REVIEWS 657 



fundamental conceptions. It is required that all constructions made shall 

 use only accessible points, so that assumptions are needed as to what points 

 are accessible when others are known to be so, e.g. for points on a line, the 

 notion of " betweenness." This Real Geometry is then generalised by the 

 introduction of postulated points, and it follows that we may regard the 

 space of the Real Geometry as part of a space in which the Propositions of 

 Incidence are completely valid, provided that postulated points, lines, and 

 planes are allowed an existence, whether they be accessible or not. Section 

 III of this chapter deduces Pappus' theorem for the case of the Real Geometry 

 amplified in this way, from the notion of an abstract order among the points 

 of a line, involving a definite assumption as to the points existing on a line. 



Chapter III resumes the Abstract Geometry of Chapter I and establishes, 

 in extension of the theory of related ranges, a theory of related spaces of any 

 number of dimensions. The algebraic symbolism then suggests a way of 

 generalising further the abstract geometry by the introduction of imaginary 

 elements ; we assign to the geometry such points that when O + cU is a 

 point there is also a point O + ^U in which z^= c. It is shown that a 

 geometrical fact corresponding to this is that it is possible to inscribe in a 

 triangle another triangle whose sides pass through given points. The 

 chapter concludes with the replacement of imaginary elements by a series 

 of real elements, a modification of the theory of von Staudt. 



The subject-matter of Vol. II is more familiar and may be dealt 

 with more briefly. The fundamental properties of conies are established 

 synthetically ; they are then considered, still without assuming any notions 

 of distance or congruence, in relation to two absolute points, giving proposi- 

 tions which, expressed in the terms current in metrical geometry, concern 

 circles, confocal conies, and so on. Co-ordinates are then introduced by 

 means of the algebraic symbolism of Vol. I and are applied in a number 

 of ways, to extensions of Feuerbach's theorem, to apolar triads, to the con- 

 sideration of the invariants of two conies. All this is familiar ground, 

 indeed, but anyone who has heard Prof. Baker lecture on elementary conies 

 will confidently expect to be inspired with new ideas and points of view by 

 reading these chapters, and he will not be disappointed. 



The last chapter deals with the theory of measurement, of length and 

 angle, by means of an absolute conic, and shows how the so-called non- 

 Euclidean geometries may be regarded as included in the general formula- 

 tion. A couple of appendices describe certain configurations leading up to 

 the complete figure for Pascal's theorem, which is best considered in four 

 dimensions. There is a page of remarks and corrections to the first volume. 

 It is unnecessary to praise the printing and general production of a book 

 published by the Cambridge University Press ; we have heard a criticism of 

 the type used in lettering the diagrams, but the latter themselves are uniformly 

 excellent, and having ourselves tried in vain to get a picture of the fifteen 

 lines of the Pascal figure, we can but gaze in awe and admiration at the 

 wonderfully clear and simple figure of the Hexagrammum Mysticum which 

 forms the frontispiece to the second volume. F. P. W. 



Multilinear Functions of Direction and their Uses in Differential Geometry. 



By E. H. Neville. [Pp. 80.] (Cambridge : at the University Press, 

 192 1. Price 8s. 6d. net.) 



An important problem in elementary differential geometry is to associate 

 the curvatures and torsions of curves on a surface with the form of the 

 surface itself ; it is a problem which may be investigated by means of 

 " moving axes." This book extends and develops this method out of all 

 recognition. The functions considered are primarily not functions of a 

 single variable direction, but functions of several independent directions. 

 By relating these originally independent directions we obtain functions of a 



