Ritchie on the Calculus -31 



imperfect faith," which he must necessarily experience when he in- 

 spects the enormous pages of formulae which may frequently meet 

 his eye, and which, in numerous instances, can be of no benefit either 

 to the student, or to him who has advanced into the depths of science ; 

 because the former cannot understand them, and the latter would 

 always prefer to make his own calculations, than to follow merely 

 the footsteps of another. 



It is no uncommon cause of the neglect of the study of the calculus 

 that, it is always placed after the most intricate and difficult parts of 

 algebra. The author, however, dissipates this error by shewing that, 

 the differential and integral calculus may be readily comprehended 

 after the pupil has acquired a knowledge of the elements of geometry 

 and the principles of algebra, as far as the end of quadratic equations. 

 He begins with an introduction explanatory of constant and variable 

 quantities, infinity, &c. in such a distinct and familiar mode, that it is 

 impossible to fail of understanding him as he proceeds. The object 

 of the differential calculus, he observes, is to determine the ratio 

 between the rate of variation of the independent variable, and that 

 of the function into which it enters. This is illustrated, as is very 

 properly done in every case, by a numerical illustration. Thus, " it 

 the side of a square increase uniformly at the rate of three feet per 

 second, at what rate is the area increasing, when the side becomes 

 10 feet ; 1 : 2 x '. : 3 : x. Hence, the rate of increase is G X 10, 

 or, at the rate of 00 square feet per second." He then explains the 

 notation of Newton and Leibnitz, adopting, very properly, that of 

 the latter ; as, for example, d x instead of x to denote the rate at 

 which the variable quantity represented by x is increasing ; although 

 we have no doubt that he yields the palm of accurate reasoning to 

 the metaphysics of the former, and admits that of the latter to be 

 much less philosophical. 



In reference to the integral calculus, he observes, that a pupil will 

 generally be at a loss to understand what is meant by finding the 

 integral of a given differential. In plain language, it only means 

 that " we have given a quantity which varies uniformly, and the 

 ratio of its rate of variation with another quantity depending on it 

 and given quantities, to find the value of that quantity." The 3d, 

 4th, 5th, and 0th sections are devoted to rules for differentiating 

 and intigrating simpler forms of functions and differentials ; of ex- 

 pressions containing two independent variables ; of functions having 

 general indices, and the reduction of differentials to known forms, 

 integration by series and definite integrals. This concludes the first 

 portion of the work. 



The second part is taken up with the application of the preceding 

 rules and principles to useful purposes, under the heads of maxima 

 and minima of quantities, curves of the second order, which is par- 

 ticularly worthy of notice, from the concise and distinct mode in 

 which their genesis and nature are treated of, normals and subnor- 

 mals, length of arcs, areas of surfaces, and surfaces and capacities 

 of solids. 



Part third treat8 of the dcvclopeinent of algebraic expressions into 

 infinite scries, differentiation of transcendental functions and Integra- 



