336 Dr. Adamson on Geometncal Demonstrations, 



The use of a converse statement is very important. It becomes 

 a criterion of the presence or existence in given circumstances of 

 the property announced as the original hypothesis, or, which is 

 the same thing, it shows that either property is found where the 

 other is, and nowhere else. If in a series of theorems converses 

 follow in immediate sequence, those to which they bear this re- 

 lation, their demonstrations will generally be indirect. Their 

 criterion character renders this advisable, even where by means 

 of preparatory theorems it might be avoided. In deducing con- 

 clusions from definitions, at the commencement of a series of 

 arguments, the mind's process is ultimately the dismissal of an 

 alternative, or is primarily indirect ; ex. gr. if angles are equal 

 their constituents will coincide ; for if not, there is a contradic- 

 tion. Evoking a contradiction will always be found to be the 

 seal of certainty on the mind's convictions, for certainty is the 

 impossibility of disbelieving. 



Oppositeness of conditions must necessarily be distinguished 

 from converseness of statement. Where such opposition gives 

 origin to two theorems, each may have its converse, and all may 

 require demonstration. 



This is exemplified when we have equality among constituents 

 in the hypothesis leading to equality among other constituents 

 in the conclusion, combined with a definite relation of inequality 

 in the data leading to a similar relation in the conclusion. Such 

 cases we have in the relations of the sides and angles of dificrent 

 forms of triangle. 



If, however, the relations of inequality are indefinite, though 

 there may be four statements, two demonstrations only are 

 needful. Take for instance the direct affirmative : " Rectangular 

 parallelograms have equal diagonals ; " to that we may have the 

 direct negative ; " Non-rectangular parallelograms have unequal 

 diagonals," without determining the inequality. There will be, 

 of course, a converse affirmative, and a converse negative. Now 

 we shall find that if we demonstrate any two which are either 

 converse to each other, or which are both direct, or both con- 

 verse, afibrding four possible combinations, we shall have ample 

 criteria, from any one of these dualisms, to determine all cases ; 

 while the other two possible arrangements do not serve that 

 purpose. It is therefore sufficient in such instances to prove 

 the direct affirmative and its converse. This is the proceeding 

 which the mutual relation of such truths always suggests. 



If we assume as the defining characteristic of proportion that 

 property which consists in the equality of fractions or of quo- 

 tients, we do not find this criterion to be readily applicable to 

 geometrical magnitudes. Euclid's criterion, in his definition, is 

 easily and elegantly applicable to them, for a reason already 



