• vv. .tsoft *^ developing the Theory of Parallel Lines, 453 



Mr. Stevelly's position is by no means a novel one. In 

 looking over the voluminous history of the subject, one is struck 

 by the large number of persons who object to anything like a 

 full and adequate definition. The objection is generally made 

 on the ground that the definition "assumes^^ something; or that 

 it contains a " property ;" or that it " involves a petitio prin- 

 cipii.'^ It is hardly necessary to say, that such objections imply 

 a misapprehension of the first principles of formal logic. In 

 framing a definition, something must be assumed; and that 

 something must be a property. When the property is incor- 

 porated in the definition, it becomes its differentia. In speak- 

 ing, as logicians, of the properties of any conception that we 

 may have thus defined, the difi*erentia is no longer included. 

 The properties are all susceptible of being deduced from the 

 definition; but, to expect that its own differentia should be 

 deduced from it in a similar manner, is, undoubtedly, a grave 

 mistake. Such a mistake will arise from a confusion between 

 the difi'erentia and the propria; and it appears to have, not 

 unfrequently, impeded the proper settlement of this question. 

 For example, Mr. Stevelly says, referring to the definition in 

 which a case of the 29th proposition is made the differentia 

 (that the two interior angles on the same side will be together 

 equal to two right angles), — " The 29th proposition of Euclid^s 

 Elements calls on me to prove that ' any^ line, which cuts those 

 [two parallel lines], also makes the two interior angles on the 

 same side together equal to two right angles. This is a relation 

 of these two lines to all others which cut them which requires 

 to be proved, and which neither the above definition nor any 

 other definition can warrant us to assume as a truth without 

 proof. It is a 'property^* which must be shown to be an 

 essential concomitant of that which has already fixed the rela- 

 tion of those two lines to one another, and to all others which 

 cut them, and which we have by the definition merely agreed 

 shall entitle us to denominate them 'parallel lines ^ in any 

 individual case in which we find it to obtain.^' Here, in 

 consequence of a confusion of the differentia and propria, the 

 remarkable oversight is committed of expecting a proof of the 

 very thing that has been assumed, and on which any demon- 

 stration must rest. It would be quite impossible to give a 

 demonstration of the differentia and propria of any complete 

 geometrical conception. The proprium chosen for a differentia 

 becomes an ultimate premiss in the sorites. As far as suscepti- 

 bility of proof is concerned, it is altogether outside the domain 

 of demonstration. 



* Had Mr. Stevelly called it by its right name, the " differentia," he 

 would at once have seen his mistake. , . , 



