626 



SCIENCE. 



[N. S. Vol. V. No. 120. 



phosphorus, srlicon trichlorid and many othei- 

 substances. It is a question whether this is 

 justifiable. It seems irrational to put rhombic 

 and monoclinic sulphur in one class and the 

 two modifications of phosphorus in another ; 

 but it is certainly interesting, and the applica- 

 tion of his theory to the point of reaction, to re- 

 action velocities and to explosions deserves 

 careful attention. It will interest many to note 

 that Duhem's view of a mixture of hydrogen 

 and oxygen as being actually in equilibrium at 

 low temperatures is not reconcilable with the 

 Ostwald-Nernst idea that it is not a case of 

 equilibrium at all, but rather of immeasurably 

 low reaction velocity. A fairly strong argu- 

 ment can be made out for either view, and the 

 scientific world owes thanks to Duhem for mak- 

 ing the question a live one. 



Wilder D. Banckoft. 



Trigonometry for Beginners. Rev. J. B. Lock, 

 M. A. Revised and enlarged by John A. 



MiLLEK, A. M., Indiana University. New 



York, The Macmillan Company. 1896. 200 



pp. Price, $1.10. 



Trigonometry, of all elementary branches of 

 mathematics, might easily substantiate its right 

 to be considered the most congenial and popular 

 subject that necessarily claims the attention of 

 engineers and practical men, otherwise but little 

 inclined to sympathize with the purest in the 

 science. Led on by its numerous and interesting 

 applications, many a student, without being 

 aware of it, has taken his first step in the theory 

 of functions, acknowledged the results to be fas- 

 cinating as well as eminently practical, and gone 

 his way to rail at the higher theory, quite un- 

 conscious of the spectacle he thereby makes of 

 himself. The natural result of this favoritism 

 has been a steady improvement in the quality 

 of the text-books produced in trigonometry 

 until such works as that of Chauvenet and, 

 to mention a less ambitious book, that of Wells, 

 challenge competition successfully for a series 

 of years. 



With a new edition of Lock's Trigonometry, 

 -The Macmillan Company enters the field, and 

 with its usual business sagacity have secured 

 its revision by an American. Professor Miller 

 has certainly earned the right to have his name 



on the cover, indeed, because of additions and 

 improvements far less necessary and funda- 

 mental many a man would have called the 

 volume his own. As claimed, the new edition 

 corrects the fundamental weakness of its prede- 

 cessor by carefully emphasizing the necessity 

 for proofs for all relations, especially for the 

 addition formula, that are rigidly correct for all 

 values of the angles involved. To this end, as 

 is necessary, we find clear demonstrations of 

 such relations as, for example, sin (90° -\- A) = 

 cos A for all values of A. This enables the 

 author in §79, while attempting to generalize 

 the addition formula, to write, 

 sin [90°+(^' + B)]= cos(^' + B), 0<^'< 



90°, 0<B<90°. 

 He then remarks since A' and B are now both 

 less than 90° we may write 

 cos {A' -\- B) = cos A' cos B — sin A' sin B, 

 ( I 76). 

 There is certainly a flaw in the general ac- 

 curacy of the argument up to this, the 

 pivotal point of trigonometric analysis, for 

 g 76 does not completely justify this ex- 

 pansion, since the demonstration to which 

 reference is made depends upon a figure 

 representing both A' and B, as in the first 

 quadrant, and their sum also as in the first 

 quadrant, whereas, in the case before us, we 

 have no means of knowing whether A' -{■ B is 

 greater or less than ninety degrees. In fact, 

 before analytic demonstrations for particular 

 cases can be accurately defended, geometric 

 demonstrations must be given for 0° < J. < 90°, 

 0° < £ <90° for the four possible cases under 

 this head, namely, 

 (4 + B)<90°, {A + B)>QO°, {A — B)yO°, 



(4-B)<0°. 

 On the other hand, it must be admitted that the 

 author, in paragraph seventy-eight, calls atten- 

 tion to the fact that a similar construction will 

 apply to all possible cases, and even gives a 

 third example, different from either, of those 

 that are necessary. The criticism is, however, 

 that, since up to this point unusual effort has 

 been made to demonstrate the addition theo- 

 rems, this is certainly not a good place to leave 

 necessary steps to the student. 



Like most English text-books, the present 



