Excerpt from On the Geometry of Surfaces
The last edition of Salmon's Analytic Geometry of Three Dimensions, which was published in 1884, has been out of print for some years; and although there are several excellent works on Quadric Surfaces and other special branches of the subject, such as those of Mr Blythe on Cubic Surfaces and of the late Mr Hudson on Kummer's Quartic Surface, yet there is no British treatise exclusively devoted to the theory of surfaces of higher degree than the second. I have therefore endeavoured to supply this want in the present work.
The Theory of Surfaces is an extensive one, and a thoroughly comprehensive treatise would necessarily be voluminous. I have therefore decided to limit this work to the more elementary portions of the subject, and have abstained from introducing investigations which require a knowledge of the Theory of Functions and of the higher branches of Modern Algebra. The ordinary methods of Analytical Geometry are quite sufficient to enable the properties of cubic and quartic surfaces and twisted curves, and also the point and plane singularities of surfaces, to be discussed with tolerable completeness, and to demonstrate a number of interesting and important theorems connected with them; but for the purpose of confining this treatise within a moderate compass, I have abstained from any general discussion of surfaces of higher degree than the fourth.
The properties of a point singularity may usually be examined by means of a surface of low degree just as Well as by one of the nth degree; but if the degree is less than a certain limit, which depends on the character of the singularity, the latter appears in an incomplete form on the surface. Thus the properties of a triple line cannot be fully investigated without employing a surface of the seventh degree, and this fact has rendered it necessary to partially discuss surfaces of higher degree than a quartic.
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