304 THE GRAMMAR OF SCIENCE 



accelerations are inversely as masses. The validity of 

 this statement is demonstrated in precisely the same 

 manner as the fourth law of motion. We note that if 

 unity be taken as representing the mass of the standard 

 corpuscle^ Q, the definition of mass on p. 302 may be 

 replaced by the formula : — 



Acceleration of Q due to A _ Mass of A 

 Acceleration of A due to Q ~ Mass of Q ' ' ^ '^ 



a result in perfect accordance with the law just stated. 



Now this law may be put into a slightly different form. 

 By a well-known proposition ^ the product of the means 

 in any proportion is equal to that of the extremes. Hence 

 it follows that : — 



Mass of A X Acceleration of A due to B 

 = Mass of B X Acceleration of B due to A. 



We will, then, give a name to this product of mass into 

 acceleration ; we will term the product of the mass of A 

 into the acceleration of A due to the presence of B, the 

 force of B on A. This force will be considered to have 

 the direction and sense of the acceleration of A due to B, 

 while its magnitude will be obtained by multiplying the 

 number of units in the acceleration of A due to B by the 

 number of units in the mass of A. Thus the proper 

 measure of a force will be its number of units of mass- 

 acceleration. Remembering that the accelerations of A 

 and B are of opposite sense, we can now restate our fifth 

 law in new language, thus : — 



The force of B on A is equal and opposite to the force of 

 K on^\ 



Or, as it was originally stated by Newton himself: — 



^'■Action and Reaction are always equal and opposite " ■' . . (e). 



Now it is clear that with our definition force is a 

 certain measure of Jiow a corpuscle is dancing relative to 



1 That is, the ratio of the mutual accelerations of Q and an absolutely 

 identical corpuscle. These accelerations must by symmetry be exactly equal, 

 and hence their ratio, the mass of Q, must be taken as unity, 



2 Euclid vi. 1 6, interpreted arithmetically. 



3 " Actioni contrariam se?nper ei aqualem esse reactionem," 



