Cleomedes: how big is the earth?

Some time between Posidonius and Ptolemy, i.e. between the 1st century BC and the 2nd century AD, a Greek named Cleomedes wrote a 2 book basic treatise on astronomy, De motu circulari corporum caelestium.^[1]  This was based mainly on the lost work of Posidonius, but also on others.

Cleomedes is our primary source for the calculations of Erastothenes, who measured the earth in the 3rd century BC.  I had never seen this, and I happened across a translation of the relevant portion, so I thought that it might be interesting to reproduce the passage here.

The text was edited in 1891 in the Teubner series by Herman Ziegler, who based his edition upon three manuscripts which he describes in a vague way.  I have supplemented his atrocious list from Pinakes:

•  Florence, Mediceo-Laurentianus Plutei LXIX, 13.  12th century, fol.137v-164, where Cleomedes follows Diogenes Laertius, Lives of the Philosophers.
• Leipzig Universitäts-Bibliothek gr. 16 (once 361; 250), foll. 286-298.
• Nuremberg, Stadtbibliothek, fonds principal, Cent. V. App. 37, 15th century.

He casually informs us that others exist at the Marciana library in Venice (no. CCXIV and CCCVIII, supposedly 11-12th century), and six others which he does not trouble to name.  Fortunately we have the Pinakes list of manuscripts here.

As he gives a shelfmark only for the Florence ms., we may reasonably infer that Ziegler followed the Florence manuscript, and borrowed material from others as he found it useful.  But we must remember that the task of editing a technical astronomical work may well have been considerable.

Edit (3rd Jan. 2019): A correspondent tells me that a new Teubner edition by R. B. Todd came out in 1990, with a proper account of the manuscripts.  This I have not seen, however.

Cleomedes has recently been translated into English, I find, although again I have not seen this; Alan C. Bowen, Robert B. Todd, Cleomedes’ Lectures on Astronomy. A Translation of The Heavens with an Introduction and Commentary. University of California Press, 2004.  A French translation by “R. Goulart” in 1980^[2] is described as “faulty” in the Oxford Classical Dictionary.^[3]

Here’s the relevant passage, from Heath’s translation of Cleomedes, On the orbits of the heavenly bodies, I, 10.^[4]

About the size of the earth the physicists, or natural philosophers,  have held different views, but those of Posidonius and Eratosthenes are preferable to the rest. The latter shows the size of the earth by a geometrical method; the method of Posidonius is simpler. Both lay down certain hypotheses, and, by successive inferences from the hypotheses, arrive at their demonstrations.

Posidonius says that Rhodes and Alexandria lie under the same meridian. Now meridian circles are circles which are drawn through the poles of the universe and through the point which is above the head of any individual standing on the earth. The poles are the same for all these circles, but the vertical point is different for different persons. Hence we can draw an infinite number of meridian circles. Now Rhodes and Alexandria lie under the same meridian circle, and the distance between the cities is reputed to be 5,000 stades.^[5] Suppose this to be the case.

All the meridian circles are among the great circles in the universe, dividing it into two equal parts and being drawn through the poles. With these hypotheses, Posidonius proceeds to divide the zodiac circle, which is equal to the mendian circles, because it also divides the universe into two equal parts, into forty-eight parts, thereby cutting each of the twelfth parts of it (i.e., signs) into four. If, then, the meridian circle through Rhodes and Alexandria is divided into the same number of parts, forty-eight, as the zodiac circle, the segments of it are equal to the aforesaid segments of the zodiac. For, when equal magnitudes are divided into (the same number of) equal parts, the parts of the divided magnitudes must be respectively equal to the parts. This being so, Posidonius goes on to say that the very bright star called Canopus lies to the south, practically on the Rudder of Argo. The said star is not seen at all in Greece; hence Aratus does not even mention it in his Phaenomena. But, as you go from north to south, it begins to be visible at Rhodes and, when seen on the horizon there, it sets again immediately as the universe revolves.1 But when we have sailed the 5,000 stades and are at Alexandria, this star, when it is exactly in the middle of the heaven, is found to be at a height above the honzon of one-fourth of a sign, that is,one forty-eighth part of the zodiac circle.’ It follows, therefore, that the segment of the same mendian circle which lies above the distance between Rhodes and Alexandria is one forty-eighth part of the said circle, because the honzon of the Rhodians is distant from that of the Alexandrians by one forty-eighth of the zodiac circle. Since, then, the part of the earth under this segment is reputed to be 5,000 stades, the parts (of the earth) under the other (equal) segments (of the meridian circle) also measure 5,000 stades; and thus the great circle of the earth is found to measure 240,000 stades, assuming that from Rhodes to Alexandria is 5,000 stades; but, if not, it is in (the same) ratio to the distance. Such then is Posidonius’ way of dealing with the size of the earth.

The method of Eratosthenes1 depends on a geometrical argument and gives the impression of being slightly more difficult to follow. But his statement will be made clear if we premise the following. Let us suppose, in this case too, first, that Syene and Alexandria he under the same meridian circle, secondly, that the distance between the two cities is 5,000 stades;1 and thirdly, that the rays sent down from different parts of the sun on different parts of the earth are parallel; for this is the hypothesis on which geometers proceed Fourthly, let us assume that, as proved by the geometers, straight lines falling on parallel straight lines make the alternate angles equal, and fifthly, that the arcs standing on (i e., subtended by) equal angles are similar, that is, have the same proportion and the same ratio to their proper circles—this, too, being a fact proved by the geometers. Whenever, therefore, arcs of circles stand on equal angles, if any one of these is (say) one-tenth of its proper circle, all the other arcs will be tenth parts of their proper circles.

Any one who has grasped these facts will have no difficulty in understanding the method of Eratosthenes, which is this Syene and Alexandria lie, he says, under the same mendian circle. Since meridian circles are great circles in the universe, the circles of the earth which lie under them are necessarily also great circles. Thus, of whatever size this method shows the circle on the earth passing through Syene and Alexandria to be, this will be the size of the great circle of the earth Now Eratosthenes asserts, and it is the fact, that Syene lies under the summer tropic. Whenever, therefore, the sun, being m the Crab at the summer solstice, is exactly in the middle of the heaven, the gnomons (pointers) of sundials necessarily throw no shadows, the position of the sun above them being exactly vertical; and it is said that this is true throughout a space three hundred stades in diameter.1 But in Alexandria, at the same hour, the pointers of sundials throw shadows, because Alexandria lies further to the north than Syene. The two cities lying under the 9ame meridian great circle, if we draw an arc from the extremity of the shadow to the base of the pointer of the sundial in Alexandria, the arc will be a segment of a great circle in the (hemispherical) bowl of the sundial, since the bowl of the sundial lies under the great circle (of the meridian). If now we conceive straight lines produced from each of the pointers through the earth, they will meet at the centre of the earth. Since then the sundial at Syene is vertically under the sun, if we conceive a straight line coming from the sun to the top of the pointer of the sundial, the line reaching from the sun to the centre of the earth will be one straight line. If now we conceive another straight line drawn upwards from the extremity of the shadow of the pointer of the sundial in Alexandria, through the top of the pointer to the sun, this straight line and the aforesaid straight line will be parallel, since they are straight lines coming through from different parts of the sun to different parts of the earth. On these straight lines, therefore, which are parallel, there falls the straight line drawn from the centre of the earth to the pointer at Alexandria, so that the alternate angles which it makes arc equal. One of these angles is that formed at the centre of the earth, at the intersection of the straight lines which were drawn from the sundials to the centre of the earth; the other is at the point of intersection of the top of the pointer at Alexandria and the straight line drawn from the extremity of its shadow to the sun through the point (the top) where it meets the pointer * Now on this latter angle stands the arc carried round from the extremity of the shadow of the pointer to its base, while on the angle at the centre of the earth stands the arc reaching from Syene to Alexandria. But the arcs are similar, since they stand on equal angles. Whatever ratio, therefore, the arc in the bowl of the sundial has to its proper circle, the arc reaching from Syene to Alexandria has that ratio to its proper circle. But the arc in the bowl is found to be one-fiftieth of its proper circle.’ Therefore the distance from Syene to Alexandria must necessarily be one-fiftieth part of the great circle of the earth. And the said distance is 5,000 stades; therefore the complete great circle measures 250,000 stades. Such is Eratosthenes’ method.

1. ^[1]H. Ziegler, Cleomediis de Motu Circulari Corporum Caelestium Libri, Leipzig: Teubner, 1891, at Google Books here.  Includes Latin translation.  There is an updated Teubner by R.B. Todd, 1990.↩
2. ^[2]Probably R. Goulet, Cléomède: Théorie Élémentaire. Texte présenté, traduit et commenté, Paris, 1980.↩
3. ^[3]Simon Hornblower etc, The Oxford Classical Dictionary, OUP, 2012. Article preview here on p.331.  This dates the work at ca. 360 AD, and says that Cleomedes’ account of the work of Erastothenes is “mostly fictitious”.↩
4. ^[4]Translated by T. L. Heath in Greek Astronomy, London (1932); via Cohen & Drabkin, A source book in Greek Science, 1948, p.149-153.  This contains diagrams and detailed but rather unclear comment.↩
5. ^[5]Which is actually too much↩