288 Notices respecting Nexv BooJcs. 



more logical clearness, displayed in a happy development of first 

 principles, than the suhsequent investigation commonly exhibits. At 

 ail events, it is a task in which there are more failures than are com- 

 mensurate with the difficulty 5 except it be admitted that to produce a 

 good elementary book be an undertaking of a higher order than is 

 commonly supposed. In mathematical science this is peculiarly the 

 case ; and in that branch to which Mr. Young's book is devoted, this 

 difficulty has been felt more strongly than, perhaps, in any other. The 

 universal complaints which we hear of theunintelligibility, not to say 

 inconclusiveness, of the processes and reasonings which stand at 

 the threshhold of the Differential Calculus, must convince us that 

 there is a radical defect somewhere, and stimulate our search for its 

 place and its essential character. 



Is there too wide a chasm between the processes of common alge- 

 bra as now practised, and the elementary steps of the Calculus ? Is 

 there something in the leading notions which the science involves, 

 which too greatly transcends the superficial conceptions of the undis- 

 ciplined mind ? or, finally, is there something in the doctrine itself 

 repugnant to the conclusions which the ancient geometry, or the 

 scarcely less satisfactory reasonings of the algebraic analysis has fur- 

 nished examples of? With respect to its compatibility with the vulgar 

 conclusions of " common sense," we might reply, — that the faculties 

 of uneducated mind take cognizance only of general appearances 

 of similarity, and neglect the more recondite difl^erences, as well as the 

 more recondite agreements that may subsist amongst the objects of 

 its occasional contemplation ; and therefore that its decisions are 

 valueless on topics foreign to those it is conversant with, and even on 

 its most familiar topics viewed under a novel aspect. To penetrate 

 further than this, we must do more than perceive the individual facts, 

 —we must class them, and reason on their appearances. This is 

 the first step in science. To do this, however, we must make a vi- 

 gorous effort to dethrone the host of prejudices which have usurped 

 the place of reason, for it is no easy matter to reject the hasty gene- 

 ralizations of common sense, which, being acquired during the vague- 

 ness of early and desultory speculations, are almost always ranged 

 on the side of error. Those processes of algebra, too, which are com- 

 monly studied prior to entering upon the Calculus, are too often mo- 

 dified to meet some fanciful or pertinacious objections of the man of 

 mere common sense, rather than framed as an introduction to the 

 kind of thinking which this science involves. It is presented as the 

 end, rather than the beginning of mathematical science. Such a pro- 

 cedure might have been well adapted to Goldsmith's " loveliest village 

 of the plain," and its interesting pedagogue ; but as the first step 

 towards those profound inquiries which characterize the physics of 

 the present day, it is impossible to imagine how works could be 

 worse adapted than our common treatises on algebra. This is the 

 source of much of the difficulty that is felt in laying down the princi- 

 ples of the Differential Calculus ; and considerable dexterity is re- 

 quired to enable the preceptor to lead his pupil to frame even a tole- 

 rable conception of the character of the science. Still this is not the 



only 



