NEWTON. 191 



into which the distance M M' is supposed to be divided, and this being 

 supposed to be increased without limit, it might be imagined by a be- 

 ginner that the, unlimited increase of terms in the series within the 

 brackets would increase the value of the expression ; but it must be 

 observed that the increase of n, the number of terms, implies a decrease 

 of the value of dx, for M M' divided by n is dx. It should be noticed 

 that in the symbol dx the letters do not stand in the sense of the or- 

 dinary algebraical notation, but constitute a single symbol for the 

 small increments of x. The value of the expression usually approaches 

 a fixed limit, and in the case we are supposing the limit is sufficiently 

 obvious. That limit is the area of the figure p M', P' M, and is called the 

 integral of $x dx, from x = o M to x = o M'. It is proved in works on 

 the integral calculus that the integral of a given function of a variable is 

 that other function whose differential coefficient is the given function. 

 In the science of statics and dynamics Newton effected certain great 

 improvements, which may here be named apart from his great discovery 

 of universal gravitation. Some advances in statics effected by Stevinus 

 and Galileo have already been noticed. It was these philosophers 

 who generalized the principle of the lever by showing that in the case 

 of every mechanical power, the force multiplied by its velocity is equal 

 to the weight multiplied by its velocity. Wallis in 1669 published his 

 " Mechanica^ in which a regular system of statics is established on 

 this principle. VARIGNON (1654 1722), a French mathematician, 

 treated all the problems of mechanics with a higher degree of generality 

 than had before been attained. He deduced the solution of every 

 problem in statics from the simple principle that when three forces are 

 in equilibrium each is proportional to the sine (page 61) of the angle be- 

 tween the directions of the other two. This principle, which is really 

 involved in the propositions of Stevinus, was proposed and applied by 

 Varignon in a work which he published 1687. Newton in his "Prin- 

 cipia? which appeared in 1687, enunciated principles of statics which 

 in effect were similar to those of Varignon; but these are noticed 

 merely as an introduction to the more difficult inquiries relating to 

 moving bodies. Galileo, as we have already seen, was the first who 

 made any real advance in ascertaining the laws which govern the mo- 

 tions of falling bodies and of projectiles. Newton reduced to their 

 simplest and most general expressions the facts of the motion of bodies, 

 which had indeed been known by former writers, but not explicitly 

 laid down as the fundamental principles of the science of dynamics. 

 These propositions of Newton's are the well-known " Laws of Motion." 

 He called them "Axiomata, sive Leges Motus" An axiom is usually de- 

 fined as a self-evident truth ; but it should be observed that the " laws 

 of motion" are axioms only to those whose experience of the pheno- 

 mena is wide enough to enable them to recognize the generality of the 

 truths expressed. We shall here give the laws of motion as Newton 

 laid them down : 



