xxi.] THEORY OF APPROXIMATION. 477 



that two forces, acting from the points B and C, are 

 simultaneously moving a body A. Let the force acting 

 from B be such that in one second it would move A 

 to p, and similarly let the second force, acting alone, 

 move A to r. The question A P 



arises, then, whether their joint * 

 action will urge A to q along 

 the diagonal of the parallelo- 

 gram. May we say that A will 

 move the distance Ap in the 

 direction AB, and Ar in the 

 direction AC, or, what is the 

 same thing, along the parallel 

 line pq ? In strictness we cannot say so ; for when A has 

 moved towards p, the force from G will no longer act along 

 the line AC, and similarly the motion of A towards r will 

 modify the action of the force from B. This interference 

 of one force with the line of action of the other will 

 evidently be greater the larger is the extent of motion 

 considered ; on the other hand, as we reduce the paral- 

 lelogram Apqr, compared with the distances AB and AC, 

 the less will be the interference of the forces. Accord- 

 ingly mathematicians avoid all error by considering the 

 motions as infinitely small, so that the interference be- 

 comes of a still higher order of infinite smallness, and 

 may be entirely neglected. By the resources of the differ- 

 ential calculus it is possible to calculate the motion of the 

 particle A, as if it went through an infinite number of 

 infinitely small diagonals of parallelograms. The great 

 discoveries of Newton really arose from applying this 

 method of calculation to the movements of the moon 

 round the earth, which, while constantly tending to move 

 onward in a straight line, is also deflected towards the 

 earth by gravity, and moves through an elliptic curve, 

 composed as it were of the infinitely small diagonals of 

 infinitely numerous parallelograms. The mathematician, 

 in his investigation of a curve, always treats it as made 

 up of a great number of straight lines, and it may be 

 doubted whether he could treat it in any other manner. 

 There is no error in the final results > because having ob- 

 tained the formulae flowing from this supposition, each 

 straight line is then regarded as becoming infinitely small, 



