We come now to propositions which depend on Hypothesis 5 that “the breadth of the earth’s shadow is that of two moons”. Prop. 13 is about the diameter of the circular section of the cone formed by the earth’s shadow at the place where the moon passes through it in an eclipse, and it is worth while to notice the extreme accuracy with which Aristarchus describes the diameter in question. It is with him “the straight line subtending the portion intercepted within the earth’s shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the moon move.” Aristarchus proves that the length of the straight line in question has to the diameter of the moon a ratio less than 2 but greater than 88 : 45, and has to the diameter of the sun a ratio less than 1 : 9 but greater than 22 : 225. The ratio of the straight line to the diameter of the moon is, in point of fact, 2 cos² 1° or 2 sin² 89°, and Aristarchus therefore proves the equivalent of
2 > 2 cos² 1° > ½(89/45)² or 7921/4050.
He then observes (without explanation) that 7921/4050 > 88/45 (an approximation easily obtained by developing 7921/4050 as a continued fraction (= 1 + (1 1 1)/(1 + 21 + 2))); his result is therefore equivalent to
1 > cos² 1° > 44/45.
The next propositions are the equivalents of more complicated trigonometrical formulæ. Prop. 14 is an auxiliary proposition to Prop. 15. The diameter of the shadow dealt with in Prop. 13 divides into two parts the straight line joining the centre of the earth to the centre of the moon, and Prop. 14 shows that the whole length of this line is more than 675 times the part of it terminating in the centre of the moon. With the aid of Props. 7, 13, and 14 Aristarchus is now able, in Prop. 15, to prove another of his main results, namely, that the diameter of the sun has to the diameter of the earth a ratio greater than 19 : 3 but less than 43 : 6. In the second half of the proof he has to handle quite large numbers. If A be the centre of the sun, B the centre of the earth, and M the vertex of the cone formed by the earth’s shadow, he proves that MA : AB is greater than (10125 × 7087) : (9146 × 6750) or 71755875 : 61735500, and then adds, without any word of explanation, that the latter ratio is greater than 43 : 37. Here again it is difficult not to see in 43 : 37 the continued fraction 1 + 11/(6+6); and although we cannot suppose that the Greeks could actually develop 71755875/61735500 or 21261/18292 as a continued fraction (in form), “we have here an important proof of the employment by the ancients of a method of calculation, the theory of which unquestionably belongs to the moderns, but the first applications of which are too simple not to have originated in very remote times” (Paul Tannery).
The remaining propositions contain no more than arithmetical inferences from the foregoing. Prop. 16 is to the effect that the volume of the sun has to the volume of the earth a ratio greater than 6859 : 27 but less than 79507 : 216 (the numbers are the cubes of those in Prop. 15); Prop. 17 proves that the diameter of the earth is to that of the moon in a ratio greater than 108 : 43 but less than 60 : 19 (ratios compounded of those in Props. 9 and 15), and Prop. 18 proves that the volume of the earth is to that of the moon in a ratio greater than 1259712 : 79507 but less than 216000 : 6859.
ARISTARCHUS ON THE YEAR AND “GREAT YEAR”.
Aristarchus is said to have increased by 1/1623rd of a day Callippus’s figure of 365¼ days as the length of the solar year, and to have given 2484 years as the length of the Great Year or the period after which the sun, the moon and the five planets return to the same position in the heavens. Tannery has shown reason for thinking that 2484 is a wrong reading for 2434 years, and he gives an explanation which seems convincing of the way in which Aristarchus arrived at 2434 years as the length of the Great Year. The Chaldæan period of 223 lunations was well known in Greece. Its length was calculated to be 6585⅓ days, and in this period the sun was estimated to describe 10⅔° of its circle in addition to 18 sidereal revolutions. The Greeks used the period called by them exeligmus which was three times the period of 223 lunations and contained a whole number of days, namely, 19756, during which the sun described 32° in addition to 54 sidereal revolutions. It followed that the number of days in the sidereal year was—
19756/(54 + 32/360) = 19756/(54 + 4/45) = (45 × 19756)/2434 = 889020/2434= 365¼ + 3/4868.
Now 4868/3 = 1623 - ⅓, and Aristarchus seems to have merely replaced 3/4868 by the close approximation 1/1623. The calculation was, however, of no value because the estimate of 10⅔° over 18 sidereal revolutions seems to have been an approximation based merely on the difference between 6585⅓ days and 18 years of 365¼ days, i.e. 6574½ days; thus the 10⅔° itself probably depended on a solar year of 365¼ days, and Aristarchus’s evaluation of it as 365¼ 1/1623 was really a sort of circular argument like the similar calculation of the length of the year made by Œnopides of Chios.