9. The differences from one another of the circles and the spheres in height are also given by Archimedes. He takes the perimeter of the Zodiac at 447,310,000 stadia, so that a straight line from the centre of the Earth to its extreme surface is the sixth part of the said number, and from the surface of the Earth on which we walk to the Zodiac is exactly one-sixth of the said number less 40,000 stadia which is the distance from the centre of the Earth to its surface. And from the circle of Kronos to the Earth, he says, the interval is 2,226,912,711 stadia; and from the p. 71. circle of the Fiery One to the Earth, 132,418,581; and from the Sun to the Earth, 121,604,454; from the Shining One to the Earth, 526,882,259; and from Aphrodite to the Earth, 50,815,160.[58]
10. And about the Moon we have before spoken. The distances and depths[59] of the spheres are thus given by Archimedes, but Hipparchus speaks differently about them, and Apollonius the mathematician differently again. But it is enough for us in following the Platonic theory to think of the intervals between the Wanderers as in ratios of 2 and 3. For thus is kept alive the theory of the harmonious construction of the universe in accordant ratios[60] by the same distances. But the numbers set out by Archimedes and the ratios quoted by the others concerning the distances, if they are not in accordant ratios, that is in those called by p. 72. Plato twofold and threefold, but are found to be outside the chords,[61] would not keep alive the theory of the harmonious construction of the universe. For it is neither probable nor possible that their distances should have no ratio to one another, that is, should be outside the chords and enharmonic scales. Except perhaps the Moon alone, from her waning and the shadows of the Earth, as to which planet alone you may trust Archimedes, that is to say for the distance of the Moon from the Earth. And it will be easy for those who accept this calculation to ascertain the number and the other distances according to the Platonic method by doubling and tripling as Plato demands.[62] If then, according to Archimedes, the Moon is distant from the Earth 5,544,130 stadia, it will be easy by increasing these numbers in ratios of 2 and 3 to find her distance from the rest by taking one fraction of the number of stadia by which the Moon is distant from the Earth.
But since the rest of the numbers stated by Archimedes about the distance of the Wanderers are not in accordant ratios, it is easy to know how they stand in regard to one p. 73. another and in what ratios they have been observed to be. But that the same are not in harmony and accord[63] when they are parts of the cosmos established by harmony is impossible. So then, as the first number (of stadia) by which the Moon is distant from the Earth is 5,544,130, the second number by which the Sun is distant from the Moon being 50,262,065, it is in ratio more than ninefold; and the number of the interval above this being 20,272,065 is in ratio less than one-half. And the number of the interval above this being 50,815,108 is in ratio more than twofold. And the number of the interval above this being 40,541,108 is in ratio more than one and a quarter.[64] And the number of the interval above this being 20,275,065 is in ratio more than half. And the number of the highest interval above this being 40,372,065 is in ratio less than twofold.[65]
11. These same ratios indeed—the more than ninefold, p. 74. less than half, more than twofold, less than one and a quarter, more than half, less than half and less than twofold are outside all harmonies and from them no enharmonic nor accordant system can come to pass. But the whole cosmos and its parts throughout are put together in an enharmonic and accordant manner. But the enharmonic and accordant ratios are kept alive as we have said before by the twofold and threefold intervals. If then we deem Archimedes worthy of faith on the distance given above, i. e., that from the Moon to the Earth, it is easy to find the rest by increasing it in the ratios of 2 and 3. Let the distance from the Earth to the Moon be, according to Archimedes, 5,544,130 stadia. The double of this will be the number of stadia by which the Sun is distant from the Moon, viz., 11,088,260. But from the Earth the Sun is distant 16,632,390 stadia and Aphrodite indeed from the Sun—16,632,390 stadia, but from the Earth 33,264,780. Ares indeed is distant from Aphrodite 22,176,520 stadia but from the Earth 105,338,470. But Zeus is distant from Ares 44,353,040 stadia, but from p. 75 the Earth 149,691,510. Kronos is distant from Zeus 40,691,510 stadia, but from the Earth 293,383,020.[66]
12. Who will not wonder at so much activity of mind produced by so great labour? It seems that this Ptolemy[67] who busies himself with these matters is not without his use to me. This only grieves me that as one but lately born he was not serviceable to the sons of the giants,[68] who, being ignorant of these measurements, thought they were near high heaven and began to make a useless tower. Had he been at hand to explain these measurements to them they would not have ventured on the foolishness. But if any one thinks he can disbelieve this let him take the measurements and be convinced; for one cannot have for the unbelieving a more manifold proof than this. O puffing-up of vainly-toiling soul and unbelieving belief, when Ptolemy is considered wise in everything by those trained in the like wisdom![69]
13. Certain men in part intent on these things as judging p. 76. them mighty and worthy of argument have constructed measureless[70] and boundless heresies. Among whom is one Colarbasus,[71] who undertakes to set forth religion by measures and numbers. And there are others whom we shall likewise point out when we begin to speak of those who give heed to Pythagorean reckoning as if it were powerful and neglect the true philosophy for numbers and elements, thus making vain divinations. Collecting whose words, certain men have led astray the uneducated, pretending to know the future and when they chance to divine one thing aright are not ashamed of their many failures, but make a boast of their one success. Nor shall I pass over their unwise wisdom, but when I have set forth their attempts to establish a religion from these sources, I shall refute them as being disciples of a school inconsistent and full of trickery.
2. Of Mathematicians.[72]
p. 77.