May 1 8. 1882] 



NA TURE 



59 



picture invariably declines. The physical type, too, of 

 the Babylonian statues from Tel-lo, approaches the 

 Caucasian rather than the Semitic type. 



ON SOME RECENT AMERICAN MATHEMA- 

 TICAL TEXT-BOOKS 



TN Nature (vol. xvi. p. 21) we drew attention to a 

 *■ "shaking" that was taking place among the "dry 

 bones " of the mathematical text-books in common use 

 in American colleges and schools, and upon the analysis 

 we then furnished of a few works before us we ventured 

 to predict a speedy awakening of mathematical life. Our 

 prognostications have been quickly fulfilled, and we now 

 propose to submit an account of five recent books, some 

 of which are quite fitted to hold their own, in our opinion, 

 with English text-books on the same subjects. 



" The Elements of the Integral Calculus, with a Key 

 to the Solution of Differential Equations," by Dr. W. E. 

 Byerly (Boston, iSSr), is a sequel to the volume on the 

 " Differential Calculus," previously noticed by us. This 

 work is founded upon Bertrand's classical treatise, sup- 

 plemented by free use of the allied treatises by Todhunter, 

 Boole, and Benjamin Peirce. The opening chapters give 

 a clear exposition of the use of symbols of operation and 

 of imaginaries. So early an introduction to these sub- 

 jects is novel to us in this connection, but it shows how 

 the subject of quaternions is coming to the front, and the 

 passage from the subjects of these chapters to quater- 

 nions is but a short one The main portion of the book 

 calls for no special comment. In Chapter XIV. we have 

 a treatment of mean value and probability, founded upon 

 the able contributions of Prof. M. W. Crofton, F.R.S., 

 to Mr. Williamson's treatise. 



The novelty of the book is Chapter XV., entitled " Key 

 to the Solution of Differential Equations." This key is 

 based upon Boole's work, and is a collection of concise, 

 practical rules for the solution of these equations. An 

 idea of its form will be best conveyed to some persons by 

 saying that it resembles the analytical key so frequently 

 prefixed now-a-days to handbooks of the British (and 

 other) flora. By a series of references we run the particu- 

 lar equation to ground. Thus, taking the example, 

 (l + x) ydx + (1 - y)xdy = o, it is a single equation, 

 this sends us to a number ; it involves ordinary deriva- 

 tives, this advances us a stage ; it contains two vari- 

 ables, is of the first order, and finally of the first degree. 

 The upshot is we arrive at the form Xdx-\- Ydy = o, 

 under which head we learn how to solve the equation. 

 Under this last head, as throughout the book, are given 

 numerous illustrative exercises for practice. 



Dr. A. S. Hardy's "Elements of Quaternions" (Bos- 

 ton, 1 SS 1) is intended to meet the wants of beginners. In 

 addition to the works of Sir William R. Hamilton and 

 Prof. Tait, the author has consulted the memoirs or works 

 of Bellavitis (" Calcolo dei Quaternione" and the "Ex- 

 position de la Methode des Equipollences " in Laisant's 

 translation); Hoiiel's " Ouantites Complexes ; " Argand's 

 "Essai" (1806); Laisant's "Applications me'caniques 

 du Calcul des Quaternions," and one or two other books 

 and papers in the American Journal of Mathematics, vol. 

 '• P- 379- It is a good introduction to such a work as 

 Prof. Tait's, the originality and conciseness of which, 

 however, Dr. Hardy thinks to be "beyond the time and 

 need of the beginner." 



Our next book is "An Elementary Treatise on Men- 

 suration," by G. B. Halsted (Boston, 1881). Dr. Halsted 

 is already known to mathematicians here as the author of 

 a very full " Bibliography of Hyper-space and non- 

 Euclidean Geometry," in the American Journal oj 

 Mathematics, vol. i., Nos. 3, 4. This treatise on Metrical 

 Geometry is " the outcome of work on the subject while 

 teaching it to large classes," so that it is no hastily pre- 

 pared book, but has been founded on actual teaching 



experience. The methods have a German " smell," and 

 this is justified by the author's residence, we presume as 

 a student, at Berlin. There are eight chapters : (1) on 

 the measurement of lines (triangles, method of limits, 

 rectification of the circle ; (2) on the measurement of 

 angles ; (3) of plane areas ; (4) of surfaces (he uses 

 Mantel for lateral surfaces, also Steregon and Sieradian 

 in connection with a solid angle) ; (5) of volumes (Quader 

 is new for De Morgan's "right solid''). In these last 

 two chapters the solids discussed are the prism, cylinder, 

 pyramid, cone, and sphere ; an article is also devoted to 

 Pappus's theorem. (6) The applicability of the prismoidal 

 formula; (7) approximative methods, as Weddle's me- 

 thod ; (8) on the mass-centre, with a paragraph on the 

 mass-centre of an octahedron, which gives Clifford's con- 

 struction (see Proc. Lond. Math. Soc, vol. ix. p. 28). 

 There are numerous exercises, these we have not tested. 

 The book is most effectively " got up," the printing, 

 figures, and paper being, to our mind, excellent. 



Our last two books are by Prof. Simon New-comb, so 

 well known as the author of " Popular Astronomy." The 

 first, "Algebra for Schools and Colleges" (New York, 

 1881), has already reached its second edition. It is a 

 capital book ; indeed we are disposed to rank it as the 

 best manual on the subject that we have seen for school 

 purposes. It is divided into two portions, " the first 

 adapted to well-prepared beginners, and comprising about 

 what is commonly required for admission to colleges, 

 and the second designed for the more advanced general 

 student." We shall perhaps best serve the end we have 

 in view in noticing this work by giving an analysis of the 

 author's preface. The principles of construction are (1) 

 that an idea cannot be fully grasped by the youthful 

 mind unless it is presented in a concrete form. Hence 

 numerical examples of nearly all algebraic operations and 

 theorems are given — so numbers are preferred to literal 

 symbols in many cases. The relations of positive and 

 negative algebraic quantities are represented by lines and 

 directions at the very earliest stage. " Should it appear 

 to any one that Jwe thus detract from the generality of 

 algebraic quantities, it is sufficient to reply that the system 

 is the same which mathematicians use to assist their 

 conceptions of advanced algebra, and without which they 

 would never have been able to grasp the complicated re- 

 lations of imaginary quantities." Principle (2) is that all 

 mathematical conceptions require time to become en- 

 grafted upon the mind, and the longer, the abstruser they 

 are. " It is from a failure to take account of this fact, 

 rather than from any inherent defect in the minds of our 

 youth, that we are to attribute the backward state of 

 mathematical instruction in this country, as compared 

 with the continent of Europe." Prof. Newcomb considers 

 the true method of meeting this difficulty is to adopt the 

 French and German plan of teaching algebra in a broader 

 way, and of introducing the more advanced conceptions 

 at the earliest practicable period in the course. A third 

 feature is the minute subdivision of each subject, and the 

 exercising the pupil on the details before combining them 

 inta a whole. This remark especially applies to the solu- 

 tion of the exercises. Some subjects have been omitted 

 (as G.C.D. of polynomials, square roots of binomial surds, 

 and Sturm's theorem), as they have no application "in 

 the usual course of mathematical study, nor advance the 

 student's conception of algebra," and in studying them 

 there is a waste of power. "Thoroughness" has been 

 our author's aim rather than " multiplicity of subjects." 

 There are other points of interest in this preface which 

 show that the writer is a very experienced teacher, and 

 which we commend to the consideration of teachers here, 

 but we must pass on to indicate the contents of the two 

 parts. 



Part I. embraces algebraic language and operations, 

 equation?, ratios and proportion, powers and roots, equa- 

 tions (quadratic), progressions, seven books in all. 



