Props. 32, 33, 34 are solved by considering the motion of a particle in a related conic.

Prop. 35. On the areas swept out by the vector in the related

conic.

Prop. 36. To find the time of descent to the centre of force by a body falling under such a force varying inversely as the square of the distance.

Prop. 37. To find the time to the centre of force by a body projected from a given point with a given velocity under such a force varying inversely as the square of the distance. This is equivalent to a geometrical determination of the value of the integral of {∞ / (α-∞) } .

Prop. 38. To find the relations between the time, velocity, and space described by a body falling from a given point under such a force varying directly as the distance.

Prop. 39. To find the velocity at any given point and the time required to reach that point by a body moving in a straight line under any centripetal force to a point in the line: the quadrature of certain subsidiary curvilinear figures used in the construction being assumed. SECTION VIII.-On the determination of the orbits in which bodies move when acted on by given centripetal forces.

Prop. 40. Comparison of the velocity of a body in such an orbit with the velocity of a body having a corresponding rectilinear motion.

Prop. 41. On the determination of the orbit in which a body moves when acted on by a given centripetal force, also of the position of the body at any assigned time: the quadrature of curvilinear figures being assumed to be possible.

Prop. 42. A body is projected under a given centripetal force in a given direction with a given velocity; to determine the motion.

Note. Newton adds that in the above propositions although the law of centripetal force is arbitrary, yet it is assumed that at equal distances from the centre it is everywhere the same.

[The application of the Newtonian methods given in the above Section is involved and inconvenient as compared with the analytical methods now current.]

SECTION IX.-On the motion of bodies in orbits which revolve round the centre of force, and on the motion of the apses.

Prop. 43. To find the force under which a body will move equiareally in an orbit revolving about the centre of force in the same manner as another body in the same orbit at rest.

Prop. 44.

The difference of the forces under which two bodies move equally, one in a quiescent orbit, the other in the same orbit revolving, varies inversely as the triplicate ratio of their common distances.

Prop. 45. To find the motion of the apses in orbits which are nearly circular. The result is illustrated by examples when the law of force is (i) μ, (ii) μrn-з, (iii) μrm-з+v-з; and by two corollaries on the determination of the law of centripetal force from the motion of the apses, and on how the motion of the apses is affected by the addition to the centripetal force of an extraneous force.

The second corollary is illustrated by showing that if this extraneous addition be 1/357 45th part of the force under which the body would revolve in the ellipse, then in each revolution the apse line would progrede 1° 31′ 14′′", a number which in the second edition was corrected to 1° 31′ 28′′. It seems clear from the Portsmouth papers that this was given merely as an illustration of the method, but in the third edition the words "Apsis lunae est duplo velocior circiter" were added. It may be that this remark was inserted in order to show that the corollary was not applicable to the case of the moon-in fact only one part of the sun's disturbing force is here treated-but a reader might also think that the remark was intended to point out a discrepancy between the theory and observations. As Newton had explained the similar difficulty in the case of the node, some writers suspected (ex. gr. Godfray, in his Lunar Theory, second edition, 1859, art. 68) that the scholium in the first edition to book iii. prop. 35 meant that he had found the explanation: but nowhere in the Principia does Newton explicitly give this explanation, though in book iii. prop. 25 he estimates that the total disturbing force of the sun on the moon bears to the earth's centripetal force the ratio 1 to 17823, which would make the annual progression of the apse line about what it actually is. The remark at the end of book i. prop. 45 was, however, read by many as indicative of a variance between observation and the Newtonian theory, and the explanation of a difference which had become an obstacle to the universal acceptance of the Newtonian system was first given by Clairaut. The Portsmouth papers contain Newton's original work, and show that he had found, by carrying the approximation to a sufficiently high order, that the mean annual motion of the apse line was 38° 51' 51", which is within 2° of the true value (see below, p. 109; also the Portsmouth Collection, Catalogue, pp. xi-xiii, xxvi-xxx, and section 1. division ix. numbers 7, 12).

SECTION X.-On the motion of bodies on smooth planes which do not contain the centre of force, and on the motion of pendulums. Prop. 46. To determine the motion of a body moving on a given plane, under a given centripetal force, when projected in any direction

on the plane and with any velocity: the quadrature of curvilinear figures being assumed to be possible.

Prop. 47. All bodies moving in any plane, under a centripetal force which varies directly as the distance, describe ellipses in equal periodic times; and rectilinear motion may be treated as a particular case of elliptic motion.

Scholium. On motion on curved surfaces.

Prop. 48. On the rectification of the epicycloid.
Prop. 49.

On the rectification of the hypocycloid.

Prop. 50. To make a pendulum whose bob shall oscillate in a given hypocycloid. This is effected by cycloidal checks.

Prop. 51. If the bob of a pendulum oscillate in a hypocycloid, under a centripetal force to the centre of the fixed circle and which varies directly as the distance from that centre, all oscillations are isochronous.

Prop. 52. To find the velocity of the bob of such a hypocycloidal pendulum at any assigned place, and the time occupied in describing any given arc. The second part of this proposition was rewritten in the second edition.

Cor. 1. The above propositions may be used to compare the times of all oscillating, falling, and revolving bodies.

Cor. 2. The above propositions are directly applicable to the motion of pendulums in mines. Also the results previously enunciated by Wren and Huygens concerning motion in a cycloid can be deduced as particular cases of these propositions.

Prop. 53. To find the law of force under which a body oscillating in a given curve may oscillate isochronously: the quadrature of curvilinear figures being assumed to be possible. With applications to the common (circular) pendulum and clocks.

Prop. 54. To find the time in which a body describes any arc of a given curve under a given force to a centre in the plane of the curve: the quadrature of curvilinear figures being assumed to be possible.

Prop. 55. If a body T move on a surface of revolution whose axis passes through the centre of force, and if P be the projection of T on a plane perpendicular to the axis and cutting it in O, then the area described by OP will be proportional to the time.

Prop. 56. A body is projected with a given velocity in a given direction along a given surface of revolution under a given centripetal force to a given centre on the axis of the surface; to find the orbit : the quadrature of curvilinear figures being assumed to be possible.

[In several of the above propositions Newton has given general solutions on the assumption that certain quadratures can be effected, i.e.

that certain functions can be integrated. It would seem from his draft in the Portsmouth papers* that he intended to insert at the end of this section a classification of algebraical curves whose quadrature could be effected—“ tandem ut compleatur solutis superiorum problematum adjicienda est quadratura ffiguram toties assumpta." Doubtless this is the rule to which he alluded in his letter to Collins of Nov. 8, 1676 (Macclesfield Correspondence, number cclxxii. vol. ii. pp. 403–405), and which (as far as I know) has not been hitherto published.

Curves defined by a binomial equation ∞ = Lva are obviously capable of quadrature. In the draft he next discusses curves defined by a trinomial equation of the form

where ≈ and v are current co-ordinates, d, e, f are given constants, and a, ß, €, ¿ are any numerical indices; and says that in three cases the area is a multiple of the corresponding area of another curve which he constructs. These theorems appear to be of less practical use than Newton supposed. He failed in his attempt to extend the method so as to find what curves defined by the quadrinomial equation

dva + evßæ¤ + furæ + gæn = 0

are capable of exact quadrature.]

SECTION XI.-On the motion of bodies under their mutual attractions. Newton commences this section by remarking that the previous propositions treat of the motions of bodies attracted to fixed centres, but that probably there is no such thing in nature as a fixed centre, for attractions are towards bodies, and action and reaction are equal. Moreover, in the following propositions when describing centripetal forces as attractions he uses the terms in their familiar mathematical sense, and is not to be supposed to be expressing a theory-for perhaps such forces may be more truly called impulses, "fortasse, si physice loquamur, verius dicantur impulsus."

Prop. 57. Two attracting bodies describe similar figures about their centre of gravity and about each other.

Prop. 58. If two attracting bodies revolve about their centre of gravity, then under the same forces a similar and equal orbit might be described about one of the bodies if it were fixed. Hence (cors. 1, 2, 3) the results of book i. props. 1, 10, 11, 12, 13 are applicable to such motions.

Prop. 59. If two bodies, of masses S and P, revolve round their centre of gravity C in the periodic time T, and if P would describe a

*Section 1. division v. number 5.

similar and equal orbit about S, supposed fixed, in the periodic time t, then T: t=√S: √(S+P).

Prop. 60. If S and P attract each other with forces inversely proportional to the square of their distance, then the ratio of the major axis of the ellipse described by P about S to the major axis of the ellipse which would be described in the same periodic time by P about

2

S when fixed, is equal to the ratio S+P: {S(S+P) 2 } *.

Prop. 61. Two bodies acted on by their mutual attractions will move as if attracted according to the same law of force by a certain body placed at their centre of gravity.

Prop. 62. To determine the motions of two bodies which attract each other with forces inversely proportional to the square of the distance between them, and which are let fall from given places.

Prop. 63. To determine the motions of two bodies which attract each other with forces inversely proportional to the square of the distance between them, and which are projected from given places in given directions with given velocities.

Prop. 64. To find the relative motions of a system of bodies which mutually attract each other with forces which vary directly as the distance.

Prop. 65. Bodies whose attractive forces vary inversely as the square of the distance may move relatively to one another approximately in ellipses, and the radii drawn to the foci may describe areas approximately proportional to the times of description. To this proposition were added three corollaries on perturbed orbits and disturbing forces.

Prop. 66. If three bodies, T, S, P, attract each other with forces which vary inversely as the square of the distances; and if round the greatest of them, T, the two others, P and S, revolve, and of the latter the body P describes the interior orbit; then the areas described by P round T will be more nearly proportional to the times of description, and the orbit of P about T will approximate more nearly to an ellipse. with T as focus than would be the case if T were not attracted by S and P but remained at rest, or if T were attracted (or moved) very much more or very much less.

This is proved for two cases according as to whether the orbits are in the same or different planes.

To this proposition were appended twenty-two corollaries in which it is applied to explain the chief effects of the disturbing action of a body like the sun on the motion of a body like the moon, and in particular to the motion in longitude, the motion in latitude, the annual equation, the motion of the apse line, the motion of the nodes, the