702 SCIENCE PROGRESS 



The book gives an account of the structure and physiology of each of the 

 organisms which are generally taken as types for practical study and enables the 

 student to obtain an insight into the relation of the chosen types to one another. 

 The introduction of various allied forms tends also to broaden the student's view. 

 Thus after an account has been given of the structure, etc. of the crayfish, it is 

 compared with the earthworm and the hydra, its position among the Arthropoda 

 is referred to, and an outline is given of the structure and life-history of the gnat 

 as a representative of the Insecta. 



G. A. Freeman. 



Plane Trigonometry. By H. S. Carslaw. [Pp. xviii, 293 + xi.] (London: 

 Macmillan & Co., Ltd., 1909. Price 4^. 6d.) 



It is extremely difficult to write with freshness on such a well-known subject as 

 elementary trigonometry, but Dr. Carslaw in his attempt has achieved a fair 

 measure of success. The first part of the work deals with trigonometry up to the 

 solution of triangles, and upon this portion of the book there is little to remark. 

 The examples on the very elementary parts of the subject are numerous and 

 graphical methods are freely introduced. A knowledge of logarithms is assumed, 

 and four-figure tables of the necessary functions are given at the end. The proofs 

 of the addition theorems for all values of the angles are particularly clear and good ; 

 and the "ambiguous case " in the solution of triangles is clearly set forth. 



The last chapter of Part I. deals with circular measure, and is the first distinctive 

 chapter in the book. It contains an interesting discussion on the length of a 

 curve, introducing the representation of rational and irrational numbers by points 

 along a straight line. The definition of the length of a circular arc follows on that 

 of the " limit of a sequence," and the whole subject is treated in a fresh and 

 interesting manner. 



The first chapter of Part II. deals with the triangle and its associated points, 

 lines and circles, and should be included in Part I. ; Part II. would then begin 

 with De Moivre's theorem and its applications, a usual and suitable arrangement. 

 After the factorisation of algebraic and trigonometrical functions there is a good 

 chapter on inverse notation, a function and its inverse function being well 

 illustrated graphically. In this part of the book, graphs, which can be very easily 

 overdone on the more elementary parts of the subject, serve an admirable 

 purpose as illustrations of the convergence of series. The power series, and 

 the series 



« + «, COS0 + w 2 cos 20 + . . ., 

 u + ui sin 9 + u 2 sin 2 9 + . . . , 



are thus illustrated by figures for which the author acknowledges his indebtedness 

 to Mr. Whipple. This treatment should prove stimulating to the student who is 

 anxious to learn sound methods rather than to work out complex examples in a 

 slipshod way. The introduction of this modern treatment into an elementary book 

 is a step forward and is a justification for a new trigonometry, intended for the use 

 of higher classes in secondary schools and junior students at the universities. The 

 author is careful to indicate, in the series with which he deals, points needing 

 further examination ; while all series which depend upon the use of a complex 

 variable are excluded from the scope of the work. If Dr. Carslaw will give us a 

 Part III., dealing with these series in as fresh and intelligible a manner as that in 



