Coroll. The centre of equiponderation of the figure B D C B is in the concourse of two strait lines, whereof one is parallel to the base, and divides the axis, so that the part next the base be ⅗ or 9⁄15 of the whole axis; the other is parallel to the axis, and so divides the base, that the part towards the axis be ½, or 12⁄24 of the whole base. For drawing O P parallel to the base, it will be as I N to N O, so F K to K P, that is, so 1 to 3, or 5 to 15. But A F is 5⁄15, or ⅓ of the whole A B; and A K is 6⁄15, or ⅖; and F K 1⁄15; and KP 3⁄15; and therefore A P is 9⁄15 of the axis A B. Also A H is ⅓, or 8⁄24; and A M ⅜, or 9⁄24 of the whole base; and therefore O Q being drawn parallel to the axis, M Q, which is triple to H M, will be 3⁄24. Wherefore A Q is 12⁄24, or ½ of the base A D. The excesses of the rest of the three-sided figures in the first row of the table of art. 3, chap. XVII, have their centres of equiponderation in two strait lines, which divide the axis and base according to those fractions, which add 4 to the numerators of the fractions of a parabola 9⁄15 and 12⁄24; and 6 to the denominators, in this manner:— In a parabola, the axis 9⁄15, the base 12⁄24. In the first three-sided figure, the axis 13⁄21, the base 16⁄30. In the second three-sided figure, the axis 17⁄27, the base 20⁄36, &c. And by the same method, any man, if it be worth the pains, may find out the centres of equiponderation of the excesses above their triangles of the rest of the figures in the second and third row, &c. [Sidenote: The centre of equiponderation of a solid sector is in the axis so divided, that the part next the vertex be to the whole axis, wanting half the axis of the portion of the sphere, as 3 to 4.] 14. The centre of equiponderation of the sector of a sphere, that is, of a figure compounded of a right cone, whose vertex is the centre of the sphere, and the portion of the sphere whose base is the same with that of the cone, divides the strait line which is made of the axis of the cone and half the axis of the portion together taken, so that the part next the vertex be triple to the other part, or to the whole strait line as 3 to 4. For let A B C (in fig. 9) be the sector of a sphere, whose vertex is the centre of the sphere A; whose axis is A D; and the circle upon B C is the common base of the portion of the sphere and of the cone whose vertex is A; the axis of which portion is E D, and the half thereof F D; and the axis of the cone, A E. Lastly, let A G be 3⁄4 of the strait line A F. I say, G is the centre of equiponderation of the sector A B C.