74 



na ture 



\May 2 2, 1879 



a comma to represent the sign of multiplication in 

 almost every equation, invests the expressions with a 

 hopeless obscurity ; what for instance would a young 

 student fresh from a little algebra make of the expres- 

 sion (p. 30), cos PC = cos PA, cos CA + sin P A, 

 sin CA, cos PA C whence PC? Mr. Milne's editors 

 and not Mr. Milne are of course to blame for this, 

 though most of the other mistakes alluded to are his 

 own. 



In a chapter on the projection of poles by the stereo- 

 graphic method Mr. Milne gives a proposition for finding 

 a pole [he means the projection of a pole] at given angular 

 distances from two poles lying on the circle of projection, 

 which is only a special and simple case of the more 

 general problem. The description of the process is 

 entirely " unintelligible. If, however, the meaning be 

 puzzled out from the figure it would seem that Mr. Milne 

 is proposing a construction simple and ingenious, although 

 to obtain it he has to combine the orthographic and 

 stereographic projections. His editors might have saved 

 him from using the expression " two half-hemispheres " 

 on p. 38, if not also from the statement that the mono- 

 symmetric or monoclinic system can present eight faces 

 for a single form. The chapter on crystallophysics is very 

 unsatisfactory; after one's expectations of somewhat 

 transcendental physics have been raised by being told 

 that for the correlation of the phenomena produced by 

 crystals with crystal-structure, " the most valuable hypo- 

 thesis would probably be that of molecular vortices," one 

 is certainly surprised to be told that in the orthorhombic, 

 monoclinic, and triclinic systems "there are two optic 

 axes or directions of double refraction" — or again, that 

 " sections in triclinic crystals cut perpendicularly to the 

 optic axes when viewed in a polariscope show a series of 

 rings round each axis. Between the axes these are 

 drawn together and may meet to form a lemniscate." 

 One is inclined to ask whether Mr. Milne has a distinct 

 idea as to what a lemniscate curve is, and how he cuts 

 the section presenting these phenomena ? 



In speaking of heat conductivity again, the author 

 places the rhombohedral and orthorhombic systems 

 together in one category, and the tetragonal system in 

 another. The errors, often arising in carelessness but 

 sometimes in ignorance, to which these criticisms apply, 

 have been selected merely at random. It has been 

 necessary, however, to make these criticisms in the in- 

 terest of the student, who might be repelled from a subject 

 when he finds what should be a simple statement appa- 

 rently untrue or unintelligible, whether on account of 

 misprints or of obscurity in the language, in the thought, 

 or in the author's method of demonstration. But having 

 performed this duty to the student of a beautiful but much 

 neglected science it would be ungenerous to a teacher in 

 far Japan, not to point out that it is still within his power 

 by recasting his Uttle volume to fill a decided gap in our 

 elementary scientific literature. He has the courage and 

 the ability, he needs only a little more familiarity with 

 the subject, a good deal more caution, and perhaps 

 somewhat more of modesty, to enable him to fulfil 

 the not very ambitious purpose he laid down for him- 

 self when he sent his little work to be published in 

 England. 



N. S. M. 



MATHEMATICAL PROBLEMS 



I. Mathetnatical Problems on the First and Second 

 Divisions of the Schedule of Subjects for the Cam- 

 bridge Mathematical Tripos Examination. Devised 

 and arranged by Joseph Wolstenholme, M.A. Second 

 Edition, greatly enlarged. (London : Macmillan and 

 Co., 1878.) 



II. Solutions of the Cambridge Senate-House Problems 

 and Riders for the Year 1875. Edited by A. G. Green- 

 hill, M.A. (Same Publishers, 1876.) 



III. The Same for the Year 1878. Edited by J. W. L. 

 Glaisher, F.R.S. (Same Publishers, 1879.) 



IV. Graduated Exercises in Plane Trigonometry. Com- 

 piled and arranged by J. Wilson, M.A., and J. R. 

 Wilson, B.A. (Same Publishers, 1879.) 



V. Geometrical Deductions, Riders, and Exercises, based 

 upon Euclid, Books I.— IV. (Stewart's Mathematical 

 Series, 1878.) 



A COMMON purpose pervades these five works, viz., 

 that of affording practice and aid in the solution 

 of mathematical problems. Prof. Wolstenholme, with a 

 marvellous versatility which has long placed him in the 

 foremost rank of "ten-minute conundrum" makers, 

 sends forth a volume (I.) which now contains 2,815 pro- 

 blems in place of the 1,628 which he published in 1867. 

 Further, his book has increased in all the directions in 

 which it is possible for a book to grow, and the number 

 of valuable hints scattered throughout the volume has 

 been greatly enlarged. Dipping into the book here and 

 there we are fain to cry out " Prodi-gi-ous ! " with worthy 

 Dominie Sampson, and to think this problem-compelling 

 Briareus ever 



' ' Agitates his anxious breast. 

 In solving problems mathematic." 



We have long used the earlier work with profit to ouiaHl 

 selves, and, we believe, to the advantage of our pupils^ 

 preparing for Cambridge scholarship examinations ; this 

 new edition is an improvement upon the old, and in its 

 line seems now perfect. What we would much like to 

 have is Prof. Wolstenholme's solutions of his questions, 

 but we fear the public, needed for the purchase of such a 

 work, is not yet in existence. Doubtless there are many 

 errors in the text, but these can only be found out by a 

 free and long-extended use ; however, we have noted in 

 question 443, for the second cos^^ read sin'5 ; question 

 925, for a'i 2a; p. 192, lines 2, 5, put - before A. 



In the volumes II., III., we have a welcome revival 

 of a fashion which has of late years died out ; it never 

 prevailed to any great extent, but its occurrence was 

 generally traceable to the influence of some one or two 

 enthusiasts, who, for the benefit of junior students, were 

 willing to put upon record neat solutions of elegant 

 problems, not counting the cost of publication. Such 

 collections as these are especially valuable, and the 

 volumes before us seem quite equal to their predecessors 

 in the same field. A novelty in III. is the publication of 

 additional remarks on some of the questions. For 

 instance, a concise general statement of the method of 

 least squares is given on pp. 162-169 ; on p. 8 is a note 

 on circulating decimals, and similar notes occur else- 

 where. In this work (III.) we have detected several 

 small errors, p. 13 line 14 insert - before fg a^ ; p. 14 



