REVIEWS 



MATHEMATICS 



Principles of Geometry. By H. F. Baker, Sc.D., F.R.S. Vol. I. Founda- 

 tions. [Pp. xi + 182.] Vol. II. Plane Geometry, Conies, Circles, 

 Non-Euclidean Geometry. [Pp. xv + 243.] (Cambridge : at the 

 University Press, 1922. Price, Vol. I, 12s. net ; Vol. II, 15s. net.) 



These two volumes are the first of a series which seeks to introduce the 

 reader to those parts of geometry which precede the theory of higher plane 

 curves and of irrational surfaces, and their importance cannot, I think, be 

 over-estimated. Prof. Baker believes that the long preliminary study of 

 elementary geometry to which at present so much time is devoted can largely 

 be avoided, and that after an extensive study of diagrams and models the 

 student may straightway enter upon a course such as is described in this 

 book, and learn at once general principles, which, with a moderate demand 

 on the memory, give an immense command of detail. He will learn, what 

 seems to be deliberately concealed by most English textbooks, that pro- 

 jective geometry need not — and indeed cannot without losing its essential 

 feature — be based upon metrical geometry ; he will learn the indispensable 

 ideas of geometry of more than three dimensions and of the geometry of 

 so-called imaginary points. His path will, I fear, not be an easy one, but 

 it will be secure, it will afford magnificent views of the country traversed, 

 and will be, in short, delightful. 



Vol. I deals with the necessary logical preliminaries, rejecting the con- 

 sideration of distance and of congruence as fundamental ideas and re- 

 placing them by a theory of related ranges. The first chapter. Abstract 

 Geometry, begins with a statement of the Propositions of Incidence, the 

 laws of combination of the entities called by the names point, line, and 

 plane ; Desargues' theorem and the construction for the fourth harmonic 

 point are deduced, and the theorem that a chain of perspectivities can 

 be reduced to two links. It is next shown that the Propositions of 

 Incidence alone do not sufl&ce to secure that the correspondence of the 

 points of two related ranges is unique when three points of the one are 

 given as corresponding to three points of the other ; but that this result 

 will follow from the assumption of Pappus' theorem concerning the col- 

 linearity of the intersections of the cross- joins of three points on each of 

 two intersecting straight lines. The last section of the chapter introduces 

 an algebraic symbolism which represents the Propositions of Incidence, and 

 it is shown that the introduction of the Pappus theorem corresponds to a 

 definite limiting law of combination of the symbols, namely, that their 

 multiplication is commutative. 



The point of view adopted in this chapter is general and abstract ; for 

 example, the word " line " is used in such a sense that every two straight lines 

 of a plane intersect one another ; also the adoption of Pappus' theorem may 

 appear artificial. Further, it takes no account of two notions which seem 

 to be inseparably bound up with our conception of space as derived from 

 experience, the notion of accessible as distinct from inaccessible points, and 

 the notion of the order of a set of points upon a line. The second chapter. 

 Real Geometry, is therefore interpolated to examine more concretely our 



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