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.
their authors, and need not be further referred to here, except for an addendum to the account of Apollonius’s Conics which is remarkable. Pappus has been speaking of the ‘locus with respect to three or four lines’ (which is a conic), and proceeds to say (p. 678. 26) that we may in like manner have loci with reference to five or six or even more lines; these had not up to his time become generally known, though the synthesis of one of them, not by any means the most obvious, had been worked out
added by the third turn, and so on, then (Prop. 27) Also Lastly, if E be the portion of the sector b′OC bounded by b′B, the arc b′zC of the circle and the arc BC of the spiral, and F the portion cut off between the arc BC of the spiral, the radius OC and the arc intercepted between OB and OC of the circle with centre O and radius OB, it is proved that On Plane Equilibriums, I, II. In this treatise we have the fundamental principles of mechanics established by the methods of geometry in its
I have already described in a previous chapter the remarkable system, explained in this treatise and in a lost work, ΆρΧαί, Principles, addressed to Zeuxippus, for expressing very large numbers which were beyond the range of the ordinary Greek arithmetical notation. Archimedes showed that his system would enable any number to be expressed up to that which in our notation would require 80,000 million million ciphers and then proceeded to prove that this system more than sufficed to express the
he placed his whole ambition in those speculations the beauty and subtlety of which is untainted by any admixture of the common needs of life.’5 (α) Astronomy. Archimedes did indeed write one mechanical book, On Sphere-making, which is lost; this described the construction of a sphere to imitate the motions of the sun, moon and planets.6 Cicero saw this contrivance and gives a description of it; he says that it represented the periods of the moon and the apparent motion of the sun with such
centre 0, and AB, CD are chords at right angles through E, the centre of the earth. To find OE. The arc BC is known (= α, say) as also the arc CA (= β). If BF be the chord parallel to CD, and CG the chord parallel to AB, and if N, P be the middle points of the arcs BF, GC, Ptolemy finds (1) the arc BF ( = α + β —180°), then the chord BF, crd. (α + β —180°), then the half of it, (2) the arc GC = arc (α + β— 2β) or arc (α — β), then the chord GC, and lastly half of it. He then adds the squares on