226 Notices respecting New Books. 



as Taylor's Theorem is employed from one end of the book to the 

 other, though the author comes rather near to using it in art. 46. It 

 is unnecessary to add that such a treatment of this subject requires 

 considerable powers of geometrical exposition ; and these are cer- 

 tainly possessed by the author. In fact he gives us to understand 

 that he has worked out the whole subject from his own point of 

 view with not much more aid from the labours of his predecessors 

 than what is implied in the fact of their having established by a 

 different method all the leading facts of the subject. 



The advantage gained by a purely geometrical exposition of such 

 a subject as Curvature is, that the student learns from it what 

 in actual space corresponds to his algebraical f ormulae ; and this 

 is a matter to which his attention has often to be directed. On 

 the other hand, it is liable to the somewhat serious drawback, 

 that the results are obtained by the use of an instrument of re- 

 search inadequate to the purpose, unless in a very skilful hand, 

 while they can easily be got at by other means. Suppose a student 

 to have a moderate skill in analysis, and to know the few general 

 formulae relating to the subject which are to be found in all books 

 of solid geometry ; e. g., suppose him to have mastered pp. 407- 

 417 of De Morgan's ' Differential Calculus,' it is hardly too much 

 to affirm that he would find it easier to investigate by their means 

 most of the theorems contained in M. Ruchonnet's book than to 

 make out his proofs. He would observe that such magnitudes as 

 the angles of contingence and torsion, the radius of curvature, the 



ratio of its increment to that of the arc — -^ in fact — are intrinsic 



as 



to the curve, and the relations between them independent of the 

 coordinate axes chosen. Consequently he would choose the axes so 

 as to simplify, as much as possible, the general expressions given in 

 books ; and, as a rule, the required results would then be obtained 

 without much difficulty. Take for instance the proposition quoted 

 above, that in general the curve cuts the osculating plane at the 

 point of contact. Suppose the curve to be given by the equations 

 y=~F(x) and z—f{x), that the point under consideration is taken 

 as the origin, the osculating plane as the plane of ccy, and (though 

 this is not necessary for the purpose immediately in hand) the 

 tangent as the axis of x. It follows from these suppositions that 

 F(0), F(0), /(0), /'(0), and /"(0) are severally zero. Now consider 

 a point (h, Jc, I) near the origin ; we have 



^=F(A)=iF(0) . h?+ k~F"'(0) . 7* 3 + . . ., 



i=f(h)=if"(0) . «+Jkrt(6) . w+ . . . 



The second equation shows that in general I changes its sign with 

 h, i. e. on one side of the normal plane the curve is above, and on 

 the other side below the osculating plane, which it therefore cuts 

 at the point of contact. Our author's proof of this theorem is 

 most ingenious, but, as already mentioned, is indirect, and by no 

 means easy to follow. 



