s 



G4 THE PRINCIPLES OF SCIENCE. 



^7~tuT^ery equation of the form y = 

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

 easily proved that the same equation is equivalent to one 

 of the form Az + B?/ + C = 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, 



B = BC, 



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 



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



