anybody else than Hippias of Elis', is mentioned elsewhere by Proclus and the mathematical learning of this sophist is directly attested by Plato himself. It is true that he is mentioned by Plato with a certain sarcasm. Protagoras, for instance, in his long and eloquent plea for his own teaching, is made to say "The others injure the young: for they drag them back against their will into arts which they would fain avoid, teaching them arithmetic and astronomy and geometry and music (and here he glanced at Hippias), but he who comes to me shall learn only that for which he comes." Hippias evidently was the polymath of his time and had high notions of a liberal curriculum, Proclus mentions him twice. In the first passage, he says that Nicomedes had solved the trisection problem by means of the conchoid curve, which he himself invented: that others had used for the same purpose the mixed curve called the quadratrix of Hippias and Nicomedes and that others divided an angle in any given proportion by using the spirals of Archimedes. In the second passage, he says that mathematicians have described the properties of various curves, Apollonius of the conic sections, Nicomedes of the conchoids, Hippias of the quadratrix (TETρaywvíčovσa) and Perseus of the spirals. Pappus, however, says that the quadrature of the circle was effected by Dinostratus, Nicomedes and other later geometers by means of a line which, from this use, was called the quadratrix. Here Hippias is ignored. Now Dinostratus belongs to the end of the 4th century B.C. and Nicomedes seems to be a century later. Cantor, therefore, proposes to reconcile the statements of Proclus and Pappus by supposing that Hippias, i.e. Hippias of Elis, invented a curve which was found useful for both the quadrature- and the trisection-problems, and that this curve was, by Dinostratus or Nicomedes or later, called

1 Allman (vII. p. 220) and Hankel (p. 151) deny this. Bretschneider (p. 94) and Cantor (p. 165) affirm it. The latter shows, by many instances, that Proclus was always careful to distinguish writers of the same name.

2 Hippias Maj. 285, CD. Hippias Minor, 367, 368. Protagoras, 318 E.

3 Ed. Friedlein, pp. 272, 356.

4

IV. c. xxx. ed Hultsch, p. 251. So also Simplicius loc. cit. quoting Iamblichus, names Nicomedes only in connexion with the quadratix. Bretschneider, p. 108. 5 p. 167.

the quadratrix, TETρaywvíčovσa. Originally, it may have been intended only for the trisection.

The construction of the quadratrix is thus described by Pappus (loc. cit.). "In the square

α

Ꮎ ๆ

δ

aßys, from a as centre with aß as radius, describe a quarter of a circle βεδ. The straight line aß moves evenly about its end a so that the other end ẞ moves in a given time along the whole arc Bed. The line By moves evenly in the same time, remaining always parallel to itself from the position By to the position ad. The locus of intersection of this straight line with the moving radius aß in the curve Bn, which is the quadratrix." The property of this curve consists in this, that any straight line ale drawn to the circumference of the circle, makes the ratio of the quadrant to the arc ed equal to the ratio of the straight lines Ba: 0. And since the straight line Ba can be divided into any number of parts, in any given ratio to one another, so also can the quadrant or the arc ed, and the trisection or any other section of an angle is performed. The quadrature of the circle is given by this curve, since the straight line which is equal to the quadrant Bed is a third proportional to an, nd1.

100. Theodorus of Cyrene, whom the Eudemian summary names with praise, is known to us only as the mathematical teacher of Plato'. Iamblichus says he was a Pythagorean and Plato introduces, in the Theaetetus, his discovery in effect that the square roots of numbers between 3 and 17 (except 4, 9, 16) are irrational. He does not seem to have visited Athens.

101. Hippocrates of Chios, who is mentioned with Theodorus in the summary, was one of the greatest geometers of antiquity. Like Thales, he began life as a merchant but lost his property either by piracy or through the chicanery of the

1 Pappus, Iv. 26. Bretschneider, p. 96. Hankel, p. 151. Cantor, p. 168, 213, (sub Dinostratus).

2 Diog. Laert. II. 104. Iamblichus Vita Pyth. 267. Plato, Theaet. 147 D.

Byzantine custom house'. He came to Athens to prosecute the offenders, employed his leisure in attending lectures2 and ultimately himself became a teacher of geometry. Aristotle says he had a talent for the science but was in other respects slow and stupid (Bλağ kaì äppwv). The Greeks, however, would naturally call any man a fool who was cheated of his property and Aristotle seems to have no other evidence for his criticism of Hippocrates. He may, of course, have been right. There are still extant mathematicians who are singularly deficient in ability for any studies but their own.

The most celebrated achievement of Hippocrates was that 'squaring of the lune' which the Eudemian summary attributes to him. He was, however, ardently engaged on both the quadrature and the duplication-problems and added enormously, in the course of his researches, to the geometry of the circle. He wrote also the first textbook of 'Elements,' a sufficient service in itself to the cause of the science.

The first step in Hippocrates' attempts at quadrature was the squaring of a particular lune as follows. On a given straight line AB, he described a semi-circle, and inscribed in this an isosceles triangle AFB. On the

equal sides of this triangle he described two other semicircles. Now in the right-angled triangle AгВ,

AB2 = AI2 + FB2, and (since circles A

A

or semicircles are to one another as the squares of their diameters) the semicircle ATB is equal to both the smaller

1 Aristotle, Eth. Eudem. VII. 14. Joh. Philoponus in Ar. Phys. ed. Brandis, p. 327.

2 Iamblichus (De Philos. Pyth. lib. III., Villoison, Anecdota Gr. I. p. 216) says that Hippocrates and Theodorus divulged the Pythagorean geometry. Fabricius, Bibl. Gr. 1. p. 505 (Hamburg. 1718), referring to this passage of Iamblichus, says wrongly that Hippocrates and Theodorus were expelled from the Pythagorean school for making money by teaching geometry. See All

man, Herm. vII. pp. 188, 189.

3 Simplicius in Bretschneider, pp. 102-103. Vieta (Opera, p. 386), quotes these two proofs of Hippocrates from Simplicius, and Montucla follows Vieta (Bretschn. pp. 122, 123).

4 This proposition (Euclid XII. 2) is expressly attributed to Hippocrates by Eudemus "in the second book of his History of Geometry," as quoted by Simplicius shortly afterwards (Bretschn. p. 110 top). The proposition as stated by Hippocrates seems to have

A

semicircles on Aг, TB or is double of either of them. But the semicircle AFB is also double of the quadrant ATA, which, therefore, is equal to the semicircle on AT. Take away from both the common part and it is seen that the triangle ATA is equal to the lune (μŋvíoxos) which lies outside the semicircle АГВ.

H

יד

H

K

The next step1 was as follows. In a semicircle he inscribed half of a regular hexagon, and on the three sides of this as diameters he described the semicircles THE, EOZ, ZKA. Then, since the sides TE, EZ, ZA are equal to the radius TA of the large semicircle and the semicircle on a radius is a quarter of that on a diameter of the same circle, it follows that each of the three smaller semicircles is a quarter of the large one. It follows that the three smaller semicircles together with that on the radius TA is equal to the larger semicircle. Deduct the common parts. Then the external lunes, together with the semi-circle on TA, are equal to the trapezium TEZA. But the lune has been shown, in the first step, to be equal to a rectilineal figure. Deduct therefore from TEZA the three rectilineal figures equal to the three external lunes, and the remainder is a rectilineal figure equal to the semicircle on TA, and twice this rectilineal figure is equal to the circle on TA and thus the circle is squared.

2

The fallacy here lies, as Simplicius rightly points out, in

been (see Bretschneider, p. 120, n. 1), that "similar circles are to one another as the squares of their diameters," from which it would appear that he was not quite sure that all circles are similar to one another,

1 Simplic. in Bretschn, pp. 103, 104. 2 ψευδογράφημα in Simplicius, i. e, a false delineation, a fallacy founded on a faulty diagram. The errors of Hip, pocrates, Antiphon and Bryson, in their attempts to square the circle are referred to and contrasted with one an

other by Aristotle, Soph. Elench. pp, 171 b. 172: Phys. 185, a. and also (as well as by Simplicius) by the commentators Themistius and Joh. Philoponus (Schol. in Ar, ed. Brandis, p. 327 b. 33, 211 b. 19, 30, 41, 212 a. 16), Bretschneider (p, 122) thinks that Hippocrates was too good a geometer to make the mistake here attributed to him and supposes that, in his second step, he merely said "If the lune on the side of a hexagon can be squared, so can the circle."

assuming that the lunes in the second step are the same as those in the first step, which they are not. The first step squares the lune formed on the side of an inscribed square in a circle: the second step deals with lunes formed on the sides of an inscribed hexagon. Hippocrates seems to have felt this difficulty, for he proceeded to examine other lunes which might lead to a quadrature of the circle. Simplicius quotes from Eudemus, with some additions of his own, these further attempts. It appears that Hippocrates made some important additions to his proposition that circles are to one another as the squares of their diameters. He proved' that similar segments of a circle are to one another as the squares of their chords (Báoes); that similar segments contain equal angles, and that in a segment less than a semicircle the angle is obtuse, in a segment greater than a semicircle the angle is acute. Using these propositions he squared a lune of which the exterior arc is greater than a semicircle3 and again a lune of which the exterior arc is less than a semicircle. Lastly, he squared a lune and a circle together in the following manner5. Describe two circles about a common centre K, and let the square on the diameter of the exterior circle be six times the square on that of the interior. Inscribe in the inner circle a hexagon ABгAEZ and draw the radii KA, KB, Kг and

1 Bretschneider, p. 110. Allman, Herm. vII. p. 197. Hippocrates defined similar segments as those which contained the same quotum of their respective circles, e.g. a semicircle is similar to a semicircle, a quadrant to a quadrant.

2 He uses also the props. Euclid II. 12 and 13, but it does not appear that he invented these.

3 Bretschneider, pp. 111, 112, fig. 8. Allman, vII. pp. 198, 199. This lune is obtained by the following construction. Hippocrates draws a trapezium having three equal sides and the fourth such that the square on it is three times the square on any other side. About this trapezium he described a circle, and

on its greater side he described a segment of a circle similar to those of which the three equal sides are the chords. The exterior arc of the lune so obtained is greater than a semicircle.

4 Bretschneider, pp. 114-119, fig. 9. Allman, VII. pp. 199-201 (with additions and corrections to Bretschneider). The proof and even the construction are too long and complicated to be given here. The proposition is remarkable as involving the consideration of a pentagon with a reentrant angle. This is described however as "a rectilineal figure composed of three triangles."

5 Bretschneider, pp. 119-121, fig. 10. Allman, vír. pp. 201, 202.