THEORY OF APPROXIMATION. 99 



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

 to />, and similarly let the second force, acting alone, 

 move A to r. The question arises, 

 then, whether their joint action 

 will urge A to q along the 

 diagonal of the parallelogram. 

 May we say that A will move 

 the distance Ap in the direction 



AB, and A?* in the direction c 



AC, or, what is the same thing, along the parallel line pq ? 

 In all strictness we cannot say so ; for when A has moved 

 towards p, the force from C'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 P^pqr, 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 small parallelograms. The mathematician, in 

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

 made up of a great number of short straight lines, and it 



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