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Chapter 7
Two Treatments of Geometry

There are two approaches to Euclidean geometry, and the more recent one has precedence[1].

[1] This perspective is noted in the foreword by Harry Goheen, page 1, in the reprint of the work Foundations of Geometry of David Hilbert, Open Court Publishing Company, ISBN 0-875-164-7. This perspective is also expressed in College Geometry by David C. Kay, Holt, Rhinehart and Wilson, 1969 ISBN 0-03-073100-3.

1. Euclidean geometry in the plane can be presented and based on assumptions about points and lines in the plane, and associated geometric constructions based on circles and line segments[2]. The axioms of Euclidean geometry dealt with points, lines and loci — the curves traced out by points. This represents the synthetic (constructive), non-analytic perspective begun in the work of Euclid, two thousand years before anyone thought of axioms for real numbers or set theory. It represents the first codification of geometry, and the first model for rule-based thought. It is coordinate-free. Its original form was developed before coordinates were even dreamt of.

[2] The word line itself originally referred to a taut rope or string. Geometry (land measurement) in the first instance physically employed taut ropes to measure and mark rectangular and circular regions on the earth surface and in construction. On paper, the use of taut lines or strings may be replaced by a compass and a straight edge. The question of what points could be reached in the plane via ruler- and compass-based construction was the initial object of Galois theory in algebra.

2. The axioms for real numbers provide an newer framework for Euclidean geometry. The framework is called analytic geometry. It is based on coordinates. Within this framework, the drawing of lines and circles and the location of points correspond to the (parametric) solutions of equations and the formation of sets of ordered pairs or triplets. The latter serve as coordinates of points in a plane or in space. Note that solutions of equations can be obtained without reliance on the physical senses and without ruler and compass. Paradoxes due to imprecisely drawn diagrams and reliance on the physical senses (except for the use of pencil and paper to record thoughts) are thus avoided in the analytic approach or codification.

The older coordinate-free, synthetic, approach gives motivation (but no warranty) for the definitions and calculations of the newer analytic, coordinate-based, exposition of Euclidean geometry[3].

[3] Technical Note: In the set theoretic framework for arithmetic-based mathematics, the axioms for real numbers provide a foundation for analytic geometry including the theory of surfaces. Some examples of non-Euclidean geometry are given by study of curved surfaces. Particular examples are given by the surfaces of a sphere, donuts or torii, ellipsoids (or footballs), and Mobius strips. On such curved surfaces, paths followed by taut strings yield the smallest distances between points. These taut strings define line segments which are curved — not necessarily straight. Moreover, for triangles drawn on these surfaces using three taut strings, the sum of interior angles need not be 180 degrees. The sum in fact depends on the curvature, the departure from flatness, of the surface area enclosed by the taut strings. The mathematical adept can modify this statement to include surfaces formed by the bending without stretching of flat surfaces — the ruled or developable case of zero Gaussian curvature.

The physical or geo-measurement assumption that ordered pairs and triplets of real numbers correspond to points in the plane or space gives the correspondence between the newer analytic, coordinate-based approach and the older synthetic approaches to Euclidean geometry. The applications of analytic or coordinate geometry in physics, engineering, technology, etc., all depend on this assumption. In particular, this physical assumption is required for the drawing of diagrams and graphs.

Mixing the analytic and the older synthetic approaches to and representations of Euclidean geometry or not distinguishing between the two, is convenient in the relaxed or elementary development of mathematics and its applications in computational disciplines. The mix may also be present in the initial exposition of trigonometry and calculus [4].

[4] The high school exposition of sines and cosines relies on the physical identification of angles of triangles of angles spanned by sectors of circles. One proof of the angle sum formula for cosines relies on the rotation of an isosceles triangle and the physical or geometry assumption of rigidity (preservation of lengths and angles) under rotation. The geometric constructions here departs from the purely analytic treatment or codification of mathematics. In college calculus, geometric arguments leading to formulas for the slope or derivative of the cosine function may fall into the same category. (For a purely analytic treatment without diagrams, a very succinct one understandable to a college student specializing in mathematics, see the text Principles of Mathematical Analysis by W. Rudin, McGraw Hill, second edition 1964. It gives analytic definitions and treatments of the exponential function, the natural logarithms, sines and cosines.)

Such mixing departs from the modern mathematics ideal of having a smallest possible set of axioms for geometry, trigonometry, calculus and other parts of arithmetic-based mathematics, at least when how to supplant or replace the coordinate-free approach to Euclidean geometry is not indicated.

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