Notices respecting New Booh. 531 



be cut by a straight line in n points ! Mr. Bell has pronounced 

 his veto upon its being cut in more than two : and we must bow, on 

 Governmental authority, to his decision, — that such shall, in all after 

 times, be the mathematical creed of England. 



It would also seem, that besides a curve line we are to admit an- 

 other class of lines, the " convex " and the " concave " line : whilst 

 it would appear that a curve line is neither the one nor the other. 

 The usual mathematical criterion of convexity and concavity is to be 

 laid aside (viz. the position of the centre of curvature with respect 

 to the tangent), and the words "from" and "towards" with respect 

 to some unspecified secant, are by some hocus-pocus system of think- 

 ing, to do the business for us which in the dark ages now closing 

 have been only effected by means of a second differential. The mil- 

 lenium of science is, surely, dawning upon us at last ! 



We have not picked out subjects of comment, in the shape of 

 isolated passages taken apart from their explanatory context : we 

 have taken a single page, and that the opening one, which in a work 

 of science is the key to the whole system. We have discussed the 

 first specimen that offered itself of Mr. Bell's peculiar excellence in 

 " concise and precise " didactic composition ; and our readers, should 

 they (unlike the actors in the Hampden controversy) take the trouble 

 to read the book itself, will find that our's is no garbled version of 

 Mr. Bell's mathematical heterodoxy. From such a foundation in 

 science, what sort of a structure can be raised? We sympathise 

 with the future youth of England who may have this book " to get 

 up;" and we sympathize, too, with the future schoolmaster of 

 England who may be compelled for the sake of the " government 

 grant " to submit to 



" rear the tender thought 



And teach the young idea how to shoot." 



We might close here : but still it appears necessary to exhibit his 

 views on other matters of a still wider range than the mere defini- 

 tions of terms. 



His idea of a proposition (p. 6) is, that " it is a portion of 

 science, and is a theorem, a problem or a lemma." 



Geometers, however, class propositions differently from Mr. Bell, 

 and substitute the "porism" for his "lemma." The office of a 

 lemma, like that of a corollary, has been altogether perverted amongst 

 these "improvements." 



" A THEOiiEM is a truth which is established by a demonstration." 

 Nothing of the kind : a theorem is a theorem independently of its 

 demonstration, and would be such, had no demonstration ever been 

 given of it. A theorem merely states that if certain conditions be 

 fulfilled, certain relations amongst the component parts must exist. 

 The name has reference to the /onw of the proposition only. 



" A PROBLEM either proposes something to be effected, as the 

 construction of a figure ; or it is a question tliat requires solution." 



A distinction witiiout a diflFerence : further than this — that the 

 latter form is vague as to both the terms " question " and " solution." 

 2 M 2 



