Nov. 8, 1888] 



NATURE 



27 



Mensuration is represented by a key to Todhunter's 

 small treatise, by the Rev. Lawrence McCarthy, of St. 

 Peter's College, Agra. As a rule we have no great liking 

 for keys, but if there is any country where such a key 

 might be used with advantage, it is India. Possibly this 

 is what prompted Prof. McCarthy to perform a labour 

 which might fitly be termed an intellectual treadmill. 

 We have always looked upon the " Mensuration "itself as 

 Todhunter's least valuable work. It is full of long-winded 

 rules which are never learnt, and the use of algebraic 

 symbols by which all such rules could at once be ren- 

 dered visible to the eye, is most curiously, and, as 

 we think, unreasonably, avoided. The same spirit cha- 

 racterizes the key. Everything is worked out at full 

 length with needless repetitions of figures, especially the 

 oft-recurring 3'i4i6, which we have counted no less than 

 twenty-nine and thirty-four times respectively on two 

 pages chosen at random. Surely the symbol ir might 

 have been substituted with advantage here. Barring 

 such defects, the work appears to have been well done, 

 and will doubtless be of use, more especially to those 

 who are unable to get tutorial assistance. 



An " Explanatory Arithmetic," by Mr. G. E. Spicker- 

 nell, which has reached a third edition, does not strike 

 us as any improvement on existing treatises. Brevity, 

 which is one of the points aimed at, is certainly secured, 

 but at the expense of both elegance and lucidity. The 

 rules read like excised telegrams, and are liable to be 

 misconstrued in much the same way. Thus, to take an 

 example of the telegram purporting to explain compound 

 subtraction: "Take like from like; and whenever it is 

 necessary, in order to make subtraction possible"; and 

 a longer one for the subtraction of fractions runs thus : 

 " Reduce minuend and subtrahend to equivalent fractions 

 having their least common denominator ; and then, 

 having like parts of integers, take less number from 

 greater, and write in figures, under remaining parts, 

 their name." Similar highly cacophonous and ambigu- 

 ous paragraphs are to be found scattered through the 

 book, and give one the impression that they will fre- 

 quently necessitate as much explanation as the principles 

 they are intended to embody. Occasionally the author 

 employs a definition which is palpably partial, thus : 

 " When an integer or whole thing is divided into a 

 number of equal parts, those parts are called fractions." 



The entire book strikes us as being of the empirical 

 cramming style, as opposed to the rational and scien- 

 tific style so well exemplified in Brook Smith's treatise 

 and the smaller one by Lock, in which rules are avoided 

 as much as possible. On the other hand, it contains 

 copious and very well assorted collections of examples 

 and exammation papers, with answers which can be 

 readily removed from it if desired. These might be 

 used with advantage, and the teacher, if a good one, 

 could translate and expand the telegrams into a more 

 rational and elegant form, or, still better, do without 

 them. 



Trigonometry, plane and spherical, is represented by 

 two booiis — one, comprising the elements of both sub- 

 jects, chiefly for the use of junior naval ufficers, by H. B. 

 Goodwin, Naval Instructor ; while the other is Part 2 of a 

 "Treatise on Higher Spherical Trigonometry and Geo- 

 metry," by Messrs. McClelland and Preston, of Dublin. 



Mr. Goodwin's work is intended to give, in one volume, 

 the course usually required for an acting sub-lieutenant 

 — which heretofore he has had to pick out of a variety of 

 elementary works— and appears to admirably fulfil its 

 author's intention. It is marked by simplicity of treat- 

 ment, the avoidance of cumbrous rules, those bugbears 

 of our ancient text-books, and a separation of the subject 

 into distinct parts, each of which is complete in itself. 



Messrs. McClelland and Preston's boo'c is a new de- 

 parture, in so far as, with the exception of the well-known 

 treatise of Mulcahy, it is the first time that spherical geo- 

 metry, as distinct from trigonometry, has been seriously 

 put into text-book form. The authors are to be con- 

 gratulated on their bold, clear, and systematic treatment 

 of this too-much-neglected and really useful branch of 

 mathematics. The work throughout is characterized by 

 lucidity and originality of treatment, and is subdivided 

 into complete chapters. Spherical and stereographic 

 projection are also carefully explained, and their power 

 as methods amply exhibited. 



We cannot help thinking that both spherical trigono- 

 metry and geometry are far too much neglected in our 

 educational curricula. In consequence, it is astonishing 

 what errors are committed when even the simplest proper- 

 ties of a spherical surface are in question. ,The curvature 

 of the earth is realized by few, and some who ought to know 

 better have not yet grasped the fact that the latitude of 

 Cairo approximately bisects the area of the northern hemi- 

 sphere. Nothing but polar projections, or, better still, 

 globes themselves, will ever correct the false impressions 

 which we get from that terrible flat-ruled distortion en- 

 titled " Mercator's projection," from which all approach 

 to curvature has been so carefully extracted. 



The theory of the trade-winds, moreover, which has 

 survived up to date in some of the text-books, takes no 

 account of the shortening of the radius in considering 

 the gain in eastward motion by the transference of the 

 air at relative rest on the equator to higher latitudes. 

 Thus, according to Loomis's " Meteorology," the gain is 

 simply found by subtracting the linear velocity at the 

 higher latitude from that at the equator ; whereas when 

 the shortened radius is considered, it amounts, in latitude 

 60', to 1554 instead of 518 miles per hour, and at the 

 Pole itself to 00 instead of 1036 miles per hour. These 

 are only a few of the most patent errors which arise from 

 a neglect of spherical principles, and might be multiplied 

 almost indefinitely. 



'' Hints for the Solution of Problems in Solid Geo- 

 metry," by Dr. Percival Frost, is a book which cannot 

 fail to be of great value to the student of this difficult 

 but important branch of mathematics. Mathematical 

 solutions have little analogy to, and, except in elementary 

 works, none of the disadvantages of, classical cribs. In 

 the present case, the execution of Dr. Frost's truly 

 laborious work has been attended by an unexpected 

 advantage, in leading to the discovery of certain errors 

 and omissions in the statement of the problems them- 

 selves, which might otherwise have escaped notice. We 

 heartily welcome Dr. Frost's hints, and trust they may 

 receive the attention they so fully deserve. 



While some branches of elementary mathematics are 

 already in danger of being congested by a plethora of 

 text-books, statics and dynamics seem to us to still pre- 



