232 THE GRAMMAR OF SCIENCE 



can determine from the speed acceleration the change in 

 speed due to this change of position. Hence we obtain 

 the speed in the new position. From the speed in the 

 new position and the normal acceleration in this position, 

 the bending in the next little element of path may be 

 deduced. This process may be repeated as often as we 

 please, till the whole path of the motion is constructed. 

 The succession of positions may be taken so close together 

 that we obtain the form of the path to any degree of 

 accuracy required. Knowing the path and the speed at 

 each point of it we are able to construct a time-chart like 

 that of our Fig. lO (p. 213). For we know from the 

 speeds the slope at each point of the Q-curve. Hence 

 we commence by drawing a little element, say P^Qg, at 

 the slope given by the initial speed ; this element by aid 

 of the horizontal Q2P2, through its terminal Q^, gives a 

 new position at distance P^P^ from the initial position ; 

 the speed in this new position determines the slope of the 

 next little element Q2Q3 of the curve ; Q^ by aid of the 

 horizontal Q3P3 gives a third position with a third speed 

 and so a slope for the third element, and this process can 

 be continued till we have constructed the time-chart by a 

 succession of little elements. By taking these elements 

 sufficiently small, we make the resulting polygonal line 

 differ as little from the true curve of the time-chart as we 

 please. Now we have seen that when the map of the 

 path and the time-chart are known, the motion has been 

 fully described. Thus we conclude that : Given the 

 velocity of a point in any positioti and the acceleration of the 

 point in all positions, the motion of the point is fully deter- 

 mined} 



This proposition really indicates the basis of the whole 

 of our mechanical description of the universe. Rightly 

 interpreted, it contains all that we can assert of the 



1 The methods by which we have shown that the initial velocity and 

 position, together with the acceleration in all positions, determine the map of 

 the path and the time-chart, are only theoretical methods of construction. 

 The practical methods of constructing these curves involve the highest refine- 

 ments of mathematical analysis. Our object here is only to show that the 

 motion is theoretically determined by a knowledge of the above quantities. 



