26 



NA TURE 



{Nov. 8, i; 



SOME RECENT MA THEM A TICAL BOOKS. 



Euclid. Part i, Books 1. and II. By H. S. Hall and 



F. H. Stephens. (Macmillan, 1887.) 

 Al^'ebraical Exercises. By H. S. Hall and S. R. Knight. 



(Macmillan, 1887.) 

 Key to Todhunters Mctisuration By Rev. L. McCarthy. 



(Macmillan, 1886.) 

 Explanatory Arithmetic. By G. E. Spickernell. Third 



Edition. (Simpkin, 1887.) 

 Platie and spherical Trigonometry. By H. B. Goodwin. 



(Longmans, 1886.) 

 Spherical Trigonometry. Part 2. By W. J. McClelland 



and T. Preston. (Macmillan, 1886.) 

 Solid Geometry : Solutions. By P.Frost. (Macmillan. 



1887.) 

 Elementary Statics. By J. Greaves. (Macmillan, 1 886.) 

 Differential Calculus. By B. Williamson. Sixth Edition. 



(Longmans, 1887.) 

 Differential Calculus. By J. Edwards. (Macmillan, 



1886.) 

 Algebra. By Oliver, Wait, and Jones. (Ithaca : Finch, 



1887.) 

 Practical Solid Geometry. By W. G. Ross. (Cassell, 



1887.) 



IF former periods of the world's history were character- 

 ized as the "Stone,"' "Bronze," and "Iron" Ages, 

 the present epoch might well be entitled the " Book" Age. 

 Amidst the flood of literature of all sorts which daily 

 pours out of the jaws of our printing leviathans, didactic 

 mathematics certainly claims its due proportion. The 

 number of Algebras, Euclid?, Arithmetics, and Trigono- 

 metries which appear, with the " rough ways made smooth 

 and the crooked ways straight," make us regard the 

 modern student with a mixture of envy and pity — envy 

 at the possession of such broad highways to knowledge 

 as we never dreamt of, and pity at the difficulty he must 

 experience in choosing out of such a multitude. We 

 cannot, moreover, help fancying that this plethora of 

 books is not entirely without compensating disadvant- 

 ages, for the very ease and tranquillity which which the 

 student glides through the cleared forest, make him care- 

 less and inattentive of the land-marks and salient features 

 that were so carefully noted by his more self-reliant, if 

 less luxuriously-equipped, predecessor. 



Of the twelve books on our table, we shall begin by 

 -noticing the most elementary. Among these is an in- 

 stalment comprising Books I. and II. of "Euclid," by 

 Messrs. Hall and Stevens, of Clifton College. For these 

 revolutionary days it is remarkably orthodox ; but certain 

 changes have been introduced, very wisely as we think, 

 ■where Euclid's enunciations were confusing, or the proofs 

 were not sufficiently comprehensive, as in Prepositions 

 8 and 26, where the identical equality of the two triangles 

 is not usually emphasized. The authors, in their preface, 

 enter very fully into the reasons which decided them to 

 avoid the use of symbols at first, and also to preserve 

 " the formal, if somewhat cumbrous, methods of Euclid," 

 and with these reasons we in the main agree. If, how- 

 ever, for " a large majority of students ' Euclid ' is 

 intended to serve, not so much as a first lesson in mathe- 

 matical reasoning, as the model of formal and rigid argu- 

 ment which most conduces to accurate and orderly 



thought in any field of study," we should welcome a 

 book of geometry brought out for the use of those whose 

 natural mathematical growth is stunted, and taste warped, 

 by a too strict adherence to the cumbersome and often 

 involved style of the ordinary text. 



We admit the dilemma for those who wish to make 

 " Euclid" serve the double purpose of an introduction to 

 logic as well as geometry, but, at the same time, we are 

 unable to see why, even for the logicians, some at least 

 of the advantages of the German method cannot be 

 introduced, such as the use of a, b, c for the sides, and 

 a, /3, y for the angles of a triangle. A great deal of con- 

 fusion arises from the use of three letters for an angle 

 and tivo for a side, and the change to the simpler method 

 would not only clear the student's mental vision, but 

 leave the logic unimpaired. 



One of the best examples of the defects arising from a 

 rigid adhesion to the formal text is in Book I., Propo- 

 sition 13. We have found many to whom this proposition 

 in its existing form is one of the most repulsive in the 

 book, and it has been almost touching to witness the joy 

 evinced by a dullard on his first realizing that the obtuse 

 angle was just as much in excess, as the acute angle was 

 in defect of a right angle — a fact which the ordinary 

 proof completely disguises. We do not think the alter- 

 native proof to Proposition 47 is likely to meet with 

 much favour, and since the authors do not altogether 

 discard alternative proofs, we should have preferred, 

 instead, the neat alternative to Proposition 48 given in 

 Casey. 



For the purpose for which it is designed we do not 

 hesitate to recommend the book. It is excellently 

 printed, the construction lines being very properly faint, 

 and the figures in all cases clearly drawn. The exercises, 

 additional theorems, and hints to solution are also un- 

 usually well arranged, and will be truly welcome to the 

 student who intends to go in for mathematics, as well as 

 train his mind into logical habits. 



" Algebraical Exercises," by Messrs. Hall and Knight, 

 is, we presume, intended to be a companion and supple- 

 ment to their excellent little " Elementary Algebra," 

 which has met with such a generally good reception. 

 While these exercises will, no doubt, be of consider- 

 able use, we think they might be improved, and ren- 

 dered more widely serviceable, (i) if Part i, devoted 

 to the earliest rules, were extended beyond a meagre six 

 pages — a range altogether out of proportion to what fol- 

 lows ; and (2) if the exercises were more gently gradu- 

 ated in Part 2. For example, linear equations in three 

 unknowns are introduced per sal turn as early as p. 10, 

 and all the papers after p. 6 strike us as being a good 

 deal harder than those encountered by the Army Pre- 

 liminary candidate, typesof which are given on pp. 146-47. 

 We also regret to find in the first ninety-seven pages an 

 almost entire absence of book-work questions— a defect 

 which is only partially made up for in the capital selec- 

 tion of typical public examination papers which follow. 

 Example-grinding is no doubt an essential art, but in 

 algebra, especially, the early parts tend to become purely 

 mechanical, unless real thinking is encouraged and stimu- 

 lated by rational questions on the processes employed. 

 In a subsequent edition a few more recent University 

 and Army papers would form a welcome addition. 



