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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