Greek Geometry

Abstract

IN the notice of the last part of “Greek Geometry from Thales to Euclid” (NATURE, vol. xxxiv. p. 548) I was uncertain whether Dr. Allman intended it to be Part vii. or not; I observe from the extract before me (Hermathena, No. xiii., 1887, vol. vi. pp. 269–78) that the present part is so entitled. The author's plan led him to the temporary omission of Theætetus of Athens, a pupil of Theodorus of Cyrene, and also a disciple of Socrates, who greatly advanced the science of geometry. How his gifts and genius impressed both Socrates and Plato is well known from the dialogue which bears his name. From an analysis which our author makes of part of this dialogue it appears that Theætetus, in addition to Eudoxus and the Pythagoreans, was one of the original thinkers to whom Euclid was most indebted in the composition of the “Elements.” Dr. Allman thus recapitulates:—” In the former parts of this paper we have seen that we owe to the Pythagoreans the substance of the first, second, and fourth books, also the doctrine of proportion and of the similarity of figures, together with the discoveries respecting the application, excess, and defect of areas, the subject-matter of the sixth book. The theorems arrived at, however, were proved for commensurable magnitudes only, and assumed to hold good for all. We have seen, further, that the doctrine of proportion, treated in a general manner, so as to include incommensurables (Book v.), and consequently the recasting of Book vi. and also the method of exhaustions (Book xii.) were the work of Eudoxus. If we are asked now: In what portion of the Elements does the work of Theætetus survive? we answer: Since Books vii., viii., and ix. treat of numbers, and our question concerns geometry; and since the substance of Book xi., containing, as it does, the basis of the geometry of volumes, is probably of ancient date, we are led to seek for the work of Theætetus in Books x. and xii.; and it is precisely with the subjects of these books that the extracts (d), (e), and (f) are concerned.”