iv.] DEDUCTIVE REASONING. 53 



For the term " matter " in either of these identities we 

 may substitute its equivalent given in the other definition. 

 Elsewhere they often employ sentences of the form exem- 

 plified in the following: 1 "The integral curvature, or 

 whole change of direction of an arc of a pJane curve, is the 

 angle through which the tangent has turned as we pass from 

 one extremity to the other." This sentence is certainly of 

 the form 



The integral curvature = the whole change of direc- 

 tion, &c. = the angle through which the tangent t 

 has turned, &c. 



Disguised cases of the same kind of inference occur 

 throughout all sciences, and a remarkable instance is found 

 in algebraic geometry. Mathematicians readily show that 

 every equation of the form y = mx 4- c corresponds to or 

 represents a straight line ; it is also easily proved that the 

 same equation is equivalent to one of the general form 

 Ax + ~By 4- C = o, and vice versd. Hence it follows that 

 every equation of the form in question, that is to say, 

 every equation of the first degree, corresponds to or 

 represents a straight line. 3 



Inference uith 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 forms 

 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 " The Mont Blanc is the highest mountain in 

 Europe, and the Mont Blanc is deeply covered with snow," 

 we infer by an obvious substitution that "The highest 

 mountain in Europe is deeply covered with snow." These 

 propositions when rigorously stated fall into the forms 

 above exhibited. 



This mode of inference is constantly employed when for 



1 Treatise on Natural Philosophy, voL i. p. 6. 



* Todhunter's Plane Co-ordinate Geometry, chap. ii. pp. II 14. 



