On the magnitudes and distances of the sun and moon (περὶ μεγεθῶν καὶ ἀποστημάτων ἡλίου καὶ σελήνης)

Aristarchus seems that he employed himself in the determination of some of the most important elements of astronomy; but none of his works remain, except a treatise on the magnitudes and distances of the sun and moon (περὶ μεγεθῶν καὶ ἀποστημάτων ἡλίου καὶ σελήνης). We do not know whether the method employed in this work was invented by Aristarchus (Suidas, s. v. φιλόσοφος, mentions a treatise on the same subject by a disciple of Plato); it is, however, very ingenious, and correct in principle. It is founded on the consideration that at the instant when the enlightened part of the moon is apparently bounded by a straight line, the plane of the circle which separates the dark and light portions passes through the eye of the spectator, and is also perpendicular to the line joining the centres of the sun and moon; so that the distances of the sun and moon from the eye are at that instant respectively the hypothenuse and side of a right-angled triangle. The angle at the eye (which is the angular distance between the sun and moon) can be observed, and then it is an easy problem to find the ratio between the sides containing it. But this process could not, unless by accident, lead to a true result; for it would be impossible, even with a telescope, to determine with much accuracy the instant at which the phaenomenon in question takes place; and in the time of Aristarchus there were no means of measuring angular distances with sufficient exactness. In fact, he takes the angle at the eye to be 83 degrees whereas its real value is less than a right angle by about half a minute only; and hence he infers that the distance of the sun is between eighteen and twenty times greater than that of the moon, whereas the true ratio is about twenty times as great, the distances being to one another nearly as 400 to 1. The ratio of the true diameters of the sun and moon would follow immediately from that of their distances, if their apparent (angular) diameters were known. Aristarchus assumes that their apparent diameters are equal, which is nearly true ; but estimates their common value at two degrees, which is nearly four times too great. The theory of parallax was as yet unknown, and hence, in order to compare the diameter of the earth with the magnitudes already mentioned, he compares the diameter of the moon with that of the earth's shadow in its neighbourhood, and assumes the latter to be twice as great as the former. (Its mean value is about 84'.) Of course all the numerical results deduced from these assumptions are, like the one first mentioned, very erroneous. The geometrical processes employed shew that nothing like trigonometry was known. No attempt is made to assign the absolute values of the magnitudes whose ratios are investigated; in fact, this could not be done without an actual measurement of the earth--an operation which seems to have been first attempted on scientific principles in the next generation. [ERATOSTHENES.] Aristarchus does not explain his method of determining the apparent diameters of the sun and of the earth's shadow; but the latter must have been deduced from observations of lunar eclipses, and the former may probably have been observed by means of the skaphium by a method described by Macrobius. (Somn. Scip. 1.20.) This instrument is said to have been invented by Aristarchus (Vitr. 9.9) : it consisted of an improved gnomon [ANAXIMANDER], the shadow being received not upon a horizontal plane, but upon a concave hemispherical surface having the extremity of the style at its centre, so that angles might be measured directly by arcsinstead of by their tangents. The gross error in the value attributed to the sun's apparent diameter is remarkable; it appears, however, that Aristarchus must afterwards have adopted a much more correct estimate, since Archimedes in the Ψαμμίτης (Wallis, Op. vol. iii. p. 515) refers to a treatise in which he made it only half a degree. Pappus, whose commentary on the book περὶ μεγεθῶν, &c. is extant, does not notice this emendation, whence it has been conjectured, that the other works of Aristarchus did not exist in his time, having perhaps perished with the Alexandrian library.

It has been the common opinion, at least in modern times, that Aristarchus agreed with Philolaus and other astronomers of the Pythagorean school in considering the sun to be fixed, and attributing a motion to the earth. Plutarch (de fac. in orb. lun. p. 922) says, that Cleanthes thought that Aristarthus ought to be accused of impiety for supposing (ὑποτιθέμενος), that the heavens were at rest, and that the earth moved in an oblique circle, and also about its own axis (the true reading is evidently Κλεάνθης ᾤετο δεῖν Ἀρίσταρχον, κ. τ. λ.); and Diogenes Laertius, in his list of the works of Cleanthes mentions one πρὸς Ἀρίσταρχον. (See also Sext. Empir. ad v. Math. p. 410c.; Stobaeus, 1.26.) Archimedes, in the ψαμμίτης (l.c.), refers to the same theory. (ὑποτίθεται γὰρ, κ. τ. λ.) But the treatise περὶ μεγεθῶν contains not a word upon the subject, nor does Ptolemy allude to it when he maintains the immobility of the earth. It seems therefore probable, that Aristarchus adopted it rather as a hypothesis for particular purposes than as a statement of the actual system of the universe. In fact, Plutarch, in another place (Plat. Quest. p. 1006) expressly says, that Aristarchus taught it only hypothetically. On this question, see Schaubach. (Gesch. d. Griech. Astronomie, p. 468, &c.) It appears from the passage in the ψαμμίτης alluded to above, that Aristarchus had much juster views than his predecessors concerning the extent of the universe. He maintained, namely, that the sphere of the fixed stars was so large, that it bore to the orbit of the earth the relation of a sphere to its centre. What he meant by the expression, is not clear : it may be interpreted as an anticipation of modern discoveries, but in this sense it could express only a conjecture which the observations of the age were not accurate enough either to confirm or refute--a remark which is equally applicable to the theory of the earth's motion. Whatever may be the truth on these points, it is probable that even the opinion, that the sun was nearly twenty times as distant as the moon, indicates a great step in advance of the popular doctrines.

Censorinus (de Die Natali, 100.18) attributes to Aristarchus the invention of the magnus annus of 2484 years.