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NATURE 



[June 25, 1891 



scholarships should be provided to enable any poor child 

 who has passed the standards to continue his education 

 either in the school or elsewhere. We do not say that 

 such scholarships should be universally provided out of 

 the present grant, but they would be a most proper 

 object to which to apply part of the surplus which will 

 be handed to many schools over and above the fee 

 equivalent. These considerations suggest another pos- 

 sible way of dealing with the surplus grants. The 

 great object of those who are interested in the develop- 

 ment of higher elementary, technical, and secondary 

 education should be to strengthen instead of weakening 

 the connection between primary and higher schools. It 

 is to be feared that any provision for freeing elementary 

 schools up to a certain point or a certain age, will tend 

 to sever rather than to unite the two grades of schools, 

 unless the flow between them is at the same time stimu- 

 lated by the establishment of free scholarships or in other 

 ways. A free (or partly free) elementary school is not 

 the ultimate ideal. We want a free road kept open to the 

 University. Is it too late to throw out the suggestion 

 that school managers receiving a fee-grant in excess of 

 the amount previously received in fees should be required 

 to use the surplus for an object akin to that contemplated 

 by the main provisions of the Bill— viz. the extension of 

 free education for selected scholars beyond the narrow 

 limits of the primary schools, in other words the provision 

 of continuation scholarships ? Up to a short time ago 

 it would have been replied that in many cases there were 

 no higher institutions accessible, but the application of 

 the Local Taxation grant to technical and secondary 

 education is fast changing all that, and a proposal which 

 a few years since would have been unfeasible is now well 

 within the range of practical politics. 



DIFFERENTIAL AND INTEGRAL CALCULUS. 

 Differential and Integral Calculus, with Applicatiojts. 

 By Alfred George Greenhill, M.A., F.R.S. Second 

 Edition. (London : Macmillan and Co., 1891.) 

 pROF. GREENHILL is known to the academic 

 -L world as an accomplished mathematician who 

 has powerfully helped to advance certain branches of 

 appHed mathematics ; he is also known to the readers 

 of Nature as a friend (militant) of the practical man. 

 We say at once, in all sincerity, that we sympathize 

 with Prof. Greenhill in both his capacities. The volume 

 on the infinitesimal calculus now before us, although 

 professedly a second edition, is in reality a new work, 

 addressed to the special needs of the practical man by 

 his mathematical friend Prof. Greenhill. 



Of many of the author's didactic innovations we highly 

 approve. The treatment of the differential and integral 

 calculus together from the very beginning is a piece of 

 sound method, the introduction of which has been delayed 

 merely by the bad but not infrequent practice of separat- 

 ing the two as examination subjects. The introduction of 

 the hyperbolic functions to systematize the integrations 

 which can be performed by means of the elementary trans- 

 cendents, has been, as we can testify from experience, a 

 great help in elementary teaching. The admirable " chap- 

 ter in the integral calculus" which was published separately 

 NO. I 1 30, VOL. 44] 



in an extended form some years ago, and is now con- 

 densed and simplified in a separate chapter at the end of 

 the work under review, is the most important addition to 

 the teaching material of the integral calculus that has 

 been made for a long time ; that chapter alone is worth 

 the price of Prof. Greenhill's book. The plan of drawing 

 the illustrations of the subject from departments of pure 

 and applied mathematics with which the learner may 

 afterwards have to do is also excellent. Finally, there 

 blows through our author's pages that inimitable fresh- 

 ness which emanates from the man who is familiar with 

 much that is newest and best in his day, who does not 

 merely make extracts from books, but who speaks of 

 things in which he has taken a part. This freshness can 

 only be compared to that agreeable odour which inland 

 people tell us comes from mariners and others who cross 

 the sea from strange lands. Like these same mariners, 

 our author produces from his pockets strange and 

 puzzling curiosities, such as reciprocants, tide predicters, 

 Schwarzian derivatives, Mehler's functions, to delight 

 and to dazzle the learner. It is true he tells but little of 

 these things ; still, it is pleasant to look at them ; and 

 they make us happy under our present toil by leading us 

 to think that we too may one day visit the country where 

 these pretty things are at home amidst their proper sur- 

 roundings. 



Where there is so much to praise we are truly sorry to 

 insinuate the bitter drop of blame ; but, much as we 

 love and follow Plato, something must be conceded to 

 truth. In the first place, we think that in this second 

 edition the introduction of heterogeneous illustration has 

 been overdone. The fundamental rules of the infinitesimal 

 calculus are really very few in number, and the practical 

 man's friend would do well to impress that upon him at 

 the outset, instead of scattering these principles through 

 a large volume, and overlaying them with thick masses of 

 disconnected application, to such an extent that poor Mr. 

 Practical-Man is in danger of losing his tools among the 

 shavings, or, to use a metaphor which Prof. Greenhill's 

 pupils might prefer, of not seeing his guns for smoke. 

 Prof. Greenhill must recollect that the man that sits 

 down to read his book is not all possible practical men 

 rolled into one, but one poor practical man— say, an 

 engineer — who wants some knowledge of the infinitesimal 

 calculus, and who will find many of the illustrations more 

 indigestible than the principles of the calculus itself. 

 Would it not be better for the practical man, as well as 

 for any other man, to have the few leading principles of the 

 calculus set before him with an adequate but moderate 

 amount of illustration of a uniform geometrical kind, and 

 not to be dazed by a flood of oracular statements about 

 soap-bubble films, tide-p edicters, &c., in the course 

 of his initiation 1 Such digressions are most useful now 

 and then in a lecture ; they serve to give picturesqueness 

 to the discourse, and help to fix the attention of the 

 hearer : but we think that too many of them destroy the 

 usefulness of a text-book, the object of which is quite 

 different from the purpose of a lecture. 



The matter we have just been criticizing may, perhaps, 

 be held to be one of taste ; and we cheerfully admit that 

 much should be allowed to a writer of strong individuality. 

 After all, we love to have the author in his book. There 

 is another matter, of more importance, on which we 



