ON ACCELERATING FORCES. 23 



instance a second, if there were no other motions than undisturbed or 

 uniform motions ; but the velocity may vary very considerably within the 

 second, and we must therefore have some other measure of it than the 

 space actually described in any finite interval of time. If, however, the 

 times be supposed infinitely short, the elements of space described may be 

 considered as the 'true measures of velocities. These elements, although 

 smaller than any assignable quantity, may yet be accurately compared with 

 each other ; and the reason that they afford a true criterion of the velo- 

 city is this, that the change produced in the velocity during so short an 

 interval of time, must be absolutely inconsiderable, in comparison with the 

 whole velocity, and the element of space becomes a true measure of the 

 temporary velocity, in the same manner as any larger portion of space may 

 be the measure of a uniform velocity. 



When the increase or diminution of the velocity of a moving body is 

 uniform, its cause is called a uniform force ; the spaces which would be 

 described in any given time with the actual velocity uniformly continued, 

 being always equally increased or diminished by the action of such a force. 

 For example, if the velocities at the beginning of any two separate 

 seconds be such that the body would describe one foot and ten feet in the 

 respective seconds, if undisturbed, and the spaces actually described 

 become two feet and eleven feet, each being increased one foot, the accele- 

 rating force must be denominated uniform. 



The power of gravitation, acting at or near the earth's surface, may, 

 without sensible error, be considered as such a force. Thus, if a body 

 begins to fall from a state of rest, it describes about 16 feet, or more cor- 

 rectly 16-jir in the first second ; if it begins a second with a velocity of 32 

 feet, it describes 32 and 16, or 48 feet in this second. The decrease of the 

 force of gravitation, in proportion to the squares of the distances from the 

 earth's centre, is barely perceptible, at any heights within our reach, by 

 the nicest tests that we can employ. 



The velocity produced by any uniformly accelerating force, is propor- 

 tional to the magnitude of the force and the time of its operation con- 

 jointly.* When the forces are the same, a little consideration will convince 

 us that, since every equal portion of time adds equally to the velocity, the 

 whole velocity produced or destroyed must be in proportion to the whole 

 time ; and when the forces differ, the velocities differ in the same ratio ; 

 for the forces are only measured by the velocities which they generate. 

 Thus a double force, in a double time, produces a quadruple velocity. 

 That a force producing a double velocity is properly called a double force, 

 may be shown from the laws of the composition of motion ; for when the 

 equal sides of a parallelogram representing two separate forces or motions, 

 approach to each other, and at last coincide in direction, the diagonal of 

 the parallelogram, representing their joint effect, becomes equal to the sum 

 of the sides. (Plate I. Fig. 10.) t 

 , The machine invented by Mr. AtwoodJ (Plate I. Fig. 11) furnishes us 



f*S^ 



* Galileo, Dialogues on Motion, Dial. III. Def. 



f Halley, Ph. Tr.xvi. (1686). 



t On the Rectilinear Motion, and the Rotation of Bodies, Camb. 1804, p. 291. 



