532 Notices respecting Nexo Booh. 



It is rarely, if ever, used in pure geometry, except by very ignorant 

 or very careless writers. 



"A lemma" — this requires modification to pass muster; and 

 more especially to justify the previous classification of theorems, 

 problems and lemmas. 



"A HYPOTHESIS is 0. foct assuDied tvithout proof, either in the 

 enunciation of a proposition, or in the course of a demonstration." 



Let him who can, interpret this : — we cannot. Mr. Bell seems to 

 have attempted to give a definition which should apply to what is 

 called an hypothesis in mathematics and to those notions which have 

 borne the same name in physics : — A good illustration of the old 

 English proverb respecting " two stools." 



" A COROLLARY is a consequence easily deduced from one or more 

 propositions." 



The ancients understood a corollary to be a distinct and separate 

 enunciation of some property which was discovered, or contained in, 

 the demonstration of another proposition. The moderns use the 

 word more loosely ; and we will give Mr. Bell the advantage of the 

 practice. 



" A SCHOLIUM is a remark on one or more propositions, which 

 explains their apjjlication, connection, limitation, extension, or some 

 other important circumstance in their nature." 



The ancient geometers had no scholia in this or any other sense. 

 The '' connections and limitations " were parts of the actual solu- 

 tion ; and are known amongst geometers worthy of the name, in 

 modern times, under the general title of " determinations." The 

 mere statement of particular circumstances relating to the history 

 or collateral relations of particular propositions was introduced by 

 the commentators of the middle ages. The slovenly practice of 

 throwing out mere hints about those determinations and limitations 

 which are partly seen and partly guessed at, amongst modern writers 

 deserves to be strongly condemned, whether they take the form of 

 corollaries or of scholia. They, however, enter more largely than 

 usual into Mr. Bell's "improvements :" but whether it be intended 

 for " conciseness " or " precision " we do not pretend to determine. 



" A DEMONSTRATION is a proccss of reasoning, and is either direct 

 or indirect. A direct demonstration is a regular process of reasoning 

 from the premises to the conclusion. An indirect demonstration 

 establishes a proposition, by proving that any hypothesis contrary to 

 it, is contradictory or absurd." 



Here, unfortunately, the terms "premises," "conclusion," "pro- 

 position " and " process of reasoning" are employed pretty much as 

 if the words had been showered out of a pepper-box, or made uj) of 

 printers' pie, to take their chance of a possible meaning. The same 

 confusion runs through the following : — 



" The DATA or premises of a proposition are the relations or con- 

 ditions granted or given, from which new relations are to be deduced, 

 or a construction to be effected." 



The " or " in this passage has either the force of signifying that 

 the words or phrases on each side of it are identical in signification ; 



