February 13, 1896] 



NATURE 



339 



to extend the aim of his volume so that it might apply to 

 the beginnings of writing all the world over ? 



Yet such an extension Mr. Hoffman has attempted in 

 the course of about 100 octavo pages printed in large type. 

 The result was inevitable. The lion's share of the little 

 space at his disposal falls naturally enough to the North 

 American Indians with whose pictographs and writing 

 he is most familiar ; but, in consequence of this mistaken 

 aim at universality, the book is peppered here and there 

 with Egyptian hieroglyphs, once or twice a cuneiform 

 character crops up, and we have even come across a 

 Christian symbol. In many places, therefore, the work 

 is patchy, and the reader is in some danger of bewilder- 

 ment. Moreover, this is not the only danger to which 

 Mr. Hoffman exposes his readers, by rashly leaving the 

 ground he knows ; for his studies have not been suffi- 

 ciently extended to enable him to act as a trustworthy 

 guide elsewhere. He is certainly wise in making con- 

 siderable use of ''The Alphabet," by Isaac Taylor, but 

 some statements made twelve years ago naturally now-a- 

 days need revision. For instance, the dates he quotes 

 at the beginning of p. 184 must now be considerably 

 altered in view of M. de Sarzec's recent discoveries at 

 Telloh in Southern Babylonia. But perhaps the most 

 misleading portions of the book are those passages in 

 which he refers to the so-called " Hittite" inscriptions. 

 Here Mr. Hoffman's authorities are not so trustworthy 

 as Mr. Taylor, and the result is disastrous. From his 

 description of these hieroglyphs, the reader would cer- 

 tainly infer that no doubt existed as to their interpreta- 

 tion, but no impression could be more erroneous. The 

 four or five systems of interpretation that have been 

 proposed, and the last of which appeared less than two 

 years ago in the Zeitschrift of the German Oriental 

 Society, differ totally from one another, and are mutually 

 exclusive ; that is to say, each interpreter has employed 

 a different method and system of interpretation, and, 

 although they all work on the same inscriptions, the 

 translations they have produced do not agree at all. Of 

 course Mr. Hoffman is at liberty to select one of these 

 systems and to say it appears to him to be correct, but to 

 take the results of one of them, and to describe them as 

 though they were universally adopted without reference 

 to the controversy which still rages, is surely misleading 

 in the highest degree. Moreover, on p. 158 occurs this 

 rather puzzling sentence : " In the Akkadian, a signifies 

 'water' . . . and is represented by the inverted triangle, 

 the prototype in Hittite being the vase or ollaP If we 

 take this sentence literally, we can only infer that Mr. 

 Hoffman considers the earliest cuneiform character to 

 be the descendant of the Hittite hieroglyphs. We think, 

 however, that he cannot seriously hold this view, and 

 that' the sentence referred to does not rightly express his 

 meaning. 



It is not necessary, however, to dwell at greater length 

 on the blemishes which mar an otherwise excellent book. 

 They might, indeed, have been avoided by the exercise 

 of a little discretion, and the book would have been none 

 the less interesting if its contents had been confined to 

 the beginnings of writing among the tribes of North 

 America. Even as it is, the greater part is both interest- 

 ing and instructive, and the excellent print and numerous 

 diagrams add considerably to its attractiveness. 

 NO. 1372, VOL. 53] 



OUR BOOK SHELF. 



Elements oj Geometry. By Prof. George C. Edwards. 



Pp. vii + 293. (London : Macmillan and Co., 1895.) 



Prof. Edwards has forsaken the general sequence of 

 parts in the study of the elements of geometry as pro- 

 pounded by Euclid, and has given us a book almost on a 

 new plan. It must have been felt by most people, when 

 in their early school-days they commenced to read Euclid's 

 Elements, that much of the first few pages was almost 

 unintelligible ; and while no one would take exception to 

 the rigidity of the methods there used, yet it would be ex- 

 ceedingly difficult for the average beginner to comprehend 

 many of the definitions from the text alone without 

 further viva voce explanation. 



The substance of Euclid's Elements may be condensed 

 into a very small fraction of its present dimensions for all 

 practical purposes without losing any of the rigidity of 

 argument or usefulness in application, and in this way a 

 student may understand the proportionality of the sides 

 of similar triangles without previously wading through 

 four or five books. Prof. Edwards has arranged his book 

 in this condensed form, and at the same time has pre- 

 sented his subject in a manner which attracts the atten- 

 tion of the reader by its simplicity and usefulness. One 

 feature of this book is the great number of diagrams and 

 figures which it contains ; and where more than one 

 figure is required with any one proposition, they are all 

 carefully and separately given. The author does not 

 overload the student with a mass of definitions at the 

 commencement, but introduces them with explanations, 

 when required, in the context. The usual mathematical 

 conventions are introduced very early into the subject- 

 matter, such as signs to denote directions ; and the usual 

 contractions are freely used, a table of them being given 

 at the beginning for reference. 



The first three chapters deal with propositions which 

 relate to triangles and parallel lines ; chapter iv. deals 

 with the properties of circles ; and chapter v. with areas, 

 and the proportional relations of similar figures. This is 

 one of the most important sections of the book, and in- 

 cludes some useful results which are generally obtained 

 from books on trigonometry. Immediately following we 

 find a chapter devoted to chords and tangents, and 

 another to polygons and limits, including finding the 

 value of TT. Then follow over thirty pages of problem 

 questions, with hints to some of their solutions, and 

 innumerable figures to illustrate them where necessary. 

 No pains appear to have been spared to make the whole 

 as clear and intelligible as possible. 



We then find a few chapters on solid geometry, which 

 include matter dealing with surfaces, volumes, inter- 

 section of surfaces, and spherical triangles. The last 

 chapter contains some useful propositions in conic 

 sections. 



Throughout the book there are a great number of 

 examples, notes, suggestions and warnings to the reader, 

 which should be of especial value to those who have not 

 pursued the study of geometry to any extent previously. 

 Altogether the book is a valuable one for any educational 

 institution, and exceptionally so to technical students. 



Elementary Mensuration. By F. H. Stevens, M.A. 



Pp. xii 4- 243. (London : Macmillan and Co., 1895.) 

 Mensuration. By the Rev. A. Dawson Clarke, M.A. 



Pp. vi + 88. (London : Rivington, Percival, and Co., 



1895.) 

 Mr. Stevens' book is divided into two parts, the first 

 only requiring a knowledge of Euclid's first book, and 

 the use of algebraic symbols, in order to understand it ; 

 while the second part necessitates a much fuller acquaint- 

 ance with geometry and algebra, and a slight knowledge 

 of trigonometry. The book is not technical, but con- 

 tains a thorough treatment of those principles upon which 



