xxi.] THEORY OF APPROXIMATION. 479 



It is a result of these principles that all small errors may 

 be assumed to vary in simple proportion to their causes a 

 new reason why, in eliminating errors, we should first of 

 all make them as small as possible. Let us suppose that 

 there is a right-angled triangle of which the two sides 

 containing the right angle are really of the lengths 3 and 

 4, so that the hypothenuse is v/3 2 + 4 2 or 5. Now, if in 

 two measurements of the first side we commit slight 

 errors, making it successively 4'OOI and 4-002, then calcu- 

 lation will give the lengths of the hypothenuse as almost 

 exactly 5*0008 and 5 '0016, so that the error in the 

 hypothenuse will seem to vary in simple proportion to 

 that of the side, although it does not really do so with 

 perfect exactness. The logarithm of a number does not 

 vary in proportion to that number nevertheless we find 

 the difference between the logarithms of the numbers 

 looooo and 100001 to be almost exactly equal to that 

 between the numbers looooi and 100002. It is thus a 

 general rule that very small differences between successive 

 values of a function are approximately proportional to 

 the small differences of the variable quantity. 



On these principles it is easy to draw up a series of 

 rules such as those given by Kohlrausch l for performing 

 calculations in an abbreviated form when the variable 

 quantity is very small compared with unity. Thus for 

 i -r (i + a) we may substitute I a; for i -7- ([ a) 

 we may put i+a; i -=- V 1 + a becomes I a, and so 

 forth. 



Four Meanings of Equality. 



Although it might seem that there are few terms more 

 free from ambiguity than the term equal, yet scientific 

 men do employ it with at least four meanings, which it 

 is desirable to distinguish. These meanings I may describe 

 as 



(1) Absolute Equality. 



(2) Sub- equality. 



(3) Apparent Equality. 



(4) Probable Equality. 



1 An Introduction to Physical Measurements, translated by Waller 

 and Procter, 1873, p. 10. 



