An Introduction to Protective Geometry and its Applications

Abstract

THIS text-book of modern projective geometry forms an admirable introduction to the subject, and should be known to all who are interested in this branch of mathematics. The first chapter deals with the general properties of projective ranges and pencils and their products; including harmonic and perspective projection, and the projective properties of the circle. Then follows an investigation of collineation in a plane, comprising1 perspective transformations, and the linear transformations of translation, rotation, and dilatation, with combinations of these. The intimate relation that exists between projective and descriptive geometry is shown. The third chapter gives the general theory of conies, the projective properties of the circle being extended to conies by perspective transformations. The next chapter deals with pencils and ranges of conies and their products, and especially with cubics, the latter being classified under the five standard types by the help of the Steinerian transformation. Throughout the book analytical and geometrical methods are employed side by side, some portions of the subject being better suited to the former treatment; moreover, the analysis affords excellent illustrations of modern analytical geometry. The main purpose of the author has been to develop the subject in regard to its practical applications in mechanics, and the last chapter is devoted to such examples. Thus we find problems in graphic statics, plane stresses, and in the stress ellipse of an elastic material, and there is an interesting account of various linkages by means of which linear and perspective transformations can be mechanically obtained. The book is excellently got up in every way, and the diagrams are quite perfect and may well serve as models of what such figures ought to be. The author is a very clever draughtsman, and his skill as a writer Is equally pronounced.

An Introduction to Protective Geometry and its Applications.

By Dr. Arnold Emch. Pp. vii + 267. (New York: Wiley and Sons; London: Chapman and Hall, Ltd., 1905.) Price 10s. 6d.