LA.WS OF GRAVITATION.] 



ASTRONOMY. 



985 



CHAPTER X. 



ON UNIVERSAL GRAVITATION. 



ALL experience shows, that every heavy body requires 

 force, effort, or support, in order to prevent its falling to 

 the earth. What are called light bodies, also require a 

 similar counteracting force when their tendency to fal 

 is not resisted by the presence of the air, as mime 

 rous experiments with the air-pump have sufficient); 

 proved. 



This influence of the earth upon all bodies suspendec 

 over it, and which renders force necessary to keep them 

 suspended, is itself called force the force of gravitation 

 Such is the name which philosophers give to the hidden 

 cause, whatever it be, which forces bodies to fall to 

 the earth when left to themselves, instead of remaining 

 suspended wherever they may be placed, or moving in 

 any other direction. 



That this force of gravitation or attraction is not ex- 

 clusively expended on terrestrial bodies, was a doctrine 

 entertained by several astronomers before Newton's 

 time. Kepler had a steady conviction that the moon 

 gravitated towards the earth ; but he conceived thai 

 tang additional force, animal or spiritual, was in opera- 

 tion to check the gravitating influence, and prevent the 

 two bodies from rushing into collision. 



It was reserved for Newton to prove that the single 

 principle of gravitation, combined w^tli one single im- 

 pulse given to each planet, was sufficient to account for 

 all the celestial movements, and to render clear and in- 

 telligible the entire mechanism of the heavens. 



In addition to the first distinct enunciation of the 

 principle, Newton also discovered the law of gravitation. 

 The principle is, tliat all particles of matter mutually 

 attract each another ; the law is, that the intensity of the 

 attraction varies inversely as the square of the distance 

 at which it acts. 



To verify these important propositions, observations 

 of a peculiar kind were indispensable. These, fortu- 

 nately, had already been supplied by Kepler ; they are 

 embodied in the three following statements, which have, 

 however, a greater degree of generality than Kepler's 

 observations strictly warrant. Hig laws, as they are 

 called, thus generalised, have been mentioned at page 

 944 : it U convenient to repeat them here. 



1. If a line be supposed to connect any planet with 

 the sun, this line will describe equal areas in equal 

 times. 



2. The planets revolve about the sun in elliptic orbits, 

 the sun occupying one focus of the ellipse. 



3. The squares of the times in which any two planets 

 perform their revolutions about the sun, are as the 

 .cubes of their mean distances from it 



The first of these three laws U not peculiarly connected 

 with astronomy, and U of no special importance in 

 establishing the physical theory of the planetary mo- 

 tions ; for, as shown by Newton, it is a universal truth, 

 that a body, moving round a point of attraction in any 

 orbit and attracted by any force, must describe equal 

 aectorial areas in equal times. This truth, therefore, is 

 quite independent of every specified law of attraction, 

 and is irrespective of all considerations as to the distance 

 of the revolving body, or the form of its orbit. If only 

 the direction of the force bo towards a fixed point, the 

 equable description of areas necessarily has place ; and, 

 conversely, if this latter have place, the direction of 

 the attractive force is always towards the same fixed 

 point. This is proved as follows, after the manner of 

 Newton : 



Suppose that a planet, moving at any distance from 

 the sun, be subjected to the influence of an attractive 

 force tending to draw it directly to that body ; and 

 imagine this force, instead of acting continuously, to act 

 by successive impulses, after certain equal intervals of 



VOL. I. 



time. Let A B (Fig. 144) be the path described by the 

 planet during one of these intervals, 

 which path will, of course, be a 

 straight line, since in describing it 

 the planet is not acted upon by any 

 force. 



Arrived at the point B, the at- 

 tractive force at S, acting merely as 

 an instantaneous impulse, changes 

 both the direction and velocity of 

 the planet, which takes another rec- 

 tilinear path, BC, described in an 

 interval of time equal to the former 

 interval ; at the end of which, 

 another impulse, in the direction 

 C S, is given to the planet ; and so 

 on. 



Now, confining our attention for the present to the 

 two sectorial areas (in this case the two triangles SAB, 

 SBC) described in the first two intervals, we readily see 

 that they must be equal For if no fresh impulse had 

 been given to the planet when at B, it would have pro- 

 ceeded onwards from B to M, in the continuation of 

 A B, and in the second interval of time would have 

 described B M equal to A B ; the impulse from S, how- 

 ever, if acting alone, would have brought the planet to 

 some point N in BS, and this impulse, combined with 

 that which acting alone would have brought it to M, 

 causes it to describe the diagonal B C of the parallelo- 

 gram M N, as the first principles of mechanics show. 



Now MC being parallel to BS, the two triangles 

 BMS, DCS are equal (Kuc., Prop. 37, B. I.); but the 

 triangle A B S is also equal to the triangle BMS; hence 

 the triangle A B S is equal to the triangle B C S ; so 

 that in the equal times, the equal triangles A B S, B C S 

 are described, however intense or however feeble the im- 

 pulse from S upon the planet at B may be. 



By increasing the number of intervals of time, and the 

 corresponding impulses from S, the path of the planet 

 would be represented by the sides of an irregular plane 

 polygon, as in the dia- rig. us. 



jrani (Fig. 145) ; and 

 however these sides 

 may differ in lengtii 

 and direction, it is 

 plain, from what is 

 shown above, that they 

 are the bases of equal 

 triangles of which the 

 mint 8 is the common 

 vertex. As each tri- 

 angle is described iu 

 .In; same time, by the 

 ine joining the planet 

 uid the sun, it follows 

 ihat equal areas are 

 always described by 

 this line in equal times. 



It is obvious that this conclusion has nothing to do 

 with the length of time which has been supposed to in- 

 .ervene between the impulses upon the planet, but only 

 upon the equality of the intervals. We may therefore 

 magine these equal intervals to be as small as we please, 

 and therefore the impulses to succeed one another with 

 .he utmost rapidity, and, in fact, to unite in one con- 

 inuous force ; in which case the sides of the polygon, 

 >ecoming shorter and shorter, must unite in one cou- 

 inuous plane curve, the centre of attraction being itself 

 n that plane. We infer, therefore, that a planet 

 noving in virtue of a primitive impulse, and diverted 

 rom its rectilinear path by an attractive force residing 



6K 



