Playfair's axiom

Last updated
Antecedent of Playfair's axiom: a line and a point not on the line Point and line.png
Antecedent of Playfair's axiom: a line and a point not on the line
Consequent of Playfair's axiom: a second line, parallel to the first, passing through the point Two Parallel lines.svg
Consequent of Playfair's axiom: a second line, parallel to the first, passing through the point

In geometry, Playfair's axiom is an axiom that can be used instead of the fifth postulate of Euclid (the parallel postulate):

Contents

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point. [1]

It is equivalent to Euclid's parallel postulate in the context of Euclidean geometry [2] and was named after the Scottish mathematician John Playfair. The "at most" clause is all that is needed since it can be proved from the first four axioms that at least one parallel line exists given a line L and a point P not on L, as follows:

  1. Construct a perpendicular: Using the axioms and previously established theorems, you can construct a line perpendicular to line L that passes through P.
  2. Construct another perpendicular: A second perpendicular line is drawn to the first one, starting from point P.
  3. Parallel Line: This second perpendicular line will be parallel to L by the definition of parallel lines (i.e the alternate interior angles are congruent as per the 4th axiom).

The statement is often written with the phrase, "there is one and only one parallel". In Euclid's Elements, two lines are said to be parallel if they never meet and other characterizations of parallel lines are not used. [3] [4]

This axiom is used not only in Euclidean geometry but also in the broader study of affine geometry where the concept of parallelism is central. In the affine geometry setting, the stronger form of Playfair's axiom (where "at most one" is replaced by "one and only one") is needed since the axioms of neutral geometry are not present to provide a proof of existence. Playfair's version of the axiom has become so popular that it is often referred to as Euclid's parallel axiom, [5] even though it was not Euclid's version of the axiom.

History

Proclus (410485 A.D.) clearly makes the statement in his commentary on Euclid I.31 (Book I, Proposition 31). [6]

In 1785 William Ludlam expressed the parallel axiom as follows: [7]

Two straight lines, meeting at a point, are not both parallel to a third line.

This brief expression of Euclidean parallelism was adopted by Playfair in his textbook Elements of Geometry (1795) that was republished often. He wrote [8]

Two straight lines which intersect one another cannot be both parallel to the same straight line.

Playfair acknowledged Ludlam and others for simplifying the Euclidean assertion. In later developments the point of intersection of the two lines came first, and the denial of two parallels became expressed as a unique parallel through the given point. [9]

In 1883 Arthur Cayley was president of the British Association and expressed this opinion in his address to the Association: [10]

My own view is that Euclid's Twelfth Axiom in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, which is the representation lying at the bottom of all external experience.

When David Hilbert wrote his book, Foundations of Geometry (1899), [11] providing a new set of axioms for Euclidean geometry, he used Playfair's form of the axiom instead of the original Euclidean version for discussing parallel lines. [12]

Relation with Euclid's fifth postulate

If the sum of the interior angles a and b is less than 180deg, the two straight lines, produced indefinitely, meet on that side. Parallel postulate en.svg
If the sum of the interior angles α and β is less than 180°, the two straight lines, produced indefinitely, meet on that side.

Euclid's parallel postulate states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. [13]

The complexity of this statement when compared to Playfair's formulation is certainly a leading contribution to the popularity of quoting Playfair's axiom in discussions of the parallel postulate.

Within the context of absolute geometry the two statements are equivalent, meaning that each can be proved by assuming the other in the presence of the remaining axioms of the geometry. This is not to say that the statements are logically equivalent (i.e., one can be proved from the other using only formal manipulations of logic), since, for example, when interpreted in the spherical model of elliptical geometry one statement is true and the other isn't. [14] Logically equivalent statements have the same truth value in all models in which they have interpretations.

The proofs below assume that all the axioms of absolute (neutral) geometry are valid.

Euclid's fifth postulate implies Playfair's axiom

The easiest way to show this is using the Euclidean theorem (equivalent to the fifth postulate) that states that the angles of a triangle sum to two right angles. Given a line and a point P not on that line, construct a line, t, perpendicular to the given one through the point P, and then a perpendicular to this perpendicular at the point P. This line is parallel because it cannot meet and form a triangle, which is stated in Book 1 Proposition 27 in Euclid's Elements. [15] Now it can be seen that no other parallels exist. If n was a second line through P, then n makes an acute angle with t (since it is not the perpendicular) and the hypothesis of the fifth postulate holds, and so, n meets . [16]

Playfair's axiom implies Euclid's fifth postulate

Given that Playfair's postulate implies that only the perpendicular to the perpendicular is a parallel, the lines of the Euclid construction will have to cut each other in a point. It is also necessary to prove that they will do it in the side where the angles sum to less than two right angles, but this is more difficult. [17]

Importance of triangle congruence

The classical equivalence between Playfair's axiom and Euclid's fifth postulate collapses in the absence of triangle congruence. [18] This is shown by constructing a geometry that redefines angles in a way that respects Hilbert's axioms of incidence, order, and congruence, except for the Side-Angle-Side (SAS) congruence. This geometry models the classical Playfair's axiom but not Euclid's fifth postulate.

Transitivity of parallelism

Proposition 30 of Euclid reads, "Two lines, each parallel to a third line, are parallel to each other." It was noted [19] by Augustus De Morgan that this proposition is logically equivalent to Playfair’s axiom. This notice was recounted [20] by T. L. Heath in 1908. De Morgan’s argument runs as follows: Let X be the set of pairs of distinct lines which meet and Y the set of distinct pairs of lines each of which is parallel to a single common line. If z represents a pair of distinct lines, then the statement,

For all z, if z is in X then z is not in Y,

is Playfair's axiom (in De Morgan's terms, No X is Y) and its logically equivalent contrapositive,

For all z, if z is in Y then z is not in X,

is Euclid I.30, the transitivity of parallelism (No Y is X).

More recently the implication has been phrased differently in terms of the binary relation expressed by parallel lines: In affine geometry the relation is taken to be an equivalence relation, which means that a line is considered to be parallel to itself. Andy Liu [21] wrote, "Let P be a point not on line 2. Suppose both line 1 and line 3 pass through P and are parallel to line 2. By transitivity, they are parallel to each other, and hence cannot have exactly P in common. It follows that they are the same line, which is Playfair's axiom."

Notes

  1. Playfair 1846 , p. 29
  2. more precisely, in the context of absolute geometry.
  3. Euclid's elements, Book I, definition 23
  4. Heath 1956 , Vol. 1, p. 190
  5. for instance, Rafael Artzy (1965) Linear Geometry, page 202, Addison-Wesley
  6. Heath 1956 , Vol. 1, p. 220
  7. William Ludlam (1785) The Rudiments of Mathematics, p. 145, Cambridge
  8. Playfair 1846 , p. 11
  9. Playfair 1846 , p. 291
  10. William Barrett Frankland (1910) Theories of Parallelism: A Historic Critique, page 31, Cambridge University Press
  11. Hilbert, David (1990) [1971], Foundations of Geometry [Grundlagen der Geometrie], translated by Leo Unger from the 10th German edition (2nd English ed.), La Salle, IL: Open Court Publishing, ISBN   0-87548-164-7
  12. Eves 1963 , pp. 385-7
  13. George Phillips (1826) Elements of Geometry (containing the first six books of Euclid), p. 3, Baldwin, Cradock, and Joy
  14. Henderson, David W.; Taimiņa, Daina (2005), Experiencing Geometry: Euclidean and Non-Euclidean with History (3rd ed.), Upper Saddle River, NJ: Pearson Prentice Hall, p. 139, ISBN   0-13-143748-8
  15. This argument assumes more than is needed to prove the result. There are proofs of the existence of parallels which do not assume an equivalent of the fifth postulate.
  16. Greenberg 1974 , p. 107
  17. The proof may be found in Heath 1956 , Vol. 1, p. 313
  18. Brown, Elizabeth T.; Castner, Emily; Davis, Stephen; O’Shea, Edwin; Seryozhenkov, Edouard; Vargas, A. J. (2019-08-01). "On the equivalence of Playfair's axiom to the parallel postulate". Journal of Geometry. 110 (2): 42. arXiv: 1903.05233 . doi:10.1007/s00022-019-0496-9. ISSN   1420-8997.
  19. Supplementary Remarks on the first six Books of Euclid's Elements in the Companion to the Almanac, 1849.
  20. Heath 1956 , Vol. 1, p. 314
  21. The College Mathematics Journal 42(5):372

Related Research Articles

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.

<span class="mw-page-title-main">Euclidean geometry</span> Mathematical model of the physical space

Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, Elements. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is proved from axioms and previously proved theorems.

<span class="mw-page-title-main">Euclidean space</span> Fundamental space of geometry

Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's Elements, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any positive integer dimension n, which are called Euclidean n-spaces when one wants to specify their dimension. For n equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics.

<span class="mw-page-title-main">Similarity (geometry)</span> Property of objects which are scaled or mirrored versions of each other

In Euclidean geometry, two objects are similar if they have the same shape, or if one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformly scaling, possibly with additional translation, rotation and reflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each is congruent to the result of a particular uniform scaling of the other.

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geometry arises by either replacing the parallel postulate with an alternative, or relaxing the metric requirement. In the former case, one obtains hyperbolic geometry and elliptic geometry, the traditional non-Euclidean geometries. When the metric requirement is relaxed, then there are affine planes associated with the planar algebras, which give rise to kinematic geometries that have also been called non-Euclidean geometry.

<span class="mw-page-title-main">Spherical geometry</span> Geometry of the surface of a sphere

Spherical geometry or spherics is the geometry of the two-dimensional surface of a sphere or the n-dimensional surface of higher dimensional spheres.

Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.

<span class="mw-page-title-main">Hyperbolic geometry</span> Non-Euclidean geometry

In mathematics, hyperbolic geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:

<span class="mw-page-title-main">Affine geometry</span> Euclidean geometry without distance and angles

In mathematics, affine geometry is what remains of Euclidean geometry when ignoring the metric notions of distance and angle.

Synthetic geometry is geometry without the use of coordinates. It relies on the axiomatic method for proving all results from a few basic properties initially called postulate, and at present called axioms.

In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. Parallel curves are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called skew lines. Line segments and Euclidean vectors are parallel if they have the same direction.

Absolute geometry is a geometry based on an axiom system for Euclidean geometry without the parallel postulate or any of its alternatives. Traditionally, this has meant using only the first four of Euclid's postulates. The term was introduced by János Bolyai in 1832. It is sometimes referred to as neutral geometry, as it is neutral with respect to the parallel postulate. The first four of Euclid's postulates are now considered insufficient as a basis of Euclidean geometry, so other systems are used instead.

<span class="mw-page-title-main">Line (geometry)</span> Straight figure with zero width and depth

In geometry, a straight line, usually abbreviated line, is an infinitely long object with no width, depth, or curvature, an idealization of such physical objects as a straightedge, a taut string, or a ray of light. Lines are spaces of dimension one, which may be embedded in spaces of dimension two, three, or higher. The word line may also refer, in everyday life, to a line segment, which is a part of a line delimited by two points.

Tarski's axioms, due to Alfred Tarski, are an axiom set for the substantial fragment of Euclidean geometry that is formulable in first-order logic with identity, and requiring no set theory. Other modern axiomizations of Euclidean geometry are Hilbert's axioms and Birkhoff's axioms.

<span class="mw-page-title-main">Saccheri quadrilateral</span> Quadrilateral with two equal sides perpendicular to the base

A Saccheri quadrilateral is a quadrilateral with two equal sides perpendicular to the base. It is named after Giovanni Gerolamo Saccheri, who used it extensively in his 1733 book Euclides ab omni naevo vindicatus, an attempt to prove the parallel postulate using the method reductio ad absurdum. Such a quadrilateral is sometimes called a Khayyam–Saccheri quadrilateral to credit Persian scholar Omar Khayyam who described them in his 11th century book Risāla fī šarḥ mā aškala min muṣādarāt kitāb Uqlīdis.

<span class="mw-page-title-main">Sum of angles of a triangle</span> Fundamental result in geometry

In a Euclidean space, the sum of angles of a triangle equals the straight angle . A triangle has three angles, one at each vertex, bounded by a pair of adjacent sides.

<span class="mw-page-title-main">Transversal (geometry)</span> Line intersecting 2 coplanar lines at 2 points

In geometry, a transversal is a line that passes through two lines in the same plane at two distinct points. Transversals play a role in establishing whether two or more other lines in the Euclidean plane are parallel. The intersections of a transversal with two lines create various types of pairs of angles: consecutive interior angles, consecutive exterior angles, corresponding angles, and alternate angles. As a consequence of Euclid's parallel postulate, if the two lines are parallel, consecutive interior angles are supplementary, corresponding angles are equal, and alternate angles are equal.

The exterior angle theorem is Proposition 1.16 in Euclid's Elements, which states that the measure of an exterior angle of a triangle is greater than either of the measures of the remote interior angles. This is a fundamental result in absolute geometry because its proof does not depend upon the parallel postulate.

Foundations of geometry is the study of geometries as axiomatic systems. There are several sets of axioms which give rise to Euclidean geometry or to non-Euclidean geometries. These are fundamental to the study and of historical importance, but there are a great many modern geometries that are not Euclidean which can be studied from this viewpoint. The term axiomatic geometry can be applied to any geometry that is developed from an axiom system, but is often used to mean Euclidean geometry studied from this point of view. The completeness and independence of general axiomatic systems are important mathematical considerations, but there are also issues to do with the teaching of geometry which come into play.

<span class="mw-page-title-main">Parallel postulate</span> Geometric axiom

In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry:

If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

References

(3 vols.): ISBN   0-486-60088-2 (vol. 1), ISBN   0-486-60089-0 (vol. 2), ISBN   0-486-60090-4 (vol. 3).