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“SQUARING THE CIRCLE”
A HISTORY OF THE PROBLEM
CAMBRIDGE UNIVERSITY PRESS
Qondor; FETTER LANE, E.C.
Cc. F. CLAY, MANAGER

 

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All rights reserved
“SQUARING THE CIRCLE”
A HISTORY OF THE PROBLEM

BY

tv

E. W. HOBSON, Sc.D., LL.D. F.R.S.

SADLEIRIAN PROFESSOR OF PURE MATHEMATICS, AND FELLOW OF
CHRIST’S COLLEGE, IN THE UNIVERSITY OF CAMBRIDGE

Cambridge :

at the University Press
1913
Cambringe :
PRINTED BY JOHN CLAY, M.A,
AT THE UNIVERSITY PRESS
PREFACE

N the Easter Term of the present year I delivered a short course
of six Professorial Lectures on the history of the problem of the
quadrature of the circle, in the hope that a short account of the
fortunes of this celebrated problem might not only prove interesting
in itself, but might also act as a stimulant of interest in the more
general history of Mathematics. It has occurred to me that, by the
publication of the Lectures, they might perhaps be of use, in the
same way, to a larger circle of students of Mathematics.

The account of the problem here given is not the result of any
independent historical research, but the facts have been taken from
the writings of those authors who have investigated various parts of
the history of the problem.

The works to which I am most indebted are the very interesting
book by Prof. F. Rudio entitled “Archimedes, Huygens, Lambert,
Legendre. Vier Abhandlungen iiber die Kreismessung” (Leipzig,
1892), and Sir T. L. Heath’s treatise “The works of Archimedes ”
(Cambridge, 1897). I have also made use of Cantor’s “Geschichte der
Mathematik,” of Vahlen’s “ Konstruktionen und Approximationen”
(Leipzig, 1911), of Yoshio Mikami’s treatise ‘“‘The development of
Mathematics in China and Japan” (Leipzig, 1913), of the translation
by T. J. McCormack (Chicago, 1898) of H. Schubert’s “ Mathematical
Essays and Recreations,” and of the article ‘The history and trans-
cendence of +” written by Prof. D. E. Smith which appeared in the
“Monographs on Modern Mathematics” edited by Prof. J. W. A.
Young. On special points I have consulted various other writings.

E. W. H.

CuHRIST’s COLLEGE, CAMBRIDGE.
October, 1913.
CONTENTS

CHAPTER I

GENERAL ACCOUNT OF THE PROBLEM

THE FIRST PERIOD .

THE SECOND PERIOD

THE THIRD PERIOD

CHAPTER II

CHAPTER III

CHAPTER IV

PAGE

13

36

43
CHAPTER I

GENERAL ACCOUNT OF THE PROBLEM

A GENERAL survey of the history of thought reveals to us the fact
of the existence of various questions that have occupied the almost
continuous attention of the thinking part of mankind for long series of
centuries. Certain fundamental questions presented themselves to the
human mind at the dawn of the history of speculative thought, and
have maintained their substantial identity throughout the centuries,
although the precise terms in which such questions have been stated
have varied from age to age in accordance with the ever varying
attitude of mankind towards fundamentals. In general, it may be
maintained that, to such questions, even after thousands of years of
discussion, no answers have been given that have permanently satisfied
the thinking world, or that have been generally accepted as final
solutions of the matters concerned. It has been said that those
problems that have the longest history are the insoluble ones.

If the contemplation of this kind of relative failure of the efforts of
the human mind is calculated to produce a certain sense of depression,
it may bea relief to turn to certain problems, albeit in a more restricted
domain, that have occupied the minds of men for thousands of years,
but which have at last, in the course of the nineteenth century,
received solutions that we have reasons of overwhelming cogency to
regard as final. Success, even in a comparatively limited field, is
some compensation for failure in a wider field of endeavour. Our
legitimate satisfaction at such exceptional success is but slightly
qualified by the fact that the answers ultimately reached are in a
certain sense of a negative character. We may rest contented with
proofs that these problems, in their original somewhat narrow form,
are insoluble, provided we attain, as is actually the case in some
celebrated instances, to a complete comprehension of the grounds,
resting upon a thoroughly established theoretical basis, upon which
our final conviction of the insolubility of the problems is founded.

H. 1
2 GENERAL ACCOUNT OF THE PROBLEM

The three celebrated problems of the quadrature of the circle, the
trisection of an angle, and the duplication of the cube, although all of
them are somewhat special in character, have one great advantage for
the purposes of historical study, viz. that their complete history as
scientific problems lies, in a completed form, before us. Taking the
first of these problems, which will be here our special subject of study,
we possess indications of its origin in remote antiquity, we are able to
follow the lines on which the treatment of the problem proceeded and
changed from age to age in accordance with the progressive develop-
ment of general Mathematical Science, on which it exercised a
noticeable reaction. We are also able to see how the progress of
endeavours towards a solution was affected by the intervention of some
of the greatest Mathematical thinkers that the world has seen, such
men as Archimedes, Huyghens, Euler, and Hermite. Lastly, we
know when and how the resources of modern Mathematical Science
became sufficiently powerful to make possible that resolution of the
problem which, although negative, in that the impossibility of the
problem subject to the implied restrictions was proved, is far from being
a mere negation, in that the true grounds of the impossibility have
been set forth with a finality and completeness which is somewhat
rare in the history of Science.

If the question be raised, why such an apparently special problem,
as that of the quadrature of the circle, is deserving of the sustained
interest which has attached to it, and which it still possesses, the
answer is only to be found in a scrutiny of the history of the
problem, and especially in the closeness of the connection of that
history with the general history of Mathematical Science. It would
be difficult to select another special problem, an account of the history
of which would afford so good an opportunity of obtaining a glimpse
of so many of the main phases of the development of general Mathe-
matics ; and it is for that reason, even more than on account of the
intrinsic interest of the problem, that I have selected it as appropriate
for treatment in a short course of lectures.

Apart from, and alongside of, the scientific history of the problem,
it has a history of another kind, due to the fact that, at all times, and
almost as much at the present time as formerly, it has attracted the
attention of a class of persons who have, usually with a very inadequate
equipment of knowledge of the true nature of the problem or of its
history, devoted their attention to it, often with passionate enthusiasm.
Such persons have very frequently maintained, in the face of all efforts
GENERAL ACCOUNT OF THE PROBLEM 3

at refutation made by genuine Mathematicians, that they had obtained
a solution of the problem. The solutions propounded by the circle
squarer exhibit every grade of skill, varying from the most futile
attempts, in which the writers shew an utter lack of power to reason
correctly, up to approximate solutions the construction of which
required much ingenuity on the part of their inventor. In some cases
it requires an effort of sustained attention to find out the precise
point in the demonstration at which the error occurs, or in which an
approximate determination is made to do duty for a theoretically
exact one. The psychology of the scientific crank is a subject with
which the officials of every Scientific Society have some practical
acquaintance. Every Scientific Society still receives from time to
time communications from the circle squarer and the trisector of
angles, who often make amusing attempts to disguise the real
character of their essays. The solutions propounded by such persons
usually involve some misunderstanding as to the nature of the con-
ditions under which the problems are to be solved, and ignore the
difference between an approximate construction and the solution of
the ideal problem. It is a common occurrence that such a person
sends his solution to the authorities of a foreign University or
Scientific Society, accompanied by a statement that the men of
Science of the writer’s own country have entered into a conspiracy to
suppress his work, owing to jealousy, and that he hopes to receive
fairer treatment abroad. The statement is not infrequently accom-
panied with directions as to the forwarding of any prize of which the
writer may be found worthy by the University or Scientific Society
addressed, and usually indicates no lack of confidence that the
bestowal of such a prize has been amply deserved as the fit reward for
the final solution of a problem which has baffled the efforts of a great
multitude of predecessors in all ages. A very interesting detailed
account of the peculiarities of the circle squarer, and of the futility of
attempts on the part of Mathematicians to convince him of his errors,
will be found in Augustus De Morgan’s Budget of Paradowes. As
early as the time of the Greek Mathematicians circle-squaring occupied
the attention of non-Mathematicians ; in fact the Greeks had a special
word to denote this kind of activity, viz. terpaywvifew, which means to
occupy oneself with the quadrature. It is interesting to remark that,
in the year 1775, the Paris Academy found it necessary to protect its
officials against the waste of time and energy involved in examining
the efforts of circle squarers. It passed a resolution, which appears

1—2
4 GENERAL ACCOUNT OF THE PROBLEM

im the Minutes of the Academy*, that no more solutions were to be
examined of the problems of the duplication of the cube, the trisection
of the angle, the quadrature of the circle, and that the same resolution
should apply to machines for exhibiting perpetual motion. An account
of the reasons which led to the adoption of this resolution, drawn up
by Condorcet, who was then the perpetual Secretary of the Academy,
is appended. It is interesting to remark the strength of the convic-
tion of Mathematicians that the solution of the problem is impossible,
more than a century before an irrefutable proof of the correctness of
that conviction was discovered.

The popularity of the problem among non-Mathematicians may
seem to require some explanation. No doubt, the fact of its com-
parative obviousness explains in part at least its popularity; unlike
many Mathematical problems, its nature can in some sense be under-
stood by anyone; although, as we shall presently see, the very terms
in which it is usually stated tend to suggest an imperfect apprehension
of its precise import. The accumulated celebrity which the problem
attained, as one of proverbial difficulty, makes it an irresistible attrac-
tion to men with a certain kind of mentality. An exaggerated notion
of the gain which would accrue to mankind by a solution of the
problem has at various times been a factor in stimulating the efforts
of men with more zeal than knowledge. The man of mystical
tendencies has been attracted to the problem by a vague idea that its
solution would, in some dimly discerned manner, prove a key to a
knowledge of the inner connections of things far beyond those with
which the problem is immediately connected.

Statement of the problem

The fact was well known to the Greek Geometers that the problems
of the quadrature and the rectification of the circle are equivalent
problems. It was in fact at an early time established that the ratio of
the length of a complete circle to the diameter has a definite value
equal to that of the area of the circle to that of a square of which the
radius is side. Since the time of Euler this ratio has always been
denoted by the familiar notation 7. The problem of “squaring the
circle” is roughly that of constructing a square of which the area is
equal to that enclosed by the circle. This is then equivalent to the
problem of the rectification of the circle, z.e. of the determination of a

* Histoire de V Académie royale, année 1775, p. 61.
GENERAL ACCOUNT OF THE PROBLEM 5

straight line, of which the length is equal to that of the circumference
of the circle. But a problem of this kind becomes definite only when it
is specified what means are to be at our disposal for the purpose of
making the required construction or determination; accordingly, in
order to present the statement of our problem in a precise form, it is
necessary to give some preliminary explanations as to the nature of
the postulations which underlie all geometrical procedure.

The Science of Geometry has two sides; on the one side, that of
practical or physical Geometry, it is a physical Science concerned with
the actual spatial relations of the extended bodies which we perceive
in the physical world. It was in connection with our interests, of a
practical character, in the physical world, that Geometry took its
origin. Herodotus ascribes its origin in Egypt to the necessity of
measuring the areas of estates of which the boundaries had been
obliterated by the inundations of the Nile, the inhabitants being com-
pelled, in order to settle disputes, to compare the areas of fields of
different shapes. On this side of Geometry, the objects spoken of,
such as points, lines, &c., are physical objects ; a point is a very small
object of scarcely perceptible and practically negligible dimensions ; a
line is an object of small, and for some purposes negligible, thickness ;
and so on. The constructions of figures consisting of points, straight
lines, circles, &c., which we draw, are constructions of actual physical
objects. In this domain, the possibility of making a particular con-
struction is dependent upon the instruments which we have at our
disposal.

On the other side of the subject, Geometry is an abstract or
rational Science which deals with the relations of objects that are no
longer physical objects, although these ideal objects, points, straight
lines, circles, &c., are called by the same names by which we denote
their physical counterparts. At the base of this rational Science there
lies a set of definitions and postulations which specify the nature of
the relations between the ideal objects with which the Science deals.
These postulations and definitions were suggested by our actual
spatial perceptions, but they contain an element of absolute exactness
which is wanting in the rough data provided by our senses. The
objects of abstract. Geometry possess in absolute precision properties
which are only approximately realized in the corresponding objects of
physical Geometry. In every department of Science there exists in a
greater or less degree this distinction between the abstract or rational
side and the physical or concrete side; and the progress of each
6 GENERAL ACCOUNT OF THE PROBLEM

department of Science involves a continually increasing amount of
rationalization. In Geometry the passage from a purely empirical
treatment to the setting up of a rational Science proceeded by much
more rapid stages than in other cases. We have in the Greek
Geometry, known to us all through the presentation of it given in that
oldest of all scientific text books, Euclid’s Elements of Geometry, a
treatment of the subject in which the process of rationalization has
already reached an advanced stage. The possibility of solving a
particular problem of determination, such as the one we are con-
templating, as a problem of rational Geometry, depends upon the
postulations that are made as to the allowable modes of determination
of new geometrical elements by means of assigned ones. The restric-
tion in practical Geometry to the use of specified instruments has its
counterpart in theoretical Geometry in restrictions as to the mode in
which new elements are to be determined by means of given ones. As
regards the postulations of rational Geometry in this respect there is
a certain arbitrariness corresponding to the more or less arbitrary
restriction in practical Geometry to the use of specified instruments.

The ordinary obliteration of the distinction between abstract and
physical Geometry is furthered by the fact that we all of us, habitually
and almost necessarily, consider both aspects of the subject at the
same time. We may be thinking out a chain of reasoning in abstract
Geometry, but if we draw a figure, as we usually must do in order to
fix our ideas and prevent our attention from wandering owing to the
difficulty of keeping a long chain of syllogisms in our minds, it is
excusable if we are apt to forget that we are not in reality reasoning
about the objects in the figure, but about objects which are their
idealizations, and of which the objects in the figure are only an
imperfect representation. Even if we only visualize, we see the images
of more or less gross physical objects, in which various qualities
irrelevant for our specific purpose are not entirely absent, and which
are at best only approximate images of those objects about which we
are reasoning.

It is usually stated that the problem of squaring the circle, or the
equivalent one of rectifying it, is that of constructing a square of an
area equal to that of the circle, or in the latter case of constructing
a straight line of length equal to that of the circumference, by a
method which involves the use only of the compass and of the ruler as
a single straight-edge. This mode of statement, although it indicates
roughly the true statement of the problem, is decidedly defective in
GENERAL ACCOUNT OF THE PROBLEM 7

that it entirely leaves out of account the fundamental distinction
between the two aspects of Geometry to which allusion has been made
above. The compass and the straight-edge are physical objects by the
use of which other objects can be constructed, viz. circles of small
thickness, and lines which are approximately straight and very thin,
made of ink or other material. Such instruments can clearly have no
direct relation to theoretical Geometry, in which circles and straight
lines are ideal objects possessing in absolute precision properties that
are only approximately realized in the circles and straight lines that
can be constructed by compasses and rulers. In theoretical Geometry,
a restriction to the use of rulers and compasses, or of other instru-
ments, must be replaced by corresponding postulations as to the
allowable modes of determination of geometrical objects. We will see
what these postulations really are in the case of Euclidean Geometry.
Every Euclidean problem of construction, or as it would be preferable
to say, every problem of determination, really consists in the deter-
mination of one or more points which shall satisfy prescribed conditions.
We have here to consider the fundamental modes in which, when a
number of points are regarded as given, or already determined, a new
point is allowed to be determined.

Two of the fundamental postulations of Euclidean Geometry are
that, having given two points A and B, then (1) a unique straight
line (A, B): (the whole straight line, and not merely the segment
between A and B) is determined such that A and B are incident on
it, and (2) that a unique circle A (8), of which A is centre and on
which B is incident, is determined. The determinancy or assumption
of existence of such straight lines and circles is in theoretical Geo-
metry sufficient for the purposes of the subject. When we know that
these objects, having known properties, exist, we may reason about
them and employ them for the purposes of our further procedure ; and
that is sufficient for our purpose. The notion of drawing or con-
structing them by means of a straight-edge or compass has no
relevance to abstract Geometry, but is borrowed from the language
of practical Geometry.

A new point is determined in Euclidean Geometry exclusively in
one of the three following ways :

Having given four points A, B, C, D, not all incident on the same
straight line, then

(1) Whenever a point P exists which is incident both on (A, B)
and on (C, D), that point is regarded as determinate.
8 GENERAL ACCOUNT OF THE PROBLEM

(2) Whenever a point P exists which is incident. both on the.
straight line (A, B) and on the circle C(D), that point is regarded-as
determinate.

(3) Whenever a point P exists which is incident on both the
circles A (B), C(D), that point is regarded as determinate.

The cardinal points of any figure determined by a Euclidean
construction are always found by means of a finite number of
successive applications of some or all of these rules (1), (2) and (8).
Whenever one of these rules is applied it must be shewn that it does
not fail to determine the point. Euclid’s own treatment is sometimes
defective as regards this requisite; as for example in the first pro-
position of his first book, in which it is not shewn that the circles
intersect one another.

In order to make the practical constructions which correspond to
these three Euclidean modes of determination, corresponding to (1) the
ruler is required, corresponding to (2) both the ruler and the compass,
and corresponding to (3) the compass only.

As Euclidean plane Geometry is concerned with the relations of
points, straight lines, and circles only, it is clear that the above system
of postulations, although arbitrary in appearance, is the system that
the exigencies of the subject would naturally suggest. It may, how-
ever, be remarked that it is possible to develop Euclidean Geometry
with a more restricted set of postulations. For example it can be
shewn that all Euclidean constructions can be carried out by means of
(3) alone*, without employing (1) or (2).

Having made these preliminary explanations we are now in a
position to state in a precise form the ideal problem of “squaring the
circle,” or the equivalent one of the rectification of the circle.

The historical problem of “squaring the circle” is that of deter-
mining a square of which the area shall equal that of a given circle,
by a method such that the determination of the corners of the square
is to be made by means of the above rules (1), (2), (3), each of which
may be applied any finite number of times. In other words, each new
point successively determined in the process of construction is to be
obtained as the intersection of two straight lines already determined,
or as an intersection of a straight line and a circle already deter-
mined, or as an intersection of two circles already determined. A

* See for example the Mathematical Gazette for March 1913, where I have
treated this point in detail in the Presidential Address to the Mathematical
Association.
GENERAL ACCOUNT OF THE PROBLEM 9

similar statement applies to the equivalent problem of the rectification
of the circle.

This mode of determination of the required figure we may speak of
shortly as a Euclidean determination.

Corresponding to any problem of Euclidean determination there is
a practical problem of physical Geometry to be carried out by actual
construction of straight lines and circles by the use of ruler and com-
passes. Whenever an ideal problem is soluble as one of Euclidean
determination the corresponding practical problem is also a feasible
one. The ideal problem has then a solution which is ideally perfect ;
the practical problem has a solution which is an approximation limited
only by the imperfections of the instruments used, the ruler and the
compass; and this approximation may be so great that there is no
perceptible defect in the result. But it is an error which accounts I
think, in large measure, for the aberrations of the circle squarer aud
the trisector of angles, to assume the converse that, when a practical
problem is soluble by the use of the instruments in such a way that
the error is negligible or imperceptible, the corresponding ideal pro-
blem is also soluble. This is very far from being necessarily the case.
It may happen that in the case of a particular ideal problem no
solution is obtainable by a finite number of successive Euclidean
determinations, and yet that such a finite set gives an approximation
to the solution which may be made as close as we please by taking the
process far enough. In this case, although the ideal problem is in-
soluble by the means which are permitted, the practical problem is
soluble in the sense that a solution may be obtained in which the
error is negligible or imperceptible, whatever standard of possible
perceptions we may employ. As we have seen, a Euclidean problem
of construction is reducible to the determination of one or more points
which satisty prescribed conditions. Let P be one such point; then
it may be possible to determine in Euclidean fashion each point of a
set P,, P,, ... Pp, -.. of points which converge to P as limiting point,
and yet the point P may be incapable of determination by Euclidean
procedure. his is what we now know to be the state of things in the
case of our special problem of the quadrature of the circle by Euclidean
determination. As an ideal problem it is not capable of solution, but
the corresponding practical problem is capable of solution with an
accuracy bounded only by the limitations of our perceptions and the
imperfections of the instruments employed. Ideally we can actually
determine by Euclidean methods a square of which the area differs
10 GENERAL ACCOUNT OF THE PROBLEM

from that of a given circle by less than an arbitrarily prescribed
magnitude, although we cannot pass to the limit. We can obtain
solutions of the corresponding physical problem which leave nothing
to be desired from the practical point of view. Such is the answer
which has been obtained to the question raised in this celebrated
historical problem of Geometry. I propose to consider in some detail
the various modes in which the problem has been attacked by people
of various races, and through many centuries; how the modes of
attack have been modified by the progressive development of Mathe-
matical tools, and how the final answer, the nature of which had
been long anticipated by all competent Mathematicians, was at last
found and placed on a firm basis.

General survey of the history of the problem

The history of our problem is typical as exhibiting in a remarkable
degree many of the phenomena that are characteristic of the history of
Mathematical Science in general. We notice the early attempts at an
empirical solution of the problem conceived in a vague and sometimes
confused manner; the gradual transition to a clearer notion of the
problem as one to be solved subject to precise conditions. We observe
also the intimate relation which the mode of regarding the problem in
any age had with the state then reached by Mathematical Science in
its wider aspect ; the essential dependence of the mode of treatment of
the problem on the powers of the existing tools. We observe the fact
that, as in Mathematics in general, the really great advances, embody-
ing new ideas of far-reaching fruitfulness, have been due to an
exceedingly small number of great men; and how a great advance has
often been followed by a period in which only comparatively small
improvements in, and detailed developments of, the new ideas have
been accomplished by a series of men of lesser rank. We observe that
there have been periods when for a long series of centuries no advance
was made; when the results obtained in a more enlightened age have
been forgotten. We observe the times of revival, when the older
learning has been rediscovered, and when the results of the progress
made in distant countries have been made available as the starting
points of new efforts and of a fresh period of activity.

The history of our problem falls into three periods marked out by
fundamentally distinct differences in respect of method, of immediate
aims, and of equipment in the possession of intellectual tools. The
first period embraces the time between the first records of empirical
GENERAL ACCOUNT OF THE PROBLEM 11

determinations of the ratio of the circumference to the diameter of a
circle until the invention of the Differential and Integral Calculus, in
the middle of the seventeenth century. This period, in which the
ideal of an exact construction was never entirely lost sight of, and was
occasionally supposed to have been attained, was the geometrical
period, in which the main activity consisted in the approximate
determination of by calculation of the sides or areas of regular
polygons in- and circum-scribed to the circle. The theoretical ground-
work of the method was the Greek method of Exhaustions. In the
earlier part of the period the work of approximation was much
hampered by the backward condition of arithmetic due to the fact
that our present system of numerical notation had not yet been
invented ; but the closeness of the approximations obtained in spite
of this great obstacle are truly surprising. In the later part of this
first period methods were devised by which approximations to the
value of + were obtained which required only a fraction of the labour
involved in the earlier calculations. At the end of the period the
method was developed to so high a degree of perfection that no
further advance could be hoped for on the lines laid down by the
Greek Mathematicians ; for further progress more powerful methods
were requisite.

The second period, which commenced in the middle of the seven-
teenth century, and lasted for about a century, was characterized by
the application of the powerful analytical methods provided by the
new Analysis to the determination of analytical expressions for the
number 7 in the form of convergent series, products, and continued
fractions. The older geometrical forms of investigation gave way to
analytical processes in which the functional relationship as applied
to the trigonometrical functions became prominent. The new methods
of systematic representation gave rise to a race of calculators of =,
who, in their consciousness of the vastly enhanced means of calcula-
tion placed in their hands by the new Analysis, proceeded to apply
the formulae to obtain numerical approximations to 7 to ever larger
numbers of places of decimals, although their efforts were quite useless
for the purpose of throwing light upon the true nature of that number.
At the end of this period no knowledge had been obtained as regards
the number z of a kind likely to throw light upon the possibility or
impossibility of the old historical problem of the ideal construction ;
it was not even definitely known whether the number is rational or
irrational. However, one great discovery, destined to furnish the clue
12 GENERAL ACCOUNT OF THE PROBLEM

to the solution of the problem, was made at this time; that of the
relation between the two numbers 7 and ¢, as a particular case of
those exponential expressions for the trigonometrical functions which
form one of the most fundamentally important of the analytical
weapons forged during this period.

In the third period, which lasted from the middle of the eighteenth
century until late in the nineteenth century, attention was turned to
critical investigations of the true nature of the number 7 itself,
considered independently of mere analytical representations. ‘The
number was first studied in respect of its rationality or irrationality,
and it was shewn to be really irrational. When the discovery was
made of the fundamental distinction between algebraic and trans-
cendental numbers, i.e. between those numbers which can be, and
those numbers which cannot be, roots of an algebraical equation with
rational coefficients, the question arose to which of these categories
the number = belongs. It was finally established by a method which
involved the use of some of the most modern devices of analytical
investigation that the number = is transcendental. When this result
was combined with the results of a critical investigation of the
possibilities of a Euclidean determination, the inference could be
made that the number z, being transcendental, does not admit of
construction either by a Euclidean determination, or even by a
determination in which the use of other algebraic curves besides the
straight line and the circle is permitted. The answer to the original
question thus obtained is of a conclusively negative character ; but it
is one in which a clear account is given of the fundamental reasons
upon which that negative answer rests.

We have here a record of human effort persisting throughout the
best part of four thousand years, in which the goal to be attained was
seldom wholly lost sight of. When we look back, in the light of the
completed history of the problem, we are able to appreciate the diffi-
culties which in each age restricted the progress which could be made
within limits which could not be surpassed by the means then avail-
able ; we see how, when new weapons became available, a new race of
thinkers turned to the further consideration of the problem with a new
outlook.

The quality of the human mind, considered in its collective
aspect, which most strikes us, in surveying this record, is its colossal
patience.
CHAPTER II

THE FIRST PERIOD

Earliest traces of the problem

THE earliest traces of a determination of 7 are to be found in the
Papyrus Rhind which is preserved in the British Museum and was
translated and explained* by Hisenlohr. It was copied by a clerk,
named Ahmes, of the king Raaus, probably about 1700 B.c., and
contains an account of older Egyptian writings on Mathematics. It is
there stated that the area of a circle is equal to that of a square whose
side is the diameter diminished by one ninth; thus 4 =(8)d’, or
comparing with the formula A =}7d?, this would give

m= 256 = 3-1604....

No account is given of the means by which this, the earliest determina-
tion of 7, was obtained; but it was probably found empirically.

The approximation 7 = 8, less accurate than the Egyptian one, was
known to the Babylonians, and was probably connected with their
discovery that a regular hexagon inscribed in a circle has its side
equal to the radius, and with the division of the circumference into
6 x 60 = 360 equal parts.

This assumption (r=38) was current for many centuries; it is
implied in the Old Testament, 1 Kings vii. 23, and in 2 Chronicles iv.
2, where the following statement occurs :

“ Also he made a molten sea of ten cubits from brim to brim, round
in compass, and five cubits the height thereof; and a line of thirty
cubits did compass it round about.”

The same assumption is to be found in the Talmud, where the
statement is made ‘‘that which in circumference is three hands broad
is one hand broad.”

* Bisenlohr, Ein mathematisches Handbuch der alten Agypter (Leipzig, 1877).
14 THE FIRST PERIOD

The earlier Greek Mathematicians

It is to the Greek Mathematicians, the originators of Geometry as
an abstract Science, that we owe the first systematic treatment of the
problems of the quadrature and rectification of the circle. The oldest
of the Greek Mathematicians, Thales of Miletus (640—548 B.c.) and
Pythagoras of Samos (580 —500 B.c.), probably introduced the Egyptian
Geometry to the Greeks, but it is not known whether they dealt with
the quadrature of the circle. According to Plutarch (in De ewilio),
Anaxagoras of Clazomene (500—428 B.c.) employed his time during
an incarceration in prison on Mathematical speculations, and constructed
the quadrature of the circle. He probably made an approximate
construction of an equal square, and was of opinion that he had
obtained an exact solution. At all events, from this time the problem
received continuous consideration.

About the year 420 3.c. Hippias of Elis invented a curve known as
the terpaywviLovoa or Quadratrix, which is usually connected with the
name of Dinostratus (second half of the fourth century) who studied
the curve carefully, and who shewed that the use of the curve gives
a construction for 7.

‘~~ This curve may be described as follows, using modern notation.
Let a point @ starting at A describe the circular quadrant AB
with uniform velocity, and let a point R 2
starting at O describe the radius OB with
uniform velocity, and so that if @ and R
start simultaneously they will reach the
point B simultaneously. Let the pomt P
be the intersection of OQ with a line perpen-
dicular to OB drawn from R. The locus of *
P is the quadratrix. Letting ~QOA=6, |
and OR=y, the ratio y/@ is constant, and ©
equal to 2a/z, where a denotes the radius of
the circle. We have

ft

 

 

Fia. 1.

= = Ty
v=ycotd, or x y cot x7

the equation of the curve in rectangular coordinates. The curve will
intersect the w axis at the point

=i Te is
n= linn (y cot *Y) 2a/m.
THE FIRST PERIOD 15

If the curve could be constructed, we should have a construction for
the length 2a/7, and thence one for 7. It was at once seen that the ,
construction of the curve itself involves the same difficulty as that of 7.

The problem was considered by some of the Sophists, who made
futile attempts to connect it with the discovery of “cyclical square
numbers,” ¢.e. such square numbers as end with the same cipher as the
number itself, as for example 25 = 5’, 36 =6?; but the right path to
a real treatment of the problem was discovered by Antiphon and
further developed by Bryson, both of them contemporaries of Socrates
(469—399 B.c.). Antiphon inscribed a square in the circle and passed
on to an octagon, l6agon, &c., and thought that by proceeding far
enough a polygon would be obtained of which the sides would be so
small that they would coincide with the circle. Since a square can
always be described so as to be equal to a rectilineal polygon, and
a circle can be replaced by a polygon of equal area, the quadrature of
the circle would be performed. That this procedure would give only
an approximate solution he overlooked. The important improvement
was introduced by Bryson of considering circumscribed as well as
inscribed polygons; in this procedure he foreshadowed the notion of
upper and lower limits in a limiting process. He thought that the
area of the circle could be found by taking the mean of the areas of
corresponding in- and circum-scribed polygons.

Hippocrates of Chios who lived in Athens in the second half of
the fifth century B.c., and wrote the first text book on Geometry, was
the first to give examples of curvilinear areas which admit of exact
quadrature. These figures are the menisci or lunulae of Hippocrates.

If on the sides of a right-angled triangle ACB semi-circles are
described on the same side, the sum of
the areas of the two lunes ANC, BDC _
is equal to that of the triangle ACB.
If the right-angled triangle is isosceles,
the two lunes are equal, and each of
them is half the area of the triangle.
Thus the area of a lunula is found. Fic. 2.

If AC=CD=DB =radius OA (see Fig. 3), the semi-circle ACH
is 4 of the semi-circle ACDB. We have now

DAB-—3DAC = ACDB-3. meniscus ACE,

and each of these expressions is {DAB or half the circle on
14B as diameter. If then the meniscus AHC were quadrable

     
  

Do

B
16 THE FIRST PERIOD

so also would be the circle on }4B as diameter. Hippocrates
recognized the fact that the meniscus is not quadrable, and he made
attempts to find other quadrable
lunulae in order to make the quad-
rature of the circle depend on that

of such quadrable lunulae. The
question of the existence of various E
kinds of quadrable lunulae was
taken up by Th. Clausen* in 1840,
who discovered four other quad-
rable lunulae in addition to the
one mentioned above. The question was considered in a general
manner by Professor Landaut of Gottingen in 1890, who pointed out
that two of the four lunulae which Clausen supposed to be new were
already known to Hippocrates.

From the time of Plato (429—348 B.c.), who emphasized the
distinction between Geometry which deals with incorporeal things or
images of pure thought and Mechanics which is concerned with things
in the external world, the idea became prevalent that problems such as
that. with which we are concerned should be solved by Euclidean
determination only, equivalent on the practical side to the use of two
instruments only, the ruler and the compass.

Fie. 3.

The work of Archimedes

The first really scientific treatment of the problem was undertaken
by the greatest of all the Mathematicians of antiquity, Archimedes -
(287—2123.c.). In order to understand the mode in which he
actually established his very important approximation to the value of
it is necessary for us to consider in some detail the Greek method of
dealing with problems of limits, which in the hands of Archimedes
provided a method of performing genuine integrations, such as his
determination of the area of a segment of a parabola, and of a con-
siderable number of areas and volumes.

This method is that known as the method of exhaustions, and
rests on a principle stated in the enunciation of Euclid x. 1, as follows :

“Two unequal magnitudes being set out, if from the greater there
be subtracted a magnitude greater than its half, and from that which

* Journal fiir Mathematik, vol. 21, p. 375.
+ Archiv Math. Physik (3) 4 (1903).
THE FIRST PERIOD 17

is left a magnitude greater than its half, and if this process be repeated
continually, there will be left some magnitude which will be less than
the lesser magnitude set out.”

This principle is deduced by Euclid from the axiom that, if there
are two magnitudes of the same kind, then a multiple of the smaller
one can be found which will exceed the greater one. This latter axiom
is given by Euclid in the form of a definition of ratio (Book v. def. 4),
and is now known as the axiom of Archimedes, although, as Archimedes
himself states in the introduction to his work on the quadrature of the
parabola, it was known and had been already employed by earlier
Geometers. The importance of this so-called axiom of Archimedes,
now generally considered as a postulate, has been widely recognized in
connection with the modern views as to the arithmetic continuum and
the theory of continuous magnitude. The attention of Mathematicians
was directed to it by O. Stolz*, who shewed that it was a consequence of
Dedekind’s postulate relating to ‘‘sections.” ‘The possibility of dealing
with systems of numbers or of magnitudes for which the principle does
not hold has been considered by Veronese and other Mathematicians,
who contemplate non-Archimedean systems, i.e. systems for which
this postulate does not hold. The acceptance of the postulate is
equivalent to the ruling out of infinite and of infinitesimal magnitudes
or numbers as existent in any system of magnitudes or of numbers
for which the truth of the postulate is accepted.

The example of the use of the method of exhaustions which is most
familiar to us is contained in the proof given in Euclid xm. 2, that the
areas of two circles are to one another as the squares on their diameters.
This theorem which is a presupposition of the reduction of the problem
of squaring the circle to that of the determination of a definite ratio +
is said to have been proved by Hippocrates, and the proof given by
Euclid is pretty certainly due to Eudoxus, to whom various other
applications of the method of Exhaustions are specifically attributed
by Archimedes. Euclid shews that the circle can be “exhausted” by
the inscription of a sequence of regular polygons each of which has twice
as many sides as the preceding one. He shews that the area of the
inscribed square exceeds half the area of the circle ; he then passes to
an octagon by bisecting the arcs bounded by the sides of the square.
He shews that the excess of the area of the circle over that of the
octagon is less than half what is left of the circle when the square is
removed from it, and so on through the further stages of the process.

* See Math. Annalen, vol. 22, p. 504, and vol. 39, p. 107.
18 THE FIRST PERIOD

The truth of the theorem is then inferred by shewing that a contrary
assumption leads to a contradiction.

A study of the works of Archimedes, now rendered easily accessible
to us in Sir T. L. Heath’s critical edition, is of the greatest interest
not merely from the historical point of view but also as affording
a very instructive methodological study of rigorous treatment of
problems of determination of limits. The method by which Archimedes
and other Greek Mathematicians contemplated limit problems impresses
one, apart from the geometrical form, with its essentially modern way
of regarding such problems. In the application of the method of
exhaustions and its extensions no use is made of the ideas of the
infinite or the infinitesimal ; there is no jumping to the limit as the
supposed end of an essentially endless process, to be reached by some
inscrutable saltus. This passage to the limit is always evaded by
substituting a proof in the form of a reductio ad absurdum, involving
the use of inequalities such as we have in recent times again adopted
as appropriate to a rigorous treatment of such matters. Thus the
Greeks, who were however thoroughly familiar with all the difficulties
as to infinite divisibility, continuity, &c., in their mathematical proofs
of limit theorems never involved themselves in the morass of indivisibles,
indiscernibles, infinitesimals, &c., in which the Calculus after its
invention by Newton and Leibnitz became involved, and from which
our own text books are not yet completely free.

The essential rigour of the processes employed by Archimedes, with
such fruitful results, Jeaves, according to our modern views, one point
open to criticism. The Greeks never doubted that a circle has a
definite area in the same sense that a rectangle has one; nor did they
doubt that a circle has a length in the same sense that a straight line
has one. They had not contemplated the notion of non-rectifiable
curves, or non-quadrable areas; to them the existence of areas and
lengths as definite magnitudes was obvious from intuition. At the
present time we take only the length of a segment of a straight line,
the area of a rectangle, and the volume of a rectangular parallelepiped
as primary notions, and other lengths, areas, and volumes we regard as
derivative, the actual existence of which in accordance with certain de-
finitions requires to be established in each individual case or in particular
classes of cases. For example, the measure of the length of a circle is
defined thus: A sequence of inscribed polygons is taken so that the
number of sides increases indefinitely as the sequence proceeds, and
such that the length of the greatest side of the polygon diminishes
THE FIRST PERIOD 19

indefinitely, then if the numbers which represent the perimeters of the
successive polygons form a convergent sequence, of which the arithmetical
limit is one and the same number for all sequences of polygons which
satisfy the prescribed conditions, the circle has a length represented
by this limit. It must be proved that this limit exists and is inde-
pendent of the particular sequence employed, before we are entitled to
regard the circle as rectifiable.
; In his work xvxAov pérpyots, the measurement of a circle, Archimedes
proves the following three theorems.

(1) The area of any circle is equal to a right-angled triangle in
which one of the sides about the right angle is equal to the radius,
and the other to the circumference, of the circle.

(2) The area of the circle is to the square on its diameter as 11 to
14.

(3) The ratio of the circumference of any circle to its diameter is
less than 3+ but greater than 37¢

It is clear that (2) must be regarded as entirely subordinate to (3).
In order to estimate the accuracy of the statement in (3), we observe
that

3. =3°14285..., 332=3°14084..., 7 =3°14159....

| In order to form some idea of the wonderful power displayed by
‘Archimedes in obtaining these results with the very limited means at
his disposal, it is necessary to describe briefly the details of the method
‘he employed.

His first theorem is established by using sequences of in- and
circum-scribed polygons and a reductio ad absurdum, as in Euclid xu.
2, by the method already referred to above.

In order to establish the first part of (3), Archimedes considers
a regular hexagon circumscribed to the circle.

In the figure, AC is half one of the sides of this hexagon. Then

OA 265
407% 3> ie

Bisecting the angle AOC, we obtain AD half the side of a regular

OD 591% *
circumscribed 12agon. It is then shewn that —7 DA > TER If OF is

the bisector of the angle DOA, AF is half the side of a circum-

ae OF 11723
scribed 24agon, and it is then shewn that FA? aa Next,

bisecting HOA, we obtain AJ’ the half side of a 48agon, and it
2—2

 
20 THE FIRST PERIOD

_ is shewn that Ss > ee Lastly if OG (not shewn in the figure)
"be the bisector of FOA, AG is the half side of a regular 96agon
circumscribing the circle, and it is shewn that oS =, 1TRS and
ne AG” 153°
thence that the ratio of the diameter to the perimeter of the 96agon

CG
D
E
F
Gy fe
Fie. 4.
. _ 46734 ps :
is > Tiga? and it is deduced that the circumference of the circle,

which is less than the perimeter of the polygon, is < 3+ of the diameter.
The second part of the theorem is obtained in a similar manner by
determination of the side of a regular 96agon inscribed in the circle.

In the course of his work, Archimedes assumes and employs,
without explanation as to how the approximations were obtained, the
following estimates of the values of square roots of numbers :

issp> V3 > 285, 30133>/9082321, 1838,%,> 3380929,
10093 > 71018405, 2017} > /40692842,, 5912 </349450,
1172} <V 137394338, 2339} < 54721322.

In order to appreciate the nature of the difficulties in the way of
obtaining these approximations we must remember the backward
condition of Arithmetic with the Greeks, owing to the fact that they
possessed a system of notation which was exceedingly inconvenient for
the purpose of performing arithmetical calculations.

The letters of the alphabet together with three additional signs
were employed, each letter being provided with an accent or with
a short horizontal stroke; thus the nine integers

1, 2, 3, 4, 5, 6, 7, 8, 9 were denoted by a’, B', y', 8, €, =’, 2, 1’, @,
the multiples of 10,
10, 20, 30, ... 90 were denoted by v', «’, X’, pw’, v’, £0, 7’, 1,
THE FIRST PERIOD 21

the multiples of 100,
100, 200, ... 900 by p', o', rv, b,x’, Wo, Y’.

The intermediate numbers were expressed by juxtaposition, repre-
senting here addition, the largest number being placed on the left, the
next largest following, and so on in order. There was no sign for
zero. Thousands were represented by the same letters as the first
nine integers but with a small dash in front and below the line; thus
for example 5’ was 4000, and 1913 was expressed by a Auy’ or a Ay.
10000 and higher numbers were expressed by using the ordinary
numerals with M or Mv as an abbreviation for the word pvpias ; the
number of myriads, or the multiple of 10000, was generally written

as
over the abbreviation, thus 349450 was M fwv’. <A variety of devices
were employed for the representation of fractions*.

The determinations of square roots such as ./3 by Archimedes
were much closer than those of earlier Greek writers. ‘There has been
much speculation as to the method he must have employed in their
determination. There is reason to believe that he was acquainted
with the method of approximation that we should denote by

aby > \éEb> at.

Various alternative explanations have been suggested; some of these
suggest that a method equivalent to the use of approximation by
continued fractions was employed.

A full discussion of this matter will be found in Sir T. L. Heath’s
work on Archimedes.

The treatise of Archimedes on the measurement of the circle must
be regarded as the one really great step made by the Greeks towards
the solution of the problem ; in fact no essentially new mode of attack
was made until the invention of the Calculus provided Mathematicians
with new weapons. In a later writing which has been lost, but which
is mentioned by Hero, Archimedes found a still closer approximation
to =.

The essential points of the method of Archimedes, when generalized
and expressed in modern notation, consist of the following theorems :

(1) The inequalities sin 6 <6 <tan 6.

* For an interesting account of the Arithmetic of Archimedes, see Heath’s
Works of Archimedes, Chapter rv.
22 THE FIRST PERIOD

(2) The relations for the successive calculation of the perimeters
and areas of polygons inscribed and circumscribed to a, circle.

Denoting by p,, @ the perimeter and area of an inscribed regular
polygon of m sides, and by P,, A» the perimeter and area of a circum-
scribed regular polygon of x sides, these relations are

Pin=N PrP n= Gn An,
P, - 27nPa _ 2m An
yeh Bgl”
Thus the two series of magnitudes
Pry Pay Pony Pons Pan; Pans ney
An, Am, Arn, Any Aans Uns +5

 

are calculated successively in accordance with the same law. In each
case any element is calculated from the two preceding ones by taking
alternately their harmonic and geometric means. This system of
formulae is known as the Archimedean Algorithm ; by means of it the
chords and tangents of the angles at the centre of such polygons as
are constructible can be calculated. By methods essentially equivalent
to the use of this algorithm the sines and tangents of small angles
were obtained to a tolerably close approximation. For example,
Aristarchus (250 B.c.) obtained the limits 2, and J, for sin 1°.

The work of the later Greeks

Among the later Greeks, Hipparchus (180—125 B.c.) calculated
the first table of chords of a circle and thus founded the science of
Trigonometry. But the greatest step in this direction was made by
Ptolemy (87—165 .D.) who calculated a table of chords in which the
chords of all angles at intervals of 4° from 0° to 180° are contained,
and thus constructed a trigonometry that was not surpassed for
1000 years. He was the first to obtain an approximation to ~ more
exact than that of Archimedes; this was expressed in sexagesimal
measure by 3° 8’ 30” which is equivalent to

34 gy + y38— or 3745 = 314166...

The work of the Indians

We have now to pass over to the Indian Mathematicians. Aryab-
hatta (about 500 a.p.) knew the value
£2832 = 31416 for 7.

The same value in the form 332% was given by Bhaskara (born 1114 A.D.)
THE FIRST PERIOD 23

in his work The crowning of the system; and he describes this value as

xact, in contrast with the inexact value 22. His commentator Gancea
-explains that this result was obtained by calculating the perimeters of
‘polygons of 12, 24, 48, 96, 192, and 384 sides, by the use of the

formula
ao V4 = ag?

connecting the sides of inscribed polygons of 2n and n sides respectively,
the radius being taken as unity. If the diameter is 100, the side of
an inscribed 384agon is /98694 which leads to the above value* given
by Aryabhatta. Brahmagupta (born 598 a.D.) gave as the exact value
= 10. Hankel has suggested that this was obtained as the
supposed limit (1000) of /965, /981, /986, /987 (diameter 10),
the perimeters of polygons of 12, 24, 48, 96 sides, but this explanation
is doubtful. It has also been suggested that it was obtained by the
approximate formula

ee oe

= 2a+wv
which gives V10=3 +4.

The work of the Chinese Mathematicians

The earliest Chinese Mathematicians, from the time of Chou-Kong
who lived in the 12th century B.c., employed the approximation 7=3.
Some of those who used this approximation were mathematicians of
considerable attainments in other respects.

According to the Sui-shu, or Records of the Sui dynasty, there
were a large number of circle-squarers, who calculated the length of
the circular circumference, obtaining however divergent results.

Chang Hing, who died in 139 a.p., gave the rule

(circumference)? _5
(perimeter of circumscribed square)? 8’

which is equivalent to 7 = 10.

- Wang Fau made the statement that if the circumference of a
circle is 142 the diameter is 45; this is equivalent to =38'1555....
No record has been found of the method by which this result was

obtained.

* See Colebrooke’s Algebra with arithmetic and mensuration, from the Sanscrit
of Brahmagupta and Bhaskara, London, 1817.
5 ey

24 THE FIRST PERIOD
‘Liu Hui published in 263 a.p. an Arithmetic in nine sections which
‘contains a determination of 7. Starting with an inscribed regular
hexagon, he proceeds to the inscribed dodecagon, 24agon, and so on,
and finds the ratio of the circumference to the diameter to be 157 : 50,
which is equivalent to 7 = 3°14.

By far the most interesting Chinese determination was that of the
great Astronomer Zsu Ch‘ung-chih (born 430 a.p.). He found the
two values 22 and 283 (=3°1415929 ..). In fact he proved that
107 lies between 31°415927 and 317415926, and deduced the
value 335.

The value 22 which is that of Archimedes he spoke of as the
“inaccurate” value, and 225 as the “accurate value.” This latter value
was not obtained either by the Greeks or the Hindoos, and was only
rediscovered in Europe more than a thousand years later, by Adriaen
Anthonisz. The later Chinese Mathematicians employed for the

-most part the ‘‘inaccurate” value, but the “accurate” value was

rediscovered by Chang Yu-chin, who employed an inscribed polygon
‘with 2 sides.

The work of the Arabs

In the middle ages a knowledge of Greek and Indian mathematics
was introduced into Europe by the Arabs, largely by means of Arabic
translations of Euclid’s elements, Ptolemy’s ovvraéis, and treatises by
Appollonius and Archimedes, including the treatise of Archimedes on
the measurement of the circle.

/ The first Arabic Mathematician Muhammed ibn Mtis4 Alchwarizmi,
t the beginning of the ninth century, gave the Greek value z= 3},

‘and the Indian values 7 =./10, = $2832, which he states to be of
Indian origin. He introduced the Indian system of numerals which
was spread in Europe at the beginning of the 13th century by Leonardo
Pisano, called Fibonacci.

The time of the Renaissance
! The greatest Christian Mathematician of medieval times, Leonardo
Pisano (born at Pisa at the end of the 12th century), wrote a work
entitled Practica geometriae, ii 1220, in which he improved on the
‘results of Archimedes, using the same method of employing the in-

1440

‘and circum-scribed 96agons. His limits are pat = 31427 and
8
THE FIRST PERIOD 25

144
ia = 3°1410..., whereas 34 = 3'1428, 342 = 3°1408... were the values
v

given by Archimedes. From these limits he chose ca or r=3°'1418...
‘as the mean result.

During the period of the Renaissance no further progress in the
problem was made beyond that due to Leonardo Pisano ; some later
writers still thought that 31 was the exact value of 7. George
Purbach (14231461), who constructed a new and more exact table
of sines of angles at intervals of 10’, was acquainted with the
‘Archimedean and Indian values, which he fully recognized to be
approximations only. He expressed doubts as to whether an exact
value exists. Cardinal Nicholas of Cusa (1401—1464) obtained
m = 3'1423 which he thought to be the exact value. His approximations
and methods were criticized by Regiomontanus (Johannes Miiller, 1436—
1476), a great mathematician who was the first to shew how to calculate
the sides of a spherical triangle from the angles, and who calculated
extensive tables of sines and tangents, employing for the first time
the decimal instead of the sexagesimal notation.

The fifteenth and sixteenth centuries

In the fifteenth and sixteenth centuries great improvements in
trigonometry were introduced by Copernicus (1473—1543), Rheticus
(1514—1576), Pitiscus (1561—1613), and Johannes Kepler (1571—
1630).

These improvements are of importance in relation to our problem,
as forming a necessary part of the preparation for the analytical
developments of the second period.

In this period Leonardo da Vinci (1452—1519) and Albrecht
-Diirer (1471—1528) should be mentioned, on account of their celebrity,
as occupying themselves with our subject, without however adding
anything to the knowledge of it.

Orontius Finaeus (1494—1555) in a work De rebus mathematicis
hactenus desiratis, published after his death, gave two theorems which
were later established by Huyghens, and employed them to obtain the
imits 22, 245 for 7; he appears to have asserted that 24,5 is the exact

alue. His theorems when generalized are expressed in our notation

by the fact that 6 is approximately equal to (sin? 4 tan 6).

\
26 THE FIRST PERIOD

The development of the theory of equations which later became of
| fundamental importance in relation to our problem was due to the
| Italian Mathematicians of the 16th century, Tartaglia (1506—1559),
| Cardano (1501—1576), and Ferrari (1522—1565).
The first to obtain a more exact value of 7 than those hitherto
nown in Europe was Adriaen Anthonisz (1527—1607) who rediscovered
the Chinese value 7 = 355 = 3°1415929..., which is correct to 6 decimal
places. His son Adriaen who took the name of Metius (1571—1635),
published this value in 1625, and expla that’ his father had
obtained the approximations 233<<33% by the method of Archi-
medes, and had then taken the mean of the numerators and denomi-
nators, thus obtaining his value.
' he first explicit expression for + by an infinite sequence of
‘operations was obtained by Vieta (Frangois Viéte, 1540—1603). He
pe that, if two regular polygons are inscribed in a circle, the first
‘having half the number of sides of the second, then the area of the
first is to that of the second as the supplementary chord of a side of
the first polygon is to the diameter of the circle. Taking a square, an |
octagon, then polygons of 16, 32, ... sides, he expressed the supple- ;
mentary chord of the side of each, and thus obtained the ratio of the
area. of each polygon—to-that of -the next. He found that, if the
diameter be taken as unity, the area of the circle is

1
2 SS
Silhe LAV hed Vid

from which we obtain

 

q

I
2 Shale td PP vi
It may be observed that this expression is obtainable from the formula

sin 0

6= (6 <7)

COs 5 COs 7 COS ¢ --
afterwards obtained by Euler, by taking 0= 2 :

Applying the method of Archimedes, starting with a hexagon and

proceeding to a polygon of 2.6 sides, Vieta shewed that, if the

| diameter of the circle be 100000, the circumference is > 31415920335,

; and is <3'141597%5%3;5; he thus obtained 7 correct to 9 places of
decimals.

‘ +
THE FIRST PERIOD 27

Adrianus Romanus (Adriaen van Rooman, born in Lyons, 1561—
1615) by the help of a 15. 2™agon calculated to 15 places of decimals.
dolf van Ceulen (Cologne) (1539—1610), after whom the number
m is still called in Germany “Ludolph’s number,” is said to have
calculated m to 35 places of decimals. According to his wish the value
was engraved on his tombstone which has been lost. In his writing
Van den Cirkel (Delft, 1596) he explained how, by employing the
ethod of Archimedes, using in- and circum-scribed polygons up to
he 60.2*agon, he obtained + to 20 decimal places. Later, in his
ork De Arithmetische en Geometrische fondamenten he obtained the
imits given by

3 1415 9265358979323846264338327950
100000000000000000000000000000000>

me the same expression with 1 instead of 0 in the last place of the
umerator.

|

The work of Snellius and Huyghens

In a work Cyclometricus, published in 1621, Willebrod Snellius
(1580—1626) shewed how narrower limits can be determined, without
increasing the number of sides of the polygons, than in the method of
Archimedes. The two theorems, equivalent to the approximations

4 (2 sin 6 + tan 0) < 0 < 3/(2 cosec 6+ cot 6),

by which he attained this result were not strictly proved by him, and
were afterwards established by Huyghens; the approximate formula
= pon" had been already obtained by Nicholas of Cusa (1401—
1464). Using in- and circum-scribed hexagons the limits 3 and 3°464
are obtained by the method of Archimedes, but Snellius obtained from
the hexagons the limits 3°14022 and 3°14160, closer than those obtained
by Archimedes from the 96agon. With the 96agon he found the
limits 3°1415926272 and 3°1415928320. Finally he verified Ludolf’s
determination with a great saving of labour, obtaining 34 places with
the 2”agon, by which Ludolf had only obtained 14 places. Grunberger*
calculated 39 places by the help of the formulae of Snellius.
The extreme limit of what can be obtained on the geometrical lines
laid down by Archimedes was reached in the work of Christian
Huyghens (1629—1665). In his work} De circuli magnitudine inventa,

* Elementa Trigonometriae, Rome, 1630.
+ A study of the German Translation by Rudio will repay the trouble.
28 THE FIRST PERIOD

which is a model of geometrical reasoning, he undertakes by improved
methods to make a careful determination of the area of acircle. He
establishes sixteen theorems by geometrical processes, and shews that
by means of his theorems three times as many places of decimals
can be obtained as by the older method. The determination made by
Archimedes he can get from the triangle alone. The hexagon gives
him the limits 3°1415926538 and 371415926538.
The following are the theorems proved by Huyghens:

I. If ABC is the greatest triangle in a segment less than a

 

semi-circle, then B
AABC<4(AAEB+ABFO), E F
where AWB, BFC are the greatest
triangles in the segments AB, BC. =A Cc
Fig. 5.

Il AFEG>4A ABC,

where ABC is the greatest tri-
angle in the segment.

 

 

II segment ACB,

~ RACH **®
provided the segment is less than
the semi-circle.

This theorem had already been 4 c
given by Hero.

 

IV segment ACB,

~ BATO ~* a
Cc

Fie. 8.
THE FIRST PERIOD 29

V. If A, is the area of an inscribed regular polygon of n sides,
and S the area of the circle, then S> A» +3(Am— An).

VI. If A,’ is the area of the circumscribed regular polygon of
n sides, then S < 2A,’ +4An,.

VII. If C, denotes the perimeter of the inscribed polygon, and C
the circumference of the circle, then C> Om +3 (Cin — Ch).

D
E
VIIL. 20D +1EF>are CE,
where # is any point on the circle.
B AF iC
Fic. 9.

IX. C< 20,44),
where C,,’ is the perimeter of the circumscribed polygon of z sides.

X. If a, a,’ denote the sides of the in- and circum-scribed
polygons, then @2,?= dm’. $dn.

XI. C<the smaller of the two mean proportionals between C;,
and C,’.

S<the similar polygon whose perimeter is the larger of the two
mean proportionals.

\

XII. If LD equals thé radius
of the circle, then BG > are BF.

 

 

Fie. 10.
30 THE FIRST PERIOD

XII. If AC=radius of
the circle, then BL <are BE.

cD

Fie. 11.

XIV. If G@ is the centroid of
the segment, then
BG>GD and <3GD.

a 12.

B
J

oT

 

segment A BC a
LABC 3
BD
1
am <35- BBs 30D:

XV.

Bi
Fie, 13.
XVI. If a denote the arc 4

(<semi-circle), and s, s’ its sine and
its chord respectively, then
s tt gag gees as 8
3 8° Qs’ + 88°
This is equivalent, as Huyghens B
points out, to

 

Pon +Pn iP <C
4 Pm Pn Spon t+ Pa -
< Pant 3° pam + Bn
Fia. 14.
THE FIRST PERIOD 3l

where p,, is the perimeter of a regular inscribed polygon of » sides, and
C is the circumference of the circle.

The work of Gregory

The last Mathematician to be mentioned in connection with the
development of the method of Archimedes is James Gregory (1638—
1675), Professor in the Universities of St Andrews and Edinburgh,
whose important work in connection with the development of the new
Analysis we shall have to refer to later. Instead of employing the
perimeters of successive polygons, he calculated their areas, using

the formulae
Avt 2AnAn — 2An' Am |
Ant Am An + Aon’

where A,, A,’ denote the areas of in- and circum-scribed regular
n-agons ; he also employed the formula A», =<A,A,' which had been
obtained by Snellius. In his work Hvercitationes geometricae published
in 1668, he gave a whole series of formulae for approximations on the
lines of Archimedes. But the most interesting step which Gregory
took in connection with the problem was his attempt to prove, by
means of the Archimedean algorithm, that the quadrature of the circle
is impossible. This is contained in his work Vera circuli et hyperbolae
quadratura which is reprinted in the works of Huyghens (Opera varia
I, pp. 315—328) who gave a refutation of Gregory’s proof. Huyghens
expressed his own conviction of the impossibility of the quadrature,
and in his controversy with Wallis remarked that it was not even
decided whether the area of the circle and the square of the diameter
are commensurable or not. In default of a theory of the distinction
between algebraic and transcendental numbers, the failure of Gregory’s
proof was inevitable. Other such attempts were made by Lagny
(Paris Mém. 1727, p. 124), Saurin (Paris Mém. 1720), Newton
(Principia 1, 6, Lemma 28), and Waring (Proprietates algebraicarum
curvarum) who maintained that no algebraical oval is quadrable.
Euler also made some attempts in the same direction (Considerationes
cyclometricae, Novi Comm. Acad. Petrop. xvi, 1771); he observed
that the irrationality of 7 must first be established, but that this
would not of itself be sufficient to prove the impossibility of the
quadrature. Even as early as 1544, Michael Stifel, in his Arithmetic
integra, expressed the opinion that the construction is impossible. He
‘emphasized the distinction between a theoretical and a practical
, construction.

 
32 THE FIRST PERIOD

The work of Descartes

The great Philosopher and Mathematician René Descartes (1596—
1650), of immortal fame as the inventor of coordinate geometry,
regarded the problem from a new point of view. A given straight line
being taken as equal to the circumference of a circle he proposed to
determine the diameter by the following construction :

Take AB one quarter of the given straight line. On AB describe
the square ABCD; by a known process a point C, on AC produced,
can be so determined that the rectangle BC,=4ABCD. Again C,
can be so determined that rect. B,C,=4BC,; and so on indefinitely.
The diameter required is given by AB., where » is the limit to
which B, B,, B., ... converge. To see the reason of this, we can
shew that AB is the diameter
of the circle inscribed in ABCD, Ce
that AB, is the diameter of the C,
circle circumscribed by the regular
octagon having the same peri-
meter as the square; and gene-
rally that AB, is the diameter of
the regular 2"+,agon having the
same perimeter as the square. BB, B;
To ee this, let Fie. 15.

=AB,, x =AB;
then = “the construction,

 

 

 

 

 

 

1
Xn (Wn — Bn) = rg x,

ws : 4a,
and this is satisfied by z, = oF * cob rue thus

: Aa, 5 y
lim 2, = << = diameter of the circle.

This process was considered later by Schwab (Gergonne’s Annales de
Math. vol. v1), and is known as the process of isometers.
This method is nae to the use of the infinite series
4 1
= tan +5 ; tan = +jtan e+. Zig
which is a particular case the formula

1 _ z il
a he stan +i tans + gtang +...

due to Euler.
THE FIRST PERIOD 33

The discovery of logarithms

One great invention made early in the seventeenth century must
be specially referred to; that of logarithms by John Napier (1550—
1617). The special importance of this invention in relation to our
subject is due to the fact of that essential connection between the
numbers 7 and e which, after its discovery in the eighteenth century,
dominated the later theory of the number z. The first announcement
of the discovery was made in Napier’s Mirifici logarithmorum canonis
descriptio (Edinburgh, 1614), which contains an account of the nature
of logarithms, and a table giving natural sines and their logarithms
for every minute of the quadrant to seven or eight figures. ‘These
logarithms are not what are now called Napierian or natural logarithms
(i.e. logarithms to the base ¢), although the former are closely related
with the latter. The connection between the two is

L
L=10" log, 107-10". 2, or d=10’e ™,

where J denotes the logarithm to the base e, and ZL denotes Napier’s
logarithm. It should be observed that in Napier’s original theory of
logarithms, their connection with the number ¢ did not explicitly
appear. The logarithm was not defined as the inverse of an exponential
function ; indeed the exponential function and even the exponential
notation were not generally used by mathematicians till long afterwards.

Approximate constructions

A large number of approximate constructions for the rectification
and quadrature of the circle have been given, some of which give very
close approximations. It will suffice to give here a few examples of
. such constructions.

(1) The following construction for the approximate rectification
of the circle was given by Kochansky (Acta Hruditorum, 1685).

B J

 

D L
Fie. 16.

Let a length DZ equal to 3.radius be measured off on a tangent
to the circle ; let DAB be the diameter perpendicular to DL.

H. 3
34 THE FIRST PERIOD

Let J be on the tangent at B, and such that . BAJ=30°. Then
JL is approximately equal to the semi-circular arc BCD. Taking the
radius as unity, it can easily be proved that

JL = fe — J12 = 3'141533 ..

the correct value to four places of decimals.
(2) The value $25 = 3°141592 ... is correct to six decimal places.

 

 

Since om 3+=— e it can easily be constructed
113 7+ 8’ y
D
E
F.
AFH G C §
Fie. 17.

Let CD=1, CH=, ne $; and let FG be parallel to CD and

FAH to EG; then AH= 7m —

This construction was ee by Jakob de Gelder (Griimert’s Archiv,

vol. 7, 1849).
(3) At A make AB=(2 +24) radius on the tangent at A and let

BC =}. radius.

 

Fie. 18.
THE FIRST PERIOD 35

On the diameter through A take AD=OB, and draw DE parallel
to OC. Then

2
a =46 = 2: therefore AH =r. gl + (=) =r. Bs /146 ;
thus AH=r. 62831839 ..., so that AZ is less than the circumference
of the circle by less than two millionths of the radius.

The rectangle with sides equal to A# and half the radius r has
very approximately its area equal to that of the circle. This construc-
tion was given by Specht (Credle’s Journal, vol. 3, p. 83).

(4) Let AOB be the diameter of a given circle. Let

OD=%r, OF =8r, OH=hr.
Describe the semi-circles DG#, AHF with DE and AF as diameters ;
and let the perpendicular to AB through O cut them in G and H

respectively. The square of which the side is GH is approximately of
area equal to that of the circle.

   

H
Fie. 19.

We find that GH =r.1°77246..., and since /=1'77245 we see
that GH is greater than the side of the square whose area is equal to
that of the circle by less than two hundred thousandths of the radius.
CHAPTER III
THE SECOND PERIOD

The new Analysis

Tne foundations of the new Analysis were laid in the second half
of the seventeenth century when Newton (1642—1727) and Leibnitz
\(1646—1716) founded the Differential and Integral Calculus, the
ground having been to some extent prepared by the labours of
Huyghens, Fermat, Wallis, and others. By this great invention of
Newton and Leibnitz, and with the help of the brothers James Bernouilli
(1654—1705) and John Bernouilli (1667—1748), the ideas and
methods of Mathematicians underwent a radical transformation which
naturally had a profound effect upon our problem. The first effect
of the new Analysis was to replace the old geometrical or semi-
geometrical methods of calculating + by others in which analytical
expressions formed according to definite laws were used, and which
could be employed for the calculation of + to any assigned degree
of approximation.

The work of John Wallis

The first result of this kind was due to John Wallis (1616—1708),
Undergraduate at Emmanuel College, Fellow of Queens’ College, and
afterwards Savilian Professor of Geometry at Oxford. He was the first
to formulate the modern arithmetic theory of limits, the fundamental
importance of which, however, has only during the last half century
received its due recognition ; it is now regarded as lying at the very
foundation of Analysis. Wallis gave in his Arithmetica Infinitorum

the expression
7 224466 8 8

TSS Be 773
for + as an infinite product, and he shewed that the approximation
obtained by stopping at any fraction in the expression on the right is
in defect or in excess of the value : according as the fraction is proper

or improper. This expression was obtained by an ingenious method
THE SECOND PERIOD 37

depending upon the expression for 3 the area of a semi-circle of

1
diameter 1 as the definite integral i Va-2 dx. The expression has

the advantage over that of Vieta that the operations required by it
are all rational ones.
Lord Brouncker (1620—1684), the first President of the Royal
Society, communicated without proof to Wallis the expression
4 1 1 9 25 49

oe 2+ 049494?
a proof of which was given by Wallis in his Arithmetica Infinitorum.
It was afterwards shewn by Euler that Wallis’ formula could be
obtained from the development of the sine and cosine in infinite
products, and that Brouncker’s expression is a particular case of much
more general theorems.

The calculation of « by series

‘ The expression from which most of the practical methods of
calculating + have been obtained is the series which, as we now write
it, is given by

tan v=xv-4ta*+haP-... (-1ls@#1).

This series was discovered by Gregory (1670) and afterwards indepen-

ently by Leibnitz (1673). In Gregory’s time the series was written as

a=t—- = + = is

\ 37° Brt "
where a, t, r denote the length of an arc, the length of a tangent at
one extremity of the arc, and the radius of the circle; the definition
of the tangent as a ratio had not yet been introduced.

The particular case

agit ed

\ 4 = 3 5 eae
is known as Leibnitz’s series ; he discovered it in 1674 and published
it in 1682, with investigations relating to the representation of 7, in
his work “De vera proportione circuli ad quadratum circumscriptum
in numeris rationalibus.” The series was, however, known previously

to Newton and Gregory.

wT wr FR s ; ‘
68’ io’ 19 2 Gregory’s series, the
calculation of z up to 72 places was carried out by Abraham Sharp under
instructions from Halley (Sherwin’s Mathematical Tables, 1705, 1706).

By substituting the values
38 THE SECOND PERIOD

The more quickly convergent series
digeepetat 2 y eat
23 2.45 :
discovered by Newton, is troublesome for purposes of calculation,
owing to the form of the coefficients. By taking #=4, Newton
himself calculated 7 to 14 places of decimals.

Euler and others occupied themselves in deducing from Gregory’s
series formulae by which z could be calculated by means of rapidly
converging series.

Euler, in 1737, employed special cases of the formula

 

tan? _ tan rae + tan? ae 5
P p+q pet+pgt+l
and gave the general expression
tan?” =tan-? @—4 + tan- oe +tan7! aS Fiske
y ay+2 ab+1 cb+1

from which more such formulae could be obtained. As an example,
we have, if a, 6, c, ... are taken to be the uneven numbers, and
a
y = 1 1 1

a ai at * ice

Z tan 5 + tan 9.4 ¢ tan as

In the year 1706, Machin (1680—1752), Professor of Astronomy
in London, employed the series

Sei (ba )
4 °\5 3.5% 5.55 7.57 °°"

= (35 pet ag )
239 3.2399 5.2399 7.2397 °—'/?
which follows from the relation

ee oe ae ae
i 4 tan 5 tan 239”

to calculate z to 100 places of decimals. This is a very convenient

expression, because in the first series z ie ... can be replaced by
4 64 eo cs :
700° 1000000’ &c., and the second series is very rapidly convergent.
In 1719, de Lagny (1660—1734), of Paris, determined in two
different ways the value of 7 up to 127 decimal places. Vega (1754—
1802) calculated to 140 places, by means of the formulae
* T

ea iat al 13 _ -1 al
\ i 5 tan 77 2 tan 79 7 2 tan 3 + tan 7?
THE SECOND PERIOD 39

due to Euler, and shewed that de Lagny’s determination was correct
with the exception of the 113th place, which should be 8 instead of 7.

Clausen calculated in 1847, 248 places of decimals by the use of
Machin’s formula and the formula

az aed al
i 2 tan 3 t tan 7

In 1841, 208 places, of which 152 are correct, were calculated by
Rutherford by means of the formula
a =4 tan’ ; — tan a + tan aa"
In 1844 an expert reckoner, Zacharias Dase, employed the formula
z = tan" 3 + tan" ; +tan7? z
supplied to him by Prof. Schultz, of Vienna, to calculate + to 200
places of decimals, a feat which he performed in two months.
In 1853 Rutherford gave 440 places of decimals, and in the same
year W. Shanks gave first 530 and then 607 places (Proc. R. S., 1853).
Richter, working independently, gave in 1853 and 1855, first 333,
then 400 and finally 500 places.
Finally, W. Shanks, working with Machin’s formula, gave (1873—
74) 707 places of decimals.
Another series which has also been employed for the calculation
of zw is the series

t ae afte Fat (Ge) tER (GS) \
SN TER 8148 6.5 ee) S894

This was given in the year 1755 by Euler, who, applying it in the

formula
aw = 20 tan-'# + 8 tan“ 3,

calculated + to 20 places, in one hour as he states. The same series
was also discovered independently by Ch. Hutton (Phil. Trans.,
1776). It was later rediscovered by J. Thomson and by De Morgan.

An expression for 7 given by Euler may here be noticed; taking
the identity

x * da * gdx fe oda

tan7? =2 + +
2-2 90 4+24 9 4+a1 0 442’

 

 

he developed the integrals in series, then put z=4, a=}, obtaining
series for tan-14, tan-14, which he substituted in the formula

Bs ial al
q = 2 tan 3 + tan 7°
40 THE SECOND PERIOD

In China a work was published by Imperial order in 1718 which
contained a chapter on the quadrature of the circle where the first
19 figures in the value of w are given.

At the beginning of the eighteenth century, analytical methods
were introduced into China by Tu Té-mei (Pierre Jartoux) a French
missionary ; it is, however, not known how much of his work is
original, or whether he borrowed the formulae he gave directly from
European Mathematicians.

One of his series

2 2 2 2 2 2
raa(1+ 24 17.8 ee )

 

4.6 4.6.8.10 4.6.8.10.12.14 *”

was employed at the beginning of the nineteenth century by Chu-
Hung for the calculation of 7. By this means 25 correct figures were
obtained.

Tséng Chi-hung, who died in 1877, published values of « and 1/r
to 100 places. He is said to have obtained his value of 7 in a month,
by means of the formula

z = tan” 4 + tan“?
and Gregory’s series.

In Japan, where a considerable school of Mathematics was
developed in the eighteenth century, 7 was calculated by Takebe in
1722 to 41 places, by employment of the regular 1024agon. It was
calculated by Matsunaga in 1739 to 50 places by means of the same
series as had been employed by Chu-Hung.

The rational values r= $$22351 7 =428224599349304 correct to
12 and 30 decimal places respectively, were given by Arima in 1766.

Kurushima Yoshita (died 1757) gave for «? the approximate
values 33°, "oso: iz» “oees

Tanyem Shékei published in 1728 the series

i. 1.4. 14.9
a — Pe panes
a8 (145 +5 te + gag og te)

2 2.8 2.8.18
m= feet ey eee
a(14; Bulb tanga)
due to Takebe, and ultimately to Jartoux.
The following series published in 1739 by Matsunaga may be
mentioned :
PP 1? 1?! 2733?
r= ——— &
o(i+57 oe ‘page? )
THE SECOND PERIOD 41

The work of Euler

Developments of the most far-reaching importance in connection
with our subject were made by Leonhard Euler, one of the greatest
Analysts of all time, who was born at Basel in 1707 and died at
St Petersburg in 1783. With his vast influence on the development
of Mathematical Analysis in general it is impossible here to deal, but
some account must be given of those of his discoveries which come
into relation with our problem.

The very form of modern Trigonometry is due to Euler. He
introduced the practice of denoting each of the sides and angles of a
triangle by a single letter, and he introduced the short designation of
the trigonometrical ratios by sin a, cos u, tan a, &e. Before Euler’s
time there was great prolixity in the statement of propositions, owing
to the custom of denoting these expressions by words, or by—letters”
specially introduced in the statement. The habit of denoting the
ratio of the circumference to the diameter of a circle by the letter =,
and the base of the natural system of logarithms by e, is due to the:
influence of the works of Euler, although the notation 7 appears as:
early as 1706, when it was used by William Jones in the Synopsis
palmariorum Matheseos. In Euler’s earlier work he frequently used
p instead of 7, but by about 1740 the letter + was used not only:
by Euler but by other Mathematicians with whom he was in)
correspondence. }

A most important improvement which had a great effect not only
upon the form of Trigonometry but also on Analysis in general was the
introduction by Euler of the definition of the trigonometrical ratios in
order to replace the old sine, cosine, tangent, &c., which were the
lengths of straight lines connected with the circular arc. Thus these
trigonometrical ratios became functions of an angular magnitude, and
therefore numbers, instead of lengths of lines related by equations
with the radius of the circle. This very important improvement was
not generally introduced into our text books until the latter half of
the nineteenth century.

This mode of regarding the trigonometrical ratios as analytic
functions led Euler to one of his greatest discoveries, the connection
of these functions with the exponential function. On the basis of the
definition of e* by means of the series

1 2 #
PR og tay hte
42 THE SECOND PERIOD

he set up the relations
eg + e-® ie ee

aS sin @=—p—

 

COs @ =

which can also be written
é==cosxt+isina, e”=cosa—isina@.

The relation ¢”=—1, which Euler obtained by putting x=7, is
the fundamental relation between the two numbers = and e which was
indispensable later on in making out the true nature of the number 7.

In his very numerous memoirs, and especially in his great work,
Introductio in analysin infinitorum (1748), Euler displayed the most
wonderful skill in obtaining a rich harvest of results of great interest,
largely dependent on his theory of the exponential function. Hardly
any other work in the history of Mathematical Science gives to the
reader so strong an impression of the genius of the author as the
Introductio. Many of the results given in that work are obtained by
bold generalizations, in default of proofs which would now be regarded
as completely rigorous; but this it has in common with a large part of all
Mathematical discoveries, which are often due to a species of divining
intuition, the rigorous demonstrations and the necessary restrictions
coming later. In particular there may be mentioned the expressions
for the sine and cosine functions as infinite products, and a great
number of series and products deduced from these expressions ; also a
number of expressions relating the number e with continued fractions
which were afterwards used in connection with the investigation of the
nature of that number.

Great as the progress thus made was, regarded as preparatory to
a solution of our problem, nothing definite as to the true nature of the
number 7 was as yet established, although Mathematicians were con-
vinced that ¢ and + are not roots of algebraic equations. Euler
himself gave expression to the conviction that this is the case. Some-
what later, Legendre gave even more distinct expression to this view
in his Eléments de Géométrie (1794), where he writes : “It is probable
that the number 7 is not even contained among the algebraical
irrationalities, ¢.e. that it cannot be a root of an algebraical equation
with a finite number of terms, whose coefficients are rational. But it
seems to be very difficult to prove this strictly.”
CHAPTER IV

THE THIRD PERIOD

The irrationality of = and ¢

Tue third and final period in the history of the problem is concerned
with the investigation of the real nature of the number 7. Owing to
the close connection of this number with the number e, the base of
natural logarithms, the investigation of the nature of the two numbers
was to a large extent carried out at the same time.

The first investigation, of fundamental importance, was that of
J. H. Lambert (1728—1777), who in his “Mémoire sur quelques
propriétés remarquables des quantités transcendentes circulaires et
logarithmiques” (Hist. de [ Acad. de Berlin, 1761, printed in 1768),
proved that ¢ and = are irrational numbers. His investigations are
given also in his treatise Vorlaufige Kenntnisse fiir die, so die
Quadratur und Rektification des Zirkels suchen, published in 1766.

He obtained the two continued fractions

f-1 1 1 1 1
+1 Ba+6/a+10/a+14/er+...’
1) dy. Ta 1

tan #@ =

La — 8/a —5]/a—7a-...’
which are closely related with continued fractions obtained by Euler,
but the convergence of which Euler had not established. As the
result of an investigation of the properties of these continued fractions,
Lambert established the following theorems :

(1) If is a rational number, different from zero, é” cannot be
a rational number.

(2) If wis a rational number, different from zero, tan x cannot be
a rational number.

If w=4n, we have tanz=1, and therefore }7, or 7, cannot be
a rational number.
“44 THE THIRD PERIOD

It has frequently been stated that the first rigorous proof of
Lambert’s results is due to Legendre (1752—1833), who proved these
theorems in his Eléments de Géométrie (1794), by the same method,
and added a proof that 7? is an irrational number. ‘The essential
rigour of Lambert’s proof has however been pointed out by Pringsheim
(Minch. Akad. Ber., Ki. 28, 1898), who has supplemented the
investigation in respect of the convergence.

A proof of the irrationality of 7 and 7? due to Hermite (Crelle’s
Journal, vol. 76, 1873) is of interest, both in relation to the proof of
Lambert, and as containing the germ of the later proof of the
transcendency of e and z.

A simple proof of the irrationality of e was given by Fourier
(Stainville, Mélanges d’analyse, 1815), by means of the series

to 3f pd

which represents the number. This proof can be extended to shew
that é is also irrational. On the same lines it was proved by Liouville
(1809—1882) (Liouville’s Journal, vol. 5, 1840) that neither ¢ nor é
can be a root of a quadratic equation with rational coefficients. This
last theorem is of importance as forming the first step in the proof
that ¢ and + cannot be roots of any algebraic equation with rational
coefficients. The probability had been already recognized by Legendre
that there exist numbers which have this property.

Existence of transcendental numbers

The confirmation of this surmised existence of such numbers was
obtained by Liouville in 1840, who by an investigation of the properties
of the convergents of a continued fraction which represents a root of
an algebraical equation, and also by another method, proved that
numbers can be defined which cannot be the root of any algebraical
equation with rational coefficients.

The simpler of Liouville’s methods of proving the existence of such
numbers will be here given.

Let a be a real root of the algebraic equation

ax” + ba") + ca"-*+...=0,
with coefficients which are all positive or negative integers. We shall
assume that this equation has all its roots unequal; if it had equal
roots we might suppose it to be cleared of them in the usual manner.
THE THIRD PERIOD 45

Let the other roots be denoted by 2, 22, ... @,_1; these may be real or

complex. If - be any rational fraction, we have

ap” + bp" q pe cp? g 3.

p.a(B—n) (2-2) ...(2—2,)

If now we have a sequence of rational fractions converging to the

ff

 

:
q

value x as limit, but none of them equal to 2, and if 5 be one of these

(2-2) (2-a) ... (Ea)

approximates to the fixed number
(a@ — a) (@ — &) ... (@— By).

We may therefore suppose that for all the fractions

o(E-s) 2) (E-

is numerically less than some fixed positive number A. Also

fractions,

P

ae

ap" + bp" q+...

is an integer numerically 2 1; therefore

 

P | 1
———) > y
iz Aq”

This mnust hold for all the fractions of such a sequence, from and

after some fixed element of the sequence, for some fixed number A.
If now a number xz can be so defined such that, however far we go in
the sequence of fractions ze and however A be chosen, there exist
Pp 1

q Aq”’
be concluded that 2 cannot be a root of an equation of degree n with
integral coefficients. Moreover, if we can shew that this is the case
whatever value may have, we conclude that # cannot be a root of

any algebraic equation with rational coefficients.
Consider a number

—-@vi<

 

fractions belonging to the sequence for which | it may

 

ky
L=
rt

ks
ee ee
46 THE THIRD PERIOD

where the integers hy, kz, ..- /m, --- are all less than the integer 7, and
do not all vanish from and after a fixed value of m.

Poh he hn
q Saal os Abie + mi

then : continually approaches w as m is increased. We have

Let

P_ Kms kime “2.
q ~ plmtiy! pinta!

1 1
<7 omen + pmo To

cw

6

Re x =H
< rae since g=r™".
It is clear that, whatever values A and x may have, if m, and

therefore g, is large enough, we have SS <ag ; and thus the

relation a? > a is not satisfied for all the fractions i The
numbers z so defined are therefore transcendental. If we take 7 = 10,
we see how to define transcendental numbers that are expressed as
decimals.

This important result provided a complete justification of the
division of numbers into two classes, algebraical numbers, and trans-
cendental numbers; the latter being characterized by the property
that such a number cannot be a root of an algebraical equation of any
degree whatever, of which the coefficients are rational numbers.

A proof of this fundamentally important distinction, depending on
entirely different principles, was given by G. Cantor (Crelle’s Journal,
vol. 77, 1874) who shewed that the algebraical numbers form an
enumerable aggregate, that is to say that they are capable of being
counted by means of the integer sequence 1, 2, 3, ..., whereas the
aggregate of all real numbers is not enumerable. He shewed how
numbers can be defined which certainly do not belong to the sequence
of algebraic numbers, and are therefore transcendental.

This distinction between algebraic and transcendental numbers
being recognized, the question now arose, as regards any particular
number defined in an analytical manner, to which of the two classes it
belongs ; in particular whether ~ and ¢ are algebraic or transcendental.
The difficulty of answering such a question arises from the fact that
the recognition of the distinction between the two classes of numbers
THE THIRD PERIOD 47

does not of itself provide a readily applicable criterion by the use of
which the question may be answered in respect of a particular number.

The scope of Euclidean determinations

Before proceeding to describe the manner in which it was finally
shewn that the number z is a transcendental number, it is desirable to
explain in what way this result is connected with the problems of the
quadrature and rectification of the circle by means of Euclidean
determinations.

The development of Analytical Geometry has made it possible to
replace every geometrical problem by a corresponding analytical one
which involves only numbers and their relations. As we have already
remarked, every Euclidean problem of what is called construction
consists essentially in the determination of one or more points which
shall satisfy certain prescribed relations with regard to a certain finite
number of assigned points, the data of the problem. Such a problem
has as its analytical counterpart the determination of a number, or
a finite set of numbers, which shall satisfy certain prescribed relations
relatively to a given set of numbers. The determination of the required
numbers is always made by means of a set of algebraical equations.

The development of the theory of algebraical equations, especially
that due to Abel, Gauss, and Galois, led the Mathematicians of the last
century to scrutinize with care the limits of the possibility of solving
geometrical problems subject to prescribed limitations as to the nature
of the geometrical operations regarded as admissible. In particular,
it has been ascertained what classes of geometrical problems are
capable of solution when operations equivalent in practical geometry
to the use of certain instruments are admitted*. The investigations
have led to the discovery of cases such as that of inscribing a regular
polygon of 17 sides in a circle, in which a problem, not previously
known to be capable of solution by Euclidean means, has been shewn
to be so.

We shall here give an account of as much of the theory of this
subject as is necessary for the purpose of application to the theory of
the quadrature and rectification of the circle.

In the first place we observe that, having given two or more points
in a plane, a Cartesian set of axes can be constructed by means of a

* An interesting detailed account of investigations of this kind will be found in
Enriques’ Questions of Elementary Geometry, German Edition, 1907.
48 THE THIRD PERIOD

Euclidean construction, for example by bisecting the segment of the
line on which two of the given points are incident, and then determining
a perpendicular to that segment. We may therefore assume that
a given set of points, the data of a Euclidean problem, are specified by
means of a set of numbers, the coordinates of these points.

The determination of a required point P is, in a Euclidean
problem, made by means of a finite number of applications of the three
processes, (1) of determining a new point as the intersection of straight
lines given each by a pair of points already determined, (2) of
determining a new point as an intersection of a straight line given by
two points and a circle given by its centre and one point on the
circumference, all four points having been already determined, and
(3) of determining a new point as an intersection of two circles which
are determined by four points already determined.

In the analytical interpretation we have an original set of numbers
(h, Gy, --. Aor given, the coordinates of the r given points; (r 2 2).
At each successive stage of the geometrical process we determine two
new numbers, the coordinates of a fresh point.

When a certain stage of the process has been completed, the data
for the next step consist of numbers (a, ds, ... Gon) containing the
original data and those numbers which have been already ascertained
by the successive stages of the process already carried out.

If (1) is employed for the next step of the geometrical process, the
new point determined by that step corresponds to numbers determined
by two equations

Av+By+C=0, A’v+ By+C’=0,

where A, B, C, A’, BY, C’ are rational functions of eight of the
numbers (a1, @2, --. Gm). Therefore x, y the coordinates of the new
point determined by this step are rational functions of a), dz, ... Gon.
In order to get the data for the next step afterwards, we have only
to add to a, Ms, -.. Gon these two rational functions of eight of them.
If case (2) is employed, the next point is determined by two
equations of the form

2
(w— dy)’ + (y — aq) = (Gy — Ag)? + (Ae — Ay),
Y=mz+n,
where m, » are rational functions of four of the numbers a, a, ... don.

On elimination of y, we have a quadratic equation for 2; and thus x
is determined as a quadratic irrational function of (a, dz, ... den), of
THE THIRD PERIOD 49

the form 4+./8, where A and B are rational functions; it is clear
that y will be determined in a similar way.

If, in the new step, (3) is employed, the equations for determining
(a, y) consist of two equations of the form

(@— ap)’ + (Y — Aq)? = (Ap — Ag)? + (4 — Gu)?

on subtracting these equations, we obtain a linear equation, and thus
it is clear that this case is essentially similar to that in which (2) is
employed, so far as the’form of x, y is concerned.

Since the determination of a required point P is to be made by

a finite number of such steps, we see that the coordinates of P are
determined by means of a finite succession of operations on

(Ay, Ue, +++ Bop)s

the coordinates of the points; each of these operations consists either
of a rational operation, or of one involving the process of taking
a square root of a rational function as well as a rational operation.

We have now established the following result :

In order that a point P can be determined by the Euclidean mode it
is necessary and sufficient that its coordinates can be expressed as such
Junctions of the coordinates (ay, 2, --- Ger) of the given points of the
problem, as involve the successive performance, a finite number of times,
of operations which are either rational or involve taking a square root of
a rational function of the elements already determined.

That the condition stated in this theorem is necessary has been
proved above; that it is sufficient is seen from the fact that a single
rational operation, and the single operation of taking a square root of
a number already known, are both operations which correspond to
possible Euclidean determinations.

The condition stated in the result just obtained may be put in
another form more immediately available for application. The ex-
pression for a coordinate x of the point P may, by the ordinary
processes for the simplification of surd expressions, by getting rid of
surds from the denominators of fractions, be reduced to the form

payee Noe res Jeti en. ees

where all the numbers

 

 

G, 5, G1, C2, «0, BY, Gi’, Ge’,
are rational functions of the given numbers (a), @2, ... Q,), and the
number of successive square roots is in every term finite. Let m be
H. 4
50 THE THIRD PERIOD

the greatest number of successive square roots in any term of a ; this
may be called the rank of z. We may then write

2=atbVB+0 VB +...,
where B, B’, ... are all of rank not greater than m—1. We can form
an equation which z satisfies, and such that all its coefficients are
rational functions of a, b, b', B, B’ ...; for VB may be eliminated by
taking («—a—b! VB’ -...)?=0°B, and this is of the form

P,+VB'P/ =0,
from which we form the biquadratic
P2-B 27 =0,

in which JB’ does not occur. Proceeding in this way we obtain an
equation in a of degree some power of 2, and of which the coefficients

are rational functions of a, b, B, B’, ..., and are therefore of rank
=m-1. This equation is of the form

Lia? + La" +... = 0,
where L,, Zz, ... are at most of rank m-1. If Z,, Z,, ... involve

a radical VK, the equation is of the form

VK (b, 2% +...) + (b/a* +...) =0,
and we can as before reduce this to an equation of degree 2°"! in which
VK does not occur; by repeating the process for each radical like

VK, we may eliminate them all, and finally obtain an equation such
that the rank of every coefficient is = m-—2. By continual repetition
of this procedure we ultimately reach an equation, such that the
coefficients are all of rank zero, 7.e. rational functions of (a, Gs, ... Mer).
We now see that the following result has been established :

In order that a point P may be determinable by Euclidean procedure
it is necessary that each of its coordinates be a root of an equation of
some degree, « power of 2, of which the coefficients are rational functions
Of (dy, Me, .. Ary), the coordinates of the points given in the data of the
problem.

From our investigation it is clear that only those algebraic equations
which are obtainable by elimination from a sequence of linear and
quadratic equations correspond to possible Euclidean problems.

The quadratic equations must consist of sets, those in the first set
having coefficients which are rational functions of the given numbers,
those in the second set having coefficients of rank at most 1; in the
next set the coefficients have rank at most 2, and so on.
THE THIRD PERIOD 51

The criterion thus obtained is sufficient, whenever it can be applied,
to determine whether a proposed Euclidean problem is a possible one
or not.

In the case of the rectification of the circle, we may assume that
the data of the problem consist simply of the two points (0, 0) and
(1, 0), and that the point to be determined has the coordinates (7, 0).
This will, in accordance with the criterion obtained, be a possible
problem only if 7 is a root of an algebraic equation with rational
coefficients, of that special class which has roots expressible by means
of rational numbers and numbers obtainable by successive operations
of taking the square roots. The investigations of Abel have shewn
that this is only a special class of algebraic equations.

As we shall see, it is now known that 7, being transcendental, is
not a root of any algebraic equation at all, and therefore in accordance
with the criterion is not determinable by Euclidean construction. The
problems of duplication of the cube, and of the trisection of an angle,
although they lead to algebraic equations, are not soluble by Euclidean
constructions, because the equations to which they lead are not in
general of the class referred to in the above criterion.

The transcendence of 7

In 1873 Ch. Hermite* succeeded in proving that the number e¢ is

transcendental, that is that no equation of the form
ae” + be” + ce" +... =0

can subsist, where m, ”, 7, .-. @, 6, ¢, ... are whole numbers. In 1882,
the more general theorem was stated by Lindemann that such an
equation cannot hold, when m, n,7r, ... a,b, ¢,... are algebraic numbers,
not necessarily real; and the particular case that e+ 1=0 cannot be
satisfied by an algebraic number z, and therefore that 7 is not algebraic,
was completely proved by Lindemannt.

Lindemann’s general theorem may be stated in the following precise

form :
Lf @y, Wa, ++» By are any real or complex algebraical numbers, all
distinct, and py, Pr,» Pn are n algebraical numbers at least one of

which is different from zero, then the sum
DO + PoO™ +... + Pen
is certainly different from zero.

* «Sur la fonction exponentielle,” Comptes Rendus, vol. 77, 1873.
+ Ber. Akad. Berlin, 1882.
52 THE THIRD PERIOD

The particular case of this theorem in which
n= 2; 2, = ix, =0, Pi = P2= 1,

shews that e+ 1 cannot be zero if 2 is an algebraic number, and thus
that, since e+ 1 =0, it follows that the number 7 is transcendental.

From the general theorem there follow also the following important
results :

(1) Let n=2, p=1, po=-4, 2 =4%, #=0; then the equation
¢—a=0 cannot hold if # and a are both algebraic numbers and 2 is
different from zero. Hence the exponential ¢” is transcendent if x is an
algebraic number different from zero. In particular e is transcendent.
Further, the natural logarithm of an algebraic number different from
zero is a transcendental number. The transcendence of iz and therefore
of x is a particular case of this theorem.

(2) Let n=3, pp=—1, po=t, pp=—2a, 4 =1%, &=—i2, a,=0;
it then follows that the equation sin z= «a cannot be satisfied if a and
x are both algebraic numbers different from zero. Hence, ¢f sin x is
algebraic, x cannot be algebraic, unless x=0, and if a is algebraic,
sina cannot be algebraic, unless a=0.

It is easily seen that a similar theorem holds for the cosine and the
other trigonometrical functions.

The fact that 7 is a transcendental number, combined with what
has been established above as regards the possibility of Euclidean
constructions or determinations with given data, affords the final
answer to the question whether the quadrature or the rectification of
the circle can be carried out in the Euclidean manner.

The quadrature and the rectification of a circle whose diameter is
given are impossible, as problems to be solved by the processes of
Euclidean Geometry, in which straight lines and circles are alone
employed in the constructions.

It appears, however, that the transcendence of 7 establishes the
fact that the quadrature or the rectification of a circle whose diameter
is given are impossible by a construction in which the use only of
algebraic curves is allowed.

The special case (2) of Lindemann’s theorem throws light on the
interesting problems of the rectification of arcs of circles and of the
quadrature of sectors of circles. If we take the radius of a circle to be
unity then 2sin 4a is the length of the chord of an are of which the
length is It has been shewn that 2sin}a and w cannot both be
algebraic, unless 7=0. We have therefore the following result :
THE THIRD PERIOD 53

Tf the chord of «@ circle bears to the diameter a ratio which is
algebraic, then the corresponding are is not rectifiable by any construction
in which algebraic curves alone are employed; neither can the quadrature
of the corresponding sector of the circle be carried out by such a con-
struction.

The method employed by Hermite and Lindemann was of a com-
plicated character, involving the use of complex integration. The
method was very considerably simplified by Weierstrass*, who gave
a complete proof of Lindemann’s general theorem.

Proofs of the transcendence of ¢ and =, progressively simple in
character, were given by Stieltjest, Hilbert, Hurwitz and Gordant,
Mertens§, and Vahlen||.

All these proofs consist of a demonstration that an equation which
is linear in a number of exponential functions, such that the coefficients
are whole numbers, and the exponents algebraic numbers, is impossible.
By choosing a multiplier of the equation of such a character that its
employment reduces the given equation to the equation of the sum of
a non-vanishing integer and a number proved to lie numerically
between 0 and 1 to zero, the impossibility is established.

Simplified presentations of the proofs will be found in Weber’s
Algebra, in Enriques’ Questions of Elementary Geometry (German
Edition, 1907), in Hobson’s Plane Trigonometry (second edition, 1911),
and in Art. 1x. of the “Monographs on Modern Mathematics,” edited
by J. W. A. Young.

Proof of the transcendence of 7

The proof of the transcendence of 7 which will here be given is
\\ founded upon that of Gordan.
\ (1) Let us assume that, if possible, + is a root of an algebraical
equation with integral coefficients ; then im is also a root of such an
equation.
Assume that iz is a root of the equation
C (@— 40) (@ ~ a)... (@ — a5) =0,
where all the coefficients
1 C, C3a, C3a,a,, ..., Cayag... as
* Ber. Akad. Berlin, 1885. + Comptes Rendus, Paris Acad. 1890.

+ These proofs are to be found in the Math. Annalen, vol. 43 (1896), by Hilbert,

Hurwitz and Gordan.
§ Wiener Ber. Kl. cv. Ila (1896). || Math. Annalen, vol. 53 (1900).

4—3
54 THE THIRD PERIOD

are positive or negative integers (including zero); thus one of the
numbers a,, ... a 18 é7. I
From Euler’s equation ¢e’” + 1 =0, we see that the relation
(1 +64) (1 +6%)...(1+6%)=0

must hold, since one of the factors vanishes. If we multiply out the
factors in this equation, it clearly takes the form

A+ f+ 2+... + Pn=0,
where A is some positive integer (2 1), being made up of 1 together
with those terms, if any, which are of the form e”t%*--, where

Oa, +ag+...=0,

(2) A symmetrical function consisting of the sum of the products
taken in every possible way, of a fixed number of the numbers
Ca,, Caz, ... Ca,, is an integer. It will be proved that the symmetrical
functions of CB,, CB., ... CB, have the same property. In order to
prove this we have need of the following lemma:

A symmetrical function consisting of the sums of the products
taken p together of a+ +y+... letters

By, Woy ++ a} Yrs Yor YB Fry Ray Sys Kl,
belonging to any number of separate sets, can be expressed in terms of
symmetrical functions of the letters in the separate sets.

It will be sufficient to prove this in the case in which there are
only two sets of letters, the extension to the general case being then
obvious.

Denote by Pal (a, y) the sum of the products which we require to

express; and denote by =P (x) the sum of the products of r dimensions
of the letters 2, %, ... @ only. In case p <a, we see that

SP (@,y)=>P (a)+3P (y) 2 P(e)+3P(y) 3 P(e)+...;

p Dp 1 p-l 2 p-2
in case p>a, we see that
=P (ay)

=3P(4) 3 P(y)+3P(@) & P(y)+3P au

SP (2) 3 P(y)+ 2 P(@) 3 PW)+3P@) 3 PY)t
and the terms on the right-hand side involve in each case only
symmetrical functions of the letters of the two separate sets ; thus the
lemma is established.

, To apply this lemma, we observe that the numbers f fall into
separate sets, according to the way they are formed from the letters a.
THE THIRD PERIOD 55

The general value of 8 consists of the sum of r of the letters
%, 4, ... a; and we consider those values of 8 that correspond to
a fixed value of r to belong to one set. It is clear that a symmetrical
function of those letters @ which belong to one and the same set is
expressible as a symmetrical function of a), a, ...a,; therefore a
symmetrical function of the products Cf, where all the 6’s belong to
one and the same set, is in virtue of what has been established in (1)
an integer. Applying the above lemma to all the m numbers Cf, we
see that the symmetrical products formed by all the numbers C@ are
integral, or zero. We have supposed those of the numbers 8 which
vanish to be suppressed and the corresponding exponentials to be
absorbed in the integer A; whether this is done before or after the
_ symmetrical functions of C8 are formed makes no difference, so that
the above reasoning applies to the numbers C@ when those of them
which vanish are removed.
(3) Let p be a prime number greater than all the numbers A, 2, C
| C” B, Bo... Bx |; and let

a np+p-1 tae =— p
$ (x)= G1 Onp+?-1 {ae — By) (@ — Br) -. (@— Bn) }?.
We observe that (2) is of the form

Cx ad he .
(CAP [Gay (Oa 4 (Ca)P~ HNO
where G1, go, --» Gn are integers. The function ¢(«) may be expressed
in the form
(a) = Cp-.0? 1 + Cp? + v0e + Cupp avr? *P,

where Cy1(p—1)!, ¢)p!, --. are integral.

We see that ¢?7?(0)=(—1)”C?"y,?, which is an integer not
divisible by p.

Also $” (0) is the value when # = 0 of

-1 a nr a1 b
pC? de [(Ca)" - q (Cx) + ...]
and is clearly an integer divisible by p. We see also that

G00), *(0), .. #PP>(0)

are all multiples of p.
Further if m<n, (Bm), $ (Bin), +++ $7 (Bm) all vanish, and

“3 gi” (Bu) "S gen) (Bil, a: "Eger (Bn)
56 THE THIRD PERIOD

‘

are all integers divisible by p. This follows from the fact that.

a CCB)” 3 is expressible in terms of those symmetrical functions which

consist of the sums of products of the numbers C,, C@,, ...; and
these expressions have integral values.
(4) Let X, denote the integer
(p-l)!q-1+plep+...+(mptp—-1)!,
which may be written in the form
'?-) (0) + Hl?) (0) +... + pine +e-1) (0).
In virtue of what has been established in (3) as to the values of
gp? (0), oH!) (0), ...
we see that K,A is not a multiple of p.
We examine the form to which the equation
A+ is Prt... 4 Bn = 0
is reduced by multiplying all the terms by X;,.

We have
r=npt+p-1
K,fm= 3% G {8n" +7Bn t+ 97 (7-1) Br? +... +7!
r=p-1
Bal Bm Eats +}
r+1 *O& +1) (7 +2)

=¢ (Bm) + ¢ (Bm) Feet PrPrPrt (Bm)

r=np+p-1 B B 2
Sep ee
- rap—1 OB? (Bs. *@F)(r+2)* -.
The modulus of the sum of the series
fn 1 Brvat
(r+1) (r+ 2)*

 

does not exceed

|Bm| [Bm |?
r+1 "(Hy (r+2)*

and this is less than ¢! 8m! ; hence we have

Br
coat (ey + + GSD a tf HF leBat lel el,

where 6, is some number whose modulus is between 0 and 1.
The oe of

_— mptp-1
5 "6, |G» Br” | ¢ | Bm | is less than e| Bml ” 2 [¢rBm | ,
THE THIRD PERIOD 57

or than
orl |B Fo" %9*-1 (|B | + | B1) (Bal + Bal) (Bul + [Ba
(p-1)! ee ee
or than 2
J BER comes (B+ |B )(B+|Bal) ~B+|BaDH

where £ denotes the greatest of the numbers | 8;|, | 8:|, ... | Bn|-
It thus appears that the modulus of

rena Bm Brn”
ae CrBn! leet (r+1) (r+ 2)* >}

is less than a number of the form PQ?/(p-1)!, where P and @ are
independent of p and of m.
We have now

K,(A+ 3 &)=K,A + "S {hl (By) +... + G"P*P-1(B,,)} + L,
m=1 m=

where K,A is not a multiple of p, the second term is an integer
divisible by p, and Z is less than nPQ?/(p—1)!. The prime p may
‘be chosen so large that nPQ?/(p—1)! is numerically less than unity.

i m=n
‘Since K,(A+ & Pm) is expressed as the sum of an integer which
m=1

i
i

‘does not vanish and of a number numerically less than unity, it is
, Impossible that it can vanish. Having now shewn that no such
» equation as

A +i + 24... + m= 0

can subsist, we see that 72 cannot be a root of an algebraic equation
with integral coefficients, and thus that 7 is transcendental.

It has thus been proved that 7 is a transcendental number, and
hence, taking into account the theorem proved on page 50, the im-
possibility of “squaring the circle” has been effectively established.

 

CAMBRIDGE: PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS
BY THE SAME AUTHOR

The Theory of Functions of a Real Variable and the Theory of
Fourier’s Series. By E. W. Hozson, Sc.D., LL.D., F.R.S., Sadleirian Professor
of Pure Mathematics, and Fellow of Christ’s College, Cambridge. Royal 8vo.
21s net.

‘*We hasten to congratulate Dr Hobson on the completion of what is, without
a doubt, a magnificent piece of work. It would be a fine piece of work even if it
were a mere compilation ; for the subject is one of which there was no systematic
account in English, and which no previous English writer had ever really mastered.
But the book is far from being a compilation, for Dr Hobson has made the subject
his own, and writes with the air of mastery that only original work can give; and
even in French, German, or Italian, there is no book which covers anything like the
same ground.’’—Nature

A Treatise on Plane Trigonometry. By E. W. Hogson, Sc.D.,
F.R.S. Third edition. Demy 8vo. 12s.

“There was a time when Prof. Hobson’s volume was the only British text-book
in which the higher portions of the subject were adequately treated. Even now it
has no serious rival, for although later works on higher analysis contain many
chapters overlapping the ground covered by the latter half of this volume, we believe
that ‘Hobson’ will continue to be the favourite text-book for those who wish to
carry the subject beyond a merely elementary course.’’—Mathematical Gazette

An Elementary Treatise on Plane Trigonometry. By E. W.
Hosson, S8c.D., F.R.S., and C. M. Jessop, M.A. Extra feap. 8vo. 4s 6d.

“This is a really excellent manual for the student with mathematical tendency.
The style of treatment is short, sharp and clear, and a very large amount of ground
is covered....A useful collection of exercises is given with each chapter, and at the
end there is a large number of more difficult miscellaneous questions, which will
thoroughly test the powers of the student in the manipulation of trigonometrical
formulae and equations.’’—Glasgow Herald

Mathematics, from the points of view of the Mathematician
and Physicist. An Address delivered to the Mathematical and Physical Society
of University College, London. By E. W. Hosson, Sc.D., F.R.S. Crown 8vo.
Paper covers. Is.

Cambridge University Press
Fetter Lane, London, B.C. C. F. Clay, Manager
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