ATTRACTION. 



85 



Prop. VII. 



Putb 

 XLIX. 

 Fig. 8. 



of IS to SA ; and the ratio of the ordinates will be- 

 come that of PS X IE" to SA X PE". But the ratio 

 of PS to SA is the subduplicate ratio of the distances 

 PS, SI ; and the ratio of IE" to PE" (because IE 

 is to PE, as IS to SA) is the subduplicate ratio of 

 the forces at the distances PS, IS. Therefore the 

 ordinates, and consequently the areas which the ordi- 

 nates describe, and the attractions proportional to 

 them, are in a ratio compounded of these subduplicate 

 ratios. 



To find the force, with which a particle, placed in 

 the centre of a spi.ere, is attracted to any segment of 

 that sphere. 



Let P be a particle in the centre of a sphere, and 

 RBSD a segment thereof, contained between the 

 plane RDS, and the spherical surface RBS. Let 

 BD be cut in F by a spherical surface EFG, descri- 

 bed frpm the centre P ; and let the segment be divi- 

 ded into the parts BREFGS, FEDG. But, let that 

 surface be not purely mathematical, but physical, ha- 

 ving a very inconsiderable thickness. Let that thick- 

 ness be called O, and the surface, according to the de- 

 monstration of Archimedes, will be as PF X DF X O. 

 Let us suppose moreover, that the attractive forces 

 of the particles of the sphere arc reciprocally as that 

 power of the distances whose index is n ;" and the 

 force, with which the surface EFG attracts the body 



_ .... DE'xO . . 2DFxO 



P, will be as ppH ; that is, as -pp=i 



DF'xO 



PF" 



Plate 

 XLIX. 

 Jiff. 5. 



Let the perpendicular FN, drawn into 



O, be proportional to this quantity ; and the curvi- 

 linear area BDI, which the ordinate FN, drawn 

 through the length DB by a continual motion, de- 

 scribes, will be as the whole force, with which the 

 whole segment RBSD attracts the body P. 

 Prop. VIII. To find the force, with which a particle, placed 

 without the centre of a sphere, in the axis of any seg- 

 ment, is attracted by that segment. 



Let the body P, placed in the axis ADB of the 

 segment EBK, be attracted by that segment. With 

 the centre P, at the interval PE, let the spherical 

 surface EFK be described; with which let the 

 segment be divided into two parts EBKFE, and 

 EFKDE. Let the force of the former part be 

 sought by Prop. V. and the force of the latter paYt by 

 Prop. VII. and the sum of the forces will be the force 

 of the whole segment EBKDE. 



If one body is attracted by another, and the at- 

 traction is very much stronger when it is contiguous 

 to the attracting body, than when they are separated 

 from each other by any interval, how small so ever ; 

 the forces of the particles of the attracting body, in 

 the recess of the body attracted, decrease in a greater 

 than the duplicate ratio of the distances from the 

 particles. 



For, if the forces decrease in a duplicate ratio of 

 the distances from the particles ; the attraction to-, 

 wards a spherical body, being reciprocally as the 

 square of the distance of the attracted body from 

 the centre of the sphere, will not be sensibly in- 

 creased by the contact : and it will be still less in- 

 creased by the contact, if the attraction in the re- 

 cess of the body attracted, decreases in a less ratio. 



Prop. IX. 



The proposition therefore is evident concerning at- Attraction 

 tractive spheres. And the case of concave spherical of Solids - 

 orbs attracting external bodies is the same. And it V ~~ J 



is much more evident in orbs which attract bodies 

 placed within them ; because the attractions, diffused 

 every where through the cavities of the orbs, are de- 

 stroyed by contrary attractions, (See Chap. V. Phy- 

 sical Astronomy,) and therefore have no effect even 

 in contact. But, if from these spheres and spherical 

 orbs any parts remote from the place of contact are 

 taken away, and new parts are added any where ; the 

 figures of these attractive bodies may be changed at 

 pleasure ; and yet the parts added or taken away, 

 being remote from the place of contact, will not re- 

 markably increase the excess of attraction which 

 arises from the contact. The proposition therefore 

 is evident in bodies of all figures. 



If the forces of the particles, of which an attractive Prop. X. 

 body is composed, decrease, in the recess of the at- 

 tracted body, in a triplicate or more than a triplicate 

 ratio of the distances of the particles, the attraction 

 will be very much stronger in contact, than when the 

 attracting and attracted bodies are separated from 

 each other by any interval, how small soever. 



For it appeai-3 by the solution of Prop. V. exhi- 

 bited in the second and third examples, that the at- 

 traction is indefinitely increased, when an attracted 

 particle approaches to an attracting sphere of this 

 kind. The same thing is also easily collected, by 

 comparing those examples, and Prop. VI. toge- 

 ther, concerning the attractions of bodies towards 

 concavo-convex orbs, whether the attracted bodies 

 are placed without the orbs, or within their cavities. 

 But the proposition will also be universally evident 

 concerning all bodies, by adding or taking away from 

 these spheres and orbs any attractive matter, any 

 where without the place of contact, so that the at- 

 tractive bodies may assume any assigned figure. 



If two bodies, similar to each other, and consisting Prop. XI. 

 of matter equally attractive, attract separately parti- 

 cles proportional to, and similarly situated with re- 

 spect of themselves ; the accelerative attractions of the % 

 particles towards the whole bodies will be, as the ac- 

 celerative attractions of those particles towards par- 

 ticles of the bodies proportional to the whole, and si- 

 milarly situated in them. 



For, if the bodies are divided into particles, which 

 are proportional to the whole bodies, and similarly 

 situated in them ; it will be, as the attraction towards 

 any particle of one body is to the attraction towards 

 the corresponding particle of the other body, so are 

 the attractions towards the several particles of the 

 first body, to the attractions towards the several cor- 

 responding particles of the other body : and, by com- 

 position, so is the attraction towards the first whole 

 body, to the attraction towards the second whole 

 body. 



Cor. 1. Therefore, if the attractive forces of par- 

 ticles, by increasing the distances of the attracted par- 

 ticles, decrease in the ratio of any power of the dis- 

 tances ; the accelerative attractions towards*he whole 

 bodies will be, as the bodies directly, and those 

 powers of the distances inversely. As, if the forces of 

 particles decrease in a duplicate ratio of the distances 



