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Synopsis
Elements Of Analytical Geometry by William Smyth. This treatise presents the foundations of analytical geometry as the systematic application of algebra to geometry, tracing the historical shift from the ancient Geometrical Analysis to the modern Algebraic Analysis. The work explains how to translate geometric problems into algebraic equations, construct algebraic expressions, and solve problems by algebra, while also deriving geometric properties from equations. It surveys the circle, parabola, ellipse, and hyperbola, and explains the general equation of the second degree, transformations of coordinates, and polar coordinates. It develops the analytical geometry of two dimensions, then extends to three dimensions, treating points in space, lines, planes, and surfaces such as spheres, cylinders, and cones, and discusses the intersection of curves and surfaces, tangents, and other classical problems. The design is to illuminate the origin and nature of analytical geometry, show the power and limits of the algebraic method, and lay groundwork for later calculus. The book was originally published in 1855.
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