NEWTON. 193 



is plainly seen ; but the reaction, namely, the fact that the needle, 

 with precisely the same force, attracts the magnet, is only recognized 

 after experience and reflection. We have only to give the magnet 

 and the needle equal freedom of motion to perceive that each moves 

 towards the other in a manner which proves that the same force is the 

 cause of the motion. The velocity of motion and the space through 

 which each moves will be unequal if the masses are unequal ; but 

 the momenta i.e., the products of the mass into the space will be 

 equal. 



Newton gives a demonstration of the composition of forces, as a 

 corollary to these laws of motion. He supposes a body acted upon 

 by an impulsive force, by virtue of which it would continue to move 

 uniformly in a straight line in a certain direction ; while at the same 

 time another force acts upon it, which would of itself draw the body 

 in a direction inclined to the former at a given angle. It is proved that 

 the body will follow neither of these directions, but will take an inter- 

 mediate course, which is determined by drawing the diagonal of a 

 parallelogram, two sides of which represent, in magnitude and direc- 

 tion, the two forces. Newton's proof is deduced from the laws of 

 motion, which declare that the body obeys each force unaltered in 

 its own direction, and that at any given time the position of the body, 

 measured on a line parallel to the direction of each force, is precisely 

 the same as if that force alone had acted. From this simple prin- 

 ciple Newton proceeds to deduce his whole theory of central forces. 

 The theorem by which he begins is a very interesting one, and the 

 reasoning can easily be followed by any one conversant with ttfe 

 geometrical fact that triangles on the same base and between the 

 same parallels are equal in area. Let 

 a body be moving unacted on by any 

 force along the line A D, Fig. 85 ; in 

 that line its positions at any equal in- 

 tervals of time would be certain equi- 

 distant points, say A, B, c, D. Suppose 

 that when the body had arrived at B, 

 a force for an instant acted upon it 

 which would have of itself caused the 

 body to move along the line B s, its 

 position at the end of the interval of 

 time in which it would otherwise have FIG. 85. 



arrived at c will be found by drawing 



from c the line c E parallel to B s, and equal to the space that the 

 impulsive force alone would have carried the body along B s in the 

 same time. The_body would therefore pass along the diagonal BE 

 of parallelogram of which E c, c B are the sides. Draw the lines s A, 

 s c. Now the areas of the triangles B s E, B c s, which are in the same 

 base, s B, and between the same parallels, are equal to each other. 



13 



