40 LECTURE VI. 



its distance are equal : thus two multiplied by one is equal to one multi- 

 plied by two. (Plate II. Fig. 29.) 



This point is most commonly called the centre of gravity ; it has also 

 sometimes been denominated the centre of position. Since it has many 

 properties independent of the consideration of gravity, it ought not to derive 

 its name from gravitation, [but as custom has familiarized the term, we 

 deem it better to retain it.] 



The centre of inertia [gravity] of any two bodies initially at rest, remains 

 at rest, notwithstanding any reciprocal action of the bodies ; that is, not- 

 withstanding any action which affects the single particles of both equally, 

 in increasing or diminishing their distance. For it may be shown, from 

 the principles of the composition of motion, that any force, acting in this 

 manner, will cause each of the two bodies to describe a space proportional 

 to the magnitude of the other body : thus a body of one pound will move 

 through a space twice as great as a body of two pounds weight, and the 

 remaining parts of the original distance will still be divided in the same 

 proportion by the original centre of inertia [gravity], which therefore still 

 remains the centre of inertia [gravity], and is at rest. And it follows also, 

 that if the centre of inertia [gravity] is at first in motion, its motion will 

 not be affected by any reciprocal action of the bodies. 



This important property is very capable of experimental illustration ; 

 first observing, that all known forces are reciprocal, and among the rest the 

 action of a spring ; we place two unequal bodies so as to be separated when 

 a spring is set at liberty, and we find that they describe, in any given 

 interval of time, distances which are inversely as their weights ; and 

 that consequently the place of the centre of inertia [gravity] remains un- 

 altered. They may either be made to float on water, or may be suspended 

 by long threads ; the spring may be detached by burning a thread that 

 confines it, and it may be observed whether or no they strike at the same 

 instant two obstacles, placed at such distances as the theory requires ; or if 

 they are suspended as pendulums, the arcs which they describe may be 

 measured, the velocities being always nearly proportional to these arcs, and 

 accurately so to their chords. (Plate II. Fig. 30.) 



The same might also be shown of attractive as well as of repulsive forces. 

 For instance, if we placed ourselves in a small boat, and pulled a rope tied 

 to a much larger one, we should draw ourselves towards the large boat with 

 a motion as much more rapid than that of the large boat, as its weight is 

 greater than that of our own boat ; and the two boats would meet in their 

 common centre of inertia [gravity], supposing the resistance of the water 

 inconsiderable. 



Having established this property of the centre of inertia [gravity] as a 

 law of motion, we may derive from it the true estimate of the quantity of 

 motion in different bodies, in a much more satisfactory manner than it has 

 usually been explained. For since the same reciprocal action produces, in 

 a body weighing two pounds, only half the velocity that it produces in 

 a body weighing one pound, the cause being the same, the effects must be 

 considered as equal, and the quantity of motion must always be measured 

 by the joint ratio of mass to mass, and velocity to velocity ; that is, by the 



