December 22, 1910] 



NATURE 



2\\ 



We would remark also that in his chapter on the 

 "Theories of Microscopical Vision," Mr. Conrady adds 

 a few paragraphs giving a short account of the con- 

 nection between N.A. and the vision of minute 

 objects of dimensions below the resolution limit, 

 whether self-luminous or opaque. 



The more important additions are those which deal 

 with the extension of dark ground illumination to 

 high powers, and the description of Siedentopf's 

 apparatus for viewing ultra-microscopic particles. 

 Dark ground illumination at high powers is obtained 

 bv the use of a condenser or illuminator of special type, 

 which brings the light, usually with the aid of side 

 reflection, to a focus on the specimen at a very oblique 

 angle. Tvpes of such condensers by Lcitz, Zeiss, and 

 Beck are described, and the method will no doubt be 

 of value to the bacteriologist. 



The Siedentopf method for illuminating ultra-micro- 

 scopic particles is well known. The subject perhaps 

 lies outside the range of the ordinary microscopist. 



Finallv, it may be mentioned that the already excel- 

 kmt series of photomicrographs has been extended by 

 the introduction of four or five interesting photo- 

 graphs of amphipleura. Unfortunately, in the copy 

 we have seen, the printers have made the mistake of 

 printing' the descriptive text on the wrong side of the 

 thin paper separating the plates, with the result of 

 mafking it somewhat difficult to read. 



GEOMETRY OF SURFACES. 

 .1 Treatise on the Geometry of Surfaces. By A. B. 



Basset, F.R.S. Pp. xvi + 291. (Cambridge: 



Deighton Bell and Co. ; London : G. Bell and Sons, 



1910.) Price los. 6d. 



\ CCORDING to his preface, Mr. Basset intends 

 -V this book to supply a want in English works on 

 solid geometry, namely, an adequate account of sur- 

 faces other than quadrics, the existing gap being due 

 to the fact that Salmon's "Geometrv- of Three 

 Dimensions " is now out of print. 



The greater part of the book seems to be devoted 

 to a detailed examination of the various types of 

 singularities which can occur in surfaces of order not 

 higher than the fourth ; such a lengthy investigation 

 cannot be properly criticised except at the cost of 

 great labour. But, for reasons given below, it is 

 doubtful if the method adopted for resolving higher 

 singularities is really sufficient to do all that is 

 claimed by the author. 



It is not altogether clear, either, for what class of 

 readers the book is intended ; the greater part of the 

 results will interest none but specialists in geometry. 

 And one may imagine that such specialists might be 

 tempted to ask why the analytical machinery is 

 developed purely from metrical definitions, when the 

 properties to be established are mainly projective (or 

 descriptive) in character. Thus, reciprocation seems 

 always to refer to a spliere, and homogeneous co- 

 ordinates are defined (§3) only as perpendiculars on 

 the faces of a tetrahedron. It is not quite easy to 

 see how Mr. Basset would justify the use of co- 

 ordinates such as x + iy, x — iy, on the last definition. 



However, there is probably a wider circle of readers, 

 not claiming to be geometrical specialists, who would 

 NO. 2147, VOL. 85] 



take an intelligent interest in an account of the pro- 

 perties of cubic and quartic curves and surfaces, and 

 particularly in results which are related to work in 

 other subjects. Such readers might also find it use- 

 ful to have information as to various models available 

 for the illustration of the shapes of the figures ; doubt- 

 less the expert geometer disdains these mental 

 crutches, and relies on his powers of intuition. But 

 those of us who confess to finding it difficult to 

 visualise surfaces from their equations, are able to 

 point to geometrical experts who have been led to 

 unexpected results by the consideration of models; one 

 need only mention Rummer's model of the surface of 

 centres of an ellipsoid (Salmon, "Geometr}- of Three 

 Dimensions," p. 273), and Henrici's models of mov- 

 able hyperboloids. Even expert analysts may make 

 slips in their work, and may find occasionally some 

 difticulty in detecting such slips, while an examination 

 of a diagram or model will often indicate the mistake 

 at once. .\n illustration may be drawn from Mr. 

 Basset's statement (§142) that the circles of 

 curvature at the ends of the minor axis of an ellipse 

 can intersect at points which lie on the circles of 

 curvature at the ends of the major axis ; a moment's 

 glance at a figure will show that the former circles 

 lie wholly outside, the latter wholly inside the ellipse, 

 for all values of the eccentricity. 



Those who wish for an introductory account of the 

 simpler properties of cubic and quartic curves will 

 find Mr. Basset's provision for them rather scanty. 

 His theorems (and proofs) occupy but little more space 

 than the summary (of results only) given in Pascal's 

 ■' Repertorio," t. ii. (ist edition); and some of Pascal's 

 references are omitted from the list (for cubic curves) 

 given on p. 100. A good deal of light would be 

 thrown on the classification of quartics of the first 

 species by a reference to the Sylvester-Weierstrass 

 method of invariant factors. The same method 

 would prove useful in handling cyclides (quartic sur- 

 faces), and leading up to Darboux's pentaspherical 

 coordinates; as Darboux's coordinates are not intrcv- 

 duced at all, Mr. Basset is unable to prove that 

 confocal cyclides cut orthogonally, and various other 

 theorems given in Salmon's account of cyclides have 

 to be omitted also. 



Nor will the inquirer after the arrangement of the 

 twentv-seven lines on a cubic surface fare much better. 

 Mr. Basset gives half a page to proving their exist- 

 ence, and that of forty-five triple tangent-planes, but 

 he has no illustration to give us of even the simplest 

 example of a double-six. Details of the singularities 

 of the twenty-three different types of cubic surfaces 

 are enumerated; but we are not told that, say, the 

 cubic with a nodal line (of the first kind) can be 

 illustrated by the familiar cylindroid. models of which 

 are amongst the commonest examples of ruled skew 

 surfaces. 



The resolution of compound singularities (chapters 

 iv. and v.) is discussed first for the case of plane 

 curves ; the method appears in all cases to rest on the 

 assumption (see, for instance, §165) that the most 

 general singularity of order ' p can be found on a curve 



1 We have not succeeded in finding a precise definition of what Mr. Basset 

 m»ans by this term : it would seem to be a singular point with/ tangents 

 (some or all of which may coincide). 



