332 THE PRINCIPLES OF SCIENCE. 



a moving slip of paper, so that equal intervals of time are 

 represented by equal lengths. There is, perhaps, a ten- 

 dency to reduce all comparisons to the comparison of space 

 magnitudes, but in any case one of the senses must be the 

 ultimate judge of coincidence or non-coincidence. 



Since the equation to be established may exist between 

 any multiples or submultiples of the quantities compared, 

 there naturally arise several different modes of comparison 

 adapted to different cases. Let p be the magnitude to 

 be measured, and q that in terms of which it is to be 

 expressed. Then we wish to find such numbers x and y, 



that the equation p = ~q may be true. Now this same 



equation may be presented in four slightly different forms, 

 namely : 



First Form. Second Form. Third Form. Fourth Form. 

 x y P Q 



P = -q p- = q py = qx - - - 



y x x y 



Each of these modes of expressing the same equation cor- 

 responds to one mode of effecting a measurement. 



When the standard quantity is greater than that to be 

 measured, we often adopt the first mode, and subdivide 

 the unit until we get a magnitude equal to that measured. 

 The angles observed in surveying, in astronomy, or in 

 goniometry are usually smaller than a whole revolution, 

 and the measuring circle is divided by the use of the 

 microscope and screw, until we obtain an angle undistin- 

 guishable from that observed. The dimensions of minute 

 objects are determined by subdividing the inch or centi- 

 metre, the screw micrometer being the most accurate 

 means of subdivision. Ordinary temperatures are esti- 

 mated by division of the standard interval between the 

 freezing and boiling points of water, as marked on a 

 thermometer tube. 



In a still greater number of cases, perhaps, we multiply 



