H 



ATTRACTION. 



Attraction from the particles attra-ted, and the bodies are as A J 



jf Solid 



Prop. XII. 



Plate 

 XLIX. 



and B' ; and therefore, both the cubic sides of the 

 bodies, and the distances of the attracted particles 

 from the bodies, are as A and B ; the accelerative at- 



A' B3 

 tractions towards tht; bodies will be as-j-j- and-j.,; 



that is, as A and B the cubic sides of the bodies. If 

 the forces of particles decrease in the triplicate ratio 

 of the distances from the attracted particles, the ac- 

 celerative attractions towards the whole bodies will be 



as 4- and ~ ; that is, equal. If the forces decrease 



A 3 B 3 * 



in a quadruplicate ratio, the attractions towards the 



bodies will be as -r- and frri that is reciprocally 



A 4 U 4 



as the cubic sides A and B. And so on in other cases. 



Cor. '2. Hence, on the contrary, the ratio of the 

 decrease of the forces of attractive particles, in the 

 recess of the attracted particle, may be collected from 

 the forces, with which similar bodies attract particles 

 similarly situated ; if only that decrease is directly or 

 inversely in any ratio of the distances. 



If the attractive forces of equal particles of any 

 body are as the distances of the places from the par- 

 ticles ; the forces of the whole body will tend to its 

 centre of gravity ; and will be the same with the force 

 of a globe, consisting of similar and equal matter, and 

 having its centre in the centre of gravity. 



Let the particles A, B, of the body RSTV attract 

 any particle Z with forces, which, if the particles are 

 equal to each other, are as the distances AZ, BZ ; 

 but, if the particles are supposed unequal, are as those 

 particles, and their distances AZ, BZ, jointly ; or, 

 as those particles drawn into their distances AZ, BZ, 

 respectively. And let these forces be expressed by 

 those contents, A X AZ and B X BZ. Let AB be 

 joined, and let it be cut in G, so that AG may be to 

 BG, as the particle B to the particle A ; and G will 

 be the common centre of gravity of the particles A 

 and B. The force A X AZ is resolved into the forces 

 A X GZ and A X AG ; and the force B X BZ into the 

 forces B X GZ and B X BG. But the forces A X AG 

 and B X BG, because A is to B as BG to AG, are 

 equal ; and therefore, when they are directed towards 

 contrary parts, destroy each other. The forces 

 A X GZ and B X GZ remain. These tend from Z 

 towards the centre G, and compose the force A -f- B 

 X GZ ; that is, the same force, as if the attractive 

 particles A and B were placed in their common cen- 

 tre of gravity G, composing there a globe. 



By the same reasoning, if a third particle C is add- 

 ed, and the force of this is compounded with the 

 force A + BxGZ, tending to the centre G ; the 

 force thence arising will tend to the common centre 

 of gravity of that globe in G, and of the particle C ; 

 that is to the common centre of gravity of the three 

 particles A, B, C -, and will be the same as if that 

 globe and the particle C were placed in that common 

 centre, composing there a greater globe. And thus 

 we may go on continually. Therefore, the whole 

 force of all the particles of any body RSTV is the 

 same as if that body, preserving its centre of gravity, 

 was to assume the figure of a globe. 



Cor. Hence, the motion of the attracted body Z Attraction 

 will be the same, as if the attracting body RSTV of So1k 

 was spherical : and therefore, if that attracting body 

 is either at rest, or proceeds uniformly in a right line, 

 the body attracted will move in an ellipsis, having it9 

 centre in the centre of gravity of the attracting body. 



If there are several bodies consisting of equal par- Prop. XIl 

 tides, whose forces are as the distances of the places 

 from each ; the force, compounded of the forces of 

 all, by which any particle is attracted, will tend to 

 the common centre of gravity of the attracting bo- 

 dies ; and will be the same as if those attracting bo- 

 dies, preserving their common centre of gravity, 

 should unite there, and be formed into a globe. 



This is demonstrated in the same manner as the 

 foregoing proposition. 



Cor. Therefore the motion of the attracted body 

 will be the same, as if the attracting bodies, preser- 

 ving their common centre of gravity, should unite 

 there, and be formed into a globe. And therefore, if 

 the common centre of gravity of the attracting bodies 

 is either at rest, or proceeds uniformly in a right line ; 

 the body attracted will move in an ellipsis, having its 

 centre in that common centre of gravity of the at- 

 tracting bodies. 



If equal centripetal forces tend to the several points Prop. XIV. 

 of any circle, increasing or decreasing in any ratio of 

 the distances ; it is required to find the force with 

 which a particle is attracted, placed any where in a 

 right line, which stands perpendicularly to the plane 

 of the circle at its centre. 



Suppose a circle to be described about the centre Plate 

 A, with any interval AD, in a plane, to which the XLIX. 

 right line AP is perpendicular ; and let it be requi- Fl - 1 ^ 

 red to find the force, with which any particle P is at- 

 tracted towards the same. Let the right line PE be 

 drawn to the attracted particle P from any point E of 

 the circle. In the right line PA let PF be taken 

 equal to PE, and let the perpendicular FK be erect- 

 ed, which may be as the force with which the point 

 E attracts the particle P. And let I KL be the curve 

 line, which the point K continually touches. Let 

 that curve meet the plane of the circle in L. In PA 

 let PH be taken equal to PD ; and let the perpendi- 

 cular HI be erected, meeting the curve in I ; and the 

 attraction of the particle P towards the circle will be 

 as the area AHIL, multiplied into the altitude AP. 



For, let a very small line Ee be taken in AE. Let 



Pe be joined; and let PC, P/" be taken in PE, PA, 



equal to Ye. And since the force, with which any 



point E of the annulus described about the centre A, 



with the interval AE in the aforesaid plane, attracts 



the body P towards itself, is supposed to be as FK ; 



and therefore the force, with which that point at- 



, , , , a APxFK 

 tracts the body P towards A, is as =p ; and 



the force, with which the whole annulus attracts the 



, , , j APxFK 



body P towards A, is as the annulus and ==; 



jointly : but that annulus is as the rectangle under the 

 radius AE and the breadth Ec ; and this rectangle 

 (because PE and AE, Ee and CE are proportion- 

 al) is equal to the rectangle PE X CE or PE X Ef; 

 the force, with which that annulus attracts the body 



