82 



A TTIIACTIO N. 



planes and the axis : and H any point in the plane 

 EF. The centripetal force of the point H upon 

 the particle P, exerted in the direction of the line 

 PH, is as the distance PH ; and according to the 

 direction of the line PG, or towards the centre S, 

 is as the length PG. Therefore, the force of all 

 the points in the plane EF, that is, the force of the 

 whole plane, by which the particle P is attracted 

 towards the centre S, is as the distance PG multi- 

 plied by the number of those points; that is, as the 

 solid which is contained under that plane EF and 

 the distance PG. And, in like manner, the force of 

 the plane ef, with which the particle P is attracted 

 towards the centre S, is as that plane multiplied into 

 its distance P, or as the equal plane EF multiplied 

 into that distance P#: and the sum of the forces of 

 both planes as the plane EF multiplied into the sum 

 of the distances PG-f-Pg; that is, as that plane mul- 

 tiplied, into double the distance PS of the centre and 

 the particle ; that is, as double the plane EF multi- 

 plied into the distance PS ; or as the sum of the equal 

 planes EF-j-e/ multiplied into the same distance. 

 And, by a like reasoning, the forces of all.the planes 

 in the whole sphere, equally distant on each side from 

 the centre of the sphere, are as the sum of the planes 

 multiplied into the distance PS ; that is, as the whole 

 sphere and as the distance PS jointly. 



Case 2. Let the particle P now attract the sphere 

 AEBF. And, by the same reasoning, it will be 

 proved, that the force, with which that sphere is at- 

 tracted, is as the distance PS. 



Case 3. Let another sphere be now composed of 

 innumerable particles P ; and, since the force, with 

 which each particle is attracted, is as the distance of 

 the particle from the centre of the first sphere, and as 

 the same sphere jointly ; and therefore is the same, as 

 if the whole proceeded from one particle in the centre 

 of the sphere ; the whole force, with which all the 

 particles in the second sphere are attracted, that is, 

 with which that whole sphere is attracted, will be the 

 same, as if that sphere was attracted by a force pro- 

 seeding from one particle in the centre of the first 

 sphere ; and therefore is proportional to the distance 

 between the centres of the spheres. 



Case 4. Let the spheres attract each other mu- 

 tually, and the force being doubled will preserve the 

 former proportion. 



Case 5. Let the particle p be now placed within 

 the sphere AEBF ; and, since the force of the plane 

 ef upon the particle 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 p G ; the force compound- 

 ed of both will be as the difference of the solids ; that 

 is, as the sum of the equal planes multiplied into half 

 the difference of the distances ; that is, as that sum 

 multiplied into p S, the distance of the particle from 

 the centre of the sphere. And, by a like reasoning, 

 the attraction of all the planes EF, efin the whole 

 sphere, that is, the attraction of the whole sphere is 

 jointly as the sum of all the planes, or as the whole 

 sphere, and as pS the distance of the particle from 

 the centre of the sphere 



Case 6. And, if a new sphere be composed of in- 

 numerable particles/;, placed within the former sphere 



AEBF; it may be proved as before, that either the Attraction 

 single attraction of one towards the other, or the mu- of Sol:<i >- 

 tual attraction of both towards each other, will be as v 



the distance of the centres pSt 



If spheres are dissimilar and inequable in proceed- Prop. II. 

 ing directly from the centre to the circumference ; 

 but are every where similar at every given distance 

 in a circumference around ; and the attractive force 

 of every point is as the distance of the attracted 

 body : the whole force, with which two spheres of 

 this kind attract each other, is proportional to the 

 distance between the centres of the spheres. 



This is demonstrated from the preceding proposi- 

 tion, in the same manner as the Proposition in Chap. V. 

 p. 691. col. 1. of Physical Astiionomy was demon- 

 strated. 



Cor. Those things which are demonstrated of the 

 motion of bodies round the centres of conic sections, 

 take place, when all the attractions are made by the 

 force of spherical bodies of the quality already de- 

 scribed, and the attracted bodies are spheres of the 

 same kind. 



If any circle AEB is described with the centre S ; Lemma, 

 and two circles EF, ej are described with the centre p LATE 

 P, cutting the former in E, c, and the line PS in XUX. 

 F/; and ED, ed be let fall perpendicular to PS; Fig-<> 

 then, if the distance of the arcs EF, c/'is supposed to 

 be continually diminished, the limit of the ratios of 

 the variable line Dd to the variable line Y f is the 

 same as the ratio of the line PE to the line PS. 



For, if the line P<? cuts the arc EF in q; and the 

 right line Ee, which approaches nearer than by any 

 assignable difference to the arc Ee, be produced, and 

 meet the right line PS in T ; and SG be let fall from 

 S, perpendicular to PE : because of the similar tri- 

 angles DTE, dTe, DES, Dd will be to Ee, as DT 

 to TE, or DE to ES : and, because of the similar 

 triangles Eey, ESG, Ee will be to eq or F/J as ES 

 to SG ; and, ex cequo, D d to Yf, as DE to SG ; 

 that is, because of the similar triangles PDE, PGS, 

 as PE to PS. 



If EF/e, considered as a surface, by reason of its Prop. III. 

 breadth being indefinitely diminished, describes a sphe- 

 rical concavo-convex solid by its revolution round the 

 axis PS, to the several equal particles of which there 

 tend equal centripetal forces ; the force, with which 

 that solid attracts a particle placed in P, is in a ratio 

 compounded of the ratio of the solid DE ! X F/i and 

 the ratio of the force, with which a given particle in 

 the place F/* would attract the same particle in P. 



For, if we first consider the force of the spherical 

 surface FE, which is generated by the revolution of the 

 arc FE, and is any where cut in r by the line dc ; the 

 annular part of the surface, generated by the revolu- 

 tion of the arc rE, will be as the small line Dd, the 

 radius of the sphere PE remaining the same ; as Ar- 

 chimedes has demonstrated in his book concerning 

 the sphere and cylinder. And the force of this, ex- 

 erted in the direction of the lines PE or IV, placed 

 around in a conical surface, is as this annular surface 

 itself; that is, as the line Dd ; or, which is the same, 

 as the rectangle under the given radius PE of '.he 

 sphere, and that line Dd : but that force, acting in 

 the direction of the line PS tending to the centre S, 

 is lew, in the ratio of PD to PE, and therefore as 



