NA TURE 



[December 5, 1901 



beautifully illustrated article on medical ophthalmology, 

 by Ur. James Taylor, and one on the medical appli- 

 cations of Electricity, by Dr. Bertram Abrahams. 



Dr. Allchin's third volume is, in our opmion," highly to 

 be recommended. We know of no book in the language 

 upon this subject which will be more « orth the student's, 

 and indeed the practitioner's, while to read and to possess. 



K. W. T. 



PRACTICAL MA THEM A TICS. 

 Practical Mathematics for Beginners. By Frank Castle, 

 M.I.M.E. Pp. ix + 313. (London: Macmillan and 

 Co., Ltd., igoi.) Price zs.bd. 



THIS little book deserves the title of Practical 

 Mathematics better than any work that we have 

 seen. The subjects dealt with are arithmetic, plane 

 geometry, algebra, mensuration and analytic geometry. 

 The chapters on arithmetic deal with those operations 

 in which this subject is most nearly related to algebra- 

 such as the theory of fractions, ratio and the extraction 

 of the square root. The part on geometry is strictly 

 limited to constructions with rule, compass, &c., and 

 explains the use of simple and diagonal scales ; it is in 

 nj sense a course of deductive geometry such as we have 

 in the books of Euclid. The part of the book dealing 

 with algebra is more extensive, but still very elementary ; 

 it does not, for example, include a discussion of quadratic 

 equations, although it shows how a quadratic expression 

 in X can, in very simple cases, be resolved into factors. 

 While noticing this part of the book we may point out 

 some corrections which should be made in the next 

 edition. Thus, in p. 76, where it is proposed to resolve 

 ;r2 -gijr + 20 into factors, we find the statement, " Hence 

 jr = 4, or ;r-4 = o is a factor." The beginner should be 

 put on his guard against such a loose mode of expression. 

 In the next example on the same page we find, " Next 

 put -r = + 5, and it is found to be a factor." The factor 

 referred to is .v-5. In p. 77 we have the incorrect ex- 

 pression, " When required to add, subtract or compare 

 fractional expressions, it is necessary that they shall all 

 have a common denominator." In p. 88, ;' is described 

 as 32'2 " feet per second " instead of 322 " feet per second 

 per second," which the majority of mathematicians have 

 at last been forced to acknowledge as the only correct 

 mode of speaking. 



These, however, are minor blemishes which are very 

 easily removed. 



It is a cardinal aim with the author to make all his 

 examples illustrative of (|uestions relating to various 

 branches of physics, and for a certain class of students 

 (those who have already come into contact with such 

 practical matters) this is a very good plan, because it 

 enlists the interest of the learner in convincing him that 

 he is applying his mathematics to something real. It is 

 doubtful if the plan has as much value for the ordinary 

 schoolboy who is, under our precious system of education, 

 a complete stranger to everything in the domain of 

 physics. Hence such questions as that in example 5, p. 88, 

 relating to the arrangement of a number of Grove's cells, 

 will not convey much meaning to any but students of 

 physics. There arc useful little chapters on logarithms, 

 NO. 1675, VOL. 65] 



showing their use and illustrating several things in which 

 beginners are very apt to make mistakes. After this we 

 come to an explanation of the slide rule and its applica- 

 tions ; and the remainder of the book is that which most 

 entitles it to the name of Practical Mathematics, this 

 portion being of value to the student who wishes to be 

 able to apply his pure mathematics to the representation 

 of physical results. Here there is a great deal of graphic 

 work done by means of squared paper, and a considerable 

 portion of the analytic geometry of right lines, circles and 

 higher curves is expounded, the accompanying illustra- 

 tions being all drawn from physics. The fundamental 

 notions of the differential calculus are very well and 

 simply explained by this same system of plotting on 

 squared paper ; and the ease with which the processes 

 can be followed and understood even by beginners who 

 have nothing but a knowledge of arithmetic and 

 elementary algebra to go upon shows that, in our 

 ordinary course of mathematical teaching, the differential 

 calculus is very unnecessarily postponed — that, in other 

 words, our mathematical course for beginners should be 

 made eclectic in character, a portion of any subject being 

 introduced when the mind of the student is in a state to 

 understand it. Our present system is essentially different ; 

 we feel constrained to finish each subject before beginning 

 another, although the finish of one subject may be much 

 more difficult than the preliminary portion of that which 

 is postponed ; and we thus lose sight of the fact that our 

 present divisions of mathematics are only artificial, and 

 that mathematics is, in reality, one connected whole. 



In the part of the work dealing with mensuration two 

 planimeters are described — the Hatchet and Amslers. 



The work gives an excellent epitome of the various 

 branches of mathematics dealt with, and it will serve as a 

 store of very good exercises in elementary methods for 

 all students who desire to make a practical use of their 

 mathematical knowledge in picturing the relations 

 between various physical quantities. 



OUR BOOK SHELF. 



Memorial Lectures delivered before the Chemical Society, 

 ■893-1900. Pp.560. With fourteen portraits. (London: 

 Gurney and Jackson, 1901.) Price Ts. 6d.. 

 The Chemical Society has done an important service to 

 chemists and to students of chemistry by collecting these 

 memorial lectures into one volume, and issuing it under 

 conditions which render it accessible to readers of whom 

 some may not be Fellows of the Society and conse- 

 quently have not enjoyed the advantage of hearing the 

 lectures when delivered or of reading them in the pages 

 of the Transactions. 



The lives of the men whose work and achievements 

 are commemorated in this volume link us with the now 

 long-distant past, and remind us of the immense strides 

 which have been made in consequence of their discoveries 

 and the discoveries of their contemporaries since the 

 days when Berzelius and, later, Liebig were the dominant 

 authorities. They remind us of the great and almost 

 sudden advance which was accomplished between 1850 

 and 1865, when the modern system of atomic weights, 

 definite ideas of valency and constitutional formuUt were 

 finally established. The student who aspires to under- 

 stand by what methods and with what laborious etTort 

 the greatest degree of scientific accuracy is alone attain- 

 able must read about the work of Stas on atomic 



