382 Notices respecting New Books. 



A daring pioneer, of the Perry type, is required to bring 

 Integration into its proper priority of historical order, as senior 

 to Differentiation by a thousand years. The idea of integration 

 is less abstract and easier for the mind to grasp, wheu treated 

 from first principles as the total growth of a quantity; instead 

 of its rate of growth at any instant as expressed by the differential 

 coefficient. 



But first principles are difficult of application, and require 

 special treatment for each particular case, so that for rapidity of 

 progress in the applications, Integration is taught as a process of 

 Anti-differentiation, the method inculcated in Chapter VI. 



The picture of the process of Integration as a Quadrature 

 follows in Chapter VII, explaining the operation from first 

 principles in its proper historical order ; and detailed geometrical 

 applications follow in Chapter VIII, exemplified by the calculation 

 of area, in Cartesian and polar coordinates, and by use of the 

 Planimeter, of volume and surface, of the rectification of an arc, 

 of mean centre, our old friend centre of gravity ; the theorems 

 also of Pappus, a.d. 300. 



The student at this stage will begin to find himself in the 

 subject, after working at the carefully selected collection of 

 examples. 



Juggling with curves (instead of letters) is the scornful name 

 sometimes given to the geometrical applications by the pure analyst, 

 anxious to get to work on his favourite detail of the Failure of 

 Taylor's Theorem. But where else is the student to learn 

 about this subject, so important for its applications in the higher 

 branches, except in a systematic treatment such as given here in 

 Chapters IX, X. 



The last quarter of the volume takes up Differential Equations, 

 and so renders unnecessary a separate study in another volume, 

 of terrifying size and association, looming in store ahead. 



Most of this terror could be cured, or would not arise, if the 

 author would introduce at the earliest stage the mere name 

 Differential Equation, and the associated elementary ideas. Then 

 the beginner could be assured that he had been working uncon- 

 sciously at differential equations from the start; and that the 

 laws of mechanical nature are revealed to us in the Differential 

 Equation. The separate individual problem merely assigns the 

 constants of the equation. 



On this plan the simplest integrations would be asked for 

 as the solution of a differential equation ; as for instance of the 

 differential equations 



with a drawing of the graph, and of the singular solution, as a 

 tac, cusp, or node locus ; and so on. 



