Book: A Treatise on the Analytic Geometry of Three Dimensions

Overview

Arthur Cayley develops a systematic, algebraic account of geometry in three dimensions, casting the classical elements of solid geometry into a unified coordinate framework. Fundamental figures such as planes, straight lines, spheres, cones, and cylinders receive treatment through general equations and manipulation of symbols, while attention is paid to surfaces of arbitrary algebraic degree. The exposition emphasizes methods that make geometric properties accessible to algebraic calculation and transformation.

Core content

The book begins with the elementary equation of a plane and of a straight line, setting down the coordinate relations and conditions that determine mutual position, intersection, and coplanarity. Spheres are introduced by their quadratic equations and studied for contacts and mutual intersections, while cones and cylinders are handled as families of lines generated from conic directrices or axis conditions. A large portion is devoted to second-degree surfaces (quadrics): ellipsoids, hyperboloids, paraboloids and their canonical reductions under allowable coordinate changes.

Techniques and methods

Algebraic elimination, manipulation of determinants, and systematic use of coordinate transformation are the principal tools. Cayley develops procedures for reducing general equations to simpler canonical forms, extracting invariant quantities that determine geometric type, and computing tangents and normals by algebraic differentiation and polar constructions. The polar and reciprocal relations with respect to a given quadric are exploited to convert questions about points and planes into dual problems about planes and points, facilitating determination of tangency, conjugacy, and focal properties.

Higher-degree surfaces and singularities

Beyond quadrics, general algebraic surfaces are introduced by polynomial equations of higher degree and examined for factorization, singular points, and multiple components. Methods for analyzing intersections of surfaces, common tangents, and asymptotic directions are developed by algebraic elimination and by studying resultant conditions. Singularities are characterized by vanishing of partial derivatives, and their local structure is related to the factorization of the defining polynomial, enabling classification of ordinary double points and cuspidal features within the algebraic framework.

Applications and examples

Concrete coordinate examples illustrate reduction to principal axes, computation of intersection curves between standard surfaces, and determination of generators on ruled surfaces. Cones and cylinders are treated both as loci defined by angular or parallel constraints and as envelopes of families of planes, with explicit criteria for when a surface is ruled or developable. The analytic approach yields formulas for centers, axes, and principal radii that connect directly with geometric constructions.

Significance and influence

The algebraic viewpoint advanced here helped bridge classical Euclidean geometry and the emerging algebraic and projective methods of the 19th century. By emphasizing transformations, invariants, and duality, the exposition prefigures later developments in algebraic geometry of surfaces and in the study of invariants of forms. The techniques for canonical reduction, polar methods, and singularity analysis provided a foundation that influenced subsequent work on the classification and manipulation of algebraic surfaces.