Christiani Hugenii ...Theoremata De Quadratura Hyperboles, Ellipsis Et Circuli, Ex Dato Portionum Gravitatis Centro

parallel,) and in the same is the center of gravity of the smaller portion A E C, as was just shown; the center of gravity of the remaining portion A B C will also be on B D E; which was to be demonstrated.

8. lib. 1. Archim. on Equilibrium.

LEMMA.

Let there be a line E B, to which two equal lines E S and B P are added at each end, and in addition another line P D. I say that the amount by which the rectangle E D B exceeds E P B is equal to the rectangle S D P. For the rectangle E D B is

A geometric diagram illustrates a line segment marked with points P, e, B, e, p, S, and d, demonstrating the relationship between different rectangular areas.

equal to these two: the rectangle E D P and the rectangle under E D and P B; the latter of which exceeds the rectangle E P B by the rectangle D P B. Therefore, the excess of the rectangle E D B over the rectangle E P B is equal to these two: the rectangle E D P and D P B. But the rectangle E D P added to the rectangle D P B, that is, the rectangle under E S and D P, is equal to the rectangle S D P. It is manifest, therefore, that the excess of the rectangle E D B over E P B is equal to the rectangle S D P.

Again, let there be a line E B, from which two equal lines E S and B P are taken away at each end, and in addition another line P D. I say again that the amount by which the rectangle E D B exceeds E P B is equal to the rectangle S D P. For the rectangle

A diagram shows a horizontal line segment labeled E, S, D, P, B.

E D B is equal to these two: the rectangle E D P and the rectangle under E D and P B; however, of these, E D P is again equal to two: namely, the rectangle S D P and that which is contained under E S and D P, or the rectangle D P B. Therefore the rectangle