CHAPTER XII.

THREE CLASSICAL GEOMETBICAL PROBLEMS.
Among the more interesting geometrical problems of anti- quity are three questions which attracted the special attention of the early Greek mathematicians. Our knowledge of geometry is derived from Greek sources, and thus these questions have attained a classical position in the history of the subject. The three questions to which I refer are (i) the duplication of a cube, that is, the determination of the side of a cube whose volume is double that of a given cube ; (ii) the triscction of an angle ; and (iii) the squaring of a circle, that is, the deter- mination of a square whose area is equal to that of a given circle — each problem to be solved by a geometrical construction involving the use of straight lines and circles only, that is, by Euclidean geometry.
This limitation to the use of straight lines and circles implies that the only instruments available in Euclidean geometry are compasses and rulers. But the compasses must be capable of opening as wide as is desired, and the ruler must be of un- limited length. Further the ruler must not be graduated, for if there were two fixed marks on it we can obtain constructions equivalent to those obtained by the use of the conic sections.
With the Euclidean restriction all three problems are in- soluble*. To duplicate a cube the length of whose side is a, '
* See F. C. Klein, Vortrage iiher ausgeiocihlte Fragen der Elementargeometrie, Leipzig, 1895. It is said that the earliest rigorous proof that the problems ■ were insoluble by Euclidean geometry was given by P. L. Wantzell in 1837. J
CH. XIl] THREE GEOMETRICAL PROBLEMS 285
we have to find a line of length x, such that a? = 2a^ Again, to trisect a given angle, we may proceed to find the sine of the angle, say a, then, if x is the sine of an angle equal to one-third of the given angle, we have 4^^ =Hx — a. Thus the first and second problems, when considered analytically, require the solu- tion of a cubic equation ; and since a construction by means of circles (whose equations are of the form a^ -h y"^ -\- ax -\- hi/ + c = 0) and straight lines (whose equations are of the form or^+^y -f 7= 0) cannot be equivalent to the solution of a cubic equation, it is inferred that the problems are insoluble if in our constructions we are restricted to the use of circles and straight lines. If the use of the conic sections is permitted, both of these questions can be solved in many ways. The third problem is different in character, but under the same restrictions it also is insoluble.
I propose to give some of the constructions which have been proposed for solving the first two of these problems. To save space I shall not draw the necessary diagrams, and in most cases I shall not add the proofs : the latter present but little difficulty. I shall conclude with some historical notes on approximate solutions of the quadrature of the circle.
TJie Duplication of the Cube*.
The problem of the duplication of the cube was known in ancient times as the Delian problem, in consequence of a legend that the Delians had consulted Plato on the subject. In one form of the story, which is related by Philoponusf, it is asserted that the Athenians in 430 B.C., when sufferinsf from the plague of eruptive typhoid fever, consulted the oracle at Delos as to how they could stop it. Apollo replied that they must double the size of his altar which was in the form of a cube. To the unlearned suppliants nothing seemed more easy, and a new altar was constructed either having each of its edges
* See Hiatorla Prohlematis cle Cubi BupUcatione by N. T. Reimer, Gottingen, 1798; and Historia Prohlematis Cubi DupUcandi by C. H. Biering, Copenhagen, 1844 : also Das Delische Problem, by A. Sturm, Linz, 1895-7. Some notes on the subject are given in my History of Mathematics.
t Philoponus ad Aristotelis Analytica Posteriora, bk. i, chap. vii.
286 THREE GEOMETRICAL PROBLEMS [CH. XII
double that of the old one (from which it followed that the volume was increased eight-fold) or by placing a similar cube altar next to the old one. Whereupon, according to the legend, the indignant god made the pestilence worse than before, and informed a fresh deputation that it was useless to trifle with him, as his new altar must be a cube and have a volume exactly double that of his old one. Suspecting a mystery the Athenians applied to Plato, who referred them to the geometricians. The insertion of Plato's name is an obvious anachronism. Eratos- thenes* relates a somewhat similar story, but with Minos as the propounder of the problem.
In an Arab work, the Greek legend was distorted into the following extraordinarily impossible piece of history, which I cite as a curiosity of its kind. "Now in the days of Plato," says the writer, " a plague broke out among the children of Israel. Then came a voice from heaven to one of their prophets, saying, ' Let the size of the cubic altar be doubled, and the plague will cease ' ; so the people made another altar like unto the former, and laid the same by its side. Nevertheless the pestilence continued to increase. And again the voice spake unto the prophet, saying, ' They have made a second altar like unto the former, and laid it by its side, but that does not pro- duce the duplication of the cube.' Then applied they to Plato, the Grecian sage, who spake to them, saying, * Ye have been neglectful of the science of geometry, and therefore hath God chastised you, since geometry is the most sublime of all the sciences.' Now, the duplication of a cube depends on a rare problem in geometry, namely...." And then follows the solu- tion of Apollonius, which is given Jater.
If a is the length of the side of the given cube and x that of the required cube, we have a:^ = 2a^, that is, a; : a= ^2 : 1. It is probable that the Greeks were aware that the latter ratio is incommensurable, in other words, that no two integers can be found whose ratio is the same as that of \/2 : 1, but it did not therefore follow that they could not find the ratio by
* Archimedis Oj^era cum Eutocii Commentariis, ed. Torelli, Oxford, 17S2, p. 144; ed. Heiberg, Leipzig, 1880-1, vol. iii, pp. 104 — 107.
CH. Xli] THREE GEOMETRICAL PROBLEMS 287
geometry: in fiict, the side and diagonal of a square are instances of lines whose numerical measures are incommen- surable.
I proceed now to give some of the geometrical constructions which have been proposed for the duplication of the cube*. With one exception, I confine myself to those which can be effected by the aid of the conic sections.
Hippocrates f (circ. 420 B.C.) was perhaps the earliest mathe- matician who made any progress towards solving the problem. He did not: give a geometrical construction, but he reduced the question to that of finding two means between one straight line (a), and another twice as long (2a). If these means are a? and y, we have a : x — x : y = y : 2a, fi:om which it follows that cc^ = 2a^. It is in this form that the problem is always presented now. Formerly any process of solution by finding these means was called a mesolabum.
One of the first solutions of the problem was that given by Archytasj in or about the year 400 B.C. His construction is equivalent to the following. On the diameter OA of the base of a right circular cylinder describe a semicircle whose plane is perpendicular to the base of the cylinder. Let the plane con- taining this semicircle rotate round the generator through 0, then the surface traced out by the semicircle will cut the cylinder in a tortuous curve. This curve will itself be cut by a right cone, whose axis is OA and semi-vertical angle is (say) 60°, in a point P, such that the projection of OP on the base of the cylinder will be to the radius of the cylinder in the ratio of the side of the required cube to that of the given cube. Of course the proof given by Archytas is geometrical ; and it is interesting to note that in it he shows himself familiar with the results of the propositions Euc. ill, 18, ill, 35, and xi, 19. To
* On the application to this problem of the traditional Greek methods of analysis by Hero and Philo (leading to the solution by the use of Apollonius's circle), by Nicomedcs (leading to the solution by the use of the conchoid), and by Pappus (leading to the solution by the use of the cissoid), see Geometrical Analysis by J. Leslie, Edinburgh, second edition, 1811, pp. 247 — 250, 453.
t Proclus, ed. Friedlein, pp. 212—213.
X Archimcdis Opera, ed. Torelli, p. 14S ; ed. Heiberg, vol. iii, pp. 93 — 103.
288 THREE GEOMETRICAL PROBLEMS [CH. XII
show analytically that the construction is correct, take OA as the axis of x, and the generator of the cylinder drawn through 0 as axis of z, then with the usual notation, in polar co-ordinates, if a is the radius of the cylinder, we have for the equation of the surface described by the semicircle r=2a sin 0 ; for that of the cylinder r sin 0 =2a cos cf) ; and for that of the cone sin 6 cos (t>=2' These three surfaces cut in a point such that sin^ 0 = ^, and therefore (r sin Oy = 2a^. Hence the volume of the cube whose side is r sin 6 is twice that of the cube w^hose side is a.
The construction attributed to Plato* (circ. 360 B.C.) de- pends on the theorem that, if CAB and DAB are two right- angled triangles, having one side, AB, common, their other sides, AD and BC, parallel, and their hypothenuses, AC and BD, at right angles, then if these hypothenuses cut in P, we have PC :PB = FB : PA = PA : PD. Hence, if such a figure can be constructed having PD = 2P(7, the problem will be solved. It is easy to make an instrument by which the figure can be drawn.
The next writer whose name is connected with the problem is Menaechmusf, who in or about 340 B.C. gave two solutions of it.
In the first of these he pointed out that two parabolas having a common vertex, axes at right angles, and such that the latus rectum of the one is double that of the other, will intersect in another point whose abscissa (or ordinate) will give a solution. If we use analysis this is obvious; for, if the equations of the parabolas are y"^ — 2ax and x^ = ay, they intersect in a point whose abscissa is given hj c(^ = 2aK It is probable that this method was suggested by the form in which Hippocrates had cast the problem : namely, to find x and y so that a : X = X : y = y : 2a, whence we have x^ = ay and y^ = 2ax.
The second solution given by Menaechmus was as follows. Describe a parabola of latus rectum I. Next describe a rect- angular hyperbola, the length of whose real axis is 4^, and
* ArcJdmedis Opera, ed. Torelli, p. 135 ; ed. Heiberg, vol. iii, pp. 66 — 71. t IMd., ed. Torelli, pp. 141—143 ; ed. Heiberg, vol. iii, pp. 92—99.
CH. Xll] THREE GEOMETRICAL PROBLEMS 289
having for its asymptotes the tangent at the vertex of the parabola and the axis of the parabola. Then the ordinate and the abscissa of the point of intersection of these curves are the mean proportionals between I and 21. This is at once obvious by analysis. The curves are x^ = ly and xy — 2l\ These cut in a point determined by x^ = 21^ and y^ = 4^^ Hence
I : X = X : y = y : 21.
The solution of Apollonius*, which was given about 220 B.C., was as follows. The problem is to find two mean proportionals between two given lines. Construct a rectangle OADB, of which the adjacent sides OA and OB are respectively equal to the two given lines. Bisect ^jB in G. With G as centre describe a circle cutting OA produced in a and cutting OB produced in h, so that aDh shall be a straight line. If this circle can be so described, it will follow that OA : Bb = Bb : Aa — Aa : OB, that is, Bb and Aa are the two mean proportionals between OA and OB. It is impossible to construct the circle by Euclidean geometry, but Apollonius gave a mechanical way of describing it.
The only other construction of antiquity to which I will refer is that given by Diodes and Sporusf. It is as follows. Take two sides of a rectangle OA, OB, equal to the two lines between which the means are sought. Suppose OA to be the greater. With centre 0 and radius OA describe a circle. Let OB produced cut the circumference in G and let A 0 produced cut it in D. Find a point E on BG so that if BE cuts AB produced in F and cuts the circumference in G, then FE = EG. If E can be found, then OE is the first of the means between OA and OB. Diodes invented the cissoid in order to determine E, but it can be found equally conveniently by the aid of conies.
In more modern times several other solutions have been suggested. I may allude in passing to three given by HuygensJ,
* Archimedis Opera, ed. Torelli, p. 137 ; ed. Heiberg, vol. iii, pp. 76 — 79. The solution is given in my History of Mathematics, London, 1901, p. 84.
t Ibid., ed. Torelli, pp. 138, 139, 141 ; ed. Heiberg, vol. m, pp. 78—84, 90-93.
X Opera Varia, Leyden, 1724, pp. 393—396.
B. R, 19
290 THREE GEOMETRICAL PROBLEMS [CH. XII
but I will enunciate only those proposed respectively by Vieta, Descartes, Gregory of St Vincent, and Newton.
Vieta's construction is as follows*. Describe a circle, centre 0, whose radius is equal to half the length of the larger of the two given lines. In it draw a chord AB equal to the smaller of the two given lines. Produce AB to E so that BE = AB. Through A draw a line AF parallel to OE. Through 0 draw a line DOCi^G, cutting the circumference in D and G, cutting AF in F, and cutting BA produced in G, so that GF= GA. If this line can be drawn then AB : GG =GG : GA = GA : GB.
Descartes pointed outf that the curves
x^ = ay and x^ + y^= ay + bx
cut in a point {x, y) such that a : x = x : y == y : h. Of course this is equivalent to the first solution given by Menaechmus, but Descartes preferred to use a circle rather than a second conic.
Gregory's construction was given in the form of the following theorem |. The hyperbola drawn through the point of inter- section of two sides of a rectangle so as to have the two other sides for its asymptotes meets the circle circumscribing the rectangle in a point whose distances from the asymptotes are the mean proportionals between two adjacent sides of the rect- angle. This is the geometrical expression of the proposition that the curves xy = ab and x^ + y^ = ay + bx cut in a point (x, y) such that a : x = x : y=y : b.
One of the constructions proposed by Newton is as folio ws§. Let GA be the greater of two given lines. Bisect GA in B. With centre 0 and radius GB describe a circle. Take a point C on the circumference so that BG is equal to the other of the two given lines. From 0 draw GDE cutting A G produced in D, and BG produced in E, so that the intercept DE = GB. Then
♦ Opera Mathematica, ed. Schooten, Leyden, 1646, prop, v, pp. 242—243.
t Geometria, bk. iii, ed. Schooten, Amsterdam, 1659, p. 91.
X Gregory of St Vincent, Opus Geometricum Quadraturae Circuli, Antwerp, 1647, bk. VI, prop. 138, p. 602.
§ Arithmetica Universalis, Ralphson's (second) edition, 1728, p. 242; see also pp. 2i3, 245.
CH. Xll] THREE GEOMETRICAL PROBLEMS 291
BC : OD = OD : CE = GE : OA. Hence OD and CE are two mean proportionals between any two lines BC and OA.
The Trisection of an Angle*.
The trisection of an angle is the second of these classical problems, but tradition has not enshrined its origin in romance. The following two constructions are among the oldest and best known of those which have been suggested; they are quoted by Pappus f, but I do not know to whom they were due originally.
The first of them is as follows. Let A OB be the given angle. From any point P in OB draw FM perpendicular to OA. Through P draw Pi2 parallel to 0^. On il/P take a point Q so that if OQ is produced to cut PR in R then QR = 2 . OP. If this construction can be made, then AOR = ^AOB. The solution depends on determining the position of R. This was effected by a construction which may be expressed analytically thus. Let the given angle be tan~^ (b/a). Construct the hyper- bola xy — ab, and the circle (x — of + {y — hf = ^ (a^ -I- 6^). Of the points where they cut, let x be the abscissa which is greatest, then PR = x— a, and tan~^ (^/^) = i tan~^ (^/
The second construction is as follows. Let AOB be the given angle. Take OB = OA, and with centre 0 and radius OA describe a circle. Produce AO indefinitely and take a point G on it external to the circle so that if GB cuts the circumference in D then GD shall be equal to OA. Draw OE parallel to GDB. Then, if this construction can be made, A OE = ^A OB. The ancients determined the position of the point G by the aid of the conchoid : it could be also found by the use of the conic sections.
I proceed to give a few other solutions, confining myself to those effected by the aid of conies.
* On the bibliography of the subject see the supplements to VIntermidiaire des Mathematiciens, Paris, May and June, 1904.
t Pappus, Mathematicae CoUectiones, bk. iv, props. 32, '6'6 (cd. Commandino, Bonn, 1670, pp. 97 — 99). On the application to this problem of the traditional Greek methods of analysis see Geometrical Analysis, by J. Leslie, Edinburgh, second edition, 1811, pp. 245—247.
19—2
292 THREE GEOMETRICAL PROBLEMS [CH. XII
Among other constructions given by Pappus* I may quote the following. Describe a hyperbola whose eccentricity is two. Let its centre be G and its vertices A and A'. Produce CA' to >S^ so that A'S=GA\ On AS describe a segment of a circle to contain the given angle. Let the orthogonal bisector oi AS cut this segment in 0. With centre 0 and radius OA or OS describe a circle. Let this circle cut the branch of the hyperbola through A' in P. Then SOP = ^SOA.
In modern times one of the earliest of the solutions by a direct use of conies was suggested by Descartes, who effected it by the intersection of a circle and a parabola. His con- struction f is equivalent to finding the points of intersection, other than the origin, of the parabola y^ = ios and the circle rjQ^^y'i — 13.x + ^ay = 0. The ordinates of these points are given by the equation 4y = 3y — a. The smaller positive root is the sine of one- third of the angle' whose sine is a. The demonstra- tion is ingenious.
One of the solutions proposed by Newton is practically equivalent to the third one which is quoted above from Pappus. It is as follows |. Let A be the vertex of one branch of a hyperbola whose eccentricity is two, and let S be the focus of the other branch. On AS describe the segment of a circle con- taining an angle equal to the supplement of the given angle. Let this circle cut the S branch of the hyperbola in P. Then PAS will be equal to one-third of the given angle.
The following elegant solution is due to Clairaut§. Let A OB be the given angle. Take OA — OB, and with centre 0 and radius OA describe a circle. Join AB, and trisect it in H, K, so that AH = HK = KB. Bisect the angle A OB by 00 cutting AB in L. Then AH — 2.HL. With focus A, vertex if, and directrix OC, describe a hyperbola. Let the branch of
* Pappus, Mathematicae Collectiones^ bk. iv, prop. 34, pp. 99 — 104.
+ Geometria, bk. in, ed. Schooten, Amsterdam, 1659, p. 91.
X Arithmetica Universalis, problem xlii, Ralphson's (second) edition, London, 1728, p. 148 ; see also pp. 243—245.
§ I believe that this was first given by Clairaut, but I have mislaid my reference. The construction occurs as an example in the Geometry of GoiiicSf by G. Taylor, Cambridge, 1881, No. 308, p. 126.
CH. XIl] THREE GEOMETR[CAL PROBLEMS 293
this hyperbola which passes through H cut the circle in P. Draw PM perpendicular to OC and produce it to cut the circle in Q. Then by the focus and directrix property we have AP : PM=AH :HL=2:1,.'.AP = 2.PM = PQ. Hence, by symmetry, AP=PQ= QR. /. AOP = POQ = QOR
I may conclude by giving the solution which Chasles* regards as the most fundamental. It is equivalent to the following proposition. If OA and OB are the bounding radii of a circular arc AB, then a rectangular hyperbola having OA for a diameter and passing through the point of intersection of OB with the tangent to the circle at A will pass through one of the two points of trisection of the arc.
Several instruments have been constructed by which mechanical solutions of the problem can be obtained.
The Quadrature of the Circle f.
The object of the third of the classical problems was the determination of a side of a square whose area should be equal to that of a given circle.
The investigation, previous to the last two hundred years, of this question was fruitful in discoveries of allied theorems, but in more recent times it has been abandoned by those who are able to realize what is required. The history of this subject has been treated by competent writers in such detail that I shall content myself with a very brief allusion to it.
Archimedes showed J (what possibly was known before) that the problem is equivalent to finding the area of a nght-angled
* Traite des sections coniques, Paris, 1865, art. 37, p. 36.
+ See Montucla's Histoire des Recherches sur la Quadrature du Cercle, edited by P. L. Lacroix, Paris, 1831 ; also various articles by A. De Morgan, and especially his Budget of Paradoxes, London, 1872. A popular sketch of the subject has been compiled by H. Schubert, Die Quadratur des Zirkels, Hamburg, 1889 ; and since the publication of the earlier editions of these Recreations Prof. F. Rudio of Zurich has given an analysis of the arguments of Ajchimedes, Huygens, Lambert, and Legendre on the subject, with an intro- duction on the history of the problem, Leipzig, 1892.
X Archimedit Opera, KvkKov fiirpricis, prop, i, ed. Torelli, pp. 203 — 205 ; ed. Heiberg, vol. i, pp. 258—201, vol. iii, pp. 2G9— 277.
294 THEEE GEOMETRICAL PROBLEMS [CH. XII
triangle whose sides are equal respectively to the perimeter of the circle and the radius of the circle. Half the ratio of these lines is a number, usually denoted by tt.
That this number is incommensurable had been long sus- pected, and has been now demonstrated. The earliest analytical proof of it was given by Lambert* in 1761 ; in 1803 Legendref extended the proof to show that tt^ was also incommensurable ; and recently Lindemann| has shown that tt cannot be the root of a rational algebraical equation.
An earlier attempt by James Gregory to give a geometrical demonstration of this is worthy of notice. Gregory proved § that the ratio of the area of any arbitrary sector to that of the inscribed or circumscribed polygons is not expressible by a finite number of algebraical terms. Hence he inferred that the quadrature was impossible. This was accepted by Montucla, but it is not conclusive, for it is conceivable that some particular sector might be squared, and this particular sector might be the whole circle.
In connection with Gregory's proposition above cited, I may add that Newton || proved that in any closed oval an arbitrary sector bounded by the curve and two radii cannot be expressed in terms of the co-ordinates of the extremities of the arc by a finite number of algebraical terms. The argument is condensed and difficult to follow: the same reasoning would show that a closed oval curve cannot be represented by an algebraical equation in polar co-ordinates. From this proposition no
* Memoires de VAcademie de Berlin for 1761, Berlin, 17G8, pp. 265— 322.
t Legendre's Geometry, Brewster's translation, Edinburgh, 1824, pp. 239—
245.
X Ueber die Zahl tt, Mathematische Annalen, Leipzig, 1882, vol. xx, pp. 213 — 225. The proof leads to the conclusion' that, if re is a root of a rational integral algebraical equation, then e' cannot be rational : hence, if iri was the root of such an equation, e^* could not be rational ; but e"^ is equal to - 1, and therefore is rational ; hence iri cannot be the root of such an algebraical equation, and therefore neither can tt.
§ Vera Circuli et Hyperbolae Quadratura, Padua, 1668 : this is reprinted in Huygens's Opera Varia, Leyden, 1724, pp. 405— 4G2.
II Principia, bk. i, section vi, lemma xxviii.
CH. XIl] THREE GEOMETRICAL PROBLKMS 295
conclusion as to the quadrature of the circle is to be drawn, nor did Newton draw any. In the earlier editions of this work I expressed an opinion that the result presupposed a particular definition of the word oval, but on more careful reflection I think that the conclusion is valid without restriction.
With the aid of the quadratrix, or the conchoid, or the cissoid, the quadrature of the circle is easy, but the construction of those curves assumes a knowledge of the value of tt, and thus the question is begged.
I need hardly add that, if tt represented merely the ratio of the circumference of a circle to its diameter, the deter- mination of its numerical value would have but slight interest. It is however a mere accident that tt is defined usually in that way, and it really represents a certain number which would enter into analysis from whatever side the subject was approached.
I recollect a distinguished professor explaining how different would be the ordinary life of a race of beings born, as easily they might be, so that the fundamental processes of arithmetic, algebra and geometry were different to those which seem to us so evident, but, he added, it is impossible to conceive of a universe in which e and tt should not exist.
I have quoted elsewhere an anecdote, w^hich perhaps will bear repetition, that illustrates how little the usual definition of TT suggests its properties. De Morgan was explaining to an actuary what was the chance that a certain proportion of some group of people would at the end of a given time be alive ; and quoted the actuarial formula, involving tt, which, in answer to a question, he explained stood for the ratio of the circumference of a circle to its diameter. His acquaintance, who had so far listened to the explanation with interest, interrupted him and exclaimed, " My dear friend, that must be a delusion, what can a circle have to do with the number of people alive at the end of a given time ? " In reality the fact that the ratio of the length of the circumference of a circle to its diameter is the number denoted by ir does not afford the best analytical defini- tion of TT, and is only one of its properties.
296 THREE GEOMETRICAL PROBLEMS [cH. Xll
The use of a single symbol to denote this number 3'14159... seems to have been introduced about the beginning of the eighteenth century. William Jones* in 1706 represented it by tt; a few years laterf John Bernoulli denoted it by c; Euler in 1734 used p, and in 1736 used c; Christian Goldback in 1742 used tt; and after the publication of Euler's Analysis the symbol tt was generally employed.
The numerical value of tt can be determined by either of two methods with as close an approximation to the truth as is desired.
The first of these methods is geometrical. It consists in calculating the perimeters of polygons inscribed in and circum- scribed about a circle, and assuming that the circumference of the circle is intermediate between these perimeters^. The ap- proximation would be closer if the areas and not the perimeters were employed. The second and modern method rests on the determination of converging infinite series for tt.
We may say that the 7r-calculators who used the first method regarded ir as equivalent to a geometrical ratio, but those who adopted the modern method treated it as the symbol for a certain number which enters into numerous branches of mathematical analysis.
It may be interesting if I add here a list of some of the approximations to the value of tt given by various writers §. This will indicate incidentally those who have studied the sub- ject to the best advantage.
* Synopsis Palmariorum Matheseos, London, 1706, pp. 243, 263 et seq. t See notes by G. Enestrom in the Bibliotheca Mathematica, Stockholm, 1889, vol. m, p. 28; Ibid., 1890, vol. iv, p. 22.
:;: The history of this method has been written by K. E. I. Selander, Uistorik ajver Ludolphska Talet, Upsala, 1868.
§ For the methods used in classical times and the results obtained, see the notices of their authors in M. Canter's Geschichte der Mathematik, Leipzig, vol. I, 1880. For medieval and modern approximations, see the article by A. De Morgan on the Quadrature of the Circle in vol. xix of the Penny Cyclopaedia, London, 1841; with the additions given by B. de Haan in the Verhandelingen of Amsterdam, 1858, vol. iv, p. 22 : the conclusions were tabulated, corrected, and extended by Dr J. W. L. Glaisher in the Messenger of Mathematics, Cambridge, 1873, vol. ii, pp. 119—128; and Ihid., 1874, vol. in, pp. 27—46.
CH. XIl] THREE GEOMETRICAL PROBLEMS 297
The ancient Egyptians* took 256/81 as the value of tt, this is equal to 3'1605... ; but the rougher approximation of 3 was used by the Babylonians f and by the Jews J:. It is not unlikely that these numbers were obtained empirically.
We come next to a long roll of Greek mathematicians who attacked the problem. Whether the researches of the members of the Ionian School, the Pythagoreans, Anaxagoras, Ilippias, Antipho, and Bryso led to numerical approximations for the value of TT is doubtful, and their investigations need not detain us. The quadrature of certain lunes by Hippocrates of Chios is ingenious and correct, but a value of tt cannot be thence deduced ; and it seems likely that the later members of the Athenian School concentrated their efforts on other questions.
It is probable that Euclid §, the illustrious founder of the Alexandrian School, was aware that tt was greater than 3 and less than 4, but he did not state the result explicitly.
The mathematical treatment of the subject began with Archimedes, who proved that tt is less than 3f and greater than 3ff, that is, it lies between 3-1428... and 3*1408... . He established II this by inscribing in a circle and circumscribing about it regular polygons of 96 sides, then determining by geometry the perimeters of these polygons, and finally assuming that the circumference of the circle was inter- mediate between these perimeters : this leads to a result from which he deduced the limits given above. This method is equivalent to using the proposition sin ^ d — 'TT/QQ: the values of sin^ and tan ^ were deduced by Archimedes fi'om those of sin ^tt and tan ^tt by repeated bisections of the angle. With a polygon of n sides this
* Ein mathematisches Handbiich der alten Aegypter {i.e. the Ebiiid papyrus), by A. Eisenlohr, Leipzig, 1877, arts. 100—109, 117, 124.
t Oppert, Journal Asiatique, August, 1872, and October, 1874.
X 1 Kings, ch. 7, ver. 23 ; 2 Chronicles, ch. 4, vcr. 2.
§ These results can be deduced from Euc. iv, 15, and iv, 8: see also book xii, prop. 16.
II Archimedis Opera, KiJkXou iJ.hpr)(xis, prop, iii, ed. Torelli, Oxford, 1792, pp. 205 — 216; ed. Heiberg, Leipzig, 18S0, vol. i, pp. 263 — 271.
298 THREE GEOMETRICAL PROBLEMS [CH. XII
process gives a value of tt correct to at least the integral part of (2 logri — 1*19) places of decimals. The result given by Archimedes is correct to two places of decimals. His analysis leads to the conclusion that the perimeters of these polygons for a circle whose diameter is 4970 feet would lie between 15610 feet and 15620 feet— actually it is about 15613 feet 9 inches.
Apollonius discussed these results, but his criticisms have been lost.
Hero of Alexandria gave* the value 3, but he quoted f the result 22/7 : possibly the former number was intended only for rough approximations.
The only other Greek approximation that I need mention is that given by Ptolemy J, who asserted that tt = 3° 8' 80''. This is equivalent to taking tt = 3 + ^ + -g^gg = Sj^^ = 3*1416.
The Roman surveyors seem to have used 3, or sometimes 4, for rough calculations. For closer approximations they often employed 3^ instead of 3f , since the fractions then introduced are more convenient in duodecimal arithmetic. On the other hand Gerbert§ recommended the use of 22/7.
Before coming to the medieval and modern European mathe- maticians it may be convenient to note the results arrived at in India and the East.
Baudhayana|| took 49/16 as the value of tt.
Arya-BhataH, circ. 530, gave 62832/20000, which is equal to 31416. He showed that, if a is the side of a regular polygon of n sides inscribed in a circle of unit diameter, and if h is the side of a regular inscribed polygon of 2n sides, then 6^ = J — J (1 — a'^y\ From the side of an inscribed hexagon, he found successively the sides of polygons of 12, 24, 48, 96, 192,
* Blensurae, ed. Hultscli, Berlin, 1864, p. 188.
t Geometria, ed. Hultsch, Berlin, 1864, pp. 115, 136.
X Almagest, bk. vi, chap. 7 ; ed. Halma, vol. i, p. 421.
§ (Euvres de Gerbert, ed. Olleris, Clermont, 1867, p. 453.
II The Sulvasutras by G. Thibaut, Asiatic Society of Bengal, 1875, arta. 26—28.
IF Legons de calcul d'Aryabhata, by L. Rodet in the Journal Asiatique, 1879, series 7, vol. xiii, pp. 10, 21.
CH. XIl] THREE GEOMRTRICAL PROBLEMS 299
and 884 sides. The perimeter of the last is given as equal to V9*8694, from which his result was obtained by approxi- mation..
Brahmagupta*, circ. 650, gave VlO, which is equal to 81622.... He is said to have obtained this value by inscribing in a circle of unit diameter regular polygons of 12, 24, 48, and 96 sides, and calculating successively their perimeters, which he found to be V9^, Vim, V9^, V9^ respectively; and to have assumed that as the number of sides is increased indefinitely
the perimeter would approximate to VlO.
Bhaskara, circ. 1150, gave two approximations. Onef — pos- sibly copied from Arya- Bhata, but said to have been calculated afresh by Archimedes's method from the perimeters of regular polygons of 384 sides— is 3927/1250, which is equal to 31416: the otherj is 754/240, which is equal to 31416, but it is in- certain wliether this was not given only as an approximate value.
Among the Arabs the values 22/7, VIO, and 62832/20000 were given by Alkarismi§, circ. 830; and no doubt were derived from Indian sources. He described the first as an approximate value, the second as used by geometricians, and th3 third as used by astronomers.
In Chinese works the values 3, 22/7, 157/50 are said to occur: probably the last two results were copied from the Arabs. The Japanesejj approximations were closer.
Returning to European mathematicians, we have the following successive approximations to the value of tt: many of those prior to the eighteenth century having been calculated originally with the view of demonstrating the incorrectness of some alleged quadrature.
* Algebra... from Brahmegupta and Bhascara, trans, by H. T. Colebrooke, London, 1817, chap, xii, art. 40, p, 308.
t Ibid., p. 87.
X Ibid., p. 95.
§ The Algebra of Mohammed ben Musa, ed. by F. Eosen, London, 1831, pp. 71—72.
II On Japanese approximations and the methods used, see P. Harzer, Transactions of the British Association for 1905, p. 325.
300 THREE GEOMETRICAL PROBLEMS [CH. XII
Leonardo of Pisa*, in the thirteenth century, gave for tt the value 1440/458^, which is equal to 3'1418.... In the fifteenth century, Purbachf gave or quoted the value 62832/20000, which is equal to 3'1416 ; Cusa believed that the accurate value was } (V3 + \/6), which is equal to 31423...; and, in 1464, RegiomontanusJ: is said to have given a value equal to 314243.
Vieta§, in 1579, showed that ir was greater than 31415926535/10^", and less than 31415926537/10^°. This was deduced from the perimeters of the inscribed and circumscribed polygons of 6 X 2^^ sides, obtained by repeated use of the formula 2 sin^ 1^ = 1 — cos ^. He also gave|| a result equivalent to the
formula
2 _ V2 V(2 + V2) V{2 + V(2 + V2)}
7r~ 2 2 2
The father of Adrian MetiusH, in 1585, gave 355/113, which is equal to 3"14159292..., and is correct to six places of decimals. This was a curious and lucky guess, for all that he proved was that tt was intermediate between 377/120 and 333/106, whereon he jumped to the conclusion that he would obtain the true fractional value by taking the mean of fche numerators and the mean of the denominators of these fractions.
In 1593 Adrian Romanus** calculated the perimeter of the inscribed regular polygon of 1073,741824 {i.e. 2^'^) sides, from which he determined the value of ir correct to 15 places of decimals.
* Boncompagni's Scritti di Leonardo, voL ii {Practica Geometriae), Eome, 1862, p. 90.
t Appendix to the De Triangulis of Regiomontanus, Basle, 1541, p. 131.
J In his correspondence with Cardinal Nicholas de Cusa, De Quadratura Girculi, Nuremberg, 1533, wherein he proved that the cardinal's result was wrong. I cannot quote the exact reference, but the figures are given by com- petent writers and I have no doubt are correct.
§ Canon Mathematicus seu ad Triangula, Paris, 1579, pp. 56, 66 : probably this work was printed for private circulation only, it is very rare.
II Vietae Opera, ed. Schooten, Leyden, 16i6, p. 400.
^ Arithmeticae libri duo et Geometriae, by A. Metius, Leyden, 1626, pp. 88— 89. [Probably issued originally in 1611.]
** Ideae Mathematicae, Antwerp, 1593 : a rare work, which I have never been able to consult.
CH. XIl] THREE GEOMETRICAL PROBLEMS 301