Feb. II, 1 8 86 J 



NATURE 



54' 



on the Floss," and so cannot identify the particular 

 edition which produced such a wretched result, but we 

 doubt not it was one of the ordinary sni.ill text-books 

 with which youth were well acquainted, in shape at least, 

 at the time referred to. And this makes us allude to the 

 portentous dimensions of the book before us, which con- 

 sists of sone 370 large octavo pages. The book is not 

 for schoolboys, but is intended for students of larger 

 growth. It commences, as does also " The Elements of 

 Plane Geometry " (brought out by the Association for the 

 Improvement of Geometrical Teaching), with a prelimin- 

 ary chapter on Logic, which gives sufficient introduction 

 to a subject in which " the mind first finds logic a practi- 

 cal instrument of great power." 



We turn aside for a moment to state a raison d'etre for 

 the volume before us. " In America the geometries most 

 in vogue at present are vitiated by the immediate assump- 

 tion and misuse of that subtile term 'direction'; and 

 teachers who know something of the non-Euclidian, or 

 even the modern synthetic geometries, are seeing the 

 evils of this superficial ' directional ' method. More- 

 over, the attempt, in these books, to take away by 

 definition from the familiar word ' distance ' its abstract 

 character and connection with length-units, only confuses 

 the ordinary student. A reference to the article Measure- 

 ment, in the ' Encyclop. Britannica' will show that 

 around the word ' distance ' centers the most abstruse ad- 

 vance in pure science and philosophy. An elementary geo- 

 metry has no need of the words ' direction ' and ' distance.' 

 The present work, composed with special reference to use 

 in teaching, yet strives to present the elements of geometry 

 in a way so absolutely logical and compact that they may 

 be ready as rock-foundation for more advanced study." 



This lengthy extract puts our readers in possession of 

 Mr. Halsted's views : the result of his efforts is an edition 

 which will, we think, repay perusal. 



Now, in reply to old Tullivers query, " Wat's Euclid ? " 

 it was replied, " It's definitions, and axioms, and triangles, 

 and things. It's a book I have got to learn in — there's no 

 sense in it." (Such is the view of some boys of the present 

 day, as we discovered in looking over answers to a recent 

 examination paper.) 



Mr. Halsted defines a straight line thus : — It is a line 

 which pierces space evenly, so that a piece of space from 

 along one side of it will fit any side of any other portion. 

 In his definition of an angle {AOB) one of the angles 

 is said to be the expletnent of the other ; he uses the 

 term "straight angle," calls a terminated line (as in his 

 " Mensuration") a sect, and "the whole angle which a 

 sect must turn through, about one of its end points, to 

 take it all around into its first position, or, in one plane, 

 the whole amount of angle round a point, is called a 

 perigoii^' Other definitions do not call for notice, except 

 that in the definition of a circle he has, by an oversight, 

 omitted to state that the sect must revolve in a plane. 



The First Book is divided into eight chapters, and em- 

 braces the matter of Euclid's First Book, with several other 

 important propositions : the order is not that of Sim- 

 son's text, but propositions are grouped under problems, 

 inequalities, parallels, triangles, and polygons. This last 

 head is broken up into the divisions, general properties 

 (congruence), parallelograms, equivalence, and axial and 

 central symmetry. 



In Book 11. the commutative law (for addition and 

 multiplication), the associative law, and the distributive 

 law are established, and the propositions proved sym- 

 bolically. Books III., IV., \^, and VI. correspond to 

 Euclid's divisions, but the selection of propositions and 

 their arrangement and treatment agree with results we 

 have seen nearer home- 

 But we hasten to a close, remarking that the remaining 

 books treat of planes and lines (VII.) : tri-dimensional 

 spherics (VIII.) ; two-dimensional spherics (IX.); poly- 

 hedrons (.K.) ; mensuration or metrical geometry (XI.) in 

 five chapters, length, area, ratio of a circle to its diameter, 

 measurement of surfaces, space-angles, and the measure- 

 ment of volumes. The work closes with short paragraphs 

 on direction, principle of duality, linkage, and cross-ratio. 

 There are some 234 exercises grouped together at the 

 end, and also interspersed throughout the text. 



There are a few typographical errors and a few slips in 

 statement, and very many novel terms, i.e. to persons 

 who have not read the " Mensuration " referred to above. 

 We understand that the book is about to be published in 

 this country, when geometers will be easily able to pro- 

 cure a copy for an examination, which will not be unat- 

 tended, we believe, with profit. 



The figures are in the main carefully drawn, though 

 some few require correction. 



OUR BOOK .SHELF 

 The Zoolos:ical Record for 18S4. Edited by Prof. F. 



Jeffrey Bell, M.A. (London: John Van Voorst, 1885.) 

 We have to congratulate \.\\^Zoological Record Ksi.0(:\2L\\i:)n 

 on having brought out this the twenty-first volume of the 

 .^scon/within the year. The publication oi the Zoological 

 Record, begun in 1865, was continued from 1871 by this 

 Association, which well deserves every encouragement 

 that the biologist can give to it. It would be a deep 

 disgrace to our British School of Natural History if so 

 valuable a work should be allowed to come to an end 

 after having well and bravely struggled for existence for 

 one-and-twenty years. At present the Association 

 numbers only fifty-three members and seventy-one sub- 

 scribers, in addition to which several of our public libra- 

 ries no doubt take their copies from the publishers ; but 

 to make the Association a self-supporting one, it should 

 have a couple of hundred new subscribers, and such a 

 number ought to be had from among the numerous 

 students of zoology in this country. A vigorous effort 

 now made might mark this year in the history of the 

 Zoological Record 3.S one of financial success. 



While writing thus of the present, our thoughts also 

 wander to the past. .-Although none of the original writers 

 for the Record have gone to their long account, yet with 

 the present volume, the last of them. Prof E. von 

 Martens, ceases from his Record \aho\\rs, and his place is 

 to be taken by four very excellent recruits, — Prof. Herd- 

 man, Messrs. W. E. Hoyle, G. R. Vines, and G. H. 

 Fowler. The only one of the Recorders who kept in the 

 race for the whole of the twenty one years — Prof, von Mar- 

 tens—had all through the great group of the Mollusca to 

 record, and then, on the dropping out of the ranks of 

 other Recorders, he took the MoUuscoida and Crustacea. 

 It is not possible to part with such a contributor without 

 publicly recording the gi-eat debt that all interested in 

 zoology owe to him for his labours. 



One other contributor, also of long standing, now parts 

 company from his comrades. On Mr. Kirby had fallen 

 the parts of Dallas and Rye. No less than Prof, von 

 Marter.s he deserves our thanks. His part we are 



