454 Mr. Hennessy on the Importance of an Adequate Definition 



In attempting to decide on the diflFerentia of parallel lines, or 

 of any other geometrical conception, it is of importance that we 

 should cai'efully ascertain how far we are privileged to go ; that 

 is, we should ascertain how much a mathematical definition is 

 permitted, and expected, to include. It is obvious that the 

 nature of the definition will depend in a great measure on 

 whether we believe, on the one hand, that " it entitles us merely 

 to denominate certain lines parallel in any individual case;^' of 

 whether, on the other hand, there is something besides the mere 

 meaning of a term, as a nominal definition, involved in it. The 

 views of Dr. Whewell on this subject are well known. He* 

 has always upheld the real character of mathematical definition* 

 Even Mr. Mill recognizes t something more than an arbitrary name* 

 There is, however, no necessity to go further than the writings 

 of Dr. Whately to find the strongest confirmation of the views 

 I have ventured to put forward. " The term definition,^' he 

 says, " is used with some laxity ; and much confusion has thence 

 resulted. Such definitions as the mathematical, must imply every 

 attribute that belongs to the thing defined.^' Again, he says, — 

 " In mathematics there is no distinction between nominal and 

 real definition ; the meaning of the term and the nature of the 

 thing being one and the same : so that no correct definition what^ 

 soever of any mathematical term can be devised,which shall not imply 

 everything which belongs to that term.'' 



The next point to determine is, whether we are at liberty to 

 take any adequate proprium we choose for the difierentia. That 

 we are entitled to do so there cannot be the slightest doubt* 

 On the plainest logical principles, it would be absurd to suppose 

 that any property, possessing sufficient comprehension and 

 clearness, could be, by any arbitrary arraigement whatever, 

 distinguished from the others, and set aside as incapable of 

 being employed in definition. And, on the other hand, we are 

 fully at liberty, in case any advantage appears to be gained by 

 such a course, to strip a definition of its discretive form, and to 

 place it amongst the propositions {. 



It would be indeed very difficult to point out any striking pro- 

 prium of parallels which has not at some time or another been 

 made a differentia, by the second class of geometers, alluded to 



* Philosophy of the Inductive Sciences, vol. i. p. 92; vol. ii. pp. 598,599. 



t In speaking of the definition of a circle (System of Logic, vol. i. p. 336). 



X The tery course here alluded to has been tftken by many Continental 

 geometers. In this country, amongst others, the author of the article 

 " Geometry" in the Encyclopadia Metropolitana has adopted it. After 

 giving a definition, practically the same as Wolfius's, he observes, — " We 

 have preferred the definition above given, and have made the property ot 

 parallel hnes never meeting a proposition instead of a definition. "-*-Pttre 

 Sciences, vol. i. p. 3l3. 



