HOW ARCHIMEDES EXPECTED TO MOVE THE EARTH

by

A. G. Drachmann

The story of Archimedes’s inventions for hauling and lifting, with his famous saying that he would move the earth itself, if he could have a place to stand on, has reached us in so many shapes that it is not at all easy to get to the bottom of it. That he did invent cranes of hitherto unknown power seems to be quite certain, unless we are to reject the descriptions in Polybius”, Liviuss, and Plutarchusg of his activity in the defense of Syracuse. We are told that he sent stones weighing 260 kg down on the enemy’s ships, and that he lifted the ships out of the water and dashed them against the rocks.

Of the beginning of this display of mechanics applied to warfare Plutarchuslo tells us this: “Archimedes, who was a kinsman and a friend of King Hieron (of Syracuse) had written that it is possible to lift a given weight by means of a given power, and in his youthful enthusiasm he is said to have uttered, on the strength of the proof, that if he had another earth, he would move this earth, when he had taken place on the other one.

“When Hieron wondered and wanted him to put the theory to proof in practice and show some big thing that was moved by a small power, he loaded one of the king’s three-masted freightships, which had been hauled on land with great labour and by many men, with a large crew and a full load, and then drew it along, smoothly and evenly as if it was floating in water, not with great labour, but sitting down at a distance, gently swinging with his hand the end of a compound tackle”.

Hieron then persuaded him to prepare the defense of Syracuse.

Centaurus 1958: vol. 5: no. 3-4: pp. 278-282

How ARCHIMEDES EWE- TO MOVE THE EARTH 279

Plutarchus does not tell us how far the ship was moved. Since it was a demonstration only, a few yards would be enough, and t h i s makes the story more credible. But later authors have dressed it up by combining it with the account of the big ship presented by Hieron to king Ptolemy of Egypt.

Athenaeusl, who quotes an otherwise unknown Moschion as his source, tells us that Archimedes superintended the building, and that when the hull was finished and had to be launched, and no one knew how to do it, Archimedes himself launched the ship by means of a screw, which was an invention of his own2.

Tzetzesl4 remArks that the ship could load 50,000 medimns, a medimn being a corn measure of about 54 litres. He lets Archimedes draw the ship with his left hand, by means of a triple pulley.

Proclos*2 goes one better: he tells us that Archimedes let Hieron himself launch the ship.

About the proposed moving of the earth, Simpliciusl3 gives us this account: “When Archimedes made the instrument for weighing called charistion by the proportion of that which is moving, that which is moved, and the way travelled, as the proportion went on as far as it could go, he made the well-known boast: “Somewhere to stand, and I shall move the earth”.”

Tzetzes mentions the moving of the earth in two places; in one16 he tells us that Archimedes moved the earth by means of a triple pulley; in the other15 he lets him say: “Somewhere to stand, and I shall move the earth with a charistion.”

Charistion means a steelyard; this is shown quite clearly in the Preface to the 1952 edition of the Liber Carastonist

The idea of Archimedes, the practical mechanic who could lift Roman warships bodily out of the water, offering to move the earth by means of a steelyard, or just a simple lever, is quite absurd. Since the word charistion is found in this connexion in these two places only, it seems quite likely that Tzetzesls is merely quoting Simpliciusl3; the more so since he gives the words of Archimedes in the Doric dialect, as they are given also by Simplicius. But then it would seem that he has misunder- stood his source, for Simplicius does not tell us that Archimedes boasted that he could move the earth by means of a steelyard. He says that Archimedes invented the steelyard on the principle later to be known as the Golden Rule of Mechanics, and that as a result of following this

280 A. G. DRACHMA”

principle to its furthest consequences he made his boast. But the principle is not confjned to the steelyard and the lever; it works in the pulley, the rope and drum, the gearwheel and the endless screw.

Papposs tells us that Archimedes found the law by which a given weight may be moved by a given power, and that this made him say that he could move the earth, if only he had somewhere to stand.

Are we then to suppose that Archimedes, merely on the strength of this newly found theory, boasted that he could move the earth? It is not very likely, since he was also a practical mechanic; he would have said: “It should be possible to move the earth”. But that was not what he said. His words were: “I shall move the earth!”, and when Hieron challenged him, he was prepared at once to move singlehanded a fully loaded ship, a feat that would have seemed just as impossible, at that time, as that of moving the earth. But was it possible? How did he do it?

That he did not use a steelyard, or a simple lever, is obvious. Plutar- c h d tells us that he used a multiple pulley; Tzetzes mentions a triple pulley14, and Athenaeus says that he used a screw2. All this is not quite as contradictory as it would seem.

Oribasiof writes that Apellis or Archimedes invented the triple pulley; these two men were not surgeons, but engineers, and they invented the thing for the launching of ships; they did not, however, draw the ropes by hand, but by means of windlasses. In another place6 Oribasios mentions that Archimedes invented the compound pulley. The explanation then is simply this, that Archimedes drew the ship

along by means of a compound pulley worked by a windlass containing a screw. It seems worth remembering that according to Plutarchuslo, who seems to be by far the best source, Archimedes did not launch a ship; he merely showed had he could move it. How long a way it was moved is not stated; it may have been just a few paces, enough to prove his point. This makes the story far more credible, especially in the light of the feats performed afterwards by his engines of defense.

That Archimedes invented the endless screw is generally admitted; beside Athenaeusz we have the remark in Eustathius3, that “screw” means an engine invented by Archimedes, though he may possibly refer to the watersnail, not to the endless screw. In this engine the Golden Rule of Mechanics manifests itself in its most overpowering guise. If you wind a rope round a drum with a circumference of one meter, and the drum is turned by means of an endless screw engaging a toothed wheel of

How ARCHIMEDES EXPECTED TO MOVE THE EARTH 28 1

50 teeth, and the handle of the screw travels one meter for each turn, the power is in the proportion of 150. If a pulley of 5 sheaves is added, one man will pull for 250. Even if we lose part of the power in friction, there will still be quite a lot left. But if we turn the screw by means of another endless screw giving the same proportion, we get a pull of 1:2500; another screw added will bring the proportion to 1:125,000. I think that Archimedes had made this calculation and was telling Hieron about the enormous power latent in this new form for windlass. Hieron suggested that there must be some limit, but Archimedes replied: “There is no limit. Just give me somewhere to stand, and I shall move the earth”.

This seems to be mere speculation on the strength of just one word in Athenzus2, to wit that Archimedes used a screw. But it is corroborated by Plutarchuslo. He writes that Archimedes pulled the ship along, not by strenuous labour, but sitting at ease, some way off, swinging gently the end of a compound pulley. If we take this to mean that Archimedes just made a few passes with his hand in the air and that the ship then came along as if by magic, we cannot believe it. But if it means that he turned the handle of a windlass that pulled on a compound pulley, it seems quite a good description of a man turning a windlass by means of an endless screw. And then the thing is not beyond belief.

Proclusl2 remarks that Hieron’s comment was: “From this day on we have to believe Archimedes whatever he says”. No wonder!

REFERENCES

1. Athenaeus: Deipnosophistae 5: 206d. 2. Atbenaeus: Deipnosophistae 5 : 207 b. 3. Eusthatius: Commentarii ad Homeri Iliadem (M 293). (ed. Stallbaum, Tom. 3,

Lpz. 1829). 4. Liber Carastonis p. 79-80. In: Medieval Science of Weights. Ed. by E. A. Moody &

M. Clagett. Madison 1952. 5. Livius: Ab urbe condita 24: 34. 6. Oribasius: Collectiones medicae 49: 6. (ed. Raeder, Vol. 4. Lpz. 1933, p. 12). 7. Oribasius: Collectiones medicae 49: 23: 1 (p. 33). 8. Pappos: Collectio 8: I 0 (ed. Hultsch Berl. 1876-78 p. 1060). 9. Plutarchus: Vita Marcelli 14-17.

10. Plutarchus: Vita Marcelli 14: 7-8.

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1 1. Polybios : Historis 8 : 3-7. 12. Proclos: Prologus 2, B 37. (In primurn Euclidis Elementorum librum commentarii, ed.

G. Friedlein Lpz. 1873, p. 63). 13. Simplicius p. 1110. (In Aristotelis Physicorurn libros commcntarii, ed. Dick B e d

189%. 14. Tzctzes: Chiliades 2, hist. 35: 107-8. 15. Tmtzes: Chiliades 2, hist. 35: 128. 16. Tzetzes: Chiliades 3, hist. 66: 60.