# The Mathematical Principles of Natural Philosophy (1729)/Book 1/Section 12

*Of the attractive forces of sphærical bodies.*

Proposition LXX. Theorem XXX.

*If to every point of a sphærical surface there tend equal centripetal forces decreasing in the duplicate ratio of the disstances from those points; I say that a corpuscle placed within that superficies will not be attracted by those forces any way.*

Let *HIKL* (*Pl*. 21. *Fig*. 4.) be that sphærical superficies, and *P* a corpuscle placed within. Through *P* let there be drawn to this superficies the two lines *HK*, *IL*, intercepting very small arcs *HI*, *KL*; and because (by cor. 3. lem. 7.) the triangles *HPI*, *LPK* are alike, those arcs will be proportional to the distances *HP*, *LP*; and any particles at *HI* and *KL* of the sphærical superficies, terminated by right lines passing through *P*, will be in the duplicate ratio of those distances. Therefore the forces of these particles exerted upon the body *P* are equal between themselves. For the forces are as the particles directly and the squares of the distances inversely. And these two ratio's compose the ratio of equality. The attractions therefore being made equally towards contrary parts destroy each other. And by a like reasoning all the attractions through the whole sphærical superficies are destroyed by contrary attractions. Therefore the body *P* will not be any way impelled by those attractions. *Q. E. D.*

*The same things supposed as above, I say that a corpuscle placed without the sphærical superficies is attracted towards the centre of the sphere with a force reciprocally proportional to the square of its distance from that centre.*

Let *AHKB*, *ahkb* (Pl. 21. *Fig.* 5.) be two equal sphærical superficies described about the centres *S*, *s*; their diameters *AB*, *ab*; and let *P* and *p* be two corpuscles situate without the spheres in those diameters produced. Let there be drawn from the corpuscles the lines *PHK*, *PIL*, *phk*, *pil*, cutting off from the great circles *AHB*, *ahb*, the equal arcs *HK*, *bk*, *IL*, *il*; and to those lines let fall the perpendiculars *SD*, *sd*, *SE*, *se*, *IR*, *ir*; of which let *SD*, *sd* cut *PL*, *pl* in *F* and *f*. Let fall also to the diameters the perpendiculars *IQ*, *iq*. Let now the angles *DPE*, *dpe* vanish; and because *DS* and *ds*, *ES* and *es* are equal, the lines *PE*, *PF*, and *pe*, *pf*, and the lineolæ *DF*, *df* may be taken for equal; because their last ratio, when the angles *DPE*, *dpe* vanish together, is the ratio of equality. These things then supposed, it will be, as *PI* to *PF* so is *RI* to *DF*, and, as *pf* to *pi* so is *df* or *DF* to *ri*; and *ex æquo*, as *PI* x *pf* to *PF* x *pi* so is *RI* to *ri*, that is (by cor. 3. lem. 7.) so is the arc *IH* to the arc *ih*. Again *PI* is to *PS* as *IQ* to *SE*, and *ps* ro *pi* as *se* or *SE* to *iq*; and *ex æquo* *PI* x *ps* to *PS* x *pi* as *IQ* to *iq*. And compounding the ratio's is to , as *IH* x *IQ* to *ib* x *iq*; that is, as the circular superficies which is described by the arc *IH* as the semicircle *AKB* revolves about the diameter *AB*, is to the circular superficies described by the arch *ih* as the semicircle *akb* revolves about the diameter *ab*. And the forces with which these superficies attracts the corpuscles *P* and *p* in the direction of lines tending to those superficies are by the hypothesis as the superficies themselves directly, and the squares of the distances of the superficies from those corpuscles inversely; that is, as *pf* x *ps* to *PF* x *PS*. And these forces again are to the oblique parts of them which (by the revolution of forces as in cor. 2. of the laws) tend to the centres in the directions of the lines *PS*, *ps*, as *PI* to *PQ* and *pi* to *pq*; that is (because of the like triangles *PIQ* and *PSF*, *piq* and *psf*) as *PS* to *PF* and *ps* to *pf*. Thence *ex equo*, the attraction of the corpuscle *P* towards *S* is to the attraction of the corpuscle *p* towards *s*, as is to , that is, as to . And by a like reasoning the forces with which the superficies described by the revolution of the arcs *KL*, *kl*attract those corpuscles, will be as to . And in the same ratio will be the forces of all the circular superficies into which each of the sphærical superficies may be divided by taking *sd* always equal to *SD*, and *se* equal to *SE*. And therefore by composition, the forces of the entire sphærical superficies exerted upon those corpuscles will be it: the same ratio. *Q. E. D.*

Proposition LXXII. Theorem XXXII.

*If to the sveral points: of a sphere there tend equal centripetal forces decreasing in a duplicate ratio of the distances from those points; and there be given both the density of the sphere and the ratio of the diameter of the sphere to the distance of the corpuscle from its centre; I say that the force with which the corpuscle is* *attracted is proportional to the semi-diameter of the sphere.*

For conceive two corpuscles to be severally attracted by two spheres, one by one the other by the other, and their distances from the centres of the spheres to be proportional to the diameters of the spheres respectively; and the spheres to be resolved into like particles disposed in a like situation to the corpuscles. Then the attractions of one corpuscle towards the several particles of one sphere, will be to the attractions of the other towards as many analogous particles of the other sphere in a ratio compounded of the ratio of the particles directly and the duplicate ratio, of the disŧances inversely. But the particles are as the spheres, that is in a triplicate ratio of the diameters, and the distances are as the diameters; and the first ratio directly with the last ratio taken twice inversely, becomes the ratio of diameter to diameter. *Q. E. D.*

Cor. 1. Hence if corpuscles revolve in circles about spheres composed of matter equally attracting; and the distances from the centres of the spheres be proportional to their diameters; the periodic times will be equal.

Cor. 2. And *vice versa*, if the periodic times are equal, the distances will be proportional to the diameters. These two corollaries appear from cor. 5. prop. 4.

Cor. 3. If to the several points of any two solids whatever, of like figure and equal density, there tend equal centripetal forces decreasing in a duplicate ratio of the distances from those points; the forces with which corpuscles placed in a like *situation to those two solids, will be attracted by them will be to each other as the diameters of the solids.*

*If to the sevaral points of a given sphere there tend equal centripetal forces decreasing in a duplicate ratio of the disŧances from the points; I say that a corpuscle placed within sphere is attracted by a force proportional to its disŧance from the centre.*

In the sphere *ABCD* (Pl. 21. *Fig.* 6.) described about the centre *S*, let there be placed the corpuscle *P*; and about the same centre *S*, with the interval *SP*, conceive described an interior sphere *PEQF*. It is plain (by prop. 70.) that the concentric sphærical supericies of which the difference *AEBF* of the spheres is composed, have no effect at all upon the body *P*; their attractions being destroyed by contrary attractions. There remains therefore only the attraction of the interior sphere *PEQF* And (by prop. 72.) this is as the distance *PS*. *Q. E. D.*

By the superficies of which I here imagine the solidg composed, I do not mean superficies purely

mathematical, but orbs so extreamly thin, that their thickness is as nothing; that is, the evanescent orbs; of which the sphere will at last consist, when the number of the orbs is increased, and their thickness diminished without end. In like manner, by the points of which lines, surfaces and solids are said to be composed, are to be understood equal particles whose magnitude is perfectly incossiderable.

*The same things supposed, I say that a corpuscle situate without a force reciprocally proportional to the square of its distance form the centre.*

For suppose the sphere to be divided into innumerable concentric sphærical superficies, and the attractions of the corpuscle arising from the several superficies will be reciprocally proportional to the square of the distance of the corpuscle from the centre of the sphere (by prop. 71.) And by composition, the sum of those attractions, that is, the attraction of the corpuscle towards the entire sphere, will be in the same ratio. *Q. E. D.*

Cor. 1. Hence the attractions of homogeneous spheres at equal distances from the centres will be as the spheres themselves. For (by prop. 72.) if the distances be proportional to the diameters of the spheres, the forces will be as the diameters. Let the greater distance be diminished in that ratio; and the distances now being equal, the attraction will be increased in the duplicate of that ratio; and therefore will be to the other attraction in the triplicate of that ratio; that is, in the ratio of the spheres.

Cor. 2. At any distances whatever; the attractions are as the spheres applied to the squares of the distances.

Cor. 3. If a corpuscle placed without an homogeneous sphere is attracted by a force reciprocally proportional to the square of its distance from the centre, and the sphere consists of attractive particles; the force of every particle will decrease in a duplicate ratio of the distance from each particle.

*If to the several points of a given sphere there tend equal centripetal force; decresing in a duplicate ratio of the distances from the points; I say that another similar sphere will be attracted by it with a force reciprocal proportional to the square of the disŧance of the centres.*

For the attraction of every particle is reciprocally as the square of its distance from the centre of the attracting sphere (by prop. 74.) and is therefore the same as if that whole attracting force issued from one single corpuscle placed in the centre of this sphere. But this attraction is as great, as on the other hand the attraction of the same corpuscle would be, if that were it self attracted by the several particles of the attracted sphere with the same force with which they are attracted by it. But that attraction of the corpuscle would be (by prop. 74.) reciprocally proportional to the square of its distance from the centre of the sphere; therefore the attraction of the sphere, equal thereto, is also in the same ratio. *Q. E. D.*

Cor. 1. The attractions of spheres towards other homogeneous spheres, are as the attracting spheres applied to the squares of the distances of their centres from the centres of those which they attract.

Cor. 2.. The case is the same when the attracted sphere does also attract. For the several points of the one attract the several points of the other with the same force with which they themselves are attracted by the others again; and therefore since in all attractions (by law 3.) the attracted and attracting point are both equally acted on, the force will be doubled by their mutual attractions, the proportions remaining.

Cor. 3. Those several truths demonstrated above concerning the motion of bodies about the focus of the conic sections, will take place when an attracting sphere is placed in the focus, and the bodies move without the sphere.

Cor. 4. Those things which were demonstrated before of the motion of bodies about the centre of the conic sections take place when the motions are performed within the sphere.

*If spheres be however dissimilar (as to density of matter and attractive force) in the progress right onward from the centre to the circumference; but every where similar, at every given disŧance from the centre, on all sides round about; and the attractive force of every point decreases in the duplicate ratio of the distance of the body attracted; I say that the whole force with which one of these spheres attracts the other, will be reciprocally proportional to the force of the distance of the centres.*

Imagine several concentric similar spheres, *AB*, *CD*, *EF*, &c. (Pl. 22. *Fig.* 1.) the innermost of which added to the outermost may compose a matter more dense towards the centre, or subducted from them may leave the same more lax and rare. Then by prop. 75. these spheres will attract other similar concentric spheres *GH*, *IK*, *LM*, &c, each the other, with forces reciprocally proportional to the square of the distance *SP*. And by composition or division, the sum of all those forces, or the excess of any of them above the others; that is, the entire force with which the whole sphere *AB* (composed of an concentric spheres or of their differences) will attract the whole sphere *GH* (composed of any concentric spheres or their differences) in the same ratio. Let the number of the concentric spheres be increased in infinitum, so that the density of the matter together with the attractive force may, in the progress from the circumference to the centre, increase or decrease according to any given law; and by the addition of matter not attractive let the deficient density be supplied that so the spheres may acquire any form desired; and the force with which one of these attracts the other, will be still, by the former reasoning, in the same ratio of the square of the distance inversely. *Q. E. D.*

Cor. 1. Hence if many spheres of this kind, similar in all respects, attract each other mutually; the accelerative attractions of each to each, at any equal distances of the centres, will be as the attracting spheres.

Cor. 2. And at any unequal distances, as the attracting spheres applied to the squares of the distances between the centres.

Cor. 3. The motive attractions, or the weights of the spheres towards one another will be at equal distances of the centres as the attracting and attracted spheres conjunctly; that is, as the products arising from multiplying the spheres into each other.

Cor. 4. And at unequal distances, as those products directly and the squares of the distances between the centres inversely.

Cor. 5. These proportions take place also, when the attraction arises from the attractive virtue of both spheres mutually exerted upon each other. For the attraction is only doubled by the conjunction of the forces, the proportions remaining as before.

Cor. 6. If spheres of this kind revolve about others at rest, each about each; and the distances between the centres of the quiescent and revolving bodies are proportional to the diameters of the quiescent bodies; the periodic times will be equal.

Cor. 7. And again, if the periodic times are equal, the distances will be proportional to the diameters.

Cor. 8. All those truths above demonstrated, relating to the motions of bodies about the foci of conic sections, will take place, when an attracting sphere, of any form and condition like that above described, is placed in the focus.

Cor. 9. 9. And also when the revolving bodies are also attracting spheres of any condition like that above described.

*If to the several points of spheres there tend centripetal forces proportional to the disŧances of the points from the attracted bodies; I say that the compounded force with which two spheres attract each other mutually is or the disŧance between the centres of the spberes.*

Case 1. Let *AEBF* (Pl. 22. *Fig.* 2.) be a sphere; *S* its centre; *P* a corpuscle attracted; *PASB* the axis of the sphere pulling through the centre of the corpuscle; *EF*, *ef* two planes cutting the sphere, and perpendicular to the axis, and equidistant, one on one side, the other on the other, from the centre of the sphere; *G* and *g* the intersections of the planes and the axis; and *H* any point in the plane *EF*. The centripetal force of the point *H* upon the corpuscle *P*, exerted in the direction of the line *PH* is as the distance *PH*; and (by cor. 2. of the laws) the same exerted in the direction of the line *PG*, or towards the centre *S*, is at the length *PG*. Therefore the force of all the points in the plane *EF* (that is of that whole plane) by which the corpuscle *P* is attracted towards the centre *S* is as the distance *PG* multiplied by the number of those points, that is as the solid contained under that plane *EF* and the distance *PG*. And in like manner the force of the plane *ef' by which the corpuscle* P *is attracted towards* the centre *S*, is as at plane drawn into its distance *Pg*, or as the equal plane *EF* drawn into that distance *P*; and the sum of the forces of both planes as are plane *EF* drawn into the sum of the distances *PG* + *Pg*, that is as that plane drawn into twice the distance *PS* of the centre and the corpuscle; that is, as twice the plane *EF* drawn into the distance *PS*, or as the sum of the equal planes *EF* + *ef* drawn into the same distance. And by a like reasoning the forces of all the planes in the whole sphere, equidistant on each side from the centre of the sphere, are as the sum of those planes drawn into the distance *PS*, that is, as the whole sphere and the disŧance *PS* conjunctly. *Q. E. D.*

Case 2. Let now the corpuscle *P* attract the sphere *AEBF*. And by the same reasoning it will appear that the force with which the sphere is attracted is as the distance *PS*. *Q. E. D.*

Case 3. Imagine another sphere composed of innumerable corpuscles *P*; and because the force with which every corpuscle is attracted is as the distance of the corpuscle from the centre of the first sphere, and as the same sphere conjunctly, and is therefore the same as if it all proceeded from a single corpuscle situate in the centre of the sphere; the entire force with which all the corpuscles in the second sphere are attracted, that is, with which that whole sphere is attracted, will be the same as if that sphere were attracted by a force issuing from a single corpuscle in the centre of the first sphere; and is therefore proportional to the distance between the centres of the spheres. *Q. E. D.*

Case 4. Let the spheres attract each other mutually, and the force will be doubled. but the proportion will remain. *Q. E. D.*

Case 5. Let the corpuscle be placed within the sphere *AEBF*; (*Fig.* 3.) and because the force of the plane *ef* upon the corpuscle is as the solid contained under that plane and the distance *pg*; and the contrary force of the plane *EF* as the solid contained under that plane and the distance *pG*; the force compounded of both will be as the difference of the solids, that is as the sum of the equal planes drawn into half the difference of the distance that is, as that sum drawn into *PS*, the distance of the corpuscle from the centre of the sphere. And by a like reasoning, the attraction of all the planes *EF*, *ef* throughout the whole sphere, that is, the attraction of the whole sphere, is conjunctly as the sum of all the planes, or as the whole spheres and as *pS*, the distance of the corpuscle from the centre of the sphere. *Q. E. D.*

Case 6. And if there be composed a new sphere out of innumerable corpuscles such as *p*, situate within the first sphere *AEBF*; it may be proved as before that the attraction whether, single of one sphere towards the other, or mutual of both towards each other, will be as the distance *pS* of the centres. *Q. E. D.*

*If spheres in the progression from the centre to the circumference be however dissimimar and unequable, but similar on every side round about at all given disŧances from the centre; and the attractive force of every point be as the disŧance of the attracted body; I say that the entire force with which two spheres of this kind attract each other mutually is proportional to the centres of the spheres.*

This is demonstrated from the foregoing proposition in the same manner as the 76th proposition was demonstrated from the 75th

Cor. Those things that were above demonstrated in prop. 10. and 64. of the motion of bodies round the centres of conic sections, take place when all the attractions are made by the force of sphærical bodies of the condition above described, and the attracted bodies are spheres of the same kind.

I have now explained the two principal case of attractions; to wit, when the centripetal forces decrease in a duplicate ratio of the distances, or increase in a simple ratio of the distances; causing the bodies in both cases to revolve in conic sections, and composing sphærical bodies whose centripetal forces observe the same law of increase or decrease in the recess from the centre as the forces of the particles themselves do; which is very remarkable. It would be tedious to run over the other cases, whose conclusions are less elegant and important, so particularly as I have done these. I chuse rather to comprehend and determine them all by one general method as follows.

*If about the centre* S (Pl. 22. Fig. 4.) *these le described arty circle at* AEB, *and about the centre* P *there be also described two circles* EF, *ef*, cutting the first in *R* and *e,* and the line *PS* in *F* and *f;* and the line *PS* in *F* and *f; and there be let fall to* PS *the perpendiculars* ED, ed; *I say, that, if the disŧance of the arcs* EF, ef, *be supposed to be infinitely, the last ratio of the evanescent *evanescent line *Dd* to the evanescent line *Ff* is the same as that of the line *PE* to the line *PS.*

For if the line *Pe* cut the arc *EF* in *q*; and the right line *Ee*, which coincides with the evanescent arc *Ee*, be produced and meet the right line *PS* in *T*; and there be let fall from *S* to *PE* the perpendicular *SG*; then because of the like triangles *DTE*, *dTe*, *DES*; it will be as *Dd* to *Ee* so *DT* to *TE*, or *DE* to *ES*; and because the triangles *Eeq*, *ESG* (by lem. 8. and cor. 3. lem. 7.) are similar, it will be as *Ee* to *eq* or *Ff* so *ES* to *SG*; and *ex æquo*, as *Dd* to *Ff* so *DE* to *SG*; that is (because is the similar triangles *PDE*, *PGS*) so is *PE* to *PS*. *Q. E. D.*

*Suposse a superficies as* EFfe (Pl. 22 Fig. 5.) *to have its breadth infinitely diminished, and to be just vanishing; and that the same superficies by its revolution round the axis* PS *describes a sphærical concavo-convex solid to the several equal particles of which there tend equal centripetal forces; I say that the force with which that solid attracts a corpuscle situate in* P, *is in a ratio compunded of the ratio of the solid* *and the ratio of the force with *which the given particle in the place *Ff* would attract the same corpuscle.

For if we consider first the force of the sphærical superficies *FE* which is generated by the revolution of the arc *FE*, and is cut any where, as in *r*, by the line *de*; the annular part of the superficies generated by the revolution of the arc *rE* will be as the lineola *Dd*, the radius of the sphere *PE* remaining the same; as *Archimedes* has demonstrated in his book of the sphere and cylinder. And the force of this superficies exerted in the direction of the lines *PE* or *Pr* situate all round in the conical superficies, will be as this annular superficies it self; that is as the lineola *Dd*, or which is the same as the rectangle under the given radius *PE* of the sphere and the lineola *Dd*; but that force, exerted in the direction of the line *PS* tending to the centre *S*, will be less in the ratio of *PD* to *PE*, and therefore will be as *FD* x *Dd*. Suppose now the line *DF* to be divided into innumerable little equal particles, each of which call *Dd*; and then the superficies *FE* will be divided into so many equal annuli, whose forces will be as the sum of all the rectangles *PD* x '*Dd*, that is, as , and therefore as . Let now the superficies *FE* be drawn into the altitude *Ff*; and the force of the solid *EFfe* exerted upon the corpuscle *P* will be as ; that is, if the force be given which any given particle as *Ff* exerts upon the corpuscle *P* at the disŧance *PF*. But if that forte be not given, the force of the solid *EFfe* will be as the solid and that force not given, conjunctly. *Q. E. D.*

*If to the several equal parts of a sphere* ABE, (Pl. 22. Fig. 6.) *described about the centre* S, *there tend equal centripetal forces; and from the several points* D *in the axis of the sphere* AB *in which a corpuscle, as, is placed, there be erected the perpendiculars* DE *meeting the sphere in* E, *and if in those perpendiculars the lengths* DN *be taken as the quantity* *and as the force which a particle of the sphere situate in the axis exerts at the distance* PE upon the corpuscle *P,* conjunctly; I say that the whole force with which the corpuscle *P is attracted towards the sphere is as the area* ANB, *comprehended under the axis of the sphere* AB, *and the curve line* ANB, *the locus of the point* N.

For supposing the construction in the last lemma and theorem to stand. conceive the axis of the sphere *AB* to be divided into innumerable equal particles *Dd*, and the whole sphere to be divided into so many sphærical concavo-convex laminæ '*EFfe*; and erect the perpendicular *dn*. By the last theorem the force with which the laminæ *EFfe* attracts the corpuscle *P*. is as and the force of one particle exerted at the distance *PE* or *PF*, conjunctly. But (by the last lemma) *Dd* is to *Ff* as *PE* to *PS*, and therefore *Ff* is equal to and is equal to and therefore the force of the laminæ *EFfe* and the force of particle exerted at the disŧance *PF* conjunctly; that is supposition, as *DN* x *Dd*, or as the evanescent area *DNnd*. Therefore the forces of all the laminæ exerted upon the corpuscle *P* are as all the areas *DNnd*, that is, the whole force of the sphere will be as the whole area *ANB*. *Q. E. D.*

Cor. 1. Hence if the centripetal force tending to the several particles remain always the sæme at all distances, and *DN* be made as the whole force with which the corpuscle is attracted by the sphere is as the area *ANB*.

Cor. 2. If the centripetal force of the particles be, reciprocally as the distance of the corpuscle attracted by it, and *DN* be made as the force with which the corpuscle B is attracted by the whole sphere will be as the area *ANB*.

Cor. 3. If the centripetal force of the particles be reciprocally as the cube of the distance of the corpuscle attracted by it, and *DN* be made as

the force with which the corpuscle attracted by the whole sphere will be as the area *ANB*.

Cor. 4. And universally if the centripetal force tending to the several particles of the sphere be supposed to be reciprocally as the quantity *V*; and *DN* be made as ; the force with which 5 corpuscle is attracted by the whole sphere will be as the area *ANB*.

*The things remaining as above it is required to measure the area* ANB. (Pl. 23. Fig. 1.)

From the point *P* let there be drawn the right line *PH* touching the sphere in *H*; and to the axis *PAB* letting fall the perpendicular *HI*, bisect *PI* in *L*; and (by prop. 12. book 2. elem.) is equal to . But because the triangles *SPH*, *SHI* are alike. or is equal to the rectangle *PSI*. Therefore is equal to the rectangle contained under *PS* and *PS* + *SI* + 2'SD*; that is under* PS *and 2*LD*. Moreover is equal to , or , that is, . For or (by prop. 6. book 2. elem.) is equal* to the rectangle *ALB*. Therefore if instead of we write ; the quantity , which (by cor. 4. of the foregoing prop.) is as the length of the ordinate *DN* will now resolve it self into three parts ; where if instead of *V* we write the inverse ratio of the centripetal force, and instead of *PE* the mean proportional between *PS* and 2*LD*; thos three parts will become ordinates to so many curve lines, whose areas are discovered by the common methods. *Q. E. D.*

Example 1. If the centripetal force tending to the several particles of the sphere be reciprocally as the distance; instead of *V* write *P*E the distance; then . Suppose *DN* equal to its double ; and 2*SL* the given part of the ordinate drawn into the length *AB* will describe the rectangular area 2*SL* x *AB*; and the indefinite part *LD*, drawn perpendicularly into the same length with a continued motion, in such fort as in its motion one way or another it may either by increasing or decreasing remain always equal to the length *LD*, will desrive that is, the area *SL* x *AB*; which taken from the former area 2*SL* x *AB* leaves the area *SL* x *AB*. But the third part , drawn after the same manner with a continued motion perpendicularly into the same length, will describe the area of an hyperbola, which subducted from the area *SL* x *AB* will leave *ANB* the area sought. Whence arises this

*L*,

*A*,

*B*(

*Fig*. 2.) erect the perpendiculars

*Ll*,

*Aa*,

*Bb*;

making *Aa* equal to *Ll*, and *Bb* equal to *LA*. Making *Ll*, and *LB* asymptotes, describe through the points *LA*, the hyperbolic curve *ab*. And the chord *ba* being drawn will inclose the area *aba* equal to the area sought *ANB*.

Example 2. If the centripetal force tending to the several particles of the sphere be reciprocally as the cube of the distance, or (which is the same thing) as that cube applied to any given plane; write for *V*, and 2*PS* x *LD* for ; and *DN* will become as that is (because *PS*, *AS*, *SI' are* continually proportional) as . If we draw then these three parts into the length *AB*, the first will generate the area of an hyperbole; the second , the area ; the third , the area that is . Form the first subduct the sum of the second and third, and there will remain *ANB* the area sought. Whence arisfes this

*L*, *A*, *S3*, *B*, (*Fig.* 3.) erect the perpendicualrs *Ll*, *Aa*, *Ss*, *Bb*, of which suppose *Ss* equal to *SI*; and through the point *s*, to the asymptotes *Ll*, *LB*, describe the hyperbola *asb* meeting the perpendiculars *Aa*, *Bb*, in *a* and *b*; and the rectangle 2*ASI* subducted from the hyperbola *AasbB*, will leave *ANB* the area sought.

Example 3. If the centripetal force tending to the several particles of the spheres decrease in a quadruplicate ratio of the distance from the particles; write for *V*, then for *PE*, and *DN* will become as These three parts drawn into the length *AB*, produce so many areas, viz. into ; into ; and into . And these after due reduction come forth , and . And these by subducting the last from the first become . Therefore the entire force with which the corpuscle *P* is attracted towards the centre of the sphere is as , that us reciprocally as . *Q. E. I.*

By the same method one may determine the attraction of a corpuscle situate within the sphere, but more expeditiously by the following theorem.

*In a sphere described about the centre* S (Pl. 23. Fig. 4.) *with the interval* SA, *if there be taken* SI, SA, SP *continually proportional; I say that the attraction of a corpuscle that the attraction of a corpuscle within the sphere in any place* I, *is to its attraction without the sphere in the place* P*, in a ratio compounded of the subduplicate ratio of* IS, PS the distances from the centre, and the subduplicate ratio of the centripetal forces tending to the centre in the places *P* and *I.*

As if the centripetal forces of the particles of the sphere be reciprocally as the distances of the corpuscle attracted by them; the force with which the corpuscle situate in *I* is attracted by the entire sphere, will be to the force with which it is attracted in *P*, in a ratio compounded of the subduplicate ratio of the distance *SI* to the distance *SP*, and the subduplicate ratio of the centripetal force in the place *I* arising from any particle in the centre, to the centripetal force in the place *P* arising from the same particle in the centre, that is, in the subduplicate ratio of the distances *SI*, *SP* to each other reciprocally. These two subduplicate ratio's compose the ratio of equality, and therefore the attractions in *I* and *P* produced by the whole sphere are equal. By the like calculation if the forces of the particles of the sphere are reciprocally in a duplicate ratio of the distance, it will be found that the attraction in *I* is to the attraction in *P* as the disŧance *SP* to the semi-diameter *SA* of the sphere. If those forces are reciprocally in a triplicate ratio of the distances, the attractions in *I* and *P* will be to each other as to ; if in a quadruplicate ratio as to . Therefore since the attraction in P was found in this last case to be reciprocally as , the attraction in *I* will be reciprocally as , that is, because is given, reciprocally as . And the progression is the same in infinitum. The demonstration of this theorem is as follows.

The things remaining as above constructed and a corpuscle being in any place *P*, the ordinate *DN* was found to be as . Therefore if *IE* be drawn, that ordinate For any other place of the corpuscle as *I*, will become (*mutatis mutandis*) as . Suppose the centripetal forces flowing from any point of the sphere as *E*, to be to each other at the disŧances *IE* and *PE*, as to , (where the number in *n* denotes the index of the powers of *PE* and *IE*) and those ordinates will become as and whose ratio to each other is as to . Because *SI*, *SE*, *SP* are in continued proportion, the triangles *SPE*, *SEI* are alike; and thence *IE* is to *PE* as *IS* to *SE* or *SA*. For the ratio of *IE* to *PE* write the ratio of *IS* to *SA*; and the ratio of the ordinates becomes that of to . But the ratio of *PS* to *SA* is subduplicate of that of the distances *PS*, *SI*; and the ratio of to (because *IE* is to *PE* as *IS* to *SA*) is subduplicate of that of the forces at the distances *PS*, *IS*. Therefore the ordinates, and consequently the areas which the ordinates describe, and the attractions proportional to them, are in a ratio compounded of those subduplicate ratio's. *Q. E. D.*

*To find the force with which a corpuscle placed in the centre of sphere is attracted towards any segment of that sphere whatsover.*

Let P (Pl. 23. Fig. 5.) be a body in the centre of that sphere, and *RBSD* a segment thereof contained under the plane *RDS* and the sphærical superficies *RBS*. Let *DB* be cut in *F* by a sphærical superficies *EFG* described from the centre *P*, and let the segment be divided into the parts *BREFGS*, *FEDG*. Let us suppose that segment to be not a purely mathematical, but a physical superficies, having some, but a perfectly inconsiderable thickness. Let that thickness be called *O* and (by what *Archimedes* has demonstrated) that superficies will he as *PF* x *DF* x *O*. Let us suppose besides the attractive forces of the particles of the sphere to be reciprocally as that power of the distances, of which *n* is index; and the force with which the superhcies *EFG* attracts the body *P*, will be (by prop. 79.) as, that is, as . Let the perpendicular *FN*, drawn into *O* be proportional to this quantity; and the curvilinear area *BDI*, which the ordinate *FN*, drawn through the length *DB* with a continued motion will describe, will be as the whole force with which the whole segment *RBSD* attracts the body *P*. *Q. E. I.*

*To find the force with which a corpuscle, placed without the centre of a sphere in the axis of any segment, is attracted, by that segment.*

Let the body *P* placed in the axis *ADB* of the segment *EBK* (Pl. 23. *Fig.* 6.) be attracted by that segment. About the centre *P* with the interval *PE* let the sphærical superficies *EFK* be described; and let it divide the segment into two parts *EBKFE* and *EFKDE*. Find the force of the first of those parts by prop. 81. and the force of the latter part by prop. 83. and the sum of the forces will be the force *Pf* the whole segment *EBKDE*. *Q. E. I.*

The attractions of sphærical bodies being now explained, it comes next in order to treat of the laws of attraction in other bodies consisting in like manner of attractive particles; but to treat of them particularly is not necessary to my design. It will be sufficient to subjoin some general propositions relating to the forces of such bodies, and the motions thence arising, because the knowledge of these will be of some little use in philosophical enquiries.