372 APPENDIX. [292- 



than that to be measured, the fraction (p + l)lq would measure an object 

 somewhat greater than that to be measured. When this is the case we say 

 that the measure sought is an irrational number. Now we may take two 

 series of rational numbers a lt a z , ...a nt ... and 6 15 6 2 , ... 6 n , ... such that each 

 a is greater than every b and that, by taking n sufficiently great, a n - b n shall 

 become less than any rational number assigned beforehand. All the numbers 

 a may be greater than the measure of the object, and all the numbers 6 less 

 than that measure. The two series determine an irrational number a which 

 is the common limit of the two series. In such a case we define the measure 

 of the object to be the irrational number a. 



Thus, suppose we wish to measure the diagonal of a square whose side is 

 the unit of length. The process of extracting the square root of 2 gives 

 1-41421..., and thus the series b may be taken to consist of the numbers 



1, 1-4, 1-41, 1-414, 1-4142, 1-41421, ..., 

 and the series a may be taken to consist of the numbers 



2, 1-5, 1-42, 1-415, 1-4143, 1-41422, ..., 



and the nature of the process shows that the excess of any a above the 

 corresponding b diminishes without limit. These two series define a limit, 

 which is A/2, and this irrational number is the required measure. 



293. Number and Quantity. When the unit is stated the magnitude 

 of an object is precisely determined by its measure in terms of the unit, and 

 this measure is always a number. The &quot;object&quot; may be anything which 

 we can think of as measurable in respect of any property, and the phrase 

 &quot;magnitude of an object&quot; is thus coextensive in meaning with the word 

 &quot; quantity.&quot; The quantity does not change when the unit chosen to measure 

 it changes, and thus the quantity is not identical with the number express 

 ing it. 



A number can express a quantity only when the unit of measurement 

 is stated or understood. When the unit is stated or implied the number 

 expresses the quantity. ,, 



Mathematical equations, and inequalities, are relations between numbers, 

 expressing that a certain number which has been arrived at in one way is 

 equal to, greater than, or less than, a certain number which has been arrived 

 at in another way. 



Mathematical equations, and inequalities, between numbers expressing 

 quantities are valid expressions of relations between the quantities only if 

 they hold good for all systems of units. 



294. Fundamental and derived Quantities. The fundamental Physical 

 quantities are lengths, times, and masses. In Dynamics, as considered in 

 this book, all the other magnitudes which occur are derived from these. 

 Thus, velocity is measured by a fraction Of which the numerator is a 

 number expressing a length and the denominator is a number expressing 

 an interval of time; acceleration is measured by a fraction of which the 



