A History of Greek Mathematics, Volume 2: From Aristarchus to Diophantus (Dover Books on Mathematics)
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Volume 2 of an authoritative two-volume set that covers the essentials of mathematics and features every landmark innovation and every important figure, including Euclid, Apollonius, and others.
added some things as being necessary and omitted others as unnecessary. Pappus mentions at this point an incorrect addition to Theodosius, Sphaerica, III. 6, an omission from Euclid’s Phaenomena, Prop. 2, an inaccurate representation of Theodosius, On Days and Nights, Prop. 4, and the omission later of certain other things as being unnecessary. His object is to put these mistakes right. Allusions are also found in the Book to Menelaus’s Sphaerica, e.g. the statement (p. 476. 16) that Menelaus in
if any straight line BHE through B meets CD in H and AD produced in E, and if EF be drawn perpendicular to BE meeting BC produced in F, then Draw EG perpendicular to BF. Then the triangles BCH, EGF are similar and since BC = EG) equal in all respects: therefore EF = BH. Now or But, the angles HCF, HEF being right, H, C, F, E are concyclic, and Therefore, by subtraction, Taking away the common part, BC. CF, we have Now suppose that we have to draw BHE through B in such a way that HE = k.
regular solids 420: Solid Loci (conics) 438, ii. 116, 118–19. Aristaeus of Croton 86. Aristarchus of Samos 43, 139, ii. 1–15, ii. 251: date ii. 2 : of, ii. 1: anticipated Copernicus ii. 2–3: other hypotheses ii. 3, 4: treatise On sizes and distances of Sun and Moon ii. 1, 3, 4–15, trigonometrical purpose ii. 5 : numbers in, 39: fractions in, 43. Aristonophus, vase of, 162. Aristophanes 48, 161, 220. Aristotelian treatise on indivisible lines 157, 346–8. Aristotherue 348. Aristotle 5, 120,
R3, R2, R1 respectively and BQ3 in 0, then . 2. If two similar parabolic segments with bases BQ1 BQ2 be placed as in the last proposition, and if BR1R2 be any straight line through B meeting the segments in R1, R2 respectively, . These propositions are easily deduced from the theorem proved in the Quadrature of the Parabola, that, if through E, a point on the tangent at B, a straight line ERO be drawn parallel to the axis and meeting the curve in R and any chord BQ through B in 0, then . 3.
Sphaeric, applied to one of the subjects of the quadrivium, actually meant astronomy. The subject was so far advanced before Euclid’s time that there was in existence a regular textbook containing the principal propositions about great and small circles on the sphere, from which both Autolycus and Euclid quoted the propositions as generally known. These propositions, with others of purely astronomical interest, were collected afterwards in a work entitled Sphaerica, in three Books, by THEODOSIUS.