APEIL16, 1897.] 



SCIENCE. 



627 



volume is very rich in illustrative examples 

 carefully and admirably adapted to meet the 

 requirements of both teacher and student. In- 

 deed, this feature of the book will be its strong- 

 est recomendation to many practical teachers. 

 In revising the work Professor Miller has placed 

 Chapter VII., which, in the old edition, followed 

 Chapters VIII. and IX. in its logical position, 

 and by thus developing first the Cartesian sys- 

 tem of coordinates has succeeded in making the 

 results of the later chapters general. To this 

 portion of the text other tables might be added 

 to some advantage, for example, one tabulating 

 the values of functions of angles that are multi- 

 ples of five, and one giving an expression for 

 each function in terms of the others. While 

 of undoubted worth as a means of reference, 

 such tables, it might be argued, are of doubtful 

 value from a pedagogic standpoint. At least 

 they should be required to be established by 

 every student of trigonometry, and it would 

 have been well to have required them among 

 the examples. The book is also unique in 

 respect to the absence of the time-honored 

 figures illustrating the positions and compara- 

 tive lengths of the functions other than the sine 

 and cosine in the different quadrants of a circle 

 whose radius has been assumed as the unit. 

 Here, again, the omission may be defended on 

 the ground that nine students out of ten, having 

 these figures for the special case in mind, will 

 carry to the grave the impression that the 

 functions are lines instead of ratios. Neverthe- 

 less, they are of value in mechanical drawing, 

 and especially as affording a ready means of 

 prompting the memory in the thousand and 

 one simple relations they illustrate, and by con- 

 stant emphasizing of the ratio definitions on the 

 part of the teacher they can be used without 

 confusion of ideas. 



The added chapter on inverse functions, and 

 the one that has been much improved on the 

 solution of trigonometric equations, are valuable 

 and essential parts of the new edition, the 

 importance of which will not be underestimated 

 by the advanced student of mathematics. 

 Finally the analytic portion of the plane trigo- 

 nometry is completed by the establishment of 

 the so-called tangent formula, a — 6 : a + 6 = 

 tan J (^A — S) : tan J A + B), by a direct de- 



velopment from the rigidly demonstrated addi- 

 tion theorems. Many of the old treatises prove 

 this formula, probably because of its importance, 

 geometrically, and lose thereby in generality. 

 The geometric proof is of exceeding interest, 

 however, and the present demonstration would 

 be emphasized and the book gain in pedagogic 

 strength were it given in a foot-note with a 

 corresponding valuable reference to its limita- 

 tions. 



In outline the design of the book is to discuss, 

 in the first thirteen chapters, the general theory 

 of the trigonometric functions; then, in chapter 

 fourteen, to give a short review of the theory 

 of logarithms followed by a discussion of the 

 solution of the triangle, and, finally, by two 

 short chapters giving applications to engineer- 

 ing and geometrical problems. This is the 

 time honored arrangement, certainly in the 

 hands of a good teacher sufficiently effective. 

 As our author says, ' 'the discussion of logarithms 

 belongs properly to algebra," and as a rule the 

 student has met with them during the preced- 

 ing term's work, but has, nevertheless, far from 

 mastered their application and still less their 

 theory. Why not then begin at once with a 

 review of the theory of logarithms and insure 

 a thorough mastery of their application by con- 

 stant practice from the beginning ? While the 

 student is learning about angular measurement 

 and the trigonometric ratios the class can be 

 exercised in the evolution of complicated nu- 

 merical expressions and in the solution of loga- 

 rithmic equations, and, as soon as the functions 

 are developed, a large number of the applications 

 to right triangles reserved for the last chapters 

 may be discussed immediately, affording new 

 material for logarithmic work, thus arousing 

 at the start the keen interest of the practical 

 mind. 



The last thought has been acted upon by the 

 author. Pages 10, 11, 26 and 27 are filled with 

 practical problems of a most interesting charac- 

 ter, all of which, however, are intended for 

 solution without the aid of logarithms. 

 Finally Professor Miller has added two short 

 chapters, in which he develops the theory of 

 the solution of spherical triangles. Here, as in 

 the plane trigonometry, the author is fully alive 

 to the limitations of geometric proof and care- 



