28 



NATURE 



[Nov. lo, 1887 



the student who desires to attain a clear, logical view of 

 the foundations of algebraical science. The remaining 

 sections demand less comment as presenting less novelty 



of treatment. 



The most noticeable feature in the section on equations 

 is the introduction of the notation, and a discussion of 

 some of the properties, of determinants. We cannot but 

 regret the space that is devoted in this section to the dis- 

 cussion of the processes for the extraction of square and 

 cube roots in the old traditional shape. In the chapter 

 on division the law of formation of the quotient and 

 remainder of a rational integral function of x after division 

 by x-a has been established. Starting from this, a dis- 

 cussion involving nothing more than elementary con- 

 siderations would lead up to Horner's process in all its 

 generality, which might then be exemplified in the 

 extraction, not only of square and cube roots, but of roots 

 of any degree, both for algebraical expressions and for 

 numbers. How long shall we have to wait for a due 

 recognition in elementary treatises of this comprehensive 

 method, which, whether from a theoretical or from a 

 practical point of view, is one of the most valuable results 

 of a study of algebra? 



The section on series commences with the establish- 

 ment of the usual formulas for permutations and combi- 

 nations, as preliminary to the binomial theorem. We 

 observe with satisfaction that the path to the proof of each 

 general formula is smoothed by the prior discussion of a 

 particular case, by which, as every good teacher knows, 

 the principle involved may be more distinctly brought out 

 than in the general proof, where it is too likely to be lost 

 sight of in the generality of the symbols employed. To 

 the chapter on geometrical progression is attached, as we 

 think it always should be, one of its most important 

 applications-namely, that to compound interest and 

 annuities. The chapters on the binomial theorem and 

 other series usually discussed in elementary algebra are 

 clear and satisfactory, though we think a httle more 

 prominence should have been given to the distinction of 

 convergent and divergent series, and a fuller discussion o. 

 the tests of convergency and divergency. 



The last section includes under the general heading of 

 "Arithmetical Applications," chapters on proportion, 

 continued fractions, indeterminate equations (limited to 

 those of the first degree), inequalities, notation and 

 numbers, and probabilities. It is not intended, we presume, 

 that the study of some at least of these should be post- 

 poned till after the study of the previous sections, but that 

 as Applications they do not form a necessary part of the 

 general sequence of algebraical results, though it appears 

 to us rather strange that a place for proportion, at any 

 rate, should not have been found in such sequence. 

 ' R. B. H. 



PRACTICAL BOTANY. 

 Practical Botany. By F. O. Bower and Sydney H. Vines. 

 Part 11. (London: Macmillan and Co., 1887.) 



ABOUT twelve or fourteen years ago there occurred 

 in England two events which have had so marked 

 an influence on the development of scientific botany in 

 this country that they are likely never to be lost sight of 



by our younger school of morphologists and physiologists. 

 One of these events was the introduction into this country 

 of the teaching of Prof. Sachs, of Wurzburg ; the second 

 and even more important one was the institution by Mr. 

 Thiselton Dyer of a course of botanical instruction at 

 South Kensington on a scale never before attempted. 

 Those who had the good fortune to attend Mr. Dyer's 

 courses of practical botany in the well-known laboratory 

 at what is now the Normal School of Science must 

 always carry with them the stimulating remembrance of 

 the thorough teaching there instituted ; and the effect of 

 the exact instruction and inspiriting demonstration so 

 efficient at South Kensington can be obviously traced in 

 the excellent teaching and work of the enthusiastic younger 

 botanists of to-day. The influence took effect on the early 

 development of the present productive botanical labora- 

 tories at Cambridge and elsewhere, and the stimulus has 

 since radiated thence in all directions, as is shown not 

 only by the numerous publications of the last eight or 

 ten years, but also by contributions to the new journal. 

 The Annals of Botatty, just published by the Oxford 

 Clarendon Press, and by the activity and discussions of 

 the botanists at the recent brilliant meeting of the British 

 Association in Manchester. 



The salient features of the new course of structural 

 botany were the thorough study of leading types of 

 the vegetable kingdom by means of material dissected 

 and prepared by the students themselves, and the stress 

 laid on the rule that the students should carefully draw 

 what they saw, and thus gather their ideas at first 

 hand. The method was similar to that employed by 

 Prof. Huxley in his course on animal morphology. 



It is evident that the little hand-book of practical 

 botany now under review is the outcome of experience 

 gained in continuing this important method of instruction. 

 Part I. of the present work was published two years 

 ago, and dealt with selected types of the Vascular plants. 

 Part II. is now before us, and completes the scheme. It 

 comprises studies of the chief types of lower Cryptogams, 

 from the moss downwards. 



We have already pointed out that the essential features 

 of the new teaching are the exact and thorough study of 

 types. Nothing is assumed ; but the students are urged 

 to see everything for themselves, and to draw all they see. 

 These important points decide the plan of the work under 

 review. It will be found an excellent and trustworthy guide 

 to any who use it with the types at hand : it will be all but 

 useless to the mere crammer, for there are no illustrations 

 to take the place of actual objects in producing impressions 

 on the student's mind ; no lengthy descriptions to inter- 

 fere with the directness or clearness of the impressions ; 

 and no classified " tips " to vitiate and confuse the teach- 

 ing. We regard it as an excellent sign of the progress of 

 botany in this country that an English work of this de- 

 scription should be forthcoming, and students are greatly 

 indebted to Dr. Vines and Dr. Bower, and those who 

 have contributed to this admirable little monument of 

 practical teaching. 



As special features in the book we may commend the 

 selection of types ; they are good, for the most part 

 easily obtained, and well known. The treatment of the 

 types in the book is clear, concise, and yet sufficient. The 

 usage of bolder lettering for the chief word in the para- 



