MECHANICS. 



any distance to the lowest point 0, they 

 will, when they meet there, have veloci- 

 ties proportional to the arches through 

 which they have descended.* Now, if 

 the equal balls be permitted to descend 

 through equal arches, they will, there- 

 fore, impinge each upon the other with 

 equal velocities ; and they will be found, 

 after impact, to remain quiescent, each 

 having destroyed the force of the other. 

 This proves that, when equal masses 

 have equal velocities, they have equal 

 forces ; for if their forces were not equal 

 in this case, the united masses would, 

 after impact, move in the direction of 

 that which had the greater force. 



(20.) Now suppose that the ball A 

 (fig. 6.) is double the weight of the ball 

 B ; let A be raised towards X to the divi- 

 sion 3, and let B be raised towards Y to 

 the division 6 ; when allowed to descend 

 from those positions their velocities will 

 be as 3 to 6, but their masses are as 2 

 to 1 , and therefore their forces ought to 

 be as 2 x 3 to 1x6; that is, as 6 to 6, 

 or equal. We accordingly find, that 

 after impact they will be quiescent ; the 

 equal and opposite forces having de- 

 stroyed each other. In like manner, if 

 balls, whose weights are as 2 to 3, fall 

 from distances which are as 6 and 4, 

 their forces being as 2 x 6 to 3 x 4, or 

 as 12 to 12, will be equal, and after im- 

 pact the united masses will be quiescent. 



In the same manner, however the 

 experiment may be varied, it will be 

 found that the product of the numbers 



* Strictly, the velocities are as the chords of the 

 arches; but as the arches used in this case are 

 email compared with the radius, they may be con- 

 sidered to be nearly proportional to their chords. It 

 is another property of this apparatus, that, from 

 whatever distance from the middle of the arch the 

 balls fall, they will arrive at the middle in the same 

 time. This, however, like the property just men- 

 tioned, is only true when the arches used are small 

 compared with the radius. 



representing the mass and velocity al- 

 ways truly represents the moving force. 

 (21.) To return, then, to the case of 

 the earth and a body near its surface, 

 they attract each other with equal 

 forces ; and, therefore, in their conse- 

 quent approach to each other, the 

 earth must have a velocity as many 

 times less than that of the falling body, 

 as the mass of the earth is greater than 

 that of the falling body. Since all bo- 

 dies which can be submitted to these 

 circumstances must be infinitely smaller 

 than the earth, the space through which 

 the earth approaches them in their fall 

 must be infinitely smaller than the space 

 which they fall through. 



To take a veiy improbable and ex- 

 treme case : suppose a ball of earth of 

 a diameter equal to the tenth part of a 

 mile were to be placed at an height 

 above the surface equal to the tenth 

 part of a mile; let us consider what 

 space the earth would move through 

 to meet it. The earth's diameter being 

 about 8000 miles, and spheres being as 

 the cubes of their diameters, the mass 

 of the earth would have to the mass of 

 the ball the ratio of 512,000,000,000,000 

 to 1 ; consequently, if the tenth part of 

 a mile were divided into 512 millions of 

 millions of equal parts, one of these 

 parts would be pretty nearly the space 

 through which the earth would move 

 towards the falling body. In the tenth 



Eart of a mile there are somewhat 

 ;ss than 6400 inches : if this were di- 

 vided into 512 millions of millions of 

 parts, each part would be the eighty 

 millionth part of an inch ; it is there- 

 fore through a space less than this that 

 the earth would move, under the cir- 

 cumstances which we have supposed. 



It is, therefore, quite evident that, with 

 respect to falling bodies, the earth is to 

 be considered at rest. 



(22.) We have stated that bodies at- 

 tract each other in proportion to their 

 quantities of matter. Hence the earth 

 attracts different bodies with different 

 forces. A piece of lead contains a consi- 

 derably greater quantity of matter in the 

 same bulk than a piece of cork, and ac- 

 cordingly we find that the earth attracts 

 it with a proportionally greater force ; 

 in other words, it has greater weight. 

 It is forthis reason that weight is justly 

 assumed as the measure, or exponent of 

 the quantity of matter in any substance, 

 whatever, in other respects, be the 

 species or qualities of that substance. 



(23.) But it is not alone the masses of 



