# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 8

*Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal forces.*

Proposition XI. Theorem XIII.

*If a body, acted upon by any centripetal force, if any how moved, and another body ascends or descends in a right line; and their velocities be equal in any one case of equal altitude, their velocities will be also equal at all equal altitudes.*

Let a body descend from *A* (Pl. 17. Fig. 3.) through *D* and *E*, to the centre *C*, and let another body move from *V* in the curve line *VIKk*. From the centre *C*, with any distances, describe the concentric circles *DI*, *EK*, meeting the right line *AC* in *D* and *E*, and the curve *VIK* in *I* and *K*. Draw *IC* meeting *KE* in *N*, and on *IK* let fall the perpendicular *NT*; and let the interval *DE* or *IN*, between the circumferences of the circles be very small; and imagine the bodies in *D* and *I* to have equal velocities, Then because the distances *CD* and *CI* are equal, the centripetal forces in *D* and *I* will be also equal. Let those forces be express'd by the equal lineolæ *DE* and *IN*; and let the force *IN* (by cor 2. of the laws of motion) be resolved into two others, *NT* and *IT*. Then the force *NT* acting in the direction of the line *NT* perpendicular to the path *ITK* of the body, will not at all affect or change the velocity of the body in that path, but only draw it aside from a rectilinear course, and make it deflect perpetually from the tangent of the orbit, and proceed in the curvilinear path *ITKk*. That whole force therefore will be spent in producing this effect; but the other force *IT*; acting in the direction of the course of the body, will be all employed in accelerating it; and in the least given time will produce an acceleration proportional to it itself. Therefore the accelerations of the bodies in *D* and *I* produced in equal times, are as the lines *DE*, *IT*; (if we take the first ratios of the nascent lines *DE*, *IN*, *IK*, *IT*, *NT*); and in unequal times as those lines and the times conjunctly. But the times in which *DE* and *IK* are described, are, by reason of the equal velocities (in *D* and *I*) as the spaces described *DE* and *IK*, and therefore the accelerations in the course of the bodies through the lines *DE* and *IK* are as *DE* and *IT*; and *DE* and *IK* conjunctly; that is, as the square of *DE* to the rectangle *IT* into *IK*. But the rectangle *IT* x *IK* is equal to the square of IN; that is equal to the square of *DE*. and therefore the accelerations generated in the passage of the bodies from *D* and *I* to *F* and *K* are equal. Therefore the velocities of the bodies in *E* and *K* are also equal: and by the same reasoning they will always be found equal in any subsequent equal distances. *Q. E. D.*

By the same reasoning, bodies of equal velocities and equal distances from the centre will be equally retarded in their ascent to equal distances. *Q. E. D.*

Cor. 1. Therefore if a body either oscillates by hanging to a string, or by any polished and perfectly smooth impediment is forced to move in a curve line; and another body ascends or descends in a right line, and their velocities be equal at any one equal altitude; their velocities will be also equal at all other equal altitudes. For, by the string of the pendulous body, or by the impediment of a vessel perfectly smooth, the same thing will be effected, as by the transverse force *NT*. The body is neither accelerated nor retarded by it, but only is obliged to quit its rectilinear course.

Cor. 2. Suppose the quantity P to be the greatest distance from the centre to which a body can ascend, whether it be oscillating, or revolving in a trajectory, and so the same projected upwards from any point of a trajectory with the velocity it has in that point. Let the quantity A be the distance of the body from the centre in any other point of the orbit; and let the centripetal force be always as the power of the quantity A, the index of which power *n*-1, is any number *n* diminished by unity. Then the velocity in every altitude A will be as , and therefore will be given. For by prop. 59. the velocity of a body ascending and descending in a right line is in that very ratio.

Proposition XLI. Problem XXVIII.

*Supposing a centripetal force of any kind, and granting the quadratures of curvilinear figures, it is required to find, as well the trajectories in which bodies will move, as the times of their motions in the trajectories found.*

Let any centripetal force tend to the centre *C*, (*Pl*. 17. *Fig*. 4.) and let it be required to find the trajectory *VIKk*. Let there be given the circle *VR*, described from the centre *C* with any interval *CV* and from the same centre describe any other circles *ID*, *KE* cutting the trajectory in *I* and *K*, and the right line *CV* in *D* and *E*. Then draw the right line *CNIX* cutting the circles *KE*, *VR* in *N* and *X*, and the right line *CKY* meeting the circle *VR* in *Y*. Let the points *I* and *K* be indefinitely near; and let the body go on from *V* through *I* and *K* to *k*; and let the point *A* be the place from whence another body is to fall, so as in the place *D* to acquire a velocity equal to the velocity of the first body in *I*. And things remaining as in prop. 39. the lineola *IK*, described in the least given time will be as the velocity, and therefore as the right line whose power is the area *ABFD*, and the triangle *ICK* proportional to the time will be given, and therefore *KN* will be reciprocally as the altitude *IC*; that is (if there be given any quantity *Q* and the altitude *IC* be called *A*) as . This quantity call *Z*, and suppose the magnitude of Q to be such that in some case *√ABFD* may be to *Z* as *IK* to *KM* and then in all cases, *√ABFD* will be to *Z* as *IK* to *KM* and *ABFD* to *ZZ* as to , and by division *ABFD−ZZ* to *ZZ* as to , and therefore, to *Z* or as *IN* to *KN* and therefore will be equal to . Therefore since is to as to *AA* the rectangle will be equal to . Therefore in the perpendicular *DF* let there be taken continually *Db*, *Dc* equal to , respectively, and let the curve lines *ab*, *ac*, the toci of the points *b* and *c*, be described: and from the point *V* let the perpendicular *Va* be erected to the line *AC*, cutting off the curvilinear area's *VDba*, *VDca*, and let the ordinates *Ez*, *Ex*, be erected also. Then because the rectangle *Db*×*IN* or *DbzE* is equal to half the rectangle *A*×*KN* or to the triangle *ICK*; and the rectangle *Dc*×*IN* or *DcxE* is equal to half the rectangle *YX*×*XC* or to the triangle *XCY*; that is, because the nascent particles *DbzE*, *ICK* of the area's *VDba*, *VIC* are always equal; and the nascent particles *DcxE*. *XCY* of the area's *VDca*, *VCX* are always equal; therefore the generated area *VDba* will be equal to the generated area *VIC*, and therefore proportional to the time; and the generated area *VDca* is equal to the generated sector *VCX*. If therefore any time be given during which the body has been moving from *V,* there will be also given the area proportional to it *VDba*; and thence will be given the altitude of the body *CD* or *CI*; and the area *VDca*, and the sector *VCX* equal thereto, together with its angle *VCI*. But the angle *VCI*, and the altitude *CI* being given, there is also given the place in which the body will be found at the end of that time. *Q. E. I.*

Cor. 1. Hence the greatest and least altitudes of the bodies, that is the apsides of the trajectories, may be found very readily. For the apsides are those points in which a right line *IC* drawn thro' the centre falls perpendicularly upon the trajectory *VIK*; which comes to pass when the right lines *IK* and *NK* become equal; that is, when the area *ABFD* is equal to *ZZ*.

Cor. 2. So also, the angle *KIN* in which the trajectory at any place cuts the line *IC*, may be readily found by the given altitude *IC* of the body: to wit, by making the sine of that angle to radius as *KN* to *IK*; that is as *Z* to the square root of the area *ABFD*.

Cor. 5. If to the centre *C* (*Pl.* 17. *Fig*. 5.) and the principal vertex *V* there be described a conic section *VRS*; and from any point thereof as *R*, there be drawn the tangent *RT* meeting the axe *CV* indefinitely produced, in the point *T*; and then, joining *CR*, there be drawn the right line *CP*, equal to the abscissa *CT*, making an angle *VCP* proportional to the sector *VCR*; and if a centripetal force, reciprocally proportional to the cubes of the distances of the places from the centre, tends to the centre *C*; and from the place *V* there sets out a body with a just velocity in the direction of a line perpendicular to the right line *CV*: that body will proceed in a trajectory *VPQ*. which the point *P* will always touch; and therefore if the conic section *VRS* be an hyperbola, the body will descend to the centre; but if it be an ellipsis it will ascend perpetually, and go farther and farther off in infinitum. And on the contrary, if a body endued with any velocity goes off from the place *V*, and according as it begins either to descend obliquely to the centre or ascends obliquely from it, the figure *VRS* be either an hyperbola or an ellipsis, the trajectory may be found by increasing or diminishing the angle *VCP* in a given ratio. And the centripetal force becoming centrifugal, the body will ascend obliquely in the trajectory *VPQ*, which is found by taking the angle *VCP* proportional to the elliptic sector *VRC*, and the length *CP* equal to the length *CT*, as before. All these things follow from the foregoing proposition, by the quadrature of a certain curve, the invention of which, as being easy enough. for brevity's s I omit.

Proposition XLII. Problem XXIX.

*The law of centripetal force being given, it os required to find the motion of a body setting out from a given place, with a given velocity, in the direction of a given right line.*

Suppose the same things as in the three preceding propositions; and let the body go off from the place L (Pl. 17. *Fig.* 6.) in the direction of the little line *IK*, with the same velocity as another body, by falling with an uniform centripetal force from the place *P*, may acquire in *D*; and let this uniform force be to the force with which the body is at first urged in *I*, as *DR* to *DF*. Let the body go on towards *k'; and about the centre* C with the interval *Ck*, describe the circle *ke*, meeting the right line *PD* in *e*, and let there be erected the lines *eg*, *ev*, *ew*, ordinately applied to the curves *BFg*, *abv*, *acw*. From the given rectangle *PDRQ* and the given law of centripetal force, by which the first body is acted on, the curve line *BFg* is also given, by the construction of prop. 27. and its cor. 1. Then from the given angle *CIK* is given the proportion of the nascent lines *IK*, *KN*; and thence by the construction of prob. 28. there is given the quantity *Q*, with the curve lines *abv*, *acw*; and therefore, at the end of any time *Dbve*, there is given both the altitude of the body *Ce* or *Ck*, and the area *Dcwe*, with the sector equal to it *XCy*, the angle *ICk*, and the place *k*, in which the body will then be found. *Q. E. I.*

We suppose in these propositions the centripetal force to vary in its recess from the centre according to some law, which any one may imagine at pleasure; but at equal distances from the centre to be every where the same.

I have hitherto considered the motions of bodies in immovable orbits. It remains now to add something concerning their motions in orbits which revolve round the centres of force.