xi ii.] MEASUREMENT OF PHENOMENA. 285 



every measurement we equate some multiple or submul- 

 tiple of one quantity, with some multiple or submultiple 

 of another, and equality is always the fact which we 

 ascertain by the senses. By the eye, the ear, or the touch, 

 we judge whether there is a discrepancy or not between 

 two lights, two sounds, two intervals of time, two bars of 

 metal. Often indeed we substitute one sense for the other, 

 as when the efflux of time is judged by the marks upon 

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

 represented by equal lengths. There is a tendency to 

 reduce all comparisons to the comparison of space magni- 

 tudes, but in every 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. This equation 

 may be presented in four forms, namely : 



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



P= x -q p y - = q py = q* = q - 



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 

 screw and microscope, 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. 



