of B that x is of HK. Archimedes now divides LH, HK into parts equal to x, and A B into parts equal to w, and places the w’s at the middle points of the x’s 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.