other differentials, ds, dx, dy, were to be determined. 
Thus 



dt dt dt dt 2 ' 

" d z s " being written for " d(da) and " dt 2 " for " (cfe) 2 ". Thus 
the acceleration was expressed as " the second differential of 

d 2 8 
the space divided by the square of dt." If ^ were constant^ 







THE INFINITESIMAL CALCULUS 71 

d~s 

say a, then - = a . dt : and, integrating both sides : 
at 

ds 

=fa.dt = afdt = at + b, 

CLt 

where 6 is a new constant. Integrating again, we have : 

at 2 
s=aft.dt + bfdt = + bt + c; 

J 

which is a more general form of Galileo's result. 

Thus, the infinitesimal calculus brought about a great ad- 
vance in our powers of describing nature. And this advance 
was mainly due to Leibniz's notation : Leibniz himself attri- 
buted all of his mathematical discoveries to his improvements 
in notation. Those who know something of Leibniz's work 
know how conscious he was of the suggestive and economical 
value of a good notation. And the fact that we still use and 
appreciate Leibniz's "/" and "d" even though our views as 
to the principles of the calculus are very different from those of 
Leibniz and his school, is perhaps the best testimony to the 
importance of this question of notation. This fact that Leibniz's 
notations have become permanent is the great reason why I 
have dealt with his work before the analogous and prior work 
of Newton. 



Isaac Newton (1642-1727) undoubtedly arrived at the 
principles and practice of a method equivalent to the infini- 
tesimal calculus much earlier than Leibniz, and, like Roberval, 
his conceptions were obtained from the dynamics of Galileo. 
He considered curves to be described by moving points. If 
we conceive a moving point as describing a curve, and the 
curve referred to co-ordinate axes, then the velocity of the 
moving point can be decomposed into two others parallel to 
the axes of x and y respectively ; these velocities are called 
the " fluxions " of x and y, and the velocity of the point is the 
fluxion of the arc. Reciprocally the arc is the " fluent " of 
the velocity with which it is described, From the given equa- 
tion of the curve we may seek to determine the relations 
between the fluxions and this is equivalent to Leibniz's 
problem of differentiation ; and reciprocally we may seek the 
relations between the co-ordinates when we know that between 







72 THE NATURE OF MATHEMATICS 

their fluxions, either alone or combined with the co-ordiuates 
themselves. This is equivalent to Leibniz's general problem 
of integration, and is the problem to which we saw, at the end 
of the fourth chapter, that theoretical astronomy reduces. 

Newton denoted the fluxion of x by "a;," and the fluxion of 
the fluxion (the acceleration) of x by "a." It is obvious that 
this notation becomes awkward when we have to consider 
fluxions of higher orders ; and further, Newton did not indicate 
by his notation the independent variable considered. Thus 

"y" might possibly mean either -^ or 3?. We have x = , 

dt ax dt 

fjsv* d jf* fJ^y 

x = = T- ; but a dot-notation for would be clumsy and 
dt dt- ' dt n 

inconvenient. Newton's notation for the "inverse method 
of fluxions " was far clumsier, even, and far inferior to 
Leibniz's "/". 

The relations between Newton and Leibniz were at first 
friendly, and each communicated his discoveries to the other 
with a certain frankness. Later, a long and acrimonious dis- 
pute took place between Newton and Leibniz and their 
respective partisans. Each accused unjustly, it seems the 
other of plagiarism, and mean suspicious gave rise to meanness 
of conduct, and this conduct was also helped by what is some- 
times called " patriotism." Thus, for considerably more than 
a century, British mathematicians failed to perceive the great 
superiority of Leibniz's notation. And thus, while the Swiss 
mathematicians, James Bernoulli (1654-1705), John Bernoulli 
(1667-1748), and Leonhard Euler (1707-1783), the French 
mathematicians d'Alembert (1707-1783), Clairaut (1713- 
1765), Lagrange (1736-1813), Laplace (1749-1827), Legeudre 
(1752-1833), Fourier (1768-1830), and Poisson (1781-1850), 
and many other continental mathematicians were rapidly l 

1 It is difficult for a mathematician not to think that the sudden 
and brilliant dawn on eighteenth century France of the magnificent 
and apparently all-embracing physics of Newton and the wonder- 
fully powerful mathematical method of Leibniz inspired scientific 
men with the belief that the goal of all knowledge was nearly 
reached and a new era of knowledge quickly striding towards 
perfection begun ; and that this optimism had indirectly much to 
do in preparing for the French Revolution. 







THE INFINITESIMAL CALCULUS 73 

extending knowledge by using the infinitesimal calculus in all 
branches of pure and applied mathematics, in England com- 
paratively little progress was made. In fact, it was not until 
the beginning of the nineteenth century that there was formed, 
at Cambridge, a Society to introduce and spread the use of 
Leibniz's notation among British mathematicians : to establish, 
as it was said, "the principles of pure d-ism in opposition to 
the dot-age of the university." 

The difficulties met and not satisfactorily solved by Newton, 
Leibniz, or their immediate successors, in the principles of the 
infinitesimal calculus, centre about the conception of a " limit" ; 
and a great part of the meditations of modern mathematicians, 
such as the Frenchman Cauchy (1789-1857), the Norwegian 
Abel (1802-1829), and the German Weierstrass (1815-1897), 
not to speak of many still living, have been devoted to the 
putting of this conception on a sound logical basis. 

We have seen that, if y = a; 2 , -^ = 2a?. What we do in forming 
dx 



dy . , - 

- is to form v L - , which is readily found to be 



, and then consider that, as Aa; approaches more and 
more, the above quotient approaches 2x. We express this by 
saying that the "limit, as h approaches 0," is 2x. We do 
not consider Aa; as being a fixed " infinitesimal " or as an 
absolute zero (which would make the above quotient become 

indeterminate -), nor need we suppose that the quotient reaches 

its limit (the state of Ace being 0). What we need to consider 
is that " Aa; " should represent a variable which can take 
values differing from by as little as we please. That is to 
say, if we choose any number, however small, there is a value 
which Aa; can take, and which differs from by less than that 
number. As before, when we speak of a " variable," we mean 
that we are considering a certain doss. When we speak of a 
" limit," we are considering a certain infinite class. Thus the 
sequence of an infinity of terms 1, f, , , Jg-, and so on, 
whose law of formation is easily seen, has the limit 0. In 
this case is such that any number greater than it is greater 
than some term of the sequence, but itself is not greater 







74 THE NATURE OF MATHEMATICS 

than any term of the sequence and is not a term of the 
sequence. A sequence like 1, l + , ! + + , l++J+| 
. . ., has an analogous upper limit 2. A function /(a), as the 

9 

independent variable x approaches a certain value, like 

x 
as x approaches 0, may have a value (in this case 2, though 

at 0, is indeterminate). The question of the limits of a 
cc 

function in general is somewhat complicated, but the most 

fix + Ax) f (x) 
important limit is J - -'- fv ' as Ax approaches ; this, 



if y =/(*), is *. 



Ax 



That the infinitesimal calculus, with its rather obscure " in- 
finitesimals" treated like finite numbers when we write 

dy 1 dx 

3 dx = dy and dp = 3~> and then, on occasion, neglected 

ax ay 

leads so often to correct results is a most remarkable fact, and 
a fact of which the true explanation only appeared when 
Gauchy, Gauss (1777-1855), Kiemann (1826-1866), and 
Weierstrass had developed the theory of an extensive and 
much used class of functions. These functions happen to have 
properties which make them especially easy to be worked with, 
and nearly all the functions we habitually use in mathematical 
physics are of this class. A notable thing is that the complex 
numbers spoken of in the second chapter make this theory to 
a great extent. 

Large tracts of mathematics have, of course, not been 
mentioned here. Thus, there is an elaborate theory of integer 
numbers to be referred to in a note to the seventh chapter, 
and a geometry using the conceptions of the ancient Greeks and 
methods of modern mathematical thought ; and very many men 
still regard space-perception as something mathematics deals 
with. We will return to this soon. Again, algebra has 
developed and branched off ; the study of functions in general 
and in particular has grown ; and soon a list of some of the 
many great men who have helped in all this would not be very 







THE INFINITESIMAL CALCULUS 75 

useful. Let us now try to resume what we have seen of the 
development of mathematics along what seem to be its main 
lines. 

In the earliest times men were occupied with particular 
questions the properties of particular numbers and the 
geometrical properties of particular figures, together with 
simple mechanical questions. With the Greeks, a more general 
study of classes of geometrical figures began. But traces of an 
earlier exception to this study of particulars are afforded by 
'' algebra." In it and its later form symbols like our present 
x and y took the place of numbers, so that, what is a great 
advance in economy of thought and other labour, a part of 
calculation could be done with symbols instead of numbers, so 
that the one result stated, in a manner analogous to that of 
Greek geometry, a proposition valid for a whole infinite class of 
different numbers. 

The great revolution in mathematical thought brought about 
by Descartes in 1637 grew out of the application of this 
general algebra to geometry by the very natural thought of 
substituting the numbers expressing the lengths of straight 
lines for those lines. Thus a point in a plane for instance 
is determined in position by two numbers x and y, or co- 
ordinates. Now, as the point in question varies in position, 
x and y both vary ; to every x belongs, in general, one or more 
?/'s, and we arrive at the most beautiful idea of a single 
algebraical equation between x and y representing the whole of 
a curve the one " equation of the curve " expressing the 
general law by which, given any particular x out of an infinity 
of them, the corresponding y or y's can be found. 

The problem of drawing a tangent the limiting position of 
a secant, when the two meeting points approach indefinitely 
close to one another at any point of a curve came into 
prominence as a result of Descartes' work, and this, together 
with the allied conceptions of velocity and acceleration " at an 
instant," whicli appeared in Galileo's classical investigation, 
published in 1638, of the law according to which freely falling 
bodies move, gave rise at length to the powerful and convenient 
"infinitesimal calculus" of Liebniz and the "method of 
fluxions" of Newton. Mathematically, the finding of the 







76 THE NATURE OF MATHEMATICS 

tangent at the point of a curve, and finding the velocity of 
a particle describing this curve when it gets to that point, 
are identical problems. They are expressed as finding the 
" differential quotient," or the " fluxion " at the point. It is 
now known to be very probable that the abore two methods, 
which are theoretically but not practically the same, were 
discovered independently; Newton discovered his first, and 
Leibniz published his first, in 1684. The finding of the areas 
of curves and of the shapes of the curves which moving particles 
describe under given forces showed themselves, in this calculus, 
as results of the inverse process to that of the direct process 
which serves to find tangents and the law of attraction to 
a given point from the datum of the path described by a 
particle. The direct process is called "differentiation," the 
inverse process " integration." 

Newton's fame is chiefly owing to his application of this 
method to the solution, which, in its broad outlines, he gave, 
of the problem of the motion of the bodies in the solar system, 
which includes his discovery of the law according to which all 
matter gravitates towards is attracted by other matter. 
This was given in his Principia of 1687 ; and for more than 
a century afterwards mathematicians were occupied in extend- 
ing and applying the calculus. 

Then came more modern work, more and more directed 
towards the putting of mathematical methods on a sound logical 
basis, and the separation of mathematical processes from the 
sense-perception of space with which so much in mathematics 
grew and grows up. Thus trigonometry took its place by algebra 
as a study of certain mathematical functions, and it began to 
appear that the true business of geometry is to supply beautiful 
and suggestive pictures of abstract " analytical "or " algebrai- 
cal " or even " arithmetical " as they are called processes of 
mathematics. In the next chapter, we shall be concerned with 
part of the work of logical examination and reconstruction. 







VIEWS OF LIMITS AND NUMBERS 77 

CHAPTER VI 

MODERN VIEWS OF LIMITS AND NUMBERS 

LET us try to form a clear idea of the conception which showed 
itself to be fundamental in the principles of the infinitesimal 
calculus, the conception of a limit. 

Notice that the limit of a sequence is a number which is 
already defined. We cannot prove that there is a limit to a 
sequence unless the limit sought is among the numbers already 
defined. Thus, in the system of " numbers " here we must 
refer back to the second chapter consisting of all fractions 
(or ratios), we can say that the sequence (where 1 and 2 are 
written for the ratios ^ and f) 1, l + , ! + + , . . ., has 
a limit (2), but that the sequence 

1, 1+ T V, 1 + T V + T ^,1 +^+1-^+1^, . . .,or 1-4142. . . . 

got by extracting the square root of 2 by the known process 
of decimal arithmetic, has not. In fact, it can be proved that 
there is no ratio such that it is a limit for the above sequence. 
If there were, and it were denoted by "x," we would have 
a; 2 = 2. Here we come again to the question of incommen- 
surables and "irrational numbers." The Greeks were quite 
right in distinguishing so sharply between numbers and 
magnitudes, and it was the tacit, natural, and unjustified 
not, as it happens, incorrect presupposition that the series of 
numbers, completed into the series of what are called "real 
numbers" by "irrational numbers," exactly corresponds to 
the series of points on a straight line. The series of points 
which represents the sequence last named seems undoubtedly 
to possess a limit ; this limiting point was assumed to repre- 
sent some number, and, since it could not represent an integer 
or a ratio, it was said to represent an " irrational number," 
J'2. Another irrational number is that which is represented 
by the incommensurable ratio of the circumference of a circle 
to its diameter. This number is denoted by the Greek letter 
"IT" and its value is nearly 3-1416. . . Of course, the process 
of approximation by decimals never comes to an end. 

The subject of limits forced itself into a very conspicuous 
place in the seventeenth and eighteenth centuries owing to the 







78 THE NATURE OF MATHEMATICS 

use of infinite series as a means of approximate calculation. I 
shall distinguish what I call "sequences" and "series." A 
sequence is a collection finite or infinite of numbers; a 
series is a finite or infinite collection of numbers connected by 
addition. Sequences and series can be made to correspond in 
the following way. To the sequence 1, 2, 3, 4, . . . belongs 
a series of which the terms are got by subtracting, in order, 
the terms of the sequence from the ones immediately following 
them, thus : 

(2_l) + (3-2) + (4-3) + . . . = 1 + 1 + 1+ . . .; 
and from a series a corresponding sequence can be got by making 
the sum of the first, the first two, the first three, . . . terms 
the first, second, third, . . . term of the sequence respectively. 
Thus, to the series 1 + 1 + 1 + . . . corresponds the sequence 
1, 2, 3, ... 

Now, if a series has only a finite number of terms, it is 
possible to find the sum of all the terms ; but if the series is 
unending, we evidently cannot. But in certain cases the 
corresponding sequence has a limit, and this limit is called by 
mathematicians, neither unnaturally nor accurately, " the sum 
to infinity of the series." Thus, the sequence 1, 1 + J, 1 + 1 
+ J,... has the limit 2, and so the sum to infinity of the 
series 1 + J + i + f + ... is 2. Of course, all series do not 
have a sum : thus 1 + 1 + 1 +. . . to infinity, has not the 
terms of the corresponding sequence increase continually 
beyond all limits. Notice particularly that the terms of a 
sequence may increase continually, and yet have a limit 
those of the above sequence with limit 2 so increase, but not 
beyond 2, though they do beyond any number less than 2 ; 
also notice that the terms of a sequence may increase beyond 
all limits even if the terms of the corresponding series con- 
tinually diminish, remaining positive, towards 0. The series 
l + i + + + ^- + i 8 sucn a series ; the terms of the 
sequence slowly increase beyond all limits, as we see when 
we reflect that the sums 



i are all greater than . It is very important to realise the 
fact illustrated by this example; for it shows that the con- 
ditions under which an infinite series has a sum are by no 
means as simple as they might appear at first sight. 







VIEWS OF LIMITS AND NUMBERS 79 

The logical scrutiny to which, during the last century, the 
processes and conceptions of mathematics have been subjected, 
showed very plainly that it was a sheer assumption that such 
a process as 1*4142 . . ., though all its terms are less than 2, 
for example, has any limit at all. When we replace numbers 
by points on a straight line, we feel fairly sure that there is 
one point which behaves to the points representing the above 
sequence in the same sort of way as 2 to the sequence 1, 1 + , 
1 + & + i Now, if our system of numbers is to form a 
continuum as a line seems to our thoughts to be, so that we 
can affirm that our number system is adequate, when we in- 
troduce axes in the manner of analytical geometry, to the 
description of all the phenomena of change of position which 
take place in our space, 1 then we must have a number ^2 which 
is the limit of the sequence 1*4142 . . . if 2 is of the series 
1 + i + i + , for to every point of a line must correspond 
a number which is subject to the same rules of calculation as 
the ratios or integers. Thus we must, to justify from a logical 
point of view our procedure in the great mathematical methods, 
show what irrationals are, and define them before we can prove 
that they are limits. We cannot take a series, whose law is 
evident, which has no ratio for sum, and yet such that the 
terms of the corresponding sequence all remain less than some 

fixed number (such as l + 7+T^+T7o^+f7o^4+ i when 

all the terms of the corresponding sequence are less than 3, 
for example), and then say that it "defines a limit." All 
we can prove is that if such a series has a limit, then, if the 
terms of its corresponding sequence do not decrease as we read 
from left to right (as in the preceding example), it cannot have 
more than one limit. 

Some mathematicians have simply postulated the irrationals. 
At the beginning of their discussions they have, tacitly or not, 

1 The only kind of change dealt with in the science of mechanics 
is change of position, that is, motion. It docs not seem to me to 
be necessary to adopt the doctrine that the complete description of 
any physical event is of a mechanical event ; for it is possible to 
assign and calculate with numbers of our number-continuum to 
other varying characteristics (such as temperature) of the state of 
a body besides position. 

F 







80 THE NATURE OF MATHEMATICS 

said : "In what follows we will assume that there are such 
things as fill up kinds of gaps in the system of rationals (or 
ratios)." Such a gap is shown by this. The rationals less 
than | and those greater than form two sets and | divides 
them. The rationals x such that x 2 is greater than 2 and 
those o;'s such that a? is less than 2 form two analogous sets, 
but there is only an analogue to the dividing number if we 
postulate a number ^/2. Thus by postulation we fill up these 
subtle gaps in the set of rationals and get a continuous set of 
real numbers. But we can avoid this postulation. If we 
define " ^2 " as the name of the class of rationals x such that 
x z is less than 2 and "()" as the name of the class of 
rationals x such that x is less than |. Proceeding thus, we 
arrive at a set of dosses, some of which correspond to rationals, 
as () to J, but the rest satisfy our need of a set without gaps. 
There is no reason why we should not say that these classes 
are the real numbers which include the irrationals. But we 
must notice that rationals are never real numbers ; is not (|), 
though analogous to it. We have much the same state of 
things as in the second chapter, where 2, + 2, and f were dis- 
tinguished and then deliberately confused because, with the 
mathematicians, we felt the importance of analogy in calculation. 
Here again v/e identify () with , and so on. 

Thus, integers, positive and negative " numbers," ratios, and 
real " numbers " are all different things : real numbers are 
classes, ratios and positive and negative numbers are relations. 
Integers, as we shall see, are classes. Very possibly there is 
a certain arbitrariness about this, but this is unimportant 
compared with the fact that in modern mathematics we have 
reduced the definitions of all " numbers " to logical terms. 
Whether they are classes or relations or propositions or other 
logical entities is comparatively unimportant. 

Integers can be defined as certain classes. Mathematicians 
like Weierstrass stopped before they got as far as this : they 
reduced the other numbers of analysis to logical developments 
out of the conception of integer, and thus freed analysis from 
any remaining trace of the sway of geometry. But it was 
obvious that integers had to be defined, if possible, in logical 
terms. It has long been recognised that two collections consist 
of the same number of objects if, and only if, these collections 







VIEWS OF LIMITS AND NUMBERS 81 

can be put in such a relation to one another that to every 
object of each one belongs one and only one object of the other. 
We must not think that this implies that we have already the 
idea of the number one. It is true that " one and only one " 
seems to use this idea. But : " the class a has one and only 
one member,'' is simply a short way of expressing : " x is a 
member of a, and, if y is also a member of a, then y is iden- 
tical with x." It is true, also, that we use the idea of the 
unity or the individuality of the things considered. But this 
unity is a property of each individual, while the number 1 is 
a property of a class. If a class of pages of a book is itself, 
under the name of a " volume," a member of a class of books, 
the same class of pages has a number (say 360), and a unity 
as being itself a member of a class. 

The relation spoken of above in which two classes possessing 
the same number stand to one another does not involve 
counting. Think of the fingers on your hands. If to every 
finger of each hand belongs, by some process of corre- 
spondence, one and only one remember the above meaning of 
this phrase of the other, they are said to have "the same 
number." This is a definition of what " the same number " is 
to mean for us ; the word " number " by itself is to have, as 
yet, no meaning for us ; and, to avoid confusion, we had better 
replace the phrase " the same number " by the word " similar." 
Any other word would, of course, do, but this word happens to 
be fairly suggestive and customary. Now, if the variable u is 
any class, " the number of u " is defined as short for the 
phrase : " the class whose members are classes which are 
similar to u." Thus the number of u is an entity which is 
purely logical in its nature. Some people might urge that by 
" number " they mean something different from this, and that 
is quite possible. All that is maintained by those who agree 
to the process sketched above is: (1) Classes of the kind 
described are identical in all known arithmetical properties 
with the undefined things people call "integer numbers"; 
(2) It is futile to say : "These classes are not numbers" if it 
is not also said what numbers are, that is to say, if " the 
number of" is not defined in some more satisfactory way. 
There may be more satisfactory definitions, but this one is a 
perfectly sound foundation for all mathematics, including the 







82 THE NATURE OF MATHEMATICS 

theory not touched upon here of ordinal numbers (denoted by 
"first," "second," . . .) which apply to sets arranged in some 
order, known at present. 

To illustrate (1), think of this. According to the above 
definition 2 is the general idea we call " couple." We say : 
" Mr. and Mrs. A are a couple " ; our definition would ask us 
to say in agreement with this : " The class consisting of Mr. 
and Mrs. A is a member of the class 2." We define " 2 " as 
" the class of classes u such that, if x is a u, u lacking a; is a 
1 " ; the definition of " 3 " follows that of " 2 " ; and so on. 
In the same way, we see that the class of fingers on your right 
hand and the class of fingers on your left hand are each of 
them members of the class 5. It follows that the classes of 
the fingers are similar in the above sense. 

Out of the striving of human minds to reproduce con- 
veniently and anticipate the results of experience of geometrical 
and natural events, mathematics has developed. Its develop- 
ment gave priceless hints to the development of logic, and then 
it appeared that there is no gap between the science of 
number and the science of the most general relations of objects 
of thought. As for geometry and mathematical physics, it 
becomes possible clearly to separate the logical parts from those 
parts which formulate the data of our experience. 

We have seen that mathematics has often made great strides 
by sacrificing accuracy to analogy. Let us remember that, 
though mathematics and logic are the highest forms of certainty 
within the reach of us, the process of mathematical discovery, 
which is so often confused with what is discovered, has led 
through many doubtful analogies and errors arriving from the 
great help of symbolism in making the difficult easy. Fortu- 
nately symbolism can also be used for precise and subtle 
analysis, so that we can say that it can be made to show up 
the difficulties in what appears easy and even negligible 
like 1 + 1 = 2. This is what much modern fundamental work 
does. 







THE NATURE OF MATHEMATICS 83 
CHAPTER VII 

THE NATURE OF MATHEMATICS 

IN the preceding chapters we have followed the development 
of certain branches of knowledge which are usually classed 
together under the name of "mathematical knowledge." These 
branches of knowledge were never clearly marked off from all 
other branches of knowledge : thus geometry was sometimes 
considered as a logical study and sometimes as a natural 
science the study of the properties of the space we live in. 
Still less was there an absolutely clear idea of what it was 
that this knowledge was about. It had a name Mathematics 
and few except " practical " men and some philosophers 
doubted that there was something about which things were 
known in that kind of knowledge called "mathematical." 
But what it was did not interest very many people, and there 
was and is a great tendency to think that the question as to 
what Mathematics is could be answered if we only knew all 
the facts of the development of our mathematical knowledge. 
It seems to me that this opinion is, to a great extent, due to 
an ambiguity of language : one word " mathematics " is used 
both for our knowledge of a certain kind and the thing, if 
such a thing there be, about which this knowledge is. I have 
distinguished, and will now explicitly distinguish, between 
" Mathematics," a collection of truths of which we know 
something, and " mathematics," our knowledge of Mathematics. 
Thus, we may speak of " Euclid's mathematics " or " Newton's 
mathematics," and say truly that mathematics has developed 
and therefore had history ; but Mathematics is eternal and 
unchanging, and therefore has no history it does not belong, 
even in part, to Euclid or Newton or anybody else, but is 
something which is discovered, in the course of time, by human 
minds. An analogous distinction can be drawn between 
" Logic " and " logic." The small initial indicates that we are 
writing of a psychological process which may lead to Truth ; 
the big initial indicates that we are writing of the entity the 
part of Truth to which this process leads us. The reason 
why mathematics is important is that Mathematics is not 
incomprehensible, though it is eternal and unchanging. 







84 THE NATURE OF MATHEMATICS 

Grammatical usage makes us use a capital letter even for 
"mathematics" in the psychological sense when the word 
begins a sentence, but in this case I have guarded and will 
guard against ambiguity. 

That particular function of history which I wish here to 
emphasize will now, I think, appear. In mathematics we 
gradually learn, by getting to know some things about Mathe- 
matics, to know that there is such a thing as Mathematics. 

We have, then, glanced at the mathematics of primitive 
peoples, and have seen that at first isolated properties of 
abstract things like numbers or geometrical figures and of 
abstract relations between concrete things like the relations 
between the weights and the arms of a lever in equilibrium. 
These properties were, at first, discovered and applied, of 
course, with the sole object of the satisfaction of bodily needs. 
With the ancient Greeks comes a change in point of view which 
perhaps seems to us, with our defective knowledge, as too 
abrupt. So far as we know, Greek geometry was, from its 
very beginning, deductive, general, and studied for its own 
interest, and not for any applications to the concrete world it 
might have. In Egyptian geometry, if a result was stated as 
universally true, it was probably only held to be so as a result 
of induction the conclusion from a great number of particular 
instances to a general proposition. Thus, if somebody sees 
a very large number of officials of a certain railway companj r , 
and notices that all of them wear red ties, he might conclude 
that all the officials of that company wear red ties. This 
might be probably true : it would not be certain : for certainly 
it would be necessary to know that there was some rule accord- 
ing to which all the officials were compelled to wear red ties. 
Of course, even then the conclusion would not be certain, since 
these sort of laws may be broken. Laws of Logic, however, 
cannot be broken. These laws are not, as they are sometimes 
said to be, laws of thought : for logic has nothing to do with 
the way people think, any more than poetry has to do with 
the food poets must eat to enable them to compose. Some- 
body might think that 2 and 2 make 5 : we know, by a process 
which rests on the laws of Logic, that they make 4. 

This is a more satisfactory case of induction : Format stated 
that no integral values of x, y, and s can be found such that 







THE NATURE OF MATHEMATICS 85 

x n + y n z'\ if n be an integer greater than 2. This theorem 
has been proved to be true for n = 3, 4, 5, 7, and many other 
numbers, and there is no reason to doubt that it is true. But 
to this day, no general proof of it has been given. 1 This, then, 
is an example of a mathematical proposition which has been 
reached and stated as probably true by induction. 

Now, in Greek geometry, propositions were stated and 
proved by the laws of Logic helped, as we now know, by tacit 
appeals to the conclusions which common sense draws from the 
pictorial representation in the mind of geometrical figures 
about any triangles, say, or some triangles, and thus not about 
one or two particular things but about an infinity of them. 
Then, consider any two triangles ABC and DEF. It helps 
the thinking of most of us to draw pictures of particular 
triangles, but our conclusions do not hold merely for these 
triangles. If the sides BA and AC are equal in length to the 
sides ED and DP respectively, and the angle at A is equal to 
the angle at D, then BC is equal to EF. This is proved 
rather imperfectly in the fourth proposition of the first Book 
of Euclid's Elements. 

When we examine into and complete the reasonings of 
geometricians, we find that the conception of space vanishes, 
and that we are left with logic alone. Philosophers and 
mathematicians used to think and some do now that, in 
geometry, we had to do, not with the space of ordinary life in 
which our houses stand and our friends move about, and which 
certain quaint people say is "annihilated" by electric tele- 
graphs or motor cars, but an abstract form of the same thing 
from which all that is personal or material has disappeared, 
and only things like distance and order and position have re- 
mained. Indeed, some have thought that position did not 
remain ; that, in abstract space, a circle, for example, had no 
position of its own, but only with respect to other things. 
Obviously, we can only, in practice, give the position of a thing 
with respect to other things "relatively" and not "abso- 
lutely." These " relativists " denied that position had any pro- 
perties which could not be practically discovered. Relativism, 

1 This is an example of the " theory of numbers," the study of 
the properties of integers, to which the chief contributions, per- 
haps, have been made by Fermat and Gauss. 







86 THE NATURE OF MATHEMATICS 

in a thought-out form seems quite tenable ; in a crude form, 
it seems like excluding the number 2, as distinguished from 
classes of two things, from notice as a figment of the brain, 
because it is not visible or tangible like a poker or a bit of 
radium or a mutton-chop. 

In fact, a perfected geometry reduces to a series of deduc- 
tions holding not only for figures in space, but for any abstract 
things. Spatial figures give a striking illustration of some 
abstract things ; and that is the secret of the interest which 
analytical geometry has. But it is into algebra that we must 
now look to discover the nature of Mathematics. 

We have seen that Egyptian arithmetic was more general 
than Egyptian geometry : like algebra, by using letters to 
denote unknown numbers, it began to consider propositions 
about any numbers. In algebra and algebraical geometry this 
quickly grew, and then it became possible to treat branches of 
mathematics in a systematic way and make whole classes of 
problems subject to the uniform and almost mechanical working 
of one method. Here we must again recall the economical 
function of science. 

At the same time as this rapid growth of methods algebra 
and analytical geometry and the infinitesimal calculus grew 
up from the application of mathematics to natural science, the 
new conceptions which influenced the form which mathematics 
took in the seventeenth, eighteenth, and nineteenth centuries. 
The ideas of variable and function became more and more 
prominent. These ideas were brought in by the conception of 
motion, and, unaffected by the doubts of the few logicians in 
the ranks of the mathematicians, remained to fructify mathe- 
matics. When mathematicians woke up to the necessity of 
explaining mathematics logically and finding out what Mathe- 
matics is, they found that, in mathematics the striving for 
generality had led, from very early times, to the use of a 
method of deduction used but not recognised and distinguished 
from the method usually used by the Aristotelians. I will try 
to indicate the nature of these methods, and it will be seen 
how the ideas of variable and function, in a form which does 
not depend on that particular kind of variability known as 
motion, come in. 

A proposition in logic is the kind of thing which is denoted 







THE NATURE OF MATHEMATICS 87 

by such a phrase as : " Socrates was a mortal and the husband 
of a scold." If and this is the characteristic of modern logic 
we notice that the notions of variable and function (corre- 
spondence, relation) which appeared first in a special form in 
mathematics, are fundamental in all the things which are the 
objects of our thought, we are led to replace the particular con- 
ceptions in a proposition by variables, and thus see more clearly 
the structure of the proposition. Thus: "a; is a y and has 
the relation R to z, a member of the class u " gives the general 
form of a multitude of propositions, of which the above is a 
particular case, the above proposition may be true, but it is 
not a judgment of logic, but of history or experience. The 
proposition is false if " Kant " or " Westminster Abbey " is 
substituted for "Socrates": it is neither if "x" a sign for a 
variable, is, and then becomes what we call a " prepositional 
function" of x and denote it by "x" or "fa." If more 
variables are involved, we have the notation " (x,y,z, . . . ) 
implies, for all values of the variables, i^(x,y,z, . . . ). The 
structure of the propositional functions involves only such ideas 
as are fundamental in logic, like implication, class, relation, the 
relation of a term to a class of which it is a member, and so 
on. And, of course, mathematics uses the notion of truth. 

When we say that " 1 + 1 = 2," we seem to be making a 
mathematical statement which does not come under the 
above definition. But the statement is rather mistakenly 
written : there is, of course, only one whole class of most 







88 THE NATURE OF MATHEMATICS 

classes, and the notation " 1 + 1 " makes it look as if there 
were two. Remembering that 1 is a class of certain classes, 
what the above proposition means is : If a; and y are members 
of 1, and x differs from y, then x and y together make up a 
member of 2. 

At last, then, we arrive at seeing that the nature of 
Mathematics is independent of us personally and of the world 
outside, and we can feel that our own discoveries and views do 
not affect the Truth itself, but only the extent to which we or 
others see it. Some of us discover things in science, but we do 
not really create anything in science any more than Columbus 
created America. Common sense certainly leads us astray 
when we try to use it for the purposes for which it is not 
particularly adapted, just as we may cut ourselves and not our 
beards if we try to shave with a carving knife ; but it has the 
merit of finding no difficulty in agreeing with those philosophers 
who have succeeded in satisfying themselves of the truth and 
position of Mathematics. Some philosophers have reached the 
startling conclusion that Truth is made by men, and that 
Mathematics is created by mathematicians, and that Columbus 
created America ; but common sense, it is refreshing to think, 
is at any rate above being flattered by philosophical persuasion 
that it really occupies a place sometimes reserved for an even 
more sacred Being. 



BIBLIOGRAPHY 

THE view that science is dominated by the principle of the economy 
of thought has been in part 1 very thoroughly worked out by Ernst 
Mach (see especially the translation of his Science of Mechanics, 
Chicago ; Open Court Publishing Co., 3rd ed., 1907. On the history 
of mathematics, we may mention W. W. Rouse Ball's books, A 
Primer of the History of Mathematics (3rd ed., 1906), and the fuller 
Short Account of the History of Mathematics (4th ed., 1908, both 
published in London by Macmillan), and Karl Fink's Brief History 
of Mathematics (Chicago, 3rd ed., 1910). 

As text- books of mathematics, De Morgan's books on Arithmetic, 
Algebra, and Trigonometry are still unsurpassed, and his Trigonometry j 
and Double Algebra contains one of the best discussions of complex 
numbers, for students, that there is. As De Morgan's books are 
not all easy to get, the reprints of his Elementary Illustrations of the 
Differential and Integral Calculus and his work On the Study and 
Difficulties of Mathematics (Chicago, 1899 and 1902) may be recom- 
mended. Where possible, it is best to read the works of the great 
mathematicians themselves. For elementary books, Lagrange's 
Lectures on Elementary Mathematics, of which a translation has been 
published at Chicago (2nd ed., 1901), is the most perfect specimen. 

The questions dealt with in the fourth chapter are more fully dis- 
cussed in Mach's Mechanics. An excellent collection of methods 
and problems in graphical arithmetic and algebra and so on, is 
contained in H. E. Cobb's book on Elements of Applied Mathematics 
(Boston and London : Ginn & Co., 1911). 

Finally, the best discussion of the nature of Mathematics is 
contained in B. Russell's Principles of Mathematics (Cambridge 
University Press, 1903). 

1 Cf. above, pp. 7, 12, 16, 20, 57. 







89 



INDEX 



ABACUS, 16 

Abel, 73 

Abscissa, 41 

Abstractness of mathematics, 17 

Acceleration, 58, 64, 70, 72, 75 

Achilles and the tortoise, 51 

Addition, 15, 28, 30, 33, 34, 35, 38 

Ahmes, 14, 15, 16, 19, 27 

Algebra, 15, 16, 20, 25, 27, 28, 30, 32, 

33, 38, 74, 75, 86 
Alkarismi, 29 

Ambiguous denotation, 27, 36, 68 
Analogy, 10, 31, 32, 34, 35, 39, 47, 80, 82 
Analysis, 7, 49, 76 
Analytical geometry, 21, 89, 49, 62, 53, 

79,86 

geometrical, 19, 21, 25, 27 
Anticipation of facts in thought, 12, 54, 

82 

Any, notion of, 15, 33, 48, 69, 85, 86 
Apollonius, 17 
Arabs, 16, 22, 23, 29 
Archimedes, 17, 19, 31, 51, 69 
Archytas, 17, 18, 19 
Areas, 50, 51, 52, 53, 59, 61, 65, 67, 76 
Aristotelians, 25, 86 
Aristotle, 7, 64 
Arithmetic, 20, 29,!33 
Assyrians, 10 
Astronomy, 6, 7, 17, 72 
Axes, 39, 41 

BALL, W. W. R., 89 
Barrow, 66 
Berkeley, 70 
Bernoulli, James, 72 

John, 49, 72 
Boethius, 23 
Boole, 24 

Business methods of science, 11-12 

CAUCHY, 73, 74 

Cavalieri, 52, 53 

Chinese, 14, 29 

Clairaut, 72 

Classification, 10, 11 

Cobb, H. E.,89 

Common sense, 88 

Communication of knowledge, 11 

Compensation of errors, 69 







90 



Conic sections, 42-43, 64 
Constant, 11, 24, 40, 47, 48 
Continuum, 80 
Co-ordinates, 41, 75 
Cosine, 22, 49 
Couple, 82 
Criticism, 24, 34 
Cubic equations, 37 
Curve of spaces, 58, 59, 60 

of velocities, 58, 59, 60 

D'ALEMBERT, 72 

De Morgan, 24, 89 

Decimal numeration, 16, 29 

Deduction, 17, 86 

Descartes, 25, 28, 29, 81, 33, 35, 37, 39, 

40, 42, 43, 46, 49, 52, 60, 75 
Descriptions, 9, 11, 54, 64, 71, 79 
Differential Calculus, 53, 59, 67 

quotient, 59, 76 

triangle, 66 
Differentials, 66, 67, 68 

of higher order, 70 
Differentiation, 68, 70, 76 
Diophantus, 20, 27, 28 
Dimensions, 63 
Directed quantities, SO 

EASTERN EMPIRE, 23 

Economy of thought, 6, 11, 12, 16, 18, 

20. 28, 33, 53, 69, 71, 75, 85, 89 
Egyptians, 10, 12, 13, 14, 15, 16, 17, 29, 

84,86 

Ellipse, 42, 43 

Equation of a circle, 40-43, 44-46 
Equation of a curve, 40, 71, 75 
Equations, mechanical, 62 
Euclid, 14, 17, 51, 83, 85 
Eudemus, 20 
Eudoxus, 51 
Euler, 49, 72 
Europe, 23 

Exhaustion, method of, 61, 65 
Exponents, 28 

FAITH of mathematicians 32. 34, 43, 

52 

Faraday, 10 
Fermat, 66, 85 
Fink, 89 


INDEX 







91 



Fluent, 71 

Fluxion, 72 

Fluxions, method of, 60, 65, 71, 75 

Fourier, 72 

Fractions, 14, 15, 21, 31, 32, 34, 41, 42 

French Revolution, 72 

Function, 23, 24, 36, 40, 49, 74, 86, 87 

prepositional, 87 
Functionality, 48 

GALILEO, 55, 56-62, 65, 71, 75 
Gaps in the system of rationals, 80 
Gauss, 74, 85 

Generality of algebra, 24, 38, 75, 86 
Generalisation of numbers, 31, 32 
Geometry, 83, 85 

modern, 74 

Geometrical objects, 18, 54-55 
Girard, 29 

Greek geometry, 12, 13, 16, 75, 84 
Greeks, 15, 16, 21, 23, 24, 47, 50, 51, 52, 

65, 77 
Gulliver's Travels, 32 

HARRIOT, 29, 36 
Herodotus, 13 
Hertz, 10 

Hindoos, 16, 22, 29 
Hipparchus, 22 
Hippocrates, 17 
Hyperbola, 42, 43 

IMPLICATION, 87 
Incommensurables, 20, 32, 77 
Independence of motions, 61 
Indivisibles, method of, 52, 53, 60, 65, 67 
Induction, 17, 84 
Inertia, law of, 64 
Infinite series, 78 
Infinitesimal Calculus, 46, 53 

methods, 50 

Infinitesimals, 50-51, 59, 66, 67, 73, 74 
Integers, definition of, 80-82 
Integral, 68 

Calculus, 53, 59, 60, 61 
Integration, 68, 69, 76 
Interpretation of symbols, 38 
Intersection, points of, 45 

Inverse of problem of tangents, 53, 68, 

69,76 
Irrational numbers, 32, 34, 42, 43, 77, 

79, 80 
Isosceles triangle, 18 

KEPLER, 52, 66 

Knowledge, instinctive, 10, 58 

LAQRANGE, 72, 89 

Lambert, 24 

Laplace, 72 

Laws of nature, 48, 54, 56 

of logic, 84 

of thought, 84 
Legendre, 72 







Leibniz, 24, 40, 69, 66, 67, 71, 72, 73, 75, 

86 

Leonardo of Pisa, 29 
Limit, 46, 60, 73, 77, 80 

upper, 74, 78 

Local value of symbols, 16 
Loci, 20, 21, 24, 25, 43, 46 
Logarithms, 28 

Logic, 12, 25, 34, 47, 48, 55, 82, 83, 84, 
85,86 

symbolic, 30 

modern, 5, 7, 24, 87 

Logical scrutiny of mathematics, 79 
Logicians, 47 

MACH, E., 89 
Magnitudes, 21, 77 
Mathematical discovery, 8 
Mathematics and mathematics, 8, 83 
Mathematics, use of, 5, 6, 11 

application of, 6, 7, 86 

growth of, 10 sqq. 

nature of, 9, 55, 83-88 
Maxwell, 10 
Mechanical theory, 79 
Mechanics, 19, 55, 75 
Menaechmus, 17, 18 
Methods, mathematical, 7, 8 
Mnemonic character of symbols in 

mechanics, 62, 63 
Model, 54, 55 
Moors, 23 
Motion, 24, 46, 47, 55, 86 / 

conception of, 24 
Multiplication, 15, 28, 30, 32, 38, 62, 

67 

logical, 31 

NAPIER, 28 

Natural science, 6, 9, 12, 19, 24 
Nautical Almanac, 6, 54, 55 
Need, intellectual, 9, 10, 12 

bodily, 10, 12 
Negative quantities, 41 

Newton, 10, 55, 60, 64, 65, 71, 72, 73, 

75, 76, 85 

Notation, 16, 33, 53, 71 
" Numbers," 21, 31, 33, 34, 77, 81 
Numbers, abstract, 29, 32 

complex, 38, 74 

concrete, 29, 30 

imaginary, 37 

negative, 7, 34, 35, 36 

ordinal, 82 

positive, 34, 35, 80 

representation of, by points, 31, 41, 

42,79 

ORBITS, 64, 76 
Ordinate, 41 
Origin, 41 

of geometry, 12 

PAPPUS, 43 

Parabola, 42, 43, 60, 61, 69 



INDEX 







Particular entities, 75 

Pascal, 53 

Philosophers, 9, 24, 47, 52, 83, 88 

Physics, mathematical, 65, 82 

Plato, 17, 18, 19 

Plutarch, 18 

Point, moving, 24, 47 

Poisson, 72 

Powers of Infinitesimal Calculus, 28, 

Principles of Mathematics, 7, 83-88 

Printing, invention of, 23 

Proclus, 13, 17, 20 

Projectiles, 60, 61 

Proportions, 62 

Propositions, 86 

Psychology, 8, 9 

Ptolemy, 22 

Pythagoras, 17, 20 

QUADRATIC equations, 36, 37, 45 
Quadratures, method of, 50, 65, 68 
Quantities, 21 
Quotient, 28 

EATIONALS, 80 

Ratios, 80 

Real numbers, 43, 77, 80 

Reasoning, mathematical, 7, 8, 17 

Record, 29 

Relativism, 85 

Renascence, 23 

Riemann, 74 

Roberval, 53, 65, 71 

Romans, 15, 22, 41 

Russell, 13, 89 

SCMNCE, natural, 6, 9 
formation of, 11 







Sequence, 73, 74, 77, 78 

Series, 78 

Signless numbers, 34 < 

Signs in mathematics, 5, 15, 20, 27, 35, 

36, 48, 67 
Similarity, 81 
Sines, 22, 49 
Space, 85 

Space-perception, 74, 76 
Sphere, equation of a, 46 
Straight line, equation of a. 40, 42 
Subtraction, 28, 30, 34 
Sum of a series, 78 
Surveying, 13, 18, 21, 22 
Swift, 32 

Symbolism, 20, 75, 82 
Symbols of arithmetic, 29, 30 

TANGENT to a curve, 45-46, 59, 65, 66, 

68, 69, 70, 75, 76 
Thales, 17, 18, 21 
Theory of numbers, 74, 85 
Thought, laws of, 84 
Trigonometry, 20, 22, 49, 76 

spherical, 22 

UNITS, 63 

VARIABLE, 11, 23, 24, 40, 46, 47, 48, 49, 

69, 73, 86, 87 

Velocity, 56, 57, 58, 59, 60, 71, 76 
Viete, 25 
Volumes, 50, 65 

WALIIS, 63 
Weierstrass, 73, 74, 80 

ZBNO, 24, 60, 51 







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