ON THE SIZES AND DISTANCES OF THE SUN AND MOON.

Archimedes also says that, whereas the ratio of the diameter of the sun to that of the moon had been estimated by Eudoxus at 9 : 1 and by his own father Phidias at 12 : 1, Aristarchus made the ratio greater than 18 : 1 but less than 20 : 1. Fortunately we possess in Greek the short treatise in which Aristarchus proved these conclusions; on the other matter of the apparent diameter of the sun Archimedes’s statement is our only evidence.

It is noteworthy that in Aristarchus’s extant treatise On the sizes and distances of the sun and moon there is no hint of the heliocentric hypothesis, while the apparent diameter of the sun is there assumed to be, not ½°, but the very inaccurate figure of 2°. Both circumstances are explained if we assume that the treatise was an early work written before the hypotheses described by Archimedes were put forward. In the treatise Aristarchus finds the ratio of the diameter of the sun to the diameter of the earth to lie between 19 : 3 and 43 : 6; this would make the volume of the sun about 300 times that of the earth, and it may be that the great size of the sun in comparison with the earth, as thus brought out, was one of the considerations which led Aristarchus to place the sun rather than the earth in the centre of the universe, since it might even at that day seem absurd to make the body which was so much larger revolve about the smaller.

There is no reason to doubt that in his heliocentric system Aristarchus retained the moon as a satellite of the earth revolving round it as centre; hence even in his system there was one epicycle.

The treatise On sizes and distances being the only work of Aristarchus which has survived, it will be fitting to give here a description of its contents and special features.

The style of Aristarchus is thoroughly classical as befits an able geometer intermediate in date between Euclid and Archimedes, and his demonstrations are worked out with the same rigour as those of his predecessor and successor. The propositions of Euclid’s Elements are, of course, taken for granted, but other things are tacitly assumed which go beyond what we find in Euclid. Thus the transformations of ratios defined in Euclid, Book V, and denoted by the terms inversely, alternately, componendo, convertendo, etc., are regularly used in dealing with unequal ratios, whereas in Euclid they are only used in proportions, i.e. cases of equality of ratios. But the propositions of Aristarchus are also of particular mathematical interest because the ratios of the sizes and distances which have to be calculated are really trigonometrical ratios, sines, cosines, etc., although at the time of Aristarchus trigonometry had not been invented, and no reasonably close approximation to the value of π, the ratio of the circumference of any circle to its diameter, had been made (it was Archimedes who first obtained the approximation 22/7). Exact calculation of the trigonometrical ratios being therefore impossible for Aristarchus, he set himself to find upper and lower limits for them, and he succeeded in locating those which emerge in his propositions within tolerably narrow limits, though not always the narrowest within which it would have been possible, even for him, to confine them. In this species of approximation to trigonometry he tacitly assumes propositions comparing the ratio between a greater and a lesser angle in a figure with the ratio between two straight lines, propositions which are formally proved by Ptolemy at the beginning of his Syntaxis. Here again we have proof that textbooks containing such propositions existed before Aristarchus’s time, and probably much earlier, although they have not survived.

Aristarchus necessarily begins by laying down, as the basis for his treatise, certain assumptions. They are six in number, and he refers to them as hypotheses. We cannot do better than quote them in full, along with the sentences immediately following, in which he states the main results to be established in the treatise:—

[Hypotheses.]

1. That the moon receives its light from the sun.

2. That the earth is in the relation of a point and centre to the sphere in which the moon moves.

3. That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye.

4. That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant.

5. That the breadth of the (earth’s) shadow is (that) of two moons.

6. That the moon subtends one-fifteenth part of a sign of the zodiac.

We are now in a position to prove the following propositions:—

1. The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon (from the earth); this follows from the hypothesis about the halved moon.

2. The diameter of the sun has the same ratio (as aforesaid) to the diameter of the moon.

3. The diameter of the sun has to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6; this follows from the ratio thus discovered between the distances, the hypothesis about the shadow, and the hypothesis that the moon subtends one-fifteenth part of a sign of the zodiac.

The first assumption is Anaxagoras’s discovery. The second assumption is no doubt an exaggeration; but it is made in order to avoid having to allow for the fact that the phenomena as seen by an observer on the surface of the earth are slightly different from what would be seen if the observer’s eye were at the centre of the earth. Aristarchus, that is, takes the earth to be like a point in order to avoid the complication of parallax.

The meaning of the third hypothesis is that the plane of the great circle in question passes through the point where the eye of the observer is situated; that is to say, we see the circle end on, as it were, and it looks like a straight line.

Hypothesis 4. If S be the sun, M the moon and E the earth, the triangle SME is, at the moment when the moon appears to us halved, right-angled at M; and the hypothesis states that the angle at E in this triangle is 87°, or, in other words, the angle MSE, that is, the angle subtended at the sun by the line joining M to E, is 3°. These estimates are decidedly inaccurate, for the true value of the angle MES is 89° 50′, and that of the angle MSE is therefore 10′. There is nothing to show how Aristarchus came to estimate the angle MSE at 3°, and none of his successors seem to have made any direct estimate of the size of the angle.

The assumption in Hypothesis 5 was improved upon later. Hipparchus made the ratio of the diameter of the circle of the earth’s shadow to the diameter of the moon to be, not 2, but 2½ at the moon’s mean distance at the conjunctions; Ptolemy made it, at the moon’s greatest distance, to be inappreciably less than 2⅗.

The sixth hypothesis states that the diameter of the moon subtends at our eye an angle which is 1/15th of 30°, i.e. 2°, whereas Archimedes, as we have seen, tells us that Aristarchus found the angle subtended by the diameter of the sun to be ½° (Archimedes in the same tract describes a rough instrument by means of which he himself found that the diameter of the sun subtended an angle less than 1/164th, but greater than 1/200th of a right angle). Even the Babylonians had, many centuries before, arrived at 1° as the apparent angular diameter of the sun. It is not clear why Aristarchus took a value so inaccurate as 2°. It has been suggested that he merely intended to give a specimen of the calculations which would have to be made on the basis of more exact experimental observations, and to show that, for the solution of the problem, one of the data could be chosen almost arbitrarily, by which proceeding he secured himself against certain objections which might have been raised. Perhaps this is too ingenious, and it may be that, in view of the difficulty of working out the geometry if the two angles in question are very small, he took 3° and 2° as being the smallest with which he could conveniently deal. Certain it is that the method of Aristarchus is perfectly correct and, if he could have substituted the true values (as we know them to-day) for the inaccurate values which he assumes, and could have carried far enough his geometrical substitute for trigonometry, he would have obtained close limits for the true sizes and distances.

The book contains eighteen propositions. Prop. 1 proves that we can draw one cylinder to touch two equal spheres, and one cone to touch two unequal spheres, the planes of the circles of contact being at right angles to the axis of the cylinder or cone. Next (Prop. 2) it is shown that, if a lesser sphere be illuminated by a greater, the illuminated portion of the former will be greater than a hemisphere. Prop. 3 proves that the circle in the moon which divides the dark and the bright portions (we will in future, for short, call this “the dividing circle”) is least when the cone which touches the sun and the moon has its vertex at our eye. In Prop. 4 it is shown that the dividing circle is not perceptibly different from a great circle in the moon. If CD is a diameter of the dividing circle, EF the parallel diameter of the parallel great circle in the moon, O the centre of the moon, A the observer’s eye, FDG the great circle in the moon the plane of which passes through A, and G the point where OA meets the latter great circle, Aristarchus takes an arc of the great circle GH on one side of G, and another GK on the other side of G, such that GH = GK = ½ (the arc FD), and proves that the angle subtended at A by the arc HK is less than 1/44°; consequently, he says, the arc would be imperceptible at A even in that position, and a fortiori the arc FD (which is nearly in a straight line with the tangent AD) is quite imperceptible to the observer at A. Hence (Prop. 5), when the moon appears to us halved, we can take the plane of the great circle in the moon which is parallel to the dividing circle as passing through our eye. (It is tacitly assumed in Props. 3, 4, and throughout, that the diameters of the sun and moon respectively subtend the same angle at our eye.) The proof of Prop. 4 assumes as known the equivalent of the proposition in trigonometry that, if each of the angles α, β is not greater than a right angle, and α > β, then

tan α / tan β > α/β > sin α / sin β.

Prop. 6 proves that the moon’s orbit is “lower” (i.e. smaller) than that of the sun, and that, when the moon appears to us halved, it is distant less than a quadrant from the sun. Prop. 7 is the main proposition in the treatise. It proves that, on the assumptions made, the distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth. The proof is simple and elegant and should delight any mathematician; its two parts depend respectively on the geometrical equivalents of the two inequalities stated in the formula quoted above, namely,

tan α / tan β > α/β > sin α / sin β,

where α, β are angles not greater than a right angle and α > β. Aristarchus also, in this proposition, cites 7/5 as an approximation by defect to the value of √2, an approximation found by the Pythagoreans and quoted by Plato. The trigonometrical equivalent of the result obtained in Prop. 7 is

1/18 > sin 3° > 1/20.

Prop. 8 states that, when the sun is totally eclipsed, the sun and moon are comprehended by one and the same cone which has its vertex at our eye. Aristarchus supports this by the arguments (1) that, if the sun overlapped the moon, it would not be totally eclipsed, and (2) that, if the sun fell short (i.e. was more than covered), it would remain totally eclipsed for some time, which it does not (this, he says, is manifest from observation). It is clear from this reasoning that Aristarchus had not observed the phenomenon of an annular eclipse of the sun; and it is curious that the first mention of an annular eclipse seems to be that quoted by Simplicius from Sosigenes (second century, A.D.), the teacher of Alexander Aphrodisiensis.

It follows (Prop. 9) from Prop. 8 that the diameters of the sun and moon are in the same ratio as their distances from the earth respectively, that is to say (Prop. 7) in a ratio greater than 18 : 1 but less than 20 : 1. Hence (Prop. 10) the volume of the sun is more than 5832 times and less than 8000 times that of the moon.

By the usual geometrical substitute for trigonometry Aristarchus proves in Prop. 11 that the diameter of the moon has to the distance between the centre of the moon and our eye a ratio which is less than 2/45ths but greater than 1/30th. Since the angle subtended by the moon’s diameter at the observer’s eye is assumed to be 2°, this proposition is equivalent to the trigonometrical formula

1/45 > sin 1° > 1/60.

Having proved in Prop. 4 that, so far as our perception goes, the dividing circle in the moon is indistinguishable from a great circle, Aristarchus goes behind perception and proves in Prop. 12 that the diameter of the dividing circle is less than the diameter of the moon but greater than 89/90ths of it. This is again because half the angle subtended by the moon at the eye is assumed to be 1° or 1/90th of a right angle. The proposition is equivalent to the trigonometrical formula

1 > cos 1° > 89/90.

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.