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PREFACE

IN 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 über 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 transcendence of $\pi$" 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.

Christ's College, Cambridge.
October, 1913.

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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 be a 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.

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 development 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 Mathematics; 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 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 put 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 conditions 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 accompanied 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 Paradoxes. 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. τετραγωνίζειν, 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 in the Minutes of the Academy^[1], 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 conviction 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 comparative obviousness explains in part at least its popularity; unlike many Mathematical problems, its nature can in some sense be understood 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 attraction 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 $\pi$. 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, i.e. of the determination of a 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 compelled, 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 construction 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 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 contemplating, 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 restriction 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 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 instruments, 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 determination 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 $A(B)$, 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 Geometry 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 constructing 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. (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 (3). 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 proposition 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, however, 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^[2], 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 determining 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 determined, or as an intersection of two circles already determined. A 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 compasses. 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 and 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 problem 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 insoluble 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 satisfy prescribed conditions. Let $P$ be one such point; then it may be possible to determine in Euclidean fashion each point of a set $P_{1},P_{2},\ldots P_{n},\ldots$ of points which converge to $P$ as limiting point, and yet the point $P$ may be incapable of determination by Euclidean procedure. This 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 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 Mathematical 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, embodying 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 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 $\pi$ by calculation of the sides or areas of regular polygons in- and circum-scribed to the circle. The theoretical groundwork 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 $\pi$ 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 seventeenth 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 $\pi$ 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 $\pi$, who, in their consciousness of the vastly enhanced means of calculation placed in their hands by the new Analysis, proceeded to apply the formulae to obtain numerical approximations to $\pi$ 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 $\pi$ 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 to the solution of the problem, was made at this time; that of the relation between the two numbers $\pi$ and $e$, 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 $\pi$ 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 transcendental 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 $\pi$ 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 $\pi$ 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 $\pi$, 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 difficulties which in each age restricted the progress which could be made within limits which could not be surpassed by the means then available; 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 colossa l patience.

CHAPTER II

The First Period

Earliest traces of the problem

The earliest traces of a determination of $\pi$ are to be found in the Papyrus Rhind which is preserved in the British Museum and was translated and explained^[3] by Eisenlohr. 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 $\textstyle A=({\frac {8}{9}})^{2}d^{2}$, or comparing with the formula $\textstyle A={\frac {1}{4}}\pi d^{2}$ this would give

$\textstyle \pi ={\frac {256}{81}}=3{\cdot }1604\ldots .$

No account is given of the means by which this, the earliest determination of $\pi$, was obtained; but it was probably found empirically.

The approximation $\pi =3$, 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 $6\times 60\equiv 360$ equal parts.

This assumption ($\pi =3$) 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."

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 exilio ), 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 B.C. Hippias of Elis invented a curve known as the τετραγωνίζουσα 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 $\pi$.

This curve may be described as follows, using modern notation.

Let a point $Q$ starting at $A$ describe the circular quadrant $AB$ with uniform velocity, and let a point $R$

starting at $O$ describe the radius $OB$ with uniform velocity, and so that if $Q$ and $R$ start simultaneously they will reach the point $B$ simultaneously. Let the point $P$ be the intersection of $OQ$ with a line perpendicular to $OB$ drawn from $R$. The locus of $P$ is the quadratrix. Letting $\angle QOA=\theta$, and $OR=y$, the ratio $y/\theta$ is constant, and equal to $2a/\pi$, where $a$ denotes the radius of the circle. We have

$x=y\cot {\theta }$, or $x=y\cot {\frac {\pi y}{2a}}$,

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

$x=\lim _{y=0}\left(y\cot {\frac {\pi y}{2a}}\right)=2a/\pi$.

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

The problem was considered by some of the Sophists, who made futile attempts to connect it with the discovery of "cyclical square numbers," i.e. such square numbers as end with the same cipher as the number itself, as for example 25 = 5², 36 = 6²; 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, 16agon, &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 $AEC$, $BDC/math>isequaltothatofthetriangle<math>ACB$. 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.

If $AC=CD=DB=$ radius $OA$ (see Fig. 3), the semi-circle $ACE$ is 1/4 of the semi-circle $ACDB$. We have now

$semicircleAB-3semicircleAC=ACDB-3.{\text{ meniscus }}ACE$,

and each of these expressions is $\textstyle {\frac {1}{4}}semicircleAB$ or half the circle on $AB$ as diameter. If then the meniscus $AEC$ were quadrable Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/30 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/31 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, leaves, 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 definitions 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 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 independent of the particular sequence employed, before we are entitled to regard the circle as rectifiable.

In his work κύκλου μέτρησις, 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 $\textstyle 3{\frac {1}{7}}$ but greater than $\textstyle 3{\frac {10}{71}}$.

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

$\textstyle 3{\frac {1}{7}}=3{\cdot }14285\ldots ,3{\frac {10}{71}}=3{\cdot }14084\ldots ,3{\frac {1}{7}}=3{\cdot }14159\ldots .$

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 xii. 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

${\frac {OA}{AC}}={\sqrt {3}}>{\frac {265}{153}}$.

Bisecting the angle $AOC$, we obtain $AD$ half the side of a regular circumscribed 12agon. It is then shewn that ${\frac {OD}{DA}}>{\frac {591{\frac {1}{8}}}{153}}$. If $OE$ is the bisector of the angle $DOA$, $AE$ is half the side of a circumscribed 24agon, and it is then shewn that ${\frac {OE}{EA}}>{\frac {1172{\frac {1}{8}}}{153}}$ Next, bisecting $EOA$, we obtain $AF$ the half side of a 48agon, and it Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/34 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/35 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/36 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/37 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/38 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/39 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/40 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/41 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/42 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/43 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/44 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/45

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 $C_{1}$ on $AC$ produced, can be so determined that the rectangle $\textstyle BC_{1}={\frac {1}{4}}ABCD$. Again $C_{2}$ can be so determined that rect. $\textstyle B_{1}C_{2}={\frac {1}{4}}BC_{1}$; and so on indefinitely. The diameter required is given by $AB_{\omega }$, where $B_{\omega }$ is the limit to which $B,B_{1},B_{2},\ldots$ converge. To see the reason of this, we can shew that $AB$ is the diameter of the circle inscribed in $ABCD$, that $AB_{1}$ is the diameter of the circle circumscribed by the regular octagon having the same perimeter as the square; and generally that $AB_{n}$ is the diameter of the regular $2^{n+2}$-agon having the same perimeter as the square. To verify this, let

$x_{n}=AB_{n},x_{0}=AB;$

then by the construction,

$x_{n}(x_{n}-x_{n-1})={\frac {1}{4^{n}}}x_{0}^{2},$

and this is satisfied by $x_{n}={\frac {4x_{0}}{2^{n}}}\cot {\frac {\pi }{2^{n}}}$; thus

$\lim x_{n}={\frac {4x_{0}}{\pi }}=$ diameter of the circle.

This process was considered later by Schwab (Gergonne's Annales de Math. vol. vi), and is known as the process of isometers.

This method is equivalent to the use of the infinite series

${\frac {4}{\pi }}=\tan {\frac {\pi }{4}}+{\frac {1}{2}}\tan {\frac {\pi }{8}}+{\frac {1}{4}}\tan {\frac {\pi }{16}}+\ldots ,$

which is a particular case of the formula

${\frac {1}{x}}-\cot {x}={\frac {1}{2}}\tan {\frac {x}{2}}+{\frac {1}{4}}\tan {\frac {x}{4}}+{\frac {1}{8}}\tan {\frac {x}{8}}+\ldots ,$

due to Euler.

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 $\pi$ and $e$ which, after its discovery in the eighteenth century, dominated the later theory of the number $\pi$. 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 e), although the former are closely related with the latter. The connection between the two is

$L=10^{7}\log _{e}{10^{7}}-10^{7}.l$, or $e^{l}=10^{7}e^{-{\frac {L}{10^{7}}}}$,

where $l$ denotes the logarithm to the base $e$, and $L$ denotes Napier's logarithm. It should be observed that in Napier's original theory of logarithms, their connection with the number $e$ 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 Eruditorum, 1685).

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

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

$JL={\sqrt {{\frac {40}{3}}-{\sqrt {12}}}}=3{\cdot }141533\ldots$,

the correct value to four places of decimals.

(2) The value $\textstyle {\frac {355}{113}}=3{\cdot }141592\ldots$ is correct to six decimal places.

Since ${\frac {355}{113}}=3+{\frac {4^{2}}{7^{2}+8^{2}}}$, it can easily be constructed.

Let $CD=1$, $\textstyle CE={\frac {7}{8}}$, $\textstyle AF={\frac {1}{2}}$; and let $FG$ be parallel to $CD$ and $FH$ to $EG$; then $AH={\frac {4^{2}}{7^{2}+8^{2}}}$.

This construction was given by Jakob de Gelder (Grünert's Archiv, vol. 7, 1849).

(3) At $A$ make $\textstyle AB=(2+{\frac {1}{5}})$ radius on the tangent at $A$ and let

$\textstyle BC={\frac {1}{5}}.{\text{radius}}$.

On the diameter through $A$ take $AD=OB$, and draw $DE$ parallel

to $OC$. Then ${\frac {AE}{AD}}={\frac {AC}{AO}}={\frac {13}{5}}$; therefore $AE=r.{\frac {13}{5}}{\sqrt {1+\left({\frac {11}{5}}\right)^{2}}}=r.{\frac {13}{25}}{\sqrt {146}}$; thus $AE=r.6{\cdot }2831839\ldots$, so that $AE$ is less than the circumference of the circle by less than two millionths of the radius.

The rectangle with sides equal to $AE$ and half the radius $r$ has very approximately its area equal to that of the circle. This construction was given by Specht (Crelle's Journal, vol. 3, p. 83).

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

$\textstyle OD={\frac {3}{5}}r,\qquad OF={\frac {3}{2}}r,\qquad OE={\frac {1}{2}}r$.

Describe the semi-circles $DGE$, $AHF$ 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.

Fig. 19.

We find that $GH=r.1{\cdot }77246\ldots$, and since ${\sqrt {\pi }}=1{\cdot }77245$ 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

The 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 semigeometrical methods of calculating $\pi$ by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of $\pi$ to any assigned degree of approximation.

The work of John Wallis

The first result of this kind was due to John Wallis (1616—1703), 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

${\frac {\pi }{2}}={\frac {2}{1}}.{\frac {2}{3}}.{\frac {4}{3}}.{\frac {4}{5}}.{\frac {6}{5}}.{\frac {6}{7}}.{\frac {8}{7}}.{\frac {8}{9}}\ldots$

for $\pi$ 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 ${\frac {\pi }{2}}$ according as the fraction is proper or improper. This expression was obtained by an ingenious method Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/51 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/52 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/53 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/54 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/55 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/56

CHAPTER IV

The Third Period

The irrationality of π and e

The third and final period in the history of the problem is concerned with the investigation of the real nature of the number $\pi$. 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 l'Acad. de Berlin, 1761, printed in 1768), proved that $e$ and $\pi$ are irrational numbers. His investigations are given also in his treatise Vorläufige Kenntnisse für die, so die Quadratur und Rektification des Zirkels suchen, published in 1766.

He obtained the two continued fractions

${\frac {e^{x}-1}{e^{x}+1}}={\frac {1}{2/x+}}{\frac {1}{6/x+}}{\frac {1}{10/x+}}{\frac {1}{14/x+\ldots }}$,

$\tan {x}={\frac {1}{1/x-}}{\frac {1}{3/x-}}{\frac {1}{5/x-}}{\frac {1}{7/x-\ldots }}$,

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 $x$ is a rational number, different from zero, $e^{x}$ cannot be a rational number.

(2) If $x$ is a rational number, different from zero, $\tan {x}$ cannot be a rational number.

If $\textstyle x={\frac {1}{4}}\pi$, we have $\tan {x}=1$, and therefore $\textstyle {\frac {1}{4}}\pi$, or $\pi$, cannot be a rational number.

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 Éléments de Géométrie (1794), by the same method, and added a proof that $\pi ^{2}$ is an irrational number. The essential rigour of Lambert's proof has however been pointed out by Pringsheim (Münch. Akad. Ber., Kl. 28, 1898), who has supplemented the investigation in respect of the convergence.

A proof of the irrationality of $\pi$ and $\pi ^{2}$ 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 $\pi$.

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

$1+{\frac {1}{1!}}+{\frac {1}{2!}}+{\frac {1}{3!}}+\ldots$

which represents the number. This proof can be extended to shew that $e^{2}$ is also irrational. On the same lines it was proved by Liouville (1809—1882) (Liouville's Journal, vol. 5, 1840) that neither $e^{2}$ nor $e^{2}$ 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 $e$ and $\pi$ 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 $x$ be a real root of the algebraic equation

$ax^{n}+bx^{n-1}+cx^{n-2}+\ldots =0$,

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. Let the other roots be denoted by $x_{1},x_{2},\ldots x_{n-1}$; these may be real or complex. If ${\frac {p}{q}}$ be any rational fraction, we have

${\frac {p}{q}}-x={\frac {ap^{n}+bp^{n-1}q+cp^{n-2}q^{2}+\ldots }{q^{n}.a\left({\frac {p}{q}}-x_{1}\right)\left({\frac {p}{q}}-x_{2}\right)\ldots \left({\frac {p}{q}}-x_{n}\right)}}$.

If now we have a sequence of rational fractions converging to the value $x$ as limit, but none of them equal to $x$, and if ${\frac {p}{q}}$ be one of these fractions,

$\left({\frac {p}{q}}-x_{1}\right)\left({\frac {p}{q}}-x_{2}\right)\ldots \left({\frac {p}{q}}-x_{n}\right)$

approximates to the fixed number

$(x-x_{1})(x-x_{2})\ldots (x-x_{n})$.

We may therefore suppose that for all the fractions ${\frac {p}{q}}$,

$a\left({\frac {p}{q}}-x_{1}\right)\left({\frac {p}{q}}-x_{2}\right)\ldots \left({\frac {p}{q}}-x_{n}\right)$

is numerically less than some fixed positive number $A$. Also

$ap^{n}+bp^{n-1}q+\ldots$

is an integer numerically ≥ 1; therefore

$\left|{\frac {p}{q}}-x\right|>{\frac {1}{Aq^{n}}}$.

This must hold for all the fractions ${\frac {p}{q}}$ of such a sequence, from and after some fixed element of the sequence, for some fixed number $A$. If now a number $x$ can be so defined such that, however far we go in the sequence of fractions ${\frac {p}{q}}$, and however $A$ be chosen, there exist fractions belonging to the sequence for which $\left|{\frac {p}{q}}-x\right|<{\frac {1}{Aq^{n}}}$, it may be concluded that $x$ 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 $n$ may have, we conclude that $x$ cannot be a root of any algebraic equation with rational coefficients.

Consider a number

$x={\frac {k_{1}}{r^{1!}}}+{\frac {k_{2}}{r^{2!}}}+\ldots +{\frac {k_{m}}{r^{m!}}}+\ldots$,

where the integers $k_{1},k_{2},\ldots k_{m},\ldots$ are all less than the integer r, and do not all vanish from and after a fixed value of m.

Let

${\frac {p}{q}}={\frac {k_{1}}{r^{1!}}}+{\frac {k_{2}}{r^{2!}}}+\ldots +{\frac {k_{m}}{r^{m!}}}$,

then ${\frac {p}{q}}$ continually approaches # as m is increased. We have

${\begin{aligned}x-{\frac {p}{q}}&={\frac {k_{m+1}}{r^{(m+1)!}}}+{\frac {k_{m+2}}{r^{(m+2)!}}}+\ldots \\&<r\left({\frac {1}{r^{(m+1)!}}}+{\frac {1}{r^{(m+2)!}}}+\ldots \right)\\&<{\frac {2r}{q^{m+1}}},{\text{ since }}q=r^{m!}.\end{aligned}}$

It is clear that, whatever values $A$ and $n$ may have, if $m$, and therefore $q$, is large enough, we have ${\frac {2r}{q^{m+1}}}<{\frac {1}{Aq^{n}}}$; and thus the relation $\left|{\frac {p}{q}}-x\right|>{\frac {1}{Aq^{n}}}$ is not satisfied for all the fractions ${\frac {p}{q}}$. The numbers $x$ so defined are therefore transcendental. If we take $r=10$, 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 transcendental 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 $\pi$ and $e$ 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 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/61 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/62 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/63 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/64 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/65 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/66 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/67 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/68 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/69 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/70 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/71 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/72 Page:Squaring the circle a history of the problem (IA squaringcirclehi00hobsuoft).djvu/73

1. Histoire de l'Académie royale, année 1775, p. 61.
2. 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.
3. Eisenlohr, Ein mathematisches Handbuch der alten Ägypter (Leipzig, 1877).