236 THE GRAMMAR OF SCIENCE 



and 0'.,0'^ will measure the fitting shift of the paper. 

 This carries P, clearly forward to P'^ and O to O,. Thus 

 the motion of P relative to O' may be looked at as the 

 motion of P due to two sources — a movement of P about 

 O, and a movement of the plane containing P and O ; this 

 later motion is the motion of O about O', or is equal 

 and opposite to the perfectly arbitrary motion of O' about 

 O. Thus we conclude that if a point P has two inde- 

 pendent velocities (corresponding to the limits of the 

 displacements P^P, and P.,P^o) then the actual velocity of 

 P will be found by adding these velocities geometrically. 

 This statement is usuall}- termed the parallelogram of 

 velocities. A precisely similar statement holds for inde- 

 pendent accelerations (p. 212), and is called the parallelo- 

 gram of accelerations. To these important results we 

 shall have occasion again to refer. We conclude, there- 

 fore, with the general statement that the independent 

 displacements, the independent velocities, and the inde- 

 pendent accelerations of a moving point are respectively 

 added geometrically as we add steps, or by the so-called 

 parallelogram law. 



The value of this rule of combination lies in the power 

 it gives us of building up complex cases of motion from 

 simple cases. If we find as a result of experience that 

 the perceptual antecedents ^ of a motion we describe by 

 one acceleration may be superposed on the perceptual 

 antecedents of a motion we describe by a second accelera- 

 tion — without it being necessary to alter the values of 

 these accelerations (at any rate to our degree of refine- 

 ment in appreciating change) when describing the motion 

 corresponding to the combined antecedents, — then the 

 parallelogram of accelerations will be invaluable as a 

 mode of synthesis, or of constructing the complex from 

 the simple. The law of gravitation applied to the 



1 By " perceptual antecedents of motion" we are to understand cause in 

 the scientific sense, but the word has not been used in the above paragraph, 

 because the reader might have supposed the cause of motion to be the 

 metaphysical (and imperceptible) entity force, whereas it really lies in a 

 perceptible relationship, i.e. the relativity in perceptual space (Chap. VIII. 

 § 5). 



