480 THE PRINCIPLES OF SCIENCE. [CHAP. 



By absolute equality we signify that which is complete 

 and perfect to the last degree ; but it is obvious that we 

 can only know such equality in a theoretical or hypotheti- 

 cal manner. The areas of two triangles standing upon the 

 same base and between the same parallels are absolutely 

 equal. Hippocrates beautifully proved that the area of a 

 lunula or figure contained between two segments of circles 

 was absolutely equal to that of a certain right-angled 

 triangle. As a general rule all geometrical and other 

 elementary mathematical theorems involve absolute 

 equality. 



De Morgan proposed to describe as siib-equa.1 those 

 quantities which are equal within an infinitely small 

 quantity, so that x is sub-equal to x + d x. The dif- 

 ferential calculus may be said to arise out of the neglect 

 of infinitely small quantities, and in mathematical science 

 other subtle distinctions may have to be drawn between 

 kinds of equality, as De Morgan has shown in a remarkable 

 memoir " On Infinity ; and on the sign of Equality." l 



Apparent equality is that with which physical science 

 deals. Those magnitudes are apparently equal which differ 

 only by an imperceptible quantity. To the carpenter 

 anything less than the hundredth part of an inch is non- 

 existent ; there are few arts or artists to which the hundred- 

 thousandth of an inch is of any account. Since all 

 coincidence between physical magnitudes is judged by one 

 or other sense, we must be restricted to a knowledge of 

 apparent equality. 



In reality even apparent equality is rarely to be ex- 

 pected. More commonly experiments will give only 

 probable equality, that is results will come so near to each 

 other that the difference may be ascribed to unimportant 

 disturbing causes. Physicists often assume quantities to 

 be equal provided that they fall within the limits of 

 probable error of the processes employed. We cannot 

 expect observations to agree with theory more closely 

 than they agree with each other, as Newton remarked of 

 his investigations concerning Halley's Comet. 



1 Cambridge Philosophical Transactions (1865), vol. xi. Part L 



