Notices respecting Nexv Books. 289 



only source ; for after all, there is much vagueness (even amongst 

 those who have become considerable adepts in the practice of the 

 Calculus; in the reasons which are commonly oflered for the validity 

 of its processes. This vagueness is displayed in the preliminary steps 

 of the inquiry in a striking degree. Amongst mathematicians of 

 considerable eminence, we shall find confusion and contradiction pre- 

 vail, as to the fundamental axioms, the fundamental definitions, and 

 the fundamental reasonings of the science : and it almost invariably 

 happens that the student acquires considerable practical dexterity long 

 before he is able to unravel the mysteries of prime and ultimate ratios, 

 effluxions or rates of increase, or any of the otherprinciples which have 

 been made the basis of the method. If he have courage to persevere 

 (more as a calculator than a metaphysician) till he can learn the method 

 of operating ; — if he can solve a tew problems by means of the Calcu- 

 lus thus acquired ; and if in addition he can verify these results by an 

 appeal to some geometrical properties already known ; then he learns, 

 and thus he learns, by degrees, to feel confidence in the power and 

 accuracy of his rules. He infers that because it gives right results in 

 the cases he has tested, it </jere/ore will in all. Yet u;/;?/ it should 

 give right results in anyone case, he cannot make out in any plausible 

 manner: and, indeed, the vagueness and confusion which he expe- 

 rienced at the outset, when he attempted to master the reasons, dis- 

 couraged and disgusted him too much to render it an agreeable un- 

 dertaking to attempt the investigation anew. He thus willingly takes 

 it upon trust, or on the authority which belongs to mere induction, 

 instead of laying a broad and satisfactory basis upon a course of in- 

 disputable logical demonstrations. It is a painful fact, — but there is 

 reason to fear that it is the case of nine mathematicians out of ten, — 

 that they have no clear view of the first principles of the Differential 

 Calculus, however expert they generally are in the application of its 

 rules to the solution of problems : and it is in consequence of this 

 painful fact, that we hail with pleasure the little work of Mr. Young, 

 as a panacea, in some degree, for the evil we complain of. It is 

 of the utmost importance that the young student should be inspired 

 with clear views of the nature of the relations contemplated in the 

 science : that his mind should be put in attitude to perceive the force 

 of the reasoning — and that the ulterior objects of the Calculus should 

 open upon his understanding with the utmost possible f/is/iwc/neiA-. This 

 can only be effected by an author deeply in love with truth, and whose 

 mind is enlarged by an extended survey of the subject upon which he 

 writes ; — an author who conducts his inquiries and develops his 

 conceptions with due regard to method, and who is, moreover, en- 

 dowed with a happy elegance of diction ; — such a writer will not alter 

 what has already been done well, for the mere purpose of differing 

 from others, but will avail himself of the labours of his predecessors so 

 far as they contribute to the great object he has in view. Yet he will 

 always throw them aside when they are defective, illogical, irrelevant, 

 or unnecessarily operose ; and so modify them, where they admit of 

 modification, as to alter the visual a])pearance, or even the received 

 and familiarized phraseology, as little as is compatible with a clear and 

 N.S. Vol. 10. No. 58. Or/. 1831. 2 P perspicuous 



