January 6, 1910] 



NA TURE 



275 



Shaler returned from it to find even neutral Kentucky 

 divided witiiin itself; and soon, though the chief 

 actor tells us nothing of it, "Shaler's battery " became 

 known upon the Union side. 



Shaler married in 1862 ; but his wife refers us to his 

 various writings on the war, and quotes ver}' little 

 from letters written to her from the field. The ad- 

 vance of Rosecrans on Nashville left Kentucky outside 

 the cra^h of armies, and no echo reaches us of the 

 bitter days round Chattanooga. It is of far more 

 importance to Mrs. .Shaler to record — and this was 

 probablv the feeling of all who knew her husband — 

 that in 1864 Shaler was appointed assistant in palse- 

 ontology in Harvard University. In iS6c) he became 

 professor of this subject, when only twenty-eight ; then 

 he wa^, made professor of geology; and in 1891 he w'as 

 chosen as Dean of the Lawrence Scientific School. 

 Even his position from 1874 to 18S0 as Director of the 

 Geological Survey of Kentucky did not break his con- 

 nection with the development of Harvard. For more 

 than forty years, down to his death in igo6, he was 

 one of the most familiar figures in the courts of the 

 university. 



Chapter xi.K., which deals with a visit to England, 

 contains characteristic mention of Tyndall, and of 

 several English geologists. Here, as in other places, 

 some proper names have gone astray. The Rev. Mr. 

 Simons of p. 256 — have we not made the same rural 

 journey to enjoy a meeting at his gate? — must surely 

 be the late W. S. Symonds, the friend of all naturalists 

 in the Midlands. Elsewhere we have " Renivier," 

 "Guinitz," " Geoffrys " for Jeffreys, and " Marais," as 

 we may presume, for "Marey." These slips result 

 from copying out of diaries, where the incidents of the 

 day have been set down. The incidents thus recorded, 

 page after page, seem rarely of value in themselves ; 

 yet it is clear that we may end this notice much as 

 we began. To those for whom the book is written, 

 those whom Shaler had helped or stood by as a friend, 

 nothing about Shaler will seem unworthv to be ex- 

 pressed. Grenville a. J. Cole. 



MATHEMATICAL TEXT-BOOKS. 

 (i) Geometry for Beginners. By C. Godfrey and 

 A. W. Siddons. Pp. x-l-79. (Cambridge: University 

 Press, 1909.) Price is. 



(2) The School Geometry. Parts i. and ii. By W. P. 

 Workman and A. G. Cracknell. Part i., pp. viii 4- 

 248. Part ii., pp. viii 4- (233-383). (Cambridge: 

 University Tutorial Press, Ltd., 1909.) Price 2S. 

 each part. 



(3) Coordinate Geometry. By H. B. Fine and H. D. 

 Thompson. Pp. viii 4- 300. (London: Macmillan 

 and Co., Ltd., 1909.) Price 6s. 6d. net. 



(4) Exercise Papers in Elementary Algebra. By the 

 Rev. E. M. Radford. Pp. viii4-ii2. (London: 

 J. M. Dent and Co., 1909.) 



(5) Problem Papers in Mathematics. By R. C. 

 Fawdry. Pp. vii4-24o. (London: Macmillan and 

 Co., Ltd., igog.) Price 4s. 6J. 



(i) TN the light of the experience gained in the 

 -L last eight years or more, it is now possible 

 to estimate, with considerable accuracy, the effect of 

 NO. 2097, VOL. 82] 



the numerous changes which have been made in the 

 methods of teaching elementary geometry. The 

 circular issued by the Board of Education last March 

 contains a report on this subject which is well worth 

 careful perusal. The central feature of the modern 

 movement has been an attempt to familiarise the pupil 

 with the fundamental concepts by experimental 

 methods, before providing him with formal proofs. 

 It is now suggested that this experimental stage, by 

 being made more systematic, should replace the first 

 part of the present deductive course. The properties 

 of parallel lines and congruent triangles possess a 

 characteristic which pertains to few, if any, of the 

 later theorems. Once a pupil clearly apprehends their 

 significance, which is possible only by experimental 

 work, he is convinced with absolute assurance of their 

 truth ; and this very fact only serves to increase the 

 difficulties which surround the formal proof. In the 

 words of the circular, 



"to commence the subject by proving what seems to 

 need no proof is a safe way to make boys think that 

 the whole subject is artificial and unreal. It is much 

 better to begin Euclidean, that is, deductive proofs 

 at the point where their necessity can be appreciated 

 — that is after these fundamental propositions — and 

 where, therefore, the proof is a natural process, not 

 subject to arbitrary or artificial rules." 



If then these base-theorems are incorporated in the 

 experimental stage, and if at the end of this course 

 those fundamental concepts, which have been thereby 

 assimilated, are allowed to be assumed without formal 

 proof, the course of deductive geometry w-ill open with 

 the properties of areas of triangles and parallelograms, 

 and continue with theorems on the circle. In this 

 way, at the end of his first year, a pupil will have 

 covered as much ground as at present is covered, in 

 the majority of cases, only after two or three years. 



The present excellent little volume has been com- 

 piled to cover the complete experimental course out- 

 lined above, and it follows in every respect, save one, 

 the suggestions made by the Board of Education. It 

 is, however, advised in the circular that riders should 

 be excluded entirely from the experimental stage ; 

 although in this way time may be saved and greater 

 emphasis placed upon the fundamental theorems, yet 

 simple riders so frequently serve to illustrate a theorem, 

 and, moreover, form a valuable introduction to the 

 future deductive course, that we are firmly convinced 

 that the authors are right in inserting a large number 

 of easy deductive examples in the text. We have 

 developed these considerations at some length, because 

 we consider that the change now advocated is likely 

 to exert a profound influence on the teaching of 

 geometry, and that the more it is considered in all 

 its bearings, the more advantageous it will appear. 



(2) This is an abridged edition of the authors' work, 

 entitled "Geometry: Theoretical and Practical," pub- 

 lished about two years ago. In the present book a 

 certain amount of theory which may be fairly con- 

 sidered to be beyond the range of the average school- 

 boy has been omitted. The sequence adopted in the 

 theoretical course is that of tlie Cambridge syllabus. 

 Part i. contains the substance of Euclid books i., iii. ; 

 part ii. that of books ii., iv., vi., together with those 



