64 THE PRINCIPLES OF SCIENCE. 



show that every equation of the form y = mx + c is 

 equivalent to or represented by a straight line ; it is also 

 easily proved that the same equation is equivalent to one 

 of the form Ax + By + = o, and vice versa. Hence it 

 follows that every equation of the first degree is equivalent 

 to or represents a straight line e . 



Inference with a Simple and a Partial Identity. 



A form of reasoning somewhat different from that last 

 considered consists in inference between a simple and a 

 partial identity. If we have two propositions of the 

 form 



A = B, 



we may then substitute for B in either proposition its 

 equivalent in the other, getting in both cases A = BC ; 

 in this we may if we like make a second substitution for 

 B, getting 



A = AC. 



Thus, since 'Mont Blanc is the highest mountain in 

 Europe, and Mont Blanc is deeply covered with snow/ we 

 infer by an obvious substitution that ' The highest moun- 

 tain in Europe is deeply covered with snow.' These pro- 

 positions when rigorously stated fall into the form above 

 exhibited. 



This form of inference is constantly employed when for 

 a term we substitute its definition, or vice versa. The 

 very purpose of a definition is in fact to allow a single 

 term to be employed in place of a long descriptive phrase. 

 Thus when we say ' Circles are curves of the second 

 degree/ we may substitute the definition of a circle, 

 getting ' A plane curve, all points of whose perimeter are 

 at equal distances from a certain fixed point, is a curve of 



e Todhunter's ' Plane Co-ordinate Geometry,' chap. ii. pp. 11-14. 



