The distance E A being known, and E p being assumed equal to it, the distance E a can be calculated from the known ratio E a: p a.
So far (with the exception of the mansions of the moon) the Arab writer has followed the Greek, but we have now reached a point where he diverges. Ptolemy, he says, only tells us the distances and sizes of the sun and moon, and said nothing about the other heavenly bodies; but if we suppose the greatest distance of the moon to be the same as the least distance of Mercury, and from this calculate his greatest distance (for the ratio is known), and if we proceed in the same way with Mercury and Venus, we shall find that the greatest distance of Venus equals the least distance of the sun as given by Ptolemy. Ptolemy’s least distance for the sun, which was totally wrong, was 1160 Earth-radii: the greatest distance of Venus, calculated in this way from Ptolemy’s figures, was 1150. Alfraganus takes this unlucky coincidence as an indication that there is only just sufficient space between each sphere and the next to allow their respective epicycles to pass one another, and upon this entirely erroneous assumption he proceeds to lay down the distances of each planet from the earth, and finally of the stars, which are all supposed to be at the same distance, equal to the greatest distance of Saturn.
Who first suggested this method of estimating distances we do not know: the first mention of it occurs in Europe in the fifth century a.d. The following table shows the distances obtained in this way:—
GREATEST DISTANCE.
| In Semi-Diameters of Earth. | |
|---|---|
| Moon | 64⅛ |
| Mercury | 167 |
| Venus | 1120 |
| Sun | 1220 |
| Mars | 8876 |
| Jupiter | 14,405 |
| Saturn and Stars | 20,110 |
In this, the moon’s distance is approximately correct, but the sun’s is not much more than one-twentieth of its true value. To set the stars at a distance of only twenty thousand times Earth’s semi-diameter seems to us to bring them very close,[67] but they would still be beyond measurement by naked eye methods, so it is no contradiction to Alfraganus’ earlier statement that Earth is a point compared with the heavens.
The Arabs also believed that they had succeeded in measuring the apparent diameters of the planets and even of the points of light which are all we can see of stars, so Alfraganus gives the accepted sizes of all. I give them below in descending order of size. The modern values in the third column show how false were the results obtained by this mistaken method.
DIAMETER: EARTH = 1.
| Alfraganus. | Modern Values. | ||
|---|---|---|---|
| Sun | 5½ | 109½ | |
| The 15 first-magnitude stars | 4¾ |  | Arcturus, Sirius, Spica, and others, |
| much larger than the sun. | |||
| Jupiter | 4⁹/₁₆ | 11 | |
| Saturn | 4½ | 9 | |
| Other stars, in order of | | Various. Some certainly | |
| magnitude, 2nd to 6th ... | larger than the sun. | ||
| Mars | 1⅛ | ½ | |
| Earth | 1 | 1 | |
| Venus | ³/₁₀ | ⁹/₁₀ | |
| Moon | ⁵/₁₇ | ¼ | |
| Mercury | ¹/₁₈ | ⅓ | |
In the above table the size of the moon (whose parallax had been found by the Greeks) is the only one which is nearly right. The sun is far too small, and so are the stars. We cannot yet know with certainty the diameter of any star, but they are all comparable with the sun, and many are enormously larger.[68] As their distances are all different, some of the brightest may be comparatively small, and some of the faintest the largest of all.