xui.] MEASUREMENT OF PHENOMENA. 283 



thickness. When a boundary is vague and graduated, 

 like the penumbra in a lunar eclipse, it is impossible to 

 say where the end really is, and different people will come 

 to different results. We may sometimes overcome this 

 difficulty to a certain extent, by observations repeated in 

 a special manner, as we shall afterwards see ; but when 

 possible, we should choose opportunities for measure- 

 ment when precise definition is easy. The moment of 

 occultation of a star by the moon can be observed with 

 great accuracy, because the star disappears with perfect 

 suddenness ; but there are other astronomical conjunctions, 

 eclipses, transits, &c., which occupy a certain length of 

 time in happening, and thus open the way to differences 

 of opinion. It would be impossible to observe with pre- 

 cision the movements of a body possessing no definite 

 points of reference. The colours of the complete spectrum 

 shade into each other so continuously that exact deter- 

 minations of refractive indices would have been impossible, 

 had we not the dark lines of the solar spectrum as precise 

 points for measurement, or various kinds of homogeneous 

 light, such as that of sodium, possessing a nearly uniform 

 length of vibration. 



In the second place, we cannot measure accurately 

 unless we have the means of multiplying or dividing 

 a quantity without considerable error, so that we may 

 correctly equate one magnitude with the multiple or sub- 

 multiple of the other. In some cases we operate upon the 

 quantity to be measured, and bring it into accurate coin- 

 cidence with the actual standard, as when in photometry 

 we vary the distance of our luminous body, until its 

 illuminating power at a certain, point is equal to that of a 

 standard lamp. In other cases we repeat the unit until it 

 equals the object, as in surveying land, or determining a 

 weight by the balance. The requisites of accuracy now 

 are : (i) That we can repeat unit after unit of exactly 

 equal magnitude ; (2) That these can be joined together 

 so that the aggregate shall really be the sum of the 

 parts. The same conditions apply to subdivision, which 

 may be regarded as a multiplication of subordinate units. 

 In order to measure to the thousandth of an inch, we must 

 be able to add thousandth after thousandth without error 

 in the magnitude of these spaces, or in their conjunction. 



