(1) Fermat's Last Theorem (2) The Elements of Non-Euclidean Plane Geometry and Trigonometry (3) The Algebraic Theory of Modular Systems

Abstract

(1) THE main fallacy of Mr. Cashmore's paradoxical tract is this:—Let f, φ be polynomials in x, and A a constant different from zero; then, if f, φ have a common factor (x−a),(x=a) may be regarded as a solution of f/φ = λ. Conversely, if f/φ = λ has a root a, then (x− a) must be a common factor of f and φ”. (See p. 18.) (2) By this time it is fairly well known among mathematicians that ordinary geometry is a sort of border-line between two equally consistent theories, in each of which Euclids axiom of parallels is false. In one of these the sum of the angles of a rectilinear triangle exceeds two right angles; in the other it falls short of it, and may even converge to zero. If similar triangles are defined by parallelism of sides, we have the sums of their angles differing according to a fixed law; and, similarly, if we define them by proportion of sides (generally according to a different law). These non-Euclidean geometries apply to three-dimensional space as well as to the plane, and the question for teachers is to make them intelligible to the student by intuitional methods. As regards the case when the sum of the angles of a triangle is less than two right angles, nothing can be better than to take as straight lines circles which cut a fixed ordinary sphere orthogonally, and to regard all points out- side this sphere either as non-existent or as images of accessible points within the sphere. The plane version of this is given by Prof. Carsiaw (pp. 15375) in the clearest manner conceivable; but he does not seem (in this book) to have considered the analogous theory in solido. There is no satisfactory theory of three-dimensional non-Euclidean geometry, from an intuitional point of view, unless it gives us a clear three-dimensional image in our ordinary space, assuming, of course, that our powers of “intuition” are confined to ordinary space.

(1) Fermat's Last Theorem.

By M. Cashmore. Pp. 63. (London: G. Bell and Sons, Ltd., 1916.) Price 2s. net.

(2) The Elements of Non-Euclidean Plane Geometry and Trigonometry.

By Prof. H. S. Carslaw. Pp. xii + 179. (London: Longmans, Green and Co., 1916.) Price 5s. net.

(3) The Algebraic Theory of Modular Systems.

By F. S. Macaulay. Pp. xiv + 112. (London: At the Cambridge University Press, 1916.) Price 4s. 6d. net.