May 2 1, 1908J 



NA TURE 



53 



(2) This book covers the ground of Euclid ii., iii. 

 35 fo 37' '^-1 ^""^ ^'■' together with the properties of 

 harmonic (but not of anharmonic) ranges, the nine- 

 point circle, the radical axis, poles and polars, in- 

 verse figures, &c. The text is on the whole more theo- 

 retical than practical; the examples are divided into 

 theoretical riders, practical constructions, and calcu- 

 lations, a good and sufficient collection of each being 

 given. The book contains rather more than the 

 average student will require ; the authors asterisk 

 some of the less necessary sections, and probably the 

 teacher will advise the omission of others also. The 

 properties of rectangles are developed from the geo- 

 metrical standpoint ; algebraic methods are, however, 

 also given. In the theory of proportion only commen- 

 surable quantities are dealt with at first, the extension 

 to incommensurables being given in the last chapter. 

 The book is sound and sensible throughout, and de- 

 serves to hold its own easily in the severe competition 

 which text-books on elementary geometry have to 

 face at present. 



(3) The author starts by defining integration as the 

 inverse of differentiation, and then shows how to find 

 the indefinite integrals of all the usual standard types. 

 This part of the book may be quite useful, 

 but the latter portion is not so satisfactory. 

 The author has attempted too much for the space at 

 his disposal, and sacrifices in places not only sound- 

 ness but intelligibility also. Definite integrals are 

 introduced without any adequate explanation, and the 

 connection between definite and indefinite integrals 

 is obscure. It is almost impossible to make applica- 

 tions of the calculus to geometry clear without a 

 single diagram or without proving that a definite 

 integral may be considered as the limit of a sum ; this 

 is, however, what the author attempts. In fact, the 

 argument is too condensed to be followed by the type 

 of student for whom the book is written ; for instance, 

 such a step as " -d6jd/=ii6, .'. i/6ld= -adt'' is sure to 

 give trouble if unexplained. 



Minor errors are the statement of the " test-ratio " 

 rule on p. 36, and misprints at the bottom of pp. 42, 

 43. .\ book which aims at being " practical " should 

 not calculate the temperature of a cup of tea to four 

 places of decimals. 



(4) This book is very short ; for the margins and 

 print are large, while twenty-si.x of its seventy-nine 

 pages are occupied by a somewhat superfluous intro- 

 duction, and fifteen by solutions of examples. The 

 remainder is devoted to the indefinite integral of 

 cos '"X sin ".r (m and n integral) without the aid of 

 reduction formulae, and to the integral of expressions in- 

 volving sja^-x-, sla-^x'-, &-c. This latter part is 

 d one be tter in other books ; for surely in integrating 

 sja' + x- the substitution x = a sinhfl is preferable to 

 x = 2a sine. 



(5) This book is intended to cover the mathematics 

 Jearnt in the first year of a two years' course at an 

 engineering school. The authors disregard the tradi- 

 tional division of mathematics into distinct subjects, 

 and introduce the principles of each subject as needed. 

 By thus developing algebra, analysis, and calculus 



NO. 2012, VOL. 78] 



side by side the student has his interest stimulated, 

 realises the interdependence of different parts of 

 mathematics, and learns the art of choosing the best 

 method of attacking any given problem. Against 

 tliis must be set the fact that the conventional division 

 of mathematical study has the great advantage of 

 helping the student to systematise his knowledge. If 

 once we admit the principle of no division, we could 

 hardly wish for a better book. The subjects are very 

 skilfully coordinated ; the treatment throughout is 

 sound and mathematical without ceasing to be in- 

 teresting or "practical." The examples are useful 

 and very numerous, and answers are given. In this 

 first year's course is covered a good deal of the more 

 elementary parts of the theor}' of equations, deter- 

 minants, graphs, analytical geometry as far as the 

 general equation of the second degree, and differential 

 calculus, including curvature and critical values of 

 functions, but not asymptotes. Conies, especially geo- 

 metrical conies, are treated less fully than usual ; their 

 place is partly taken by other interesting curves. E.\- 

 cellent and interesting though the book is, it makes 

 heavy demands on the reader's attention; and 

 would probably require considerable ability on his part 

 if it is to be mastered in one year. 



Ot/i? BOOK SHELF. 



ycpliilim. By William J. H. Bohannan. Pp. 236. 



(New York ; Reeve A. .Silk, 190S.) Price 1.50 dollars. 

 "Tins book is written," so we are told, "to show 

 the error of 'science,' and to point out the truth of 

 statement of the Bible concerning physical phe- 

 nomena." 



It certainly does show the error of "science," as 

 understood or misunderstood by the author of the 

 book. The following extracts are given merely as 

 examples of the style in which the book is written : — 



"The more the thermic dominates in structural 

 composition of a body, the more penetrable it is to 

 ihc magnetic entities of the field of another and the 

 less to the thermic of that field." 



" The earth's thermic entities of field, emitted from 

 her equatorial region, her southward geographical 

 pole being toward the sun, were taken outward from 

 him, under the action of the entities of his field, her 

 inseparable thermic entities of field enveloping the 

 separable of her northern regions." 



As a further example, we are informed by way of 

 correction that " Nephilim," p. 154, fourth sentence, 

 should read : " The planet Jupiter, on the other hand, 

 had three tails, two visible to human eye, passing 

 outward from the poles of the planet and from the 

 sun ; the other one magnetic and invisible, but vastly 

 greater than the visible, passing from the equator of 

 the planet toward the sun." 



Some of the chapters suggest a kind of vortex 

 theory, while others profess to deal with the theory 

 of the tides. The author of " Nephilim " would have 

 stood a better chance of recognition if he had, made 

 a careful study of the whole existing literature on 

 one or other of these two subjects. The mere quota- 

 tion of extracts from articles in popular encyclopaedias, 

 the contributors of which were probably limited to 

 1000 or 2000 words, is a very small step in that 

 direction. Such short articles were never intended to 

 give a complete explanation of all the difficulties which 

 have been studied in connection with these theories; 



