737 



PRINCIPIA. 



PRINCIPIA. 



738 



exercises in drawing conic sections through given points, or touching 

 given straight lines, &c. : the results are hardly of use, even in the 

 rest ofthe work, and a particular reference would now be of no use 

 whatever, though the sections themselves are highly interesting to the 

 geometrical inquirer. 



Section 6. On finding the motion in a given orbit. (30) To find the 

 place of a body in a parabola at the end of a given time. 3 Cor. Lemma 

 28. There is no oval figure whose area contained being any two radii, 

 can be obtained by an equation in finite terms. [QUADRATURE OF THE 

 CIRCLE.] 1 Cor. ; relates to the ellipse. (31) To find the place of a 

 body in an ellipse at the end of a given time. Scholium ; approximate 

 method. 



Section 7. On rectilinear ascent and descent. (32) Required the space 

 described in a given time by a body descending towards a centre.* 

 Three cases, derived from the three conic sections. (33) Law of the 

 velocity in the preceding, in the cases derived from the ellipse and 

 hyperbola. 2 Cor. (34) The same in the case derived from the 

 parabola. (35) An equable description of certain areas in the conic 

 sections just alluded to takes place during the motion. (36) The time 

 of the whole descent of a body from rest. (37) The time of the whole 

 descent of a projected body. (38) Velocity and time determined in 

 descent to a centre, the force being as the distance. 2 Cor. (39) 

 Granting the quadrature of curves [QUADRATURE], and the law of force 

 being any whatever, to determine the time and velocity at any point of 

 a descent. 3 Cor. 



Section 8. Determination of the orbit under any law of centripetal 

 force. (40) The velocity at a given distance is always the same both 

 in an orbit and a descent, if it be the same at any one distance in both. 

 2 Cor. (41) Granting the quadrature of curves, to find the orbit and 

 the time of describing an arc, under any law of force. 3 Cor. (42) The 

 same, the initial velocity and direction being given. 



Section 9. On the motion of bodies in moveable orbits, and on the 

 motion of the apsides. (43) How to make a body revolve equiareally 

 in both a moving orbit and in the same fixed. (44) The difference of 

 forces in the two cases is as the inverse cube of the distance. 6 Cor. ; 

 mostly exhibiting the conclusion in algebraical form. (45) To find 

 the motion of the apsides in orbits nearly circular. 3 Examples ; 

 2 Cor. 



Section 10. On the motion of bodies in given surfaces, and on pen- 

 dulous motions. (46) Qiven a plane, and a centre of force external to 

 it, to find the motion of a point parallel to that plane, the law of force 

 being any whatever. (47) The force in the last being as the distance, 

 the orbit parallel to any plane must be an ellipse, and in all such 

 ellipses the time of revolution is the same, and the same as that of a 

 double aacent and descent. Schol. (48) and (49) Rectification of the 

 epicycloid and hypocycloid. 3 Cor. (50) Way to make a body oscil- 

 late in a given hypocycloid. Cor. (51) If the force tending to the 

 centre of the fixed circle in such an oscillation be as the distance, the 

 times of all such oscillations are equal. Cor. (52) Determination of 

 the velocity and time at any point of such an oscillation. 2 Cor. ; the 

 second being an application to the common cycloid. (53) On a given 

 curve, to find the law of force which gives isochronous oscillations. 

 2 Cor. (54) A body moving on a rigid curve, under a given law of 

 centripetal force, to find the time of its oscillations. (55) If a body 

 move on a surface of revolution, the centre of force being in the axis, 

 equal areas are described in equal times on a plane perpendicular to 

 the axis. Cor. (56) To find the curve described in the last case. 



Section 11. On the motion of bodies centripetally attracted to each 

 other. (57) Two bodies, mutually attracting, describe similar figures 

 about each other and about their common centre of gravity. (58) And 

 with the same forces, the same curve may be described by either about 

 the other at rest. 3 Cor. (59) Relation of the periodic times about the 

 centre of gravity, and of one body about the other at rest. (60) In 

 the same two cases, relation of the axes' of the ellipses described. 

 (61) And for any law of force, the bodies move round their centre of 

 gravity as if a third body were placed in that centre, attracting with 

 the same law of force. (62) Determination of the descent towards 

 each other of two mutually attracting bodies. (63) Determination of 

 the orbits of two such bodies, with given initial velocity and direction. 

 (64) The force being as the distance, determination of the relative 

 motions of several bodies. (65) The force being inversely as the square 

 of the distance, and there being several bodies, one may move round 

 another in an ellipse nearly, and describe areas nearly proportional to 

 the time*. 3 Cor. (66) The celebrated proposition of the three bodies, 

 nhowing the diminution of the disturbance by the third body attracting 

 both the others. (In the corollaries following, let the earth and moon, 

 for distinctness sake, be the two bodies, and the sun the disturbing 

 body : but let it be remembered that Newton does not mention the 

 name of any planet nor hint at any application.) Cor. 1, If the earth 

 had more satellites, the same proposition would apply to one as dis- 

 turbed by the rest. Cor. 2 and 3, The moon moves quickest, casteris 

 paribua, in conjunction and opposition, and slowest in quadratures. 

 Cor. 4, The moon's orbit ia more curved in quadratures than in 

 syzygies. Cor 5, Hence, excentricity being excluded, the moon is 

 farther from the earth in quadratures than in syzygies. Cor. 6, 



* The lav of force, where not otherwise specified, is alwaj-s u the inverse 

 square of the distance. 



ABT3 ASD SCI. DIV. VOL. VI. 



Explanation of the effect of the variation of the sun's distance on the 

 moon's period. Cor 7, The moon's apsides progress and regress, but 

 the former more than the latter. Cor. 8, Effect of the position of the 

 apsides with respect to the sun. Cor. 9, Effect on the excentricity of 

 the moon's orbit. Cor. 10 and 11, Effect on the inclination and place 

 of the nodes. Cor. 12, Disturbance rather greater in conjunction than 

 in opposition. Cor. 13, The same species of effect produced whether 

 the disturbing body is the greater or the less of the three. Cor. 14, 

 15, 16, 17, On the dependence of the disturbing forces on the distance 

 of the disturbing body. Cor. 18, 19, 20, 21, 22, Explanation of pre- 

 cession of equinoxes and tides. (67) The disturbing body describes 

 areas more nearly proportional to the times about the centre of gravity 

 of the other two bodies than about either of them, and an orbit more 

 nearly elliptic. (68) And the more so on account of its attracting the 

 other bodies. Cor. (69) The attracting forces of bodies are, cseteris 

 paribus, as their masses. 3 Cor. and Scholium. 



Section 12. On the attractions of spherical bodies. (70) A particle 

 placed inside a spherical shell is in equilibrium. (71) Spherical shells 

 attract as if their whole masses were collected at their centres. (72) 

 The attractions of spheres on points similarly placed with respect to 

 them are as their diameters. 3 Cor. (73) At different internal points 

 of a solid sphere the attractions are as the distances from the centre. 

 Schol. (74) Solid spheres attract as if the whole masses were collected 

 at their centres. 3 Cor. (75) The same of spheres attracting spheres. 

 4 Cor. (76) The same of spheres consisting of concentric layers of 

 unequal density. 9 Cor. (77) The same is true when the forces of 

 particles to each other are as their distances. (78) With the same 

 law, the same is true of spheres consisting of concentric layers. Cor. 

 and Schol. : Lemma 29. (79) ; (80) 4 Cor. ; (81), 3 Exam. ; (82) : 

 these show the method of finding the attraction of any sphere on a 

 point without it, for any law of force. (83) The force being as the 

 inverse Tith power of the distance, to find the attraction of a segment 

 of a sphere on a particle at its centre. (84) The same when the par- 

 ticle is not in the centre. Schol. 



Section 13. On the attraction of non-spherical bodies. (85) If the 

 attraction of the body on a contiguous particle be much greater than 

 on one at a little distance, the attraction of the molecules of the 

 attracting body diminishes in a higher ratio than the inverse square of 

 the distance. (86) And the hypothesis of the last is a consequence, if 

 the attraction of the molecules diminishes as the inverse cube of the 

 distance, or faster. (87) If two similar bodies of the same material 

 attract two molecules proportional to themselves and similarly placed, 

 the attractions of the molecules on the two bodies will be proportional 

 to their attractions on their similar particles similarly placed. 2 Cor. 

 (88) If the particles of a body attract a molecule with forces as their 

 distances, the whole attraction of the body will be towards its centre 

 of gravity, and equal to that of a sphere equal to the body, and having 

 its centre in that centre of gravity. Cor. (89) And the same if there 

 be several bodies. Cor. (90) To determine the attraction of a circle on 

 a IK lint in its axis. 3 Cor. (91) To determine the attraction of a solid 

 of revolution on a point in its axis. 3 Cor.' relating to cylinders and 

 spheroids. (92) Given an attracting body, to find (experimentally) 

 the law of attraction of its particles. (93) If particles attract as 

 the inverse nth power, a solid bounded by a plane, but indefinitely 

 extended in all directions on one side of that plane, will attract an 

 external particle with a force proportional to the inverse (n - 3)rd 

 power of the distance from that plane. 3 Cor. Schol. 



Section 14. On the motion of particles from one medium into 

 another. (94) If a particle pass through a medium contained between 

 parallel planes, and be in its passage attracted to or repelled from the 

 boundary of the medium it has left with a force depending on the 

 distance from the boundary ; the sine of the angle of emergence is 

 always in a constant ratio to that of incidence. (95) And the velocity 

 before incidence is to that after emergence as the sine of the angle of 

 emergence to that of incidence. (96) And if the velocity must be 

 greater before than after incidence, the angle of incidence may bo 

 made so great that the particle shall be reflected, and the angles of 

 incidence and reflexion are equal. Scholium. (97) To give the boun- 

 dary separating two media such a form that all particles issuing 

 from one point may be refracted to another. 2 Cor. (98) To form 

 a lens which shall have the property mentioned in the last. Scholium. 

 THE SECOND BOOK, mostly on resisted motion, contains 9 sections 

 and 53 propositions. 



Sectimi 1. When the resistance is as the velocity. (1) The motion 

 lost is as the space described. Cor. Lemma 1. (2) When no forces 

 act but the resistance, the velocities at the beginnings of successive 

 equal times are in geometrical progression, and the spaces described 

 as the velocities. Cor. (3) To determine the resisted motion of ascent 

 or descent when the force of gravity acts. 4 Cor. (4) The same when 

 the particle ie projected obliquely. 7 Cor. and Scholium, to the effect 

 that the hypothesis is " magis mathematica quarn naturalis." 



Section 2. When the resistance is in the duplicate ratio of the velo- 

 city. (5) When no force acts, equal spaces are described in times 

 which are in increasing geometrical progression, the velocities at the 

 beginning of the times being in the inverse geometrical progression. 

 5 Cor. (6) Equal and homogeneous spheres, acted on by no forces, 

 describe equal spaces in times which are reciprocally as their initial 

 velocities, in which also they lose the same parts of their velocities. 



SB 



