Archimedes

respectively. All the w ’s are then in equilibrium about C. But all the w ’s acting at the several points along LH are equivalent to A acting as a whole at the point E. Similarly the w ’s acting at the several points on HK are equivalent to B acting at D. Therefore A, B placed at E, D respectively balance about C.

Prop. 7 deduces by reductio ad absurdum the same result in the case where A, B are incommensurable. Prop. 8 shows how to find the centre of gravity of the remainder of a magnitude when the centre of gravity of the whole and of a part respectively are known. Props. 9-15 find the centres of gravity of a parallelogram, a triangle and a parallel-trapezium respectively.

Book II., in ten propositions, is entirely devoted to finding the centre of gravity of a parabolic segment, an elegant but difficult piece of geometrical work which is as usual confirmed by the method of exhaustion.

CHAPTER VII.

HYDROSTATICS.

The science of hydrostatics is, even more than that of statics, the original creation of Archimedes. In hydrostatics he seems to have had no predecessors. Only one of the facts proved in his work On Floating Bodies, in two books, is given with a sort of proof in Aristotle. This is the proposition that the surface of a fluid at rest is that of a sphere with its centre at the centre of the earth.

Archimedes founds his whole theory on two postulates, one of which comes at the beginning and the other after Prop. 7 of Book I. Postulate 1 is as follows:—

“Let us assume that a fluid has the property that, if its parts lie evenly and are continuous, the part which is less compressed is expelled by that