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

THEOREM IV.

The center of gravity of a portion of a hyperbola, ellipse, or circle lies on the diameter of that portion.

Let there be a portion of a hyperbola, or an ellipse, or a circle, initially not greater than half the figure, A B C; let its diameter be B D. It must be shown that the center of gravity of the portion A B C is found on B D.

A geometric diagram shows a curved segment labeled A B C, with a line segment B D acting as the diameter. A series of rectangles are inscribed within the curve, reaching toward the boundary A B C. Points A, D, F, H, C are located along the horizontal base, while M, L, K, E, and other points mark the internal geometric construction used to demonstrate the center of gravity.

For if it is possible, let it be outside the diameter at E, and let E H be drawn parallel to the diameter B D. Therefore, by continuously bisecting D C, a line will eventually remain that is smaller than D H; let that be D F, and let a figure be circumscribed around the portion, orderedly composed of parallelograms whose bases are equal to the line D F, and let B A and B C be joined. The center of gravity of the figure circumscribed around the portion is therefore on the diameter B D of the portion. Let this be K, and let E K be joined, produced, and let the line A L, parallel to B D, meet it. But because the portion is larger than the triangle A B C, and the excess by which the circumscribed figure exceeds the portion is smaller than the parallelogram