102 THE PRINCIPLES OF SCIENCE. 



Four Meanings of Equality. 



Although it might seein that there are few terms more 

 free from ambiguity than the term equal, yet scientific 

 men do as a matter of fact employ it with four meanings, 

 which it is very desirable to distinguish carefully. These 

 meanings I may briefly describe as 



(1) Absolute Equality. 



(2) Sub-equality. 



(3) Apparent Equality. 



(4) Probable Equality. 



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 hypothe- 

 tical manner. The areas of two triangles standing upon 

 the same base and between the same parallels are abso- 

 lutely equal. Hippocrates beautifully proved that the 

 area of a lunula or figure contained between two seg- 

 ments 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 ab- 

 solute equality. 



De Morgan proposed to describe as sub-equal those 

 quantities which are equal within an infinitely small 

 quantity, so that x is sub-equal to x + dx. The whole of 

 the differential calculus may, as I apprehend it, be said 

 to arise out of the neglect of infinitely small quantities ; 

 with this subject however we are not in this place much 

 concerned. In mathematical science many 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 ' b . 



Apparent equality is that with which physical science 

 deals. Those magnitudes are practically equal which 



b 'Cambridge Philosophical Transactions,' [1865] vol. xi, Part I. 



