242 



NATURE 



[August 26, 1909 



The proof of the addition theorem is not very satis- 

 factory. The author proves the theorems for sin (x+y) 

 and cos (.r + y) for acute angles (using a revolving line 

 of unit length, and so denoting the sines and cosines 

 by lengths of lines instead of ratios, which seems a 

 pity), and then says " it is a fact, however, that these 

 formulae hold for angles of any magnitude, positive 

 or negative." This he illustrates by a couple of cases. 

 Then, in the next section, he says " it was shown " 

 that the formulae hold good for all angles. The proof 

 by projection now customary in the best English books 

 would have been much more satisfactory. 



The directions for solving trigonometrical equations 

 are not altogether satisfactory, and would lead to 

 difificulties in the case of such an equation as 

 sin 3.r = cos 4%. This part of the subject would need am- 

 plifying. The similar instructions for proving fdenti- 

 ties, though sometimes leading to rather heavy work, 

 would always lead to success and be useful in the 

 last resort, though not conducive to elegance. 



The solutions of triangles are well explained, and 

 there is a good chapter on the theory and use of 

 logarithms and their applications to nautical and other 

 problems, but one is sorry not to see the value of 

 the characteristic given as the distance of the highest 

 significant figure from the unit's place, plus or minus 

 according as it is to the left or right. It is more 

 fundamental and easier to remember than the old- 

 fashioned method given in the text. 



This part of the book finishes with a discussion of 

 acute angles near 0° and 90°, and a collection of 

 miscellaneous practical examples of the usual type, fol- 

 lowed by a useful recapitulation of formulse, with the 

 pages on which they are proved. 



• The spherical trigonometry assumes some previous 

 knowledge on the part of the reader, as far as the 

 properties of the polar triangle, and one or two of the 

 more advanced formulae are quoted without proof. 

 The chief features of this part are a good exposition 

 of Napier's rules for right-angled triangles, and the 

 use of the exterior angles (a, S, y) in all the formulaj 

 for oblique triangles, a most excellent innovation 

 which the reviewer has advocated for many years, but 

 has never before seen in a text-book. By this means 

 all formulae become dual without any change except 

 the interchance between a, b, c and a, /3, 7. It leads, 

 perhaps, to a preponderance of obtuse angles in the 

 practical applications, but the author in his log- 

 arithmic work, which is most excellently exemplified, 

 disposes of them by the simple device of putting («) 

 to the logarithms of negative quantities, a method 

 often used by practical computers, but not often seen 

 in text-books. The book concludes with applications 

 to astronomical and other problems, well explained 

 and illustrated by good diagrams, with a fair number 

 of e.xamples for the student to solve. 



(2) This volume completes the authors' plan of a 

 course of mathematics for students of engineering and 

 physics. The first chapter discusses infinitesimals, 

 and defines differentials of functions of a single vari- 

 able. Then come chapters on integration, with ap- 

 plications to geometry and mechanics, followed by 

 special methods of integration applicable to partial 

 fractions and trigonometric functions, including the 

 NO. 2078, VOL. 81] 



use of reduction formulae. Chapter viii. deals with 

 simple differential equations, with mechanical and geo- 

 metric examples illustrating their importance. Chapters 

 ix. and x. deal with solid geometry; chapter xi. with 

 partial differentiation ; chapters xii. and xiii. treat 

 of multiple integrals and applications, with carefully 

 drawn diagrams well illustrating the building up of 

 such integrals, in rectangular, polar, and cylindrical 

 coordinates. Then follows an introduction to line in- 

 tegrals and their connection with surface integrals 

 (Stokes's theorem). Chapter xv. is devoted to infinite 

 series, giving the easier tests of convergence, followed 

 by Maclaurin's and Taylor's series and an introduc- 

 tion to Fourier's series, and finishing with the evalua- 

 tion of indeterminate forms. Chapter xvi. contains a 

 short treatment of complex numbers and conjugate 

 functions. The remaining chapters are devoted to 

 differential equations, total and partial. 



The whole book is very solid reading, but the ex- 

 planations are well given, and when proofs are not 

 fully given references are made to other treatises. 

 The intention of the authors evidently is to take the 

 students over as much ground as possible, and intro- 

 duce them to all the functions and processes which 

 they are likely to need in their scientific work. There 

 are numerous problems for solution throughout the 

 book, and there is an index at the end to facilitate 

 reference. A. L. 



ELEMENTARY PETROLOGY AND ORE 

 FORMATION. 



(i) Rocks and Rock Minerals. A Manual of the 

 Elements of Petrology without the Use of the 

 Microscope for the Geologist, Engineer, Miner, 

 Architect, &c., or for Instruction in Colleges and 

 Schools. By L. V. Pirsson. Fifth edition. Pp. 

 vi + 414. (New York: John Wiley and Sons; 

 London : Chapman and Hall, Ltd., 1908.) Price 

 IDS. 61I. net. 



(2) Genesis of Metallic Ores and of the Rocks which 

 Enclose Them. By B. Symons. Pp. xxxiii + 494. 

 (London : The Mining Journal, 1908.) 



THE growing recognition of the economic uses of 

 petrology and the increasing complexity of 

 petrographic methods are rendering necessary the 

 development of a less technical rock nomenclature for 

 use in the field and by general geologists. No one 

 who has acquired sufficient knowledge of petrology 

 to determine the approximate chemical composition 

 and qualities of a rock from a short study of a thin 

 section is likely to discontinue the use of the micro- 

 scope. The increasing number of students of mining, 

 chemistry, engineering, and agriculture who have to 

 study rocks, but have not much time to devote to the 

 subject, is leading to the issue of special text-books 

 on petrography without the microscope, and, thanks 

 to its revelations, much can now be learnt from rocks 

 by the examination of hand specimens. 



(i) Prof. Pirsson 's " Rocks and Rock Minerals " is 

 the most advanced of the manuals of petrology with- 

 out the microscope, but it may be recommended even 

 to students who can use that instrument owing to its 

 clear statement of the principles of petrogenesis and 

 of the mode of occurrence of the sedimentary and 



