178 SCIENCE PROGRESS 



as represented by the gradient in this diagram. This brings 

 us face to face with the infinitesimal calculus. 



Mr. J. W. Mercer, of the Royal Naval College, Dartmouth, 

 has published recently an excellent Calculus for Beginners, the 

 most significant feature of which is that (in development of 

 a suggestion originally due to Prof. Perry) all the main applica- 

 tions of differentiation and integration are exemplified without 

 using any function more abstruse than x n . The first two 

 hundred and fifty-nine pages are given to x 11 and in the course 

 of these pages the reader learns how the calculus bears upon 

 velocities and accelerations, maxima and minima, relative errors, 

 definite and indefinite integrals, areas, Simpson's rule, volumes, 

 centres of gravity, moments of inertia, work done in stretching 

 strings and by expanding gases, mean values, etc. 



In all this work the manipulation is slight but the effect on 

 mental enlightenment is immense. The importance of these 

 illustrations is that they prove that a store of applications is 

 thrown open by the very simplest tools that the calculus provides. 

 The ideas, then, of the calculus and a feeling of the extra- 

 ordinary power of this new instrument are accessible to a 

 student with a modest degree of manipulative skill in algebra. 

 It is shown that it is not necessary to tread for years the weary 

 paths of highest common factor, fractions and the like before 

 becoming worthy to enter this rich country. 



Now it is not proposed that the calculus, in so far as it 

 belongs to the non-specialist course, should cover the first 

 two hundred and fifty-nine pages of Mercer's Calculus or any- 

 thing like this amount of work. The non-specialist cannot 

 integrate i/x, for this involves e. Many of the applications 

 enumerated above will be beyond his range. For all this, there 

 is much that he can do. He will be like the prospector picking 

 up the first nuggets on a new goldfield. Having been trained 

 to think functionally, he will have little difficulty in grasping 

 the idea that the gradient at any point of a curve (sayjy = x 2 ) is 

 a function of x. On a copy of the graph, lithographed on paper 

 ruled in inches and tenths, he may draw tangents with a ruler, 

 measure off the gradients by means of transparent squared 

 paper and plot these gradients as ordinates ; this will show 

 him that the gradient is a function of x and probably he can 

 now see what function. After a suitable amount of preparation 

 in this style, he may proceed to determine the gradient 



