Apeil 30, 1897.] 



SCIENCE. 



Qm 



description of its principal characters. A few 

 of the important species in each genus are quite 

 fully described and in many cases illustrated. 

 These are followed by a further enumeration of 

 a number of other species with their hosts and 

 localities, the species in many cases for Britain 

 and the United States being indicated. 



The book is very fully illustrated, a very 

 large number of the illustrations being new, 

 either from the pencil of the author or from ex- 

 cellent photographs. As foot notes, there are 

 very copious references to works even in cases 

 where space would not permit of a discussion 

 of their contents. 



Neither the author nor the translator pre- 

 tends to completeness, but modestly offer ex- 

 cuses for faults which under the conditions 

 could not be well avoided. These can well be 

 overlooked in view of the great amount of in- 

 formation contained in the volume which will 

 prove to be a very useful adjunct to reference 

 works on parasitic fungi. When a new Ger- 

 man edition shall be called for the author 

 promises to thoroughly revise it and expresses 

 the wish that those who have in the past sent 

 him copies of their investigations continue to 

 do so in order that he may make this edition as 

 complete as possible. 



Geo. F. Atkinson. 



COKNELL UNIVEESITY. 



EECENT BOOKS ON QUATERNIONS. 



1. Theorie der Quaternionen. ton De. P. Molen- 

 BEOEK. Leiden, E. J. Brill. 1891. Pp. 

 vii+284. 



2. Anwendung der Quaternionen auf die Geo- 

 metrie. By the same author. 1893. Pp. 

 xv+257. 



3. The Outlines of Quaternions. By. Lieut. - 

 Col. H. W. L. Hime. London, Longmans & 

 Co. 1894. Pp. 190. 



4. A Primer of Quaternions. By A. S. Hatha- 

 way. New York, Macmillan & Co. 1896. 

 Pp. x-l-118. 



5. Utility of Quaternions in Physics. By A. Mc- 

 AuLAY. Macmillan & Co. 1893. Pp. xiv 

 -1-107. 



The above books are all contributions to the 

 literature of the Quaternion side of space-analy- 

 sis. The first, by Dr. Molenbroek, is a care- 



fully written exposition of Hamilton's theory ;. 

 the author, if he does not examine the corre- 

 spondence of the theory with exact science and 

 established analysis, at least presents it so as to 

 be internally consistent. For instance, he ex- 

 plains the fundamental rule ij ^ i as meaning 

 that a quadrant round the axis j followed by a 

 quadrant round the axis i is equivalent to a 

 quadrant round the axis k. Consistently with 

 this, he explains the rule i * = — 1 as meaning 

 that a quadrant round the axis i followed by a 

 quadrant round the same axis is equivalent to 

 a reversal. The treatise, however, does not go 

 deep enough; for the subject of quaternion 

 logarithms and exponentials is embraced in a 

 9-page appendix, and what is there given is 

 the well-known theory of coplanar exponentials. 

 It is only when diplanar exponentials are 

 handled that problems can be attacked which 

 are insoluble, or at least not readily solved by 

 the ordinary methods of analysis. Dr. Molen- 

 broek introduces an indefinite use of n/ — 1 ta 

 signify a quadrant round some axis perpendicu- 

 lar to a given line. There are reasons for be- 

 lieving that in space-analysis \/ — 1 is scalar 

 in its nature, and that it distinguishes the hy- 

 perbolic angle from the circular angle. Any- 

 how, that is one definite meaning. 



The third book, by Col. Hime, presents a 

 very dim and imperfect outline, which it would 

 be well for the beginner to avoid. By perusing 

 it he may get his ideas confused, not only of 

 analysis, but of mechanics ; for example, at 

 p. 33 the terms ' version,' 'torsion,' 'rotation,' 

 'twist,' are all used as synonymous. This is, 

 at least, awkward, for one of the first things 

 which a student of quaternions must do is to 

 distinguish between the trigonometrical com- 

 position of angles and the mechanical composi- 

 tion of rotations. The author explains the rule 

 ij = fc by saying that j and k each signify a 

 unit vector, but i signifies a quadrantal versor 

 which turns J into k. But he fails to observe 

 that this explanation cannot apply to the com- 

 plementary rule i^ = — 1, for a quadrantal ver- 

 sor i operating on a unit vector i would leave it 

 i. Chapter Tenth is devoted to the ' Interpreta- 

 tion of Quaternion Expressions ;' thus for nine 

 chapters the reader is supposed to be dealing 

 with symbolical expressions. Would it not be 



