November 19, 1908J 



NA TURE 



65 



these functions and of square roots at the end of 

 tlie book. There is an excellent collection of ex- 

 amples, many of which are of a practical type, and, 

 therefore, in themselves more interesting than the old- 

 fashioned academic questions. The proofs of formula 

 and methods are in some cases somewhat concise, 

 and would need amplification by the teacher. In par- 

 ticular, the proof of the binomial theorem and the 

 explanation of the method of finding square roots 

 are of this character. In some cases the author 

 adopts the heuristic method, and requires the student 

 to derive formula for himself, as, for instance, in 

 finding the factors of .v' + y' and .ir''-hy" + c' — j.tvc;, 

 ;iiid in finding the meaning of fractional indices and 

 the values of logarithms. 



We should like to see less formal methods of finding 

 the H.C.F. of two algebraic expressions, based on 

 the fact that R = aA — bB contains the common factors 

 of A and B, as in many cases R can be factorised 

 and the common factors detected with much less 

 trouble and with a more direct appeal to common 

 sense than by the formal method. The chapter on 

 factors is very fully and carefully done, and this 

 method would be a natural sequel. 



The use of graphs is well exemplified and illustrated 

 by a good number of examples. The chapters on 

 ratio, proportion, and variation are good, especially 

 in the selection of interestintf practical examples. 



(2) Messrs. Barnard and Child have made a brave 

 attempt to give a log^ical development of algebra in 

 ;i form suitable for school work. They explain the 

 meaning of the laws of association, commutation, 

 and distribution as applied to addition, subtraction, 

 multiplication and division, and lead up to the solu- 

 tion of what are to a beginner quite diflicult prob- 

 lems. 



Negative numbers are not considered until part ii. 

 (p. 149), and are there explained by extending the 

 scale of natural numbers backwards. In this part 

 some uf the difficulties would seem to be too delicate 

 for tlie com])rehension uf ;i beginner, as, for example, 

 the distinction between 2 + { — 3) and ( — 3)-t-2. Every 

 teacher must, of course, use his judgment as to how 

 far to press such niceties. The explanations are 

 carefully given throughout, and the collection of ex- 

 amples is excellent. The method of factors is applied 

 to finding the H.C.F. and L.C.M. of a set of ex- 

 pressions ; in fact, one special and excellent feature 

 of the book is its early introduction to factorisation. 

 \> soon as a boy can factorise with facility, the e.x- 

 prissions have a form and interest to him which they 

 did not possess before. Fractions are introduced in 

 part iii., and theorems on equal fractions (the authors 

 avoid the use of the term " ratio ") are given in 

 chapter .xx. Graphs are introduced in chapter xxiii., 

 and illustrated by useful examples. The book ends 

 with quadratic equations and problems leading to 

 them, followed by a useful series of test papers. We 

 look forward with interest to the appearance of the 

 second volume. 



(3) This is a book on the model of Todhunter, with 

 the re-adjustments and improvements in methods of 

 proof which modern requirements demand. Proofs of 



.NO. 2038, VOL. 7()] 



index laws and of the binomial and exponential 

 theorems are given for all commensurable numbers, 

 and are assumed, perhaps, however, somewhat too 

 silently, to hold for incommensurables also, the author 

 evidently considering it wise to postpone a rigorous 

 treatment of incommensurables. Indeterminate equa- 

 tions of the first degree are introduced early, and 

 clearly illustrated by well-drawn graphs. There are 

 good chapters on permutations and combinations, and 

 on the simpler tests of convergency and divergency of 

 series, and the chapter on miscellaneous graphs forms 

 a useful introduction to curve-tracing. Continued 

 fractions and probability are not treated. The book 

 is e.xcellently printed, and there are a good number 

 of examples attached to the various chapters, but 

 rather a scanty supply of miscellaneous examples 

 (100) at the end of the book. This could be remedied 

 in a subsequent edition. 



(4) This consists of a collection of examples up to 

 quadratic equations, prefaced by a set of specimen 

 examples worked out, to secure uniformity of method, 

 and concluding with miscellaneous examples arranged 

 in short sets, and graduated in difficulty, so as to 

 test a boy's knowledge at various stages of progress. 

 There is an extensive collection of graphs, each of 

 which is accompanied by useful instructions as to 

 scale. No bookwork is given, as it is considered 

 that, in the early stages of algebra, all explanation 

 must be left to the teacher. 



OUR BOOK SHELF. 



Agriculture for Southern Schools. By J. F. Duggar. 



Pp. xi + 3(i2. (New York : The Macmillan Co. ; 



London : Macmillan and Co., Ltd., igo8.) 



Price 45. 6d. 

 This little book has been written, the author tells 

 us, as an elementary text-book on agriculture that 

 shall differ from others in having a definite and 

 limited field — the southern States of the United States. 

 While the principles underlying the subject are uni- 

 versal, their applications vary much in different locali- 

 ties, and by confining attention to a particular area it 

 becomes possible to present the subject in the concrete 

 way essential for beginners. The crops, soil condi- 

 tions, and general facts of cultivation are all within 

 the experience of the scholar or his farmer friends, 

 and he is not likely to be troubled with that bete rtoir 

 of the agricultural teacher — the principle that is per- 

 fectly sound in itself, but not economical, and there- 

 fore not applicable, in the particular district. 



.\ perusal of the book shows that the author has 

 succeeded in his somewhat difficult task. The 

 essential principles are well brought out, and the 

 illustrations are to the point. The opening chapters 

 deal with the structure of the flower and seed- 

 formation : peach, cotton, tobacco, and others being 

 chosen as examples. The conditions necessary for 

 germination and plant growth are then discussed, and 

 the author next passes on to the highly important 

 problem of adapting the crop to the soil. So much 

 money has been lost in the past through planting 

 crops unsuited to the soil that the author does well 

 to direct attention to this matter; he points out, for 

 instance, that sandy soils are best cropped with early 

 vegetables, peaches, cotton, pea-nuts, water-melons, 

 &c., while clay soils are better for hay crops, apples, 

 &c. A few chapters are then devoted to manures, and 



