And the same way may it be demonstrated, that the body having its centripetal changed into a centrifugal force, will move in the conjugate hyperbola. LEMMA XIII. The latus rectum of a parabola belonging to any vertex is quadruple the distance of that vertex from the focus of the figure. This is demonstrated by the writers on the conic sections. LEMMA XIV. The perpendicular, let fall from the focus of a parabola on its tangent, is a mean proportional between the distances of the focus from the point of contact, and from the principal vertex of the figure. For, let AP be the parabola, S its focus, A its principal vertex, P the point of contact, PO an ordinate to the principal diameter, PM the tangent meeting the principal diameter in M, and SN the perpendicular from the focus on the tangent: join AN, and because of the equal lines MS and SP, MN and NP, MA and AO, the right lines AN, OP, will be parallel; and thence the triangle SAN will be right-angled at A, and similar to the equal triangles SNM, SNP; therefore PS is to SN as SN to SA. Q.E.D. COR. 1. PS2 is to SN2 as PS to SA. COR. 2. And because SA is given, SN2 will be as PS. COR. 3. And the concourse of any tangent PM, with the right line SN, drawn from the focus perpendicular on the tangent, falls in the right line AN that touches the parabola in the principal vertex. PROPOSITION XIII. PROBLEM VIII. If a body moves in the perimeter of a parabola; it is required to find the law of the centripetal force tending to the focus of that figure. Retaining the construction of the preceding Lemma, let P be the body in the perimeter of the parabola; and from the place Q, into which it is next to succeed, draw QR parallel and QT perpendicular to SP, as also Qv parallel to the tangent, and meeting the diameter PG in v, and the distance[Pg 120] SP in x. Now, because of the similar triangles Pxv, SPM, and of the equal sides SP, SM of the one, the sides Px or QR and Pv of the other will be also equal. But (by the conic sections) the square of the ordinate Qv is equal to the rectangle under the latus rectum and the segment Pv of the diameter; that is (by Lem. XIII.), to the rectangle 4PS × Pv, or 4PS × QR; and the points P and Q coinciding, the ratio of Qv to Qx (by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Qx2, in this case, becomes equal to the rectangle 4PS × QR. But (because of the similar triangles QxT, SPN), Qx2 is to QT2 as PS2 to SN2, that is (by Cor. 1, Lem. XIV.), as PS to SA; that is, as 4PS × QR to 4SA × QR, and therefore (by Prop. IX. Lib. V., Elem.) QT2 and 4SA × QR are equal. Multiply these equals by , and will become equal to : and therefore (by Cor. 1 and 5, Prop. VI.), the centripetal force is reciprocally as ; that is, because 4SA is given, reciprocally in the duplicate ratio of the distance SP. Q.E.I.