Aristarco de Samos: Sobre os tamanhos e distâncias entre o Sol e a Lua

ARISTARCHUS OF SAMOS

HISTORIANS of mathematics have, as a rule, given too little attention to Aristarchus of Samos. The reason is no doubt that he was an astronomer, and therefore it might be supposed that his work would have no sufficient interest for the mathematician. The Greeks knew better; they called him Aristarchus ‘the mathematician’, to distinguish him from the host of other Aristarchuses; he is also included by Vitruvius among the few great men who possessed an equally profound knowledge of all branches of science, geometry, astronomy, music, &c.

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 (lit. ‘numerical’) principles. ⁽¹⁾

That Aristarchus was a very capable geometer is proved by his extant work On the sizes and distances of the Sun and Moon which will be noticed later in this chapter: in the mechanical line he is credited with the discovery of an improved sun-dial, the so-called σκάφη, 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 the 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 his master, Strato of Lampsacus; thus 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, while 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’.

Two facts enable us to fix Aristarchus’s date approximately. In 281/280 B.C. he made an observation of the summer solstice; and a book of his, presently to be mentioned, was published before the date of Archimedes’s Psammites or Sand-reckoner, a work written before 216 B.C. Aristarchus, therefore, probably lived circa 310-230 B.C., that is, he was older than Archimedes by about 25 years.

To Aristarchus belongs the high honour of having been the first to formulate the Copernican hypothesis, which was then abandoned again until it was revived by Copernicus himself. His claim to the title of ‘the ancient Copernicus’ is still, in my opinion, quite unshaken, notwithstanding the ingenious and elaborate arguments brought forward by Schiaparelli to prove that it was Heraclides of Pontus who first conceived the heliocentric idea. Heraclides is (along with one Ecphantus, a Pythagorean) credited with having been the first to hold that the earth revolves about its own axis every 24 hours, and he was the first to discover that Mercury and Venus revolve, like satellites, about the sun. But though this proves that Heraclides came near, if he did not actually reach, the hypothesis of Tycho Brahe, according to which the earth was in the centre and the rest of the system, the sun with the planets revolving round it, revolved round the earth, it does not suggest that he moved the earth away from the centre. The contrary is indeed stated by Aëtius, who says that ‘Heraclides and Ecphantus make the earth move, not in the sense of translation, but by way of turning on an axle, like a wheel, from west to east, about its own centre’ ⁽²⁾. None of the champions of Heraclides have been able to meet this positive statement. But we have conclusive evidence in favour of the claim of Aristarchus; indeed, ancient testimony is unanimous on the point. Not only does Plutarch tell us that Cleanthes held that Aristarchus ought to be indicted for the impiety of ‘putting the Hearth of the Universe in motion’ ⁽³⁾; we have the best possible testimony in the precise statement of a great contemporary, Archimedes. In the Sand-reckoner Archimedes has this passage.

You [King Gelon] are aware 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 last statement is a variation of a traditional phrase, for which there are many parallels (cf. Aristarchus’s Hypothesis 2 ‘that the earth is in the relation of a point and centre to the sphere in which the moon moves’), and is a method of saying that the ‘universe’ is infinitely great in relation not merely to the size of the sun but even to the orbit of the earth in its revolution about it ; the assumption was necessary to Aristarchus in order that he might not have to take account of parallax.)

Plutarch, in the passage referred to above, also makes it clear that Aristarchus followed Heraclides in attributing to the earth the daily rotation about its axis. The bold hypothesis of Aristarchus found few adherents. Seleucus, of Seleucia on the Tigris, is the only convinced supporter of it of whom we hear, and it was speedily abandoned altogether, mainly owing to the great authority of Hipparchus. Nor do we find any trace of the heliocentric hypothesis in Aristarchus’s extant work On the sizes and distances of the Sun and Moon. This is presumably because that work was written before the hypothesis was formulated in the book referred to by Archimedes. The geometry of the treatise is, however, unaffected by the difference between the hypotheses.

Archimedes also says that it was Aristarchus who discovered that the apparent angular diameter of the sun is about 1/720th part of the zodiac circle, that is to say, half a degree. We do not know how he arrived at this pretty accurate figure: but, as he is credited with the invention of the σκάΦη, he may have used this instrument for the purpose. But here again the discovery must apparently have been later than the treatise On sizes and distances, for the value of the subtended angle is there assumed to be 2° (Hypothesis 6). How Aristarchus came to assume a value so excessive is uncertain. As the mathematics of his treatise is not dependent on the actual value taken, 2° may have been assumed merely by way of illustration; or it may have been a guess at the apparent diameter made before he had thought of attempting to measure it. Aristarchus assumed that the angular diameters of the sun and moon at the centre of the earth are equal.

The method of the treatise depends on the just observation, which is Aristarchus’s third ‘hypothesis’, 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’; the effect of this (since the moon receives its light from the sun), is that at the time of the dichotomy the centres of the sun, moon and earth form a triangle right-angled at the centre of the moon. Two other assumptions were necessary: first, an estimate of the size of the angle of the latter triangle at the centre of the earth at the moment of dichotomy: this Aristarchus assumed (Hypothesis 4) to be ‘less than a quadrant by one-thirtieth of a quadrant’, i. e. 87°, again an inaccurate estimate, the true value being 89° 50′; secondly, an estimate of the breadth of the earth’s shadow where the moon traverses it: this he assumed to be ‘the breadth of two moons’ (Hypothesis 5).

The inaccuracy of the assumptions does not, however, detract from the mathematical interest of the succeeding investigation. Here we find the logical sequence of propositions and the absolute rigour of demonstration characteristic of Greek geometry; the only remaining drawback would be the practical difficulty of determining the exact moment when the moon ‘appears to us halved’. The form and style of the book are thoroughly classical, as befits the period between Euclid and Archimedes; the Greek is even remarkably attractive. The content from the mathematical point of view is no less interesting, for we have here the first specimen extant of pure geometry used with a trigonometrical object, in which respect it is a sort of forerunner of Archimedes’s Measurement of a Circle. Aristarchus does not actually evaluate the trigonometrical ratios on which the ratios of the sizes and distances to be obtained depend; he finds limits between which they lie, and that by means of certain propositions which he assumes without proof, and which therefore must have been generally known to mathematicians of his day. These propositions are the equivalents of the statements that,

1. if α is what we call the circular measure of an angle  and α is less than \(\frac{1}{2}\pi \), then the ratio \(\frac{{\sin \alpha }}{\alpha }\) decreases, and the ratio \(\frac{{\tan \alpha }}{\alpha }\) increases, as α increases from 0 to \(\frac{1}{2}\pi \);
2. if β be the circular measure of another angle less than \(\frac{1}{2}\pi \), then

\[\begin{array}{*{20}{l}}{\frac{{\sin \alpha }}{{\sin \beta }}}& < &{\frac{\alpha }{\beta }}& < &{\frac{{\tan \alpha }}{{\tan \beta }}}\end{array}\]

Aristarchus of course deals, not with actual circular measures, sines and tangents, but with angles (expressed not in degrees but as fractions of right angles), arcs of circles and their chords. Particular results obtained by Aristarchus are the equivalent of the following:

\[\begin{array}{*{20}{l}}{\frac{1}{{18}}}& > &{\sin 3^\circ }& > &{\frac{1}{{20}}}&{}&{}&{\left[ {{\rm{Prop}}{\rm{. 7}}} \right]}\\{\frac{1}{{45}}}& > &{\sin 1^\circ }& > &{\frac{1}{{60}}}&{}&{}&{\left[ {{\rm{Prop}}{\rm{. 11}}} \right]}\\1& > &{\cos 1^\circ }& > &{\frac{{89}}{{90}}}&{}&{}&{\left[ {{\rm{Prop}}{\rm{. 12}}} \right]}\\1& > &{{{\cos }^2}1^\circ }& > &{\frac{{44}}{{45}}}&{}&{}&{\left[ {{\rm{Prop}}{\rm{. 13}}} \right]}\end{array}\]

The book consists of eighteen propositions. Beginning with six hypotheses to the effect already indicated, Aristarchus declares that he is now in a position to prove

1. that 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;
2. that the diameter of the sun has the same ratio as aforesaid to the diameter of the moon;
3. that the diameter of the sun has to the diameter of the earth a ratio greater than 19:3, but less than 43:6.

The propositions containing these results are Props. 7, 9 and 15.

⁽¹⁾ Vitruvius, De architecture, i. 1. 16.

⁽²⁾ Aët. iii. 13. 3, Vors. i ³ , p. 341. 8.

⁽³⁾ Plutarch, De facie in orbe lunae, c. 6, pp. 922 F-923 A.

• Extraído de A HISTORY OF GREEK MATHEMATICS, BY SIR THOMAS HEATH, VOLUME II – FROM ARISTARCHUS TO DIOPHANTUS, OXFORD, AT THE CLARENDON PRESS 1921

HYPOTHESES AND PROPOSITIONS

1. That the moon receives its light front the sun.
2. That the earth is in the relation of a point and centre to the sphere in which the moon moves.⁽¹⁾
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.⁽²⁾
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.⁽³⁾
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.⁽⁴⁾

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.⁽⁵⁾
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.

⁽¹⁾ Literally ‘the sphere of the moon’.

⁽²⁾ Literally ‘verges towards our eye’, the word ‘νεύειν‘ meaning to ‘verge’ or ‘incline’. What is meant is that the plane of the great circle in question passes through the observer’s eye or, in other words, that his eye and the great circle are in one plane (cf. Aristarchus’s own explanation in the enunciation of Prop. 5).

⁽³⁾ I.e. is less than 90º by 1/30th of 90º or 3º, and is therefore equal to 87º.

⁽⁴⁾ I. e. 1/15th of 30º, or 2°. Archimedes in his Sand-reckoner (Archimedes, ed. Heiberg, ii, p. 248, 19) says that Aristarchus ‘discovered that the sun appeared to be about 1/720th part of the circle of the zodiac’; that is, Aristarchus discovered (evidently at a date later than that of our treatise) the much more correct value of 1/2° for the angular diameter of the sun or moon (for he maintained that both had the same angular diameter: cf. Prop. 8). Archimedes himself in the same place describes a rough method of observation by which he inferred that the diameter of the sun was less than 1/164th part, and greater than 1/200th part, of a right angle. Cf. pp. 311-12 ante.

⁽⁵⁾ Pappus gives this second result immediately after the first result, i. e. before the parenthesis ‘this follows from the hypothesis . . .’. This arrangement seems at first sight more appropriate, and Nizze alters his text accordingly. But I think it better to follow the above order which is that of the MSS. and Wallis. One consideration which weighs with me is that the second result does not follow from the hypothesis of the halved moon alone; it depends on another assumption also, namely, that the sun and the moon have the same apparent angular diameter (see Prop. 8).

• Extraído de Aristarchus of Samos, the ancient Copernicus; a history of Greek astronomy to Aristarchus, together with Aristarchus’s Treatise on the sizes and distances of the sun and moon, by Heath, Thomas Little, Sir, 1861-1940; Aristarchus, of Samos. On the sizes and distances of the sun and moon. English & Greek. 1913

THE PROPOSITIONS

Proposition 1.
Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres.

Proposition 2.
If a sphere be illuminated by a sphere greater than itself, the illuminated portion of the former sphere will be greater than a hemisphere.

Proposition 3.
The circle in the moon which divides the dark and the bright portions is least when the cone comprehending both the sun and the moon has its vertex at our eye.

Proposition 4.
The circle which divides the dark and the bright portions in the moon is not perceptibly different from a great circle in the moon.

Proposition 5.
When the moon appears to us halved, the great circle parallel to the circle which divides the dark and the bright portions in the moon is then in the direction of our eye; that is to say, the great circle parallel to the dividing circle and our eye are in one plane.

Proposition 6.
The moon moves (in an orbit) lower than (that of) the sun, and, when it is halved, is distant less than a quadrant from the sun.

Proposition 7.
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.

Proposition 8.
When the sun is totally eclipsed, the sun and the moon are then comprehended by one and the same cone which has its vertex at our eye.

Proposition 9.
The diameter of the sun is greater than 18 times, but less than 20 times, the diameter of the moon.

Proposition 10.
The sun has to the moon a ratio greater than that which 5832 has to 1, but less than that which 8000 has to 1.

Proposition 11.
The diameter of the moon is less than 2/45ths, but greater than 1/30th, of the distance of the centre of the moon from our eye.

Proposition 12.
The diameter of the circle which divides the dark and the bright portions in the moon is less than the diameter of the moon, but has to it a ratio greater than that which 89 has to 90.

Proposition 13.
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 is less than double of the diameter of the moon, but has to it a ratio greater than that which 88 has to 45; and it is less than 1/9th part of the diameter of the sun, but has to it a ratio greater than that which 21 has to 225. But it has to the straight line drawn from the centre of the sun at right angles to the axis and meeting the sides of the cone a ratio greater than that which 979 has to 10125.

Proposition 14.
The straight line joined from the centre of the earth to the centre of the moon has to the straight line cut off from the axis towards the centre of the moon by the straight line subtending the (circumference) within the earth’s shadow a ratio greater than that which 675 has to 1.

Proposition 15.
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.

Proposition 16.
The sun has to the earth a ratio greater than that which 6859 has to 27, but less than that which 79507 has to 216.

Proposition 17.
The diameter of the earth is to the diameter of the moon in a ratio greater than that which 108 has to 43, but less than that which 60 has to 19.

Proposition 18.
The earth is to the moon in a ratio greater than that which 1259712 has to 79507, but less than that which 216000 has to 6859.

As conclusões de Paul Tannery

No final do estudo publicado na Memória Aristarque de Samos, Paul Tannery conclui:

Je résumerai comme suit les principales conclusions de cette étude:

1.° Les anciens n’ont connu pour la détermination des distances du Soleil et de la Lune qu’une seule méthode, dont l’invention doit être attribuée à Eudoxe, qui fut le véritable fondateur de l’astronomie théorique.

2.º Cette méthode supposait la détermination de deux éléments dont l’un, le diamètre du cercle d’ombre de la Terre, peut être fixé avec assez de précision par l’observation des éclipses lunaires, mais dont l’autre, la distance angulaire du Soleil et de la Lune au moment de la dichotomie, ne put jamais être mesuré en réalité avec quelque semblant d’approximation.

3.º Comme calcul, cette méthode était de fait très simple, si l’on se contentait d’une approximation en rapport avec l’incertitude des données. Le rôle d’Âristarque fut de lui donner une rigueur géométrique, mais le défaut de la trigonométrie l’obligea à exagérer les limites des erreurs provenant du calcul.

4.º Pour déterminer les dimensions des deux astres par rapport à la Terre, il fallait de plus connaître leur diamètre apparent et le rapport de la circonférence au diamètre. Avant Ârchimède, les Grecs savaient seulement que ce rapport était compris entre 3 e 3 1/3.

5.º Du vice irrémédiable de la méthode, il s’ensuivit que la distance et les dimensions du Soleil furent toujours calculées d’une façon absolument erronée, tandis que pour la Lune ces éléments purent être déterminés avec une approximation de plus en plus satisfaisante.