SY XPX. For QP is equal to the rectangle PXxQR (Schol. Theor. 5. 4 Eu.), and therefore PX is equal to be substituted for it, in expressing the quantity of the solid, mentioned in the preceding corollary. Cor. 4. The same things being supposed, the centripetal force is, as the square of the velocity directly, and that chord inversely. For the velocity is inversely as the perpendicular SY (Cor. 1. 1 Nat. Ph). Cor. 5. Hence, if any curvilineal figure APQ be given, and in it a point S be also given, to which the centripetal force is continually directed; the law of the centripetal force may be found, by which, any body P, being continually drawn off from a rectilineal course, will be detained in the perimeter of that figure, and, in revolving, describe it. Namely, either the solid SP2x QT QR or the solid SY XPX, should be computed, as inversely proportional to this force. Scholium. Although the method of investigating centripetal forces, given in the preceding corollary, being the fifth of this proposition, is general, extending itself to any given curvilineal figure, and any point therein; yet as the principal object of these elements, is the investigation of those laws, which actually prevail in nature; and as Kepler has, from actual observation, ascertained, that the primary planets in their revolutions about the sun, describe ellipses, the sun being in one of the focuses ; see the second law discovered by him, mentioned in these elements of natural philosophy, in the preparatory observations; and as the same law has been found, as far as observations have been made, to prevail in the motions of the secondary planets round their primaries. The investigation of the law producing a motion in an ellipse, round a focus, as the centre of force, is given in the next proposition; the like investigation, as respects a motion in a hyperbola or parabola, round a focus, as the centre of force, being given in the two following propositions. PROP. V. PROB. Let a body revolve in an ellipse; the law of the centripetal force, tending to its focus, is required. meeting DK in E and Qz in x, to the tangent RPN draw QR parallel to SP, on SP and DK let fall the perpendiculars QT and PF, and draw HI parallel to DK, meeting SP in I. Because of the equals SC and CH, and the parallels EC and HI, SE is equal to EI (2. 6 Eu.), and because the angles IPR and HPN are equal (11. 1 Sup. and 15. 1 Eu.), and H1 being parallel to RN (Def. 14. 1 Sup. and 30. 1 Eu.), and therefore the angles PIH and PHI equal to their alternates IPR and HPN (29. 1 Eu.), the angles PIH and PHI are equal, and therefore the right lines PH and PI (6. 1 Eu); therefore EI is the half of SI, and IP of IP and PH together, and therefore EP is thehalf of SP and PH together, and therefore equal to the greater semiaxis CA (1. 1 Sup). The principal parameter of the ellipse being called L; Lx QR is to LxPz, as QR, or its equal (34. 1 Eu.), Px is to Pz (1. 6 Eu.), or, which is equal (2. 6 Eu.), as PE or AC is to PC; and LxPz is to the rectangle GzP, as L is to Gz (1. 6 Eu.); and the rectangle GzP is to the square of Qz, as the square of CP is to the square of CD (40. 1 Sup.); and the ratio of the square of Qz to the square of Qx, the points Q and P coming together, is the ratio of equality (Cor. 2 Lem. 7 Nat. Ph.); and the triangles QXT and PEF being, because of the right angles at T and F, and the angles at x and E equal, being alternate angles (29. 1 Eu.), equiangular, the square of Qx, or of its equal Qz is the square of QT, as the square of PE or AC is to the square of PF (4 and 22. 6 Eu.), or, which is equal (53. 1 Sup. 35. 1, and 16 and 22. 6 Eu.), as the square of CD is to the square of CB; and, compounding all these ratios, LXQR is to the square of QT, as ACxLxPC2xCD, or, ACxL being equal to 2CB (Def. 15 and 17. 1 Sup. and 17. 6 Eu.), as 2CB XPCx CD is to PCxGzxCD xCB (22. 5 Eu.), or, applying each to CB XPCXCD, which is common to both, as 2PC is to Gz; but, the points Q and P coming together, 2PC and Gz are equal; therefore LXQR and QT", which are proportional to these, are equal (Cor. 13. 5 Eu). Let these equals be drawn into SP2 QR SP3xQT9 and LxSP' is equal to 9 or (Cor. 1. 4 Nat. Ph.), in QR versely as the centripetal force; whence, L being a given quantity, the centripetal force is inversely as SP, or in an inverse duplicate ratio of the distance SP. PROP. VI. PROB. Let a body be moved in a hyperbola; the law of the centripetal force, tending to its focus, is required. Let S be the focus of the hyperbola, to which the centripetal force tends, H the other focus, C the centre, CA and CB semiaxes, GP the diameter passing through the body P, KD the diameter conjugate thereto, Q a point in the perimeter AQP at the least possible distance from P, Qz an ordinate to the diameter GP, RPN a right line touching the hyperbola in P; draw SP meeting and QZ in x and ED in E, to the tangent RPN draw QR parallel to SP, on SP and KD let fall the perpendiculars QT and PF, and draw HI parallel to KD meeting SP produced in I. Because of the equals HC and CS, and the parallels CE and HI, SE is equal to EI 2. 6 Eu.), therefore PE is equal to half the difference of PI and PS, or, the angles HPR and IPN being equal (11. 1 Sup. and 15. 1 Eu.), and therefore, RN and HI being parallel (Def. 14. 1 Sup. and 30. 1 Eu.), their alternates PHI and PIH (29. 1 Eu.), and therefore the right lines PI and PH (6. 1 Eu.), PE is equal to half the difference of HP and PS, and therefore to the transverse semiaxis CA (1. 1 Sup). The principal parameter of the hyperbola being called L; LXQR is to LxPz, as QR, or its equal (34. 1 Eu.), Px is to Pz (1.6 Eu), or, which is equal (2. 6 Eu.), as PE or AC is to PC; and LxPz is to the rectangle GzP, as L is to Gz (1.6 Eu.); and the rectangle GzP is to the square of Qz, as the square of CP is to the square of CD (40. 1 Sup.); and the ratio of the square of Qz to the square of Qx, the points Q and P coming together, is the ratio of equality (Cor. 2 Lem. 7 Nat. Ph.); and the triangles QxT and PEF being, because of the right angles at T and F, and the angles at x and E equal, the external to the internal remote on the same side (29. 1 Eu.), equiangular, the square of Qx, or of its equal Qz is to the square of QT, as the square of PE or AC is to the square of PF (4 and 22. 6 Eu.), or, which is equal (53. 1 Sup. 35. 1, and 16 and 22. 6 Eu.), as the square of CD is to the square of CB; and, compounding all these ratios, LxQR is to the square of QT, as ACxLxPC XCD, or, ACxL being equal to 2CB (Def. 15 and 17. 1 Sup. and 17. 6 Eu.), as 2CBxPC'xCD is to PCx Gzx CD xCB (22. 5 Eu.), or, applying each to CB'xPCxCD', which is common to both, as 2PC is to Gz; but the points Q and P coming together, 2PC and Gz are equal; therefore LxQ R and QT, which are proportional to them, are equal (Cor. 13. 5 Eu). Let these equals be drawn into equal to SPXQT SP9 or (Cor. 1. 4 Nat. Ph.), inversely as the centripetal force; whence, L being a given quantity, the cen= tripetal force is inversely as SP, or in an inverse duplicate ratio of the distance SP. PROP. VII. PROB. Let a body be moved in a parabola; the law of the centripetal force, tending to its focus, is required. Let AQP be the parabola, S its focus, A the principal vertex, Yz the diameter passing through the body P, Q a point in the perimeter AQP at the least possible distance from P, Qz an ordinate to the diameter Yz, MP a right line touching the parabola in P, and meeting the axis in M; join SP meeting Qz in x, draw QR to the tangent MP parallel to SP, and on SP and MP let fall the perpendiculars QT and SN. Because the angle SPM is equal to the angle YPM (11. 1 Sup.), or, which is equal (29. 1 Eu.), SMP, which is alternate to it, SP and SM are equal (6. 1 Eu.), whence, the triangles Pxz and SPM being equiangular, Px, or its equal (34. 1 Eu.), QR, is equal to Pz; but the square of Qz is equal to the rectangle under Pz and the parameter of the diameter Yz (40. 1 Sup.), or, that parameter being equal to 4PS (Def. 16. 1 Sup.), to the rectangle under Pz and 4PS, or to that under QR and 4PS; but the points P and Q coming together, the ratio Qz to Qx is the |