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 ⅘ths 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 67⅓ 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 12⅓, the diameters of the sun, earth and moon would be in the ratio of the numbers 12⅓, 1, ⅓. 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 3⅖ times that of the moon. He estimated the diameter of the sun at 18⅘ times that of the moon and therefore about 5½ times that of the earth, a figure again much inferior to that given by Hipparchus.