i8: 



NA TURE 



[August 12, 1909 



reflects much credit on those to whom it has been 

 entrusted. The size of plate adopted, it may be 

 observed, is that of the double plates issued in former 

 volumes of the Calcutta Annals, so that librarians 

 who desire to bind the illustrations in conformity with 

 the text are left free to do so. But there will probably 

 be others who may prefer to leave the plates unfolded, 

 and the editor of the series has earned the gratitude 

 of those into whose hands this volume may come for 

 his decision to issue the illustrations in a larger port- 

 folio than that which contains the letterpress, thereby 

 leaving them free to decide the course to be adopted. 

 The work before us is a valuable addition to the series 

 of volumes for the initiation of which the scientific 

 world is indebted to the late Sir George King, and 

 botanists generally will not only feel grateful to Signor 

 Beccari for its preparation, but will desire to associate 

 themselves with him in his appreciation of that 

 ■" enlightened munificence " on the part of the Govern- 

 Tnent of Bengal v.-hich has rendered its appearance 

 possible. 



GYROSCOPIC MOTION. 



An Elementary Treatment of the Theory of Spinning 

 Tops and Gyroscopic Motion. By Harold Crabtrec. 

 Pp. xii + 140 and 3 plates; with illustrations. (Lon- 

 don, New York, Bombay, and Calcutta : Longmans, 

 Green and Co., 1909.) Price 55. 6d. net. 



THIS enchanting and bewildering subject has in 

 recent years been admirably expounded in two 

 well-known books, one somewhat more severe in its 

 treatment than the other. The author has now pro- 

 vided a third, which will be valued by those who 

 already possess and take pleasure in the other two 

 even more than by those who approach the subject 

 for the first time. The mathematical treatment is far 

 more severe, so much so that the average student who 

 scoffs at the term elementary on the back of some 

 of his text-books will certainly in this case consider 

 it inconsistent with the subject-matter of the last few 

 -chapters. However, if he will afterwards read the 

 subject in, say, the " Encyclopaedia Britannica," he 

 -will realise that the term is not so misleading after 

 all. 



The method by wliich the theory is introduced is 

 admirable. In an introductory chapter, illustrated by 

 twenty-six figures, all sorts of tops and spinning 

 things, familiar and otherwise, are described, and 

 their curious behaviour in each case siniplv stated. 

 An interest is thus awakened, and the reader, if un- 

 familiar with the subject, realises at once what sort 

 of thing he is going to have presented to him. In 

 the writer's opinion this method would be advan- 

 tageous generally where a difficult subject is being 

 opened. If, for instance, before the first chapter of the 

 typical book on the integral calculus or before Euclid's 

 definitions there were a lightly written chapter giving 

 more or less familiar experiences which are puzzling, 

 but which will in due time be made clear, the reader 

 would be more encouraged than he is by the existing 

 openings. 



The author clears the ground bv giving very exact 

 NO. 2076, VOL. 81] 



ideas on the subject of rotation about a fixed axis, 

 laying stress at every point on the dimensional identity 

 of the two sides of every equation. In dealing with 

 the subject of precession, he is by no means content 

 to handle the ultimate condition of steady precession, 

 but he goes fullv into that more difficult subject which 

 may be summarised under the term gyrostatic elas- 

 ticity, and which includes the immediate displacements 

 and vibrations or wobblings of axes when disturbing 

 couples are applied or removed. 



The latter part of the book is in large part devoted 

 to the difficulties connected with the motions of the 

 axis of rotation of an ellipsoidal body within itself and 

 in space. Among matters of interest discussed will 

 be found the behaviour of celts, the self-turning of a 

 falling cat, with kinematograph views, the diabolo, 

 the Brennan mono-rail, and Schlick's gyroscopic device 

 for steadying ships. 



While criticism is misplaced, the writer would 

 suggest that those diagrams, such as Fig. 20, in which 

 bent arrows are intended to show a direction of rota- 

 tion round an axis indicated by a line, are, as drawn, 

 ambiguous, for it is impossible to tell whether the 

 arrow is intended to be in front of or behind the line. 

 If one or other were broken through the meaning 

 would be evident, as it is, for instance, in Fig. 18, 

 where the arrow is clearlv in front of a material axle. 



C. V. Boys. 



MAGIC SQUARES. 

 Easy Methods of Constructing the Various Types of 

 Magic Squares and Magic Cubes, with Symmetric 

 Designs founded Thereon. By Dr. John Willis. 

 Pp. 256. (Bradford and London : Percy Lund, 

 Humphries and Co., Ltd., 1909.) 



AM.\GIC square is an example of a problem which 

 is a particular case of another which from its 

 enunciation may be subjected to mathematical 

 analysis. The n- cells of a square of order n may 

 be supposed occupied, each of them, by one or more 

 numbers in such wise that the sums of the numbers 

 in the n rows and in the n columns have given values 

 varying from row to row and from column to column. 

 The enumeration of such squares, or more generally 

 of such rectangles, has been made the subject of 

 mathematical investigation employing algebraic sym- 

 metric functions and the allied differential operators, 

 and complete success has resulted. The absolute 

 magnitude of the numbers appearing in the cells may 

 be restricted, and any number of the cells may be 

 empty; no additional difficulties present themselves. 

 Other problems of enumeration of the magic square 

 kind, of which the simplest is known as the Latin 

 square, first examined by Euler, and by others up 

 to the time of Cayley, have also in recent years com- 

 pletely yielded to the same calculus of symmetric 

 functions. In all these cases row and column pro- 

 perties are dealt with, but directly we introduce what 

 may be termed diagonal properties the analysis fails 

 to overcome the great difliculties which are thereby 

 imported into the problems. The problem of the 

 magic square involves restrictions and limitations of 



