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Chapter 7, Two Treatments of Geometry
Volume 1B, Mathematics Curriculum Notes
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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70
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