584 



NA TURE 



[April i8, 1901 



to certain species and genera. When, however, these 

 great difficulties are taken into account, it must be 

 allowed that the author has fulfilled his task in a highly 

 creditable and satisfactory manner. 



And as regards nomenclature, classification and the 

 splittmg-up of certain old, unwieldy generic groups like 

 the squirrels into divisions of smaller size, Mr. Sclater is 

 well abreast of modern ideas. One of the most notice- 

 able of these modern changes in classification is the 

 transference of the so-called Cape jumping-hare— the 

 spring-haas of the Boers, from its old association with 

 thejerboas — to a position near the cane-rat and the 

 porcupines. Nor is this all that is noteworthy in Mr. 

 Sclater's remarks on the creature ; for we are told that, 

 in spite of the huge bounds it takes, "it is never very 

 rapid in its movements, and can be easily overtaken." 

 This information we have not found given in any of the 

 other works to which we have turned. It is a matter for 

 regret that the portrait of the spring-haas, like many of 

 the other figures in the book, has not been executed in a 

 more satisfactory style. 



An old error — to wit, that it burrows — in connection 

 with the cane-rat is also corrected, mainly on the evidence 

 of the late Prof. Peters and Captain Drummond. 

 . Among the most curious and interesting of all the 

 smaller mammals of South Africa are the elephant- 

 shrews, or jumping shrews, and the golden moles, and of 

 each of these Mr. Sclater gives an excellent account, both 

 as regards bodily characteristics and habits, although 

 further observations are stated to be required with 

 regard to the mode of life of the last-mentioned animal. 



"The golden mole," writes the author, " is exceedingly 

 common in gardens, where it makes runs in all direc- 

 tions in search of the worms and grubs on which it 

 lives. Although generally supposed to be destructive, it 

 is really a great aid to the gardener, as it destroys 

 quantities of larvae, especially those of a certain gamma 

 moth. ... A certain amount of mischief, however, is 

 done by the mole in pursuit of its prey by disturbance of 

 roots and freshly-sewn seeds." 



In addition to the Rodentia, Chiroptera and Insecti- 

 vora, the present volume also includes the South African 

 Cetacea and Edentata. Among the cetaceans special 

 interest attaches to the author's description of a specimen 

 of the lesser sperm-whale recently taken in Table Bay, 

 as the external characters of this rare whale have been 

 hitherto very imperfectly known. Of the specimen in 

 question Mr. Sclater gives a sketch, which shows the 

 characteristic shark-like mouth and small dorsal fin. 

 Certain dififerences in size which have been thought to 

 indicate specific distinction are, in the author's opinion, 

 probably due to difference of sex in the individuals 

 which have from time to time been examined. 



The aard-vark and the pangolin Mr. Sclater, although 

 with some hesitation, still retains in the same order with 

 the typical South American Edentata. And it must be 

 confessed that certain observations which have recently 

 been made with regard to the myology of these creatures 

 tends, so far as it goes, to justify this conservatism. 

 Whether there really is any close relationship between the 

 two groups is a question of the very highest importance in 

 regard to certain views that have been recently ex- 

 pressed in favour of a former connection between Africa 

 NO. 1642, VOL. 63] 



and South America. And it would greatly help matters 

 if a decisive answer could be given on this point. 



Mr. Sclater may be congratulated on the completion of 

 a very important and valuable work. R. L. 



INFINITESIMAL GEOMETRY. 

 Einfiihrung in die Theorie der Curven in der Ebene und 

 itn Raume. By Dr. Georg Schefifers. Pp. viii -1-360. 

 (Leipzig : Veit and Co., 1901.) M. 10. 



THIS volume is the first of two which will make a 

 complete work under the title " Anwendung der 

 Differential- und Integral-Rechnung auf Geometrie." The 

 subject-matter of the two volumes may be said to be, 

 roughly, the infinitesimal geometry of curves and sur- 

 faces respectively. The first volume is divided into 

 three sections, dealing with plane curves, curves in space, 

 and developable surfaces. The first section does not 

 attempt to be a complete exposition of the subject, and 

 must be regarded as an introduction to what follows, in- 

 tended to accustom readers who are already well 

 grounded in differential and integral calculus to the style 

 and methods which are employed later. The theory of 

 the curvature of plane curves is based on the definition 

 of contact of an assigned order, which is explained with 

 great exactness. The differential invariants of a curve 

 for the group of movements in the plane are fully in- 

 vestigated, and their properties established in an ele- 

 mentary manner without introducing notions of groups 

 or partial differential equations. Envelopes, evolutes, 

 singular points, and the geometrical significance of 

 differential equations of the first order and degree are 

 discussed shortly. In connection with the trajectories of 

 a family of curves, the problem is completely solved of 

 finding all curves for which the product of the normal 

 and radius of curvature is constant. The remainder of 

 the first section is devoted to an explanation of curvi- 

 linear coordinates. 



The second section contains a thorough and systematic 

 account of the curvature, torsion, and allied theory of 

 curves in space. The dual mterpretation of an ortho- 

 gonal substitution of coordinates as a change of frame of 

 reference and as a movement in space is first carefully 

 explained, and the theory of the intrinsic properties of 

 curves is built upon it. Particularly interesting are the 

 discussions of the differential invariants and of the inte- 

 gration of the intrinsic equations of a curve, in the 

 course of which an elementary account of Riccati's equa- 

 tion is given. Conditions for contact of an assigned 

 order are carefully laid down, and from them the rela- 

 tions between a curve and its osculating circle and 

 helices are deduced ; in particular we have the interest- 

 ing result that the axes of all osculating helices at any 

 point generate a cylindroid. 



In the third section the main properties of the surface 

 generated by the tangents to a curve are established. 

 The general ruled surface is introduced in order to pro- 

 vide a rigorous investigation of what is meant by saying 

 that consecutive generators intersect. The remainder 

 of the section is occupied with various loci connected with 

 a given curve, such as evolutes, involutes, parallel curves, 

 polar surface, rectifying surface, etc. The text ends with 

 a short account of minimal lines and minimal curves. 



