We are told that Aristarchus of Samos was a pupil of Strato of Lampsacus, a natural philosopher of originality, who succeeded Theophrastus as head of the Peripatetic school in 288 or 287 B.C., and held that position for eighteen years. Two other facts enable us to fix Aristarchus’s date approximately. In 281–280 he made an observation of the summer solstice; and the book in which he formulated his heliocentric hypothesis was published before the date of Archimedes’s Psammites or Sandreckoner, a work written before 216 B.C. Aristarchus therefore probably lived circa 310–230 B.C., that is, he came about seventy-five years later than Heraclides and was older than Archimedes by about twenty-five years.

Aristarchus was called “the mathematician,” no doubt in order to distinguish him from the many other persons of the same name; Vitruvius includes him among the few great men who possessed an equally profound knowledge of all branches of science, geometry, astronomy, music, etc. “Men of this type are rare, men such as were in times past Aristarchus of Samos, Philolaus and Archytas of Tarentum, Apollonius of Perga, Eratosthenes of Cyrene, Archimedes and Scopinas of Syracuse, who left to posterity many mechanical and gnomonic appliances which they invented and explained on mathematical and natural principles.” That Aristarchus was a very capable geometer is proved by his extant book, On the sizes and distances of the sun and moon, presently to be described. In the mechanical line he is credited with the invention of an improved sun-dial, the so-called scaphe, which had not a plane but a concave hemispherical surface, with a pointer erected vertically in the middle, throwing shadows and so enabling the direction and height of the sun to be read off by means of lines marked on the surface of the hemisphere. He also wrote on vision, light, and colours. His views on the latter subjects were no doubt largely influenced by the teaching of Strato. Strato held that colours were emanations from bodies, material molecules as it were, which imparted to the intervening air the same colour as that possessed by the body. Aristarchus said that colours are “shapes or forms stamping the air with impressions like themselves as it were,” that “colours in darkness have no colouring,” and that “light is the colour impinging on a substratum”.

THE HELIOCENTRIC HYPOTHESIS.

There is no doubt whatever that Aristarchus put forward the heliocentric hypothesis. Ancient testimony is unanimous on the point, and the first witness is Archimedes who was a younger contemporary of Aristarchus, so that there is no possibility of a mistake. Copernicus himself admitted that the theory was attributed to Aristarchus, though this does not seem to be generally known. Copernicus refers in two passages of his work, De revolutionibus caelestibus, to the opinions of the ancients about the motion of the earth. In the dedicatory letter to Pope Paul III he mentions that he first learnt from Cicero that one Nicetas (i.e. Hicetas) had attributed motion to the earth, and that he afterwards read in Plutarch that certain others held that opinion; he then quotes the Placita philosophorum according to which “Philolaus the Pythagorean asserted that the earth moved round the fire in an oblique circle in the same way as the sun and moon”. In Book I. c. 5 of his work Copernicus alludes to the views of Heraclides, Ecphantus, and Hicetas, who made the earth rotate about its own axis, and then goes on to say that it would not be very surprising if any one should attribute to the earth another motion besides rotation, namely, revolution in an orbit in space: “atque etiam (terram) pluribus motibus vagantem et unam ex astris Philolaus Pythagoricus sensisse fertur, Mathematicus non vulgaris”. Here, however, there is no question of the earth revolving round the sun, and there is no mention of Aristarchus. But Copernicus did mention the theory of Aristarchus in a passage which he afterwards suppressed: “Credibile est hisce similibusque causis Philolaum mobilitatem terrae sensisse, quod etiam nonnulli Aristarchum Samium ferunt in eadem fuisse sententia”.

It is desirable to quote the whole passage of Archimedes in which the allusion to Aristarchus’s heliocentric hypothesis occurs, in order to show the whole context.

“You are aware [‘you’ being King Gelon] that ‘universe’ is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while its radius is equal to the straight line between the centre of the sun and the centre of the earth. This is the common account as you have heard from astronomers. But Aristarchus brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the ‘universe’ just mentioned. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.”

The heliocentric hypothesis is here stated in language which leaves no room for doubt about its meaning. The sun, like the fixed stars, remains unmoved and forms the centre of a circular orbit in which the earth moves round it; the sphere of the fixed stars has its centre at the centre of the sun.

We have further evidence in a passage of Plutarch’s tract, On the face in the moon’s orb: “Only do not, my dear fellow, enter an action for impiety against me in the style of Cleanthes, who thought it was the duty of Greeks to indict Aristarchus on the charge of impiety for putting in motion the Hearth of the Universe, this being the effect of his attempt to save the phenomena by supposing the heaven to remain at rest and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis”.

Here we have the additional detail that Aristarchus followed Heraclides in attributing to the earth the daily rotation about its axis; Archimedes does not state this in so many words, but it is clearly involved by his remark that Aristarchus supposed the fixed stars as well as the sun to remain unmoved in space. A tract “Against Aristarchus” is mentioned by Diogenes Laertius among Cleanthes’s works; and it was evidently published during Aristarchus’s lifetime (Cleanthes died about 232 B.C.).

We learn from another passage of Plutarch that the hypothesis of Aristarchus was adopted, about a century later, by Seleucus, of Seleucia on the Tigris, a ChaldÆan or Babylonian, who also wrote on the subject of the tides about 150 B.C. The passage is interesting because it also alludes to the doubt about Plato’s final views. “Did Plato put the earth in motion as he did the sun, the moon and the five planets which he called the ‘instruments of time’ on account of their turnings, and was it necessary to conceive that the earth ‘which is globed about the axis stretched from pole to pole through the whole universe’ was not represented as being (merely) held together and at rest but as turning and revolving, as Aristarchus and Seleucus afterwards maintained that it did, the former of whom stated this as only a hypothesis, the latter as a definite opinion?”

No one after Seleucus is mentioned by name as having accepted the doctrine of Aristarchus and, if other Greek astronomers refer to it, they do so only to denounce it. Hipparchus, himself a contemporary of Seleucus, definitely reverted to the geocentric system, and it was doubtless his authority which sealed the fate of the heliocentric hypothesis for so many centuries.

The reasons which weighed with Hipparchus were presumably the facts that the system in which the earth revolved in a circle of which the sun was the exact centre failed to “save the phenomena,” and in particular to account for the variations of distance and the irregularities of the motions, which became more and more patent as methods of observation improved; that, on the other hand, the theory of epicycles did suffice to represent the phenomena with considerable accuracy; and that the latter theory could be reconciled with the immobility of the earth.

ON THE APPARENT DIAMETER OF THE SUN.

Archimedes tells us in the same treatise that “Aristarchus discovered that the sun’s apparent size is about 1/720th part of the zodiac circle”; that is to say, he observed that the angle subtended at the earth by the diameter of the sun is about half a degree.

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 p, 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 2?.

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 a, is not greater than a right angle, and a > , then

tan a/tan > a/ > sin a/sin . 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 a/tan > a/ > sin a/sin ,

where a, are angles not greater than a right angle and a > . Aristarchus also, in this proposition, cites 7/5 as an approximation by defect to the value of v2, 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 + (111)/(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 6585? days, and in this period the sun was estimated to describe 10?° 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 - ?, 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 10?° over 18 sidereal revolutions seems to have been an approximation based merely on the difference between 6585? days and 18 years of 365¼ days, i.e. 6574½ days; thus the 10?° 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.

LATER IMPROVEMENTS ON ARISTARCHUS’S FIGURES.

It may interest the reader to know how far Aristarchus’s estimates of sizes and distances were improved upon by later Greek astronomers. We are not informed how large he conceived the earth to be; but Archimedes tells us that “some have tried to prove that the circumference of the earth is about 300,000 stades and not greater,” and it may be presumed that Aristarchus would, like Archimedes, be content with this estimate. It is probable that it was Dicaearchus who (about 300 B.C.) arrived at this value, and that it was obtained by taking 24° (1/15th of the whole meridian circle) as the difference of latitude between Syene and Lysimachia (on the same meridian) and 20,000 stades as the actual distance between the two places. Eratosthenes, born a few years after Archimedes, say 284 B.C., is famous for a better measurement of the earth which was based on scientific principles. He found that at noon at the summer solstice the sun threw no shadow at Syene, whereas at the same hour at Alexandria (which he took to be on the same meridian) a vertical stick cast a shadow corresponding to 1/50th of the meridian circle. Assuming then that the sun’s rays at the two places are parallel in direction, and knowing the distance between them to be 5000 stades, he had only to take 50 times 5000 stades to get the circumference of the earth. He seems, for some reason, to have altered 250,000 into 252,000 stades, and this, according to Pliny’s account of the kind of stade used, works out to about 24,662 miles, giving for the diameter of the earth a length of 7850 miles, a surprisingly close approximation, however much it owes to happy accidents in the calculation.

Eratosthenes’s estimates of the sizes and distances of the sun and moon cannot be restored with certainty in view of the defective state of the texts of our authorities. We are better informed of Hipparchus’s results. In the first book of a treatise on sizes and distances Hipparchus based himself on an observation of an eclipse of the sun, probably that of 20th November in the year 129 B.C., which was exactly total in the region about the Hellespont, whereas at Alexandria about ?ths only of the diameter was obscured. From these facts Hipparchus deduced that, if the radius of the earth be the unit, the least distance of the moon contains 71, and the greatest 83 of these units, the mean thus containing 77. But he reverted to the question in the second book and proved “from many considerations” that the mean distance of the moon is 67? times the radius of the earth, and also that the distance of the sun is 2490 times the radius of the earth. Hipparchus also made the size (meaning thereby the solid content) of the sun to be 1880 times that of the earth, and the size of the earth to be 27 times that of the moon. The cube root of 1880 being about 12?, the diameters of the sun, earth and moon would be in the ratio of the numbers 12?, 1, ?. Hipparchus seems to have accepted Eratosthenes’s estimate of 252,000 stades for the circumference of the earth.

It is curious that Posidonius (135–51 B.C.), who was much less of an astronomer, made a much better guess at the distance of the sun from the earth. He made it 500,000,000 stades. As he also estimated the circumference of the earth at 240,000 stades, we may take the diameter of the earth to be, according to Posidonius, about 76,400 stades; consequently, if D be that diameter, Posidonius made the distance of the sun to be equal to 6545D as compared with Hipparchus’s 1245D.

Ptolemy does not mention Hipparchus’s figures. His own estimate of the sun’s distance was 605D, so that Hipparchus was far nearer the truth. But Hipparchus’s estimate remained unknown and Ptolemy’s held the field for many centuries; even Copernicus only made the distance of the sun 750 times the earth’s diameter, and it was not till 1671–3 that a substantial improvement was made; observations of Mars carried out in those years by Richer enabled Cassini to conclude that the sun’s parallax was about 9·5 corresponding to a distance between the sun and the earth of 87,000,000 miles.

Ptolemy made the distance of the moon from the earth to be 29½ times the earth’s diameter, and the diameter of the earth to be 3? times that of the moon. He estimated the diameter of the sun at 18? times that of the moon and therefore about 5½ times that of the earth, a figure again much inferior to that given by Hipparchus.