length of the constant segment describe a conchoid, cutting nß produced in к. Join λ and produce it to meet ẞa produced in μ. Then au and ye are the two mean proportionals αμ αλ . λγ = ηγ . βδ, (since y × 2 = 2y x a), .. μα = By the use of Euc. II. 6 (precisely as in the solution of Apollonius) it may be shewn that 2 = ẞK. Ky + y2 and μδ' = βμ . μα + αδ'. .. βκ . κγ + γζ = βμ . μα + αδ'. = = κγ : μα. 752 But But The conchoid was also used to solve the trisection of an angle in a way which closely resembles the 8th of the lemmas attributed to Archimedes (supra, p. 233). Proclus says that Nicomedes himself solved this problem, but Pappus claims the solution which he gives as his own1. Let aßy be the angle which it is required to trisect. From a draw ay perpendicular to By. Complete the parallelogram. η Now from B as pole, with ay as fixed straight line and 2ɑß as constant distance describe a conchoid which shall meet (a produced in e. The line Be cuts ay in 8. Bisect de in n and join an. It is then easy to see that an = ne = aß and the triangles aßn, ane are isosceles. Therefore the exterior angle anẞ= 2xen = 2nBy, and the angle αηβ = αβη = 2ηβγ. 139. Probably at the same time as Nicomedes, say 180 B.C., lived Diocles, the inventor of the cissoid or "ivy-like" curve. His date can be approximately determined only by the two 1 Proclus, p. 272; Pappus Iv. 38, p. 274 (Hultsch). facts that Geminus knew the cissoid by this name, and that Diocles lived after Archimedes, for he wrote a commentary on the unfinished problem (II. 5, supra, p. 225n) of the Sphere and Cylinder. The work in which this occurs was called Teρì TUρív or Tupelov1, whatever that may mean, and contained also a solution of the duplication problem which Eutocius cites with the rest. This solution, which involves also the definition of the cissoid, may be described as follows. Let aß and yồ be diameters of a circle at right angles to one another. On yd, at equal distances on either side of the centre λ, take the points κ and ŋ, and draw the ordinates ke, n. Join ed, cutting n in 0. The point (as also all other points similarly determined) lies on the cissoid. Also γη : ηζ=ηζ : ηδ =ηδ : ηθ. As is perpendicular to the diameter yd, it is plain that v:n5=n}:nd. For a similar reason3, yê : кe = Kе: Kd. And by similar triangles кe: кd = ŋ0 : ŋd. Therefore yê : Ke= ηθ : ηδ 1 Eutocius in Torelli, p. 171. IIupeîov (which may be the right reading), Lat. igniaria, was an instrument for making fire, by turning a pointed perpendicular stick (тpúravov) in a hole made in a flat board (éoxápa). If this was worked by strings, like a drill (see the chapter on fire-drills in Tylor's Early Hist. of Mankind), then Diocles' book may have been a treatise on some geometrical theorems suggested by the machine. 2 Torelli, p. 138. Cantor, pp. 306, 307. The solutions of Pappus and Sporus (an otherwise unknown geometer), which Eutocius gives next, are practically identical with this, though the constructions are not obtained with a cissoid. A shorter proof would run: 'And γη : ης = δκ : κε = δη : ηθ. Therefore vn: ns=ns: nd=nd: no.' and ke ky nd : no. = = = But ken, and kynd. Therefore ns: nd=nd: no. Thus n, nd are two mean proportionals to yn, no. γη, ηθ. Now, in any circle, with diameters aß, yd, at right angles to one another, draw the corresponding cissoid. On the diameter aß, take a point π such that yλ : λπ=a: b, where a and b are the two straight lines to which two mean proportionals are required. Join yπ and produce it to meet the cissoid in 0. Then ynn0=a: b. It is now necessary only to alter the lines ns, nd (which are known to be mean proportionals to yn, ne) in the ratio of yn: a, and the solution is obtained. 140. In the same century, again, perhaps about the year 150 B.C. Perseus, a geometer who treated of the sections of the σπεîρа1, seems to have lived. His date can be guessed only from the facts that he is not included in the Eudemian summary, that no notice is taken of him by the classical geometers, that Heron describes the σTeipa (110 B.C.), and that the work of Perseus was well known to Geminus. The σTeîρa is somewhat imperfectly described by Heron3 as the solid "produced by the revolution of a circle which has its centre on the circumference of another circle and which is perpendicular to the plane of that other circle. This is also called a κpixos (ring)." This solid varies in form according to the ratios between the radii of the two circles. It may resemble an anchor-ring or a modern teacake, with a dimple at the centre. Proclus describes three kinds of sections, which were obtained from it and which were the same as those described above (p. 185), à propos of the iπTоTéon of Eudoxus. Elsewhere (p. 356, 12) he seems to suggest that Perseus had treated the spiral sections as Apollonius had treated the conics. From this, perhaps, it may be inferred that whereas one or two sections of the σTЄîρa were known before and were obtained from different forms of the solid, Perseus investigated all the sections and shewed that they 1 Geminus in Proclus, pp. 111, 112. 2 The dates of Perseus, Nicomedes, Diocles, Serenus and Hypsicles are all discussed by Bretschneider (Anhang, p. 175-end who is especially severe, in this connexion, on the errors of Montucla. Bretschneider is obviously right on all the dates except that of Serenus. 3 Deff. 98, p. 27 of Hultsch's ed. could be obtained from one oπeîpa1. But the work of Perseus is wholly lost, and no extracts whatever from it are preserved by any later writer2. 141. There is not so much reason for assigning Zenodorus to the 2nd century B. C. as there is for the other writers above mentioned. He is later than Archimedes, whom he names, and is older than Quintilian (A.D. 35-95) who names him. He is supposed to be an early successor of the former merely because his style recalls the classical period. He was the author of a geometrical treatise on Figures of Equal Periphery, fourteen propositions of which are preserved both by Pappus and Theon3. Both citations are almost verbally identical, but Theon does, and Pappus does not, name Zenodorus as the author. Theon's ascription is confirmed by Proclus, who says that Zenodorus called a quadrilateral with re-entrant angle a koλoyovcov, which word occurs in Theon's extract. Of these fourteen propositions five, Nos. 1, 2, 6, 7 and 14, are worth quoting. Prop. 1 is "of regular polygons with equal periphery, that is the greatest which has most angles." Prop. 2 is "The circle has a greater area than any polygon of equal periphery." Prop. 6 is "Two similar isosceles triangles on unequal bases are together greater than two dissimilar isosceles triangles which are upon the same bases and have together the same periphery as the two similar "il serait intéressant de voir leur théorie géométrique de ces spiriques, qui sont des courbes du quatrième degré, dont l'étude semble exiger aujourd'hui des équations de surfaces et un calcul analytique assez profond." 3 Pappus v.pt. 1. p. 301 sqq. (Hultsch). Theon. Comm. Almag. ed. Halma, p. 33 sqq. reprinted by Hultsch in Pappus, pp. 1190-1211, with a prefatory note on the date of Zenodorus. The fact that both Theon and Pappus cite the same props. seems to Hultsch (Pappus, Vol. 1. p. xv.) to give colour to his theory that a large part of Theon's commentary was really taken from Pappus. triangles." Prop. 7 is "Of polygons with equal periphery the regular is the greatest." Prop. 14 is "Of segments of circles, having equal arcs, the semicircle is the greatest." It is obvious that investigations of this kind were closely connected with and suggested by the work of Archimedes and Apollonius. 142. To the same century, again, Hypsicles is assigned. To him the 14th and 15th Books added to Euclid's Elements are attributed by many MSS., but recent critics are of opinion that these are by different authors', and that only the 14th is by Hypsicles. This is certainly not Euclid's, for it has a preface which cannot have been written by Euclid, and the Elements are expressly stated by Marinus, in his prolegomena to the Data, to consist of 13 books. The preface in question, which is addressed to one Protarchus, is as follows: "Basilides of Tyre, coming to Alexandria and making the acquaintance of my father through their common love of mathematics, stayed with him during the greater part of his visit. They were discussing at one time the writings of Apollonius on the comparison of the dodecahedron and the icosahedron inscribed in the same sphere, shewing what ratio these have to one another, and they came to the conclusion that Apollonius was wrong. They therefore emended the proof, as my father used to tell. But I afterwards came across another book of Apollonius3 containing a sound proof on the subject, and was greatly incited to the investigation of the problem. The publication of Apollonius may be seen anywhere, for it has a large circulation, but I send you my lucubrations," etc. From this it is inferred, not very cogently, that Hypsicles' father died in the lifetime of Apollonius, or that, at any rate, Hypsicles cannot have lived long after the latter. But a more satisfactory determination of Hypsicles' date is obtained from the fact that his astronomical work, 'Avapopikós, does not use the trigonometry which was certainly introduced by Hipparchus, and would have been absurdly antiquated if written after Hipparchus' time (B. C. 130)*. |