# A History of Greek Mathematics Volume 2

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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1921 edition. Excerpt: ...to 90 by steps of'. (Q) Further use of proportional increase. Ptolemy carries further the principle of proportional increase as a method of finding approximately the chords of arcs containing an odd number of minutes between 0' and 30'. Opposite each chord in the Table he enters in a third column j'jjth of the excess of that chord over the one before, i.e. the chord of the arc containing 30' less than the chord in question. For example (crd. 2-) is stated in the second column of the Table as 2l 37 4." The excess of (crd. 2\$) over (crd. 2) in the Table is O/-31' 24"; ih of this is 0? l' 2" 48'," which is therefore the amount Centered in the third column opposite (crd. 2J). Accordingly, if we want (crd. 2 25'), we take (crd. 2). or 2P 5' 40" and add 25 times OP 1' 2" 48'"; or we take (crd. 2 ) or 2i 37' 4" and subtract 5 times 0' l' 2" 48'." Ptolemy adds that if, by using the approximation for 1 and J, we gradually accumulate an error, we can check the calculation by comparing the chord with that of other related arcs, e.g. the double, or the supplement (the difference between the arc and the semicircle). Some particular results obtained from the Table may be mentioned. Since (crd. 1) = 1 P 2' 50," the whole circumference = 360 (H1 2' 50"), nearly, and, the length of the diameter being 120/', the value of It is 3 (1 +y + 3r) = 3 +& +?&- which is the value used later by Ptolemy and is equivalent to 3-14166... Again, /3 = 2 sin 60 and, 2 (crd. 120) being equal to 2 (103/' 55' 23"), we have /3 = (103 +f + 3 fs) 43 55 23 which is correct to 6 places of decimals. Speaking generally, the sines obtained from Ptolemy's Table are correct to 5 places. (t) Plane...