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.
proportionals which we have given in the chapter on Special Problems (pp. 256–7). Indeed, as we said, it is certain on other grounds that the so-called Platonic solution was later than that of Eratosthenes; otherwise Eratosthenes would hardly have failed to mention it in his epigram, along with the solutions by Archytas and Menaechmus. Tannery, indeed, regards Plutarch’s story as an invention based on nothing more than the general character of Plato’s philosophy, since it took no account of the
held with its axis vertical), and so diverts the right portion of the visual current to the left and vice versa. And if you turn the mirror so that its axis is horizontal, everything appears upside down. (β) Music. In music Plato had the advantage of the researches of Archytas and the Pythagorean school into the numerical relations of tones. In the Timaeus we find an elaborate filling up of intervals by the interposition of arithmetic and harmonic means64; Plato is also clear that higher
equimultiples of a, b. By separating m, n into their units Euclid practically proves that m . na = mn . a and m . nb = mn . b. 4. If a : b = c : d, then ma : nb = mc : nd. Take any equimultiples p . ma, p . mc of ma, mc, and any equimultiples q . nb, q . nd of nb, nd. Then, by 3, these equimultiples are also equimultiples of a, c and b, d respectively, so that by Def. 5, since a : b = c : d, p . ma > = < q . nb according as p . mc > = < q . nd, whence, again by Def. 5, since p, q are any
Πάτροκλος. ‘Those then who calculate by the rule of nine take one-ninth of the sum of the pythmenes and then determine the sum of the pythmenes in the remainder. Those on the other hand who follow the “rule of seven” divide by 7. Thus the sum of the pythmenes in Πάτροκλος was found to be 34. This, divided by 7, gives 4, and since 7 times 4 is 28, the remainder is 6. …’ ‘It is necessary to observe that, if the division gives an integral quotient (without remainder), … the pythmen is the number 9
are equivalent to the geometrical solution of the quadratic equations and VI. 27 gives the condition of possibility of a solution when the sign is negative and the parallelogram falls short. This general case of course requires the use of proportions; but the simpler case where the area applied is a rectangle, and the form of the portion which overlaps or falls short is a square, can be solved by means of Book II only. The proposition II. 11 is the geometrical solution of the particular