A History of Greek Mathematics, Volume 1: From Thales to Euclid
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Volume 1 of an authoritative two-volume set that covers the essentials of mathematics and includes every landmark innovation and every important figure. This volume features Euclid, Apollonius, others.
ahead.’14 III. The Arrow. ‘If, says Zeno, everything is either at rest or moving when it occupies a space equal (to itself), while the object moved is always in the instant (ἔστι δ’ ἀὶ τὸ ςρόμνον ἐν τ νν, in the now), the moving arrow is unmoved.’15 I agree in Brochard’s interpretation of this passage,16 from which Zeller17 would banish ἢ κινται, ‘or is moved’. The argument is this. It is strictly impossible that the arrow can move in the instant, supposed indivisible, for, if it changed its
that, owing to its shape, the gnomon should then be used to describe (3) the figure which remained of a square when a smaller square was cut out of it (or the figure which, as Aristotle says, when added to a square, preserves the shape and makes up a larger square). The term is used in a fragment of Philolaus where he says that ‘number makes all things knowable and mutually agreeing in the way characteristic of the gnomon’.74 Presumably, as Boeckh says, the connexion between the gnomon and the
almost exclusively, and he has a proof of the theorem now in question by means of a figure with gnomons drawn in it, furnishing an excellent example of the geometrical algebra which is so distinctively Greek. Let AB be the side of a square AC; let and suppose BB′ = n, B′ B″ = n − 1, B″ B‴ = n − 2, and so on. Draw the squares on AB′, AB″ … forming the gnomons shown in the figure. Then the gnomon Now , therefore (gnomon BC′ D) = n. n2 = n3. Similarly (gnomon B′ C″ D′) = (n − 1)3, and
Iambl. in Nicom., p. 27. 4. 47 Cf. Arist. Metaph. Δ. 13, 1020 b 3, 4. 48 Theon of Smyrna, p. 23. 12. 49 Arist. Topics, Θ. 2, 157 a 39. 50 Eucl. VII. Def. 11. 51 Ib. Def. 13. 52 Ib. Defs. 12, 14. 53 Theon of Smyrna, p. 24. 7. 54 Nicom. i, cc. 11–13; Iambl. in Nicom., pp. 26–8. 55 Theon of Smyrna, p. 45. 56 Nicom. i. 16, 1–4. 57 Iambl. in Nicom., p. 33. 20–23. 58 The fifth perfect number may have been known to Iamblichus, though he does not give it; it was, however, known, with all its
Rivals, to both of them as having acquired a reputation for mathematics. After them came Hippocrates of Chios, the discoverer of the quadrature of the lune, and Theodorus of Cyrene, both of whom became distinguished geometers; Hippocrates indeed was the first of whom it is recorded that he actually compiled Elements. Plato, who came next to them, caused mathematics in general and geometry in particular to make a very great advance, owing to his own zeal for these studies; for every one knows that