BEGINNINGS OF MODERN MATHEMATICAL SCIENCE 293 



Kepler, and Galileo, was now to be completely formulated and 

 mathematically interpreted by Newton's crowning discovery of a 

 single mechanical principle governing the whole. 



It was now a question of verifying the correctness of this prin- 

 ciple by applying it to all measured or measurable astronomical 

 phenomena. The investigation was gradually extended to the 

 planets, the moons of Jupiter, the tides, and even the comets. 

 Everywhere the law was verified that attraction varies as the prod- 

 uct of the masses and inversely as the square of the distance. 



The whole theory was elaborated in Newton's monumental 

 Principia Philosophies Naturalis Mathematica published in 1687. 

 He begins this treatise with a series of definitions and laws : 



1. The quantity of matter is the measure of the same, arising from 

 its density and bulk conjunctly. 



2. The quantity of motion is the measure of the same, arising from 

 the velocity and quantity of matter conjunctly. 



3. The innate force of matter is a power of resisting, by which 

 every body, as much as in it lies, endeavours to persevere in its present 

 state, whether it be of rest, or of moving uniformly forward in a 

 right line. 



4. An impressed force is an action exerted upon a body, in order 

 to change its state, either of rest, or of moving uniformly forward in 

 a right line. 



5. A centripetal force is that by which bodies are drawn or im- 

 pelled, or any way tend, towards a point as to a centre. 



These and succeeding definitions are followed by the famous 

 Laws of Motion : 



I. Every body perseveres in its state of rest, or of uniform motion 

 in a right line, unless it is compelled to change that state by forces 

 impressed thereon. 



II. The alteration of motion is ever proportional to the motive 

 force impressed, and is made in the direction of the right line in 

 which that force is impressed. 



III. To every action there is always opposed an equal reaction : 

 or the mutual actions of two bodies upon each other are always equal, 

 and directed to contrary parts. Corollary I continues : A body by two 



