Postulate 1.

“Let it be supposed that a fluid is of such a character that, its parts lying evenly and being continuous, that part which is thrust the less is driven along by that which is thrust the more; and that each of its parts is thrust by the fluid which is above it in a perpendicular direction if the fluid be sunk in anything and compressed by anything else.”

Proposition 1.

If a surface be cut by a plane always passing through a certain point, and if the section be always a circumference [of a circle] whose centre is the aforesaid point, the surface is that of a sphere.

For, if not, there will be some two lines drawn from the point to the surface which are not equal.

Suppose O to be the fixed point, and A, B to be two points on the surface such that OA, OB are unequal. Let the surface be cut by a plane passing through OA, OB. Then the section is, by hypothesis, a circle whose centre is O.

Thus OA = OB; which is contrary to the assumption. Therefore the surface cannot but be a sphere.

Proposition 2.

The surface of any fluid at rest is the surface of a sphere whose centre is the same as that of the earth.

Suppose the surface of the fluid cut by a plane through O, the centre of the earth, in the curve ABCD.

“Archimedes to Dositheus greeting.

On a former occasion I sent you the investigations which I had up to that time completed, including the proofs, showing that any segment bounded by a straight line and a section of a right-angled cone [a parabola] is four-thirds of the triangle which has the same base with the segment and equal height. Since then certain theorems not hitherto demonstrated (ἀνελέγκτων) have occurred to me, and I have worked out the proofs of them. They are these: first, that the surface of any sphere is four times its greatest circle (τοῦ μεγίστου κύκλου); next, that the surface of any segment of a sphere is equal to a circle whose radius (ἡ ἐκ τοῦκέντρου) is equal to the straight line drawn from the vertex (κορυфή) of the segment to the circumference of the circle which is the base of the segment; and, further, that any cylinder having its base equal to the greatest circle of those in the sphere, and height equal to the diameter of the sphere, is itself [i.e. in content] half as large again as the sphere, and its surface also [including its bases] is half as large again as the surface of the sphere.

The cult of Hera is less manifold and less spiritual than many other Greek cults, but possesses great historic interest. It can be traced in most parts of ancient Greece, and had the strongest hold upon the sites of the oldest civilization, Argos, Mycenae, and Sparta; we can find no trace of its importation from without, no route along which it travelled into Greece; for in the islands, with the exception of Euboea and Samos where the legend connected the worship with Argos, it is nowhere prominent, nor does it appear to have had such vogue in Thessaly and along the northern shores as it had in Boeotia, Euboea, Attica, Sicyon, Corinth, and the Peloponnese19—93. We may regard the cult then as a primeval heritage of the Greek peoples, or at least of the Achaean and Ionic tribes; for its early and deep influence over these is attested by the antiquity and peculiar sanctity of the Argive and Samian worship. Whether it was alien to the Dorians in their primitive home, wherever that was, is impossible to decide; in the Peloponnese no doubt they found and adopted it, but they may have brought it with them to Cos and Crete, where we find traces of it. The Hera T∈λχινία of Rhodes, like the Spartan and Argive Hera, was probably pre-Dorian.