Abstract
Among the few choices of systems of axioms to construct a geometric model of the plane (for example, via Euclid or Hilbert) we take the least strenuous path; and, in making use of the real number system already in place, we develop real analytic plane geometry using Birkhoff’s axioms of metric geometry. One of the main purposes of this chapter is to explain what is classically known as the Cantor–Dedekind Axiom: The real number system is order-isomorphic to the linear continuum of geometry. Unlike the original approaches of Hilbert and Birkhoff, we are working here in a concrete model of the plane, built from the real number system of Chapter 2. Verifying that the Birkhoff postulates hold in our concrete model is much less demanding than the synthetic (purely axiomatic) approach. Nevertheless, our model oriented exposition still encounters some struggle, as in Sections 5.6–5.7, where the existence and properties of the circular arc length are shown using purely metric tools, and paving the way to trigonometry (Chapter 11). This also gives a precise answer to the question: “What is π?” Once again, this relies on the Least Upper Bound Property of the real number system, the main common thread with the first two chapters. A natural offspring of this technical passage is concluded with an optional section on the (often neglected) Principle of Shortest Distance, given here in full details. To ease up the complexity of the material, we make frequent side tours to develop metric properties of many geometric configurations. We determine all Pythagorean triples not by elementary number theory, but via analytic geometry: the method of rational slopes. We introduce here additional important tools that will play pivotal roles in the sequel: the Cauchy–Schwarz inequality, the AM–GM inequality, and their offsprings. Finally, still in this chapter, we present Archimedes’ duplication method to approximate π, once again with a view to algebraic formulas for many special angles given subsequently in trigonometry in Chapter 11.
“Let it have been postulated1. To draw a straight-line from any point to any point.2. And to produce a finite straight-line continuously ina straight-line. 3. And to draw a circle with any center and radius. 4. And that all right-angles are equal to one another. 5. And that if a straight-line falling across two(other) straight-lines makes internal angles on the sameside (of itself whose sum is) less than two right-angles, then the two (other) straight-lines, being produced toinfinity, meet on that side (of the original straight-line) that the (sum of the internal angles) is less than two right-angles (and do not meet on the other side).”
The five postulates in Euclid’s Elements, translated by Richard Fitzpatrick.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
The excerpts quoted here are from the English edition and translation by Richard Fitzpatrick of the Greek text of J.L. Heiberg from Euclidis Elementa, edidit et Latine interpretatus est J.L. Heiberg, in aedibus B.G. Teubneri, 1883–1885.
- 2.
The numbering follows the original translation.
- 3.
For a somewhat overly critical account on Euclid, see Russell, B., The Teaching of Euclid, The Mathematical Gazette, 2 (33) (1902) 165–167.
- 4.
Strictly speaking, the concept of plane is a definition, and the assumption that it has at least two points is a postulate.
- 5.
This corresponds to Hilbert’s order relation.
- 6.
These are axioms of the Hilbert system.
- 7.
As sets of real numbers (mod 2π).
- 8.
See Birkhoff, G.D., A Set of Postulates for Plane Geometry, Based on Scale and Protractor, Annals of Mathematics, Second Series, Vol. 33, No. 2 (1932) 329–345.
- 9.
The quote is Florian Cajori’s edition of Andrew Motte’s English translation in 1729 of Sir Isaac Newton’s Philosophiae Naturalis Principia Mathematica, published in 1687.
- 10.
Newton used this lemma to show that “if two points proceed with a uniform motion in right lines, and their distance be divided in a given ratio, the dividing point will be either at rest or proceed uniformly in a right line.”
- 11.
We will not use the vector space structure of \(\mathbb {R}^2\), nor the usual geometric concepts such as the dot product, etc.
- 12.
We changed the sign to match with the customary positive orientation of \(\mathbb {R}^2\).
- 13.
See, for example, Engel, A., Problem solving strategies, Springer, Berlin, 1997.
- 14.
This was a problem in the Asian Pacific Mathematical Competition, 1996.
- 15.
Note that this can also serve as a definition of \(\sqrt {a}\) as the equivalence class of the rational Cauchy sequence \((q_n)_{n\in \mathbb {N}}\).
- 16.
This geometric solution can be reworded to become a simple consequence of Birkhoff’s Postulates of Angle Measure and Similarity. The validity of these postulates in our model will be proved in Section 5.7. Hence, for a change, we give here a proof based on Euclid’s Elements.
- 17.
Warning: Congruence of the triangles △[P, P 0, M] and △[P, P 1, M] also follows from the observation that the lengths of the three pairs of sides of these triangles are equal, but in the Elements, this occurs after Proposition 5 of Book I.
- 18.
Compare this with primitive #15 in the Elements as stated at the beginning of this chapter.
- 19.
The words “tangent” and “secant” are derived from “tangere” and “secare,” respectively, which in Latin mean “to touch” and “to cut.”
- 20.
In this example we assume Birkhoff’s Postulates of Angle Measure and Similarity and, consequently, the Pythagorean Theorem, whose validity, in our model, will be proved in Section 5.7. This is pedagogically justified since this example is a perfect fit for our present line of argument. Alternatively, one can also refer here to Euclid’s Elements.
- 21.
Generalization of a problem in the American High School Mathematics Examination, 1995.
- 22.
Compare this with definition #18 in the Elements at the beginning of this chapter.
- 23.
As we will see below, \(\mathcal {C}\) is the shorter (arc length) circular arc with end-points P 0 and P 1.
- 24.
The circular arc \(\mathcal {C}^c\) is not the set-theoretic complement of \(\mathcal {C}\) with respect to the whole circle \(\mathbb {S}_O\) because \(\mathcal {C}\) and \(\mathcal {C}^c\) overlap in the two common end-points P 0 and P 1.
- 25.
Geometrically, the point \(Q_t\in \mathcal {C}\) is obtained from P t by radial projection from the center O.
- 26.
The opposite sides of a parallelogram have equal lengths. This follows from translation invariance of the distance as shown at the beginning of Section 5.5.
- 27.
Monotonicity changes only if d 2 > 2, and then it does across s = 0, that is, when the sign of s changes from positive to negative; s = 0 corresponds to \(Q_{2/d^2}\) and \(P_{2/d^2}\) being perpendicular to P 0.
- 28.
0 < a, b ≤ 2 implies (a + b)∕(ab) = 1∕a + 1∕b ≥ 1∕2 + 1∕2 = 1.
- 29.
In different (non-axiomatic) developments, this formula is equivalent to the so-called Law of Cosines.
- 30.
It is customary to set |P| = d(P, 0), the distance of a point P from the origin. Algebraically, |P|2 is then the sum of squares of the two coordinates of P. Translation invariance then gives d(P, Q)2 = d(P − Q, 0)2 = |P − Q|2. Using the fact that this is a quadratic form in the coordinates of the points P and Q, the computations above become more familiar.
- 31.
Note that the triple (O, A 0, B 0) is also positively oriented. Recall also that, according to our conventions, \(\mathcal {C}\) is the shorter arc length circular arc with end-points A 0 and B 0.
- 32.
The numeral refers to the G.A. Plimpton Collection in Columbia University.
- 33.
We use here the letters a, b, and c for the coefficients in the typical equation of a line, not to be confused with the same letters occurring in the Pythagorean triples above.
- 34.
This was a problem in the Nordic Mathematical Contests, 1998. Note, however, that the general solution without the upper bound is contained in Sierpiński, W., Elementary Theory of Numbers, 2nd ed. North Holland, 1985.
- 35.
The property of being simple, that is, one-to-one, excludes “self-intersections.” As we consider here only open curves and minimize the arc length, imposing this does not restrict the generality.
- 36.
In somewhat more generality, a curve on the plane is called rectifiable if it has bounded variation, that is, if the supremum above is finite. It can be shown that for a curve of bounded variation, there is always a Lipschitz parametrization as above.
- 37.
The Principle of Least Distance asserts that at C, the angle of incidence and the angle of reflection are equal. This determines the point C uniquely. This principle is usually proved in calculus using a minimization technique. In reality, it is much simpler.
- 38.
For a short history of π, see the author’s Glimpses of Algebra and Geometry, 2nd ed. Springer, New York, 2002.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Toth, G. (2021). Real Analytic Plane Geometry. In: Elements of Mathematics. Undergraduate Texts in Mathematics(). Springer, Cham. https://doi.org/10.1007/978-3-030-75051-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-75051-0_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-75050-3
Online ISBN: 978-3-030-75051-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)