Concept

Parallel postulate

Overview

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

If a straight line intersects two other 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.

This may be also formulated as:

If a straight line intersects two other straight lines, the two interior angles on the same side add to less than two right angles if and only if the two lines, if extended indefinitely, meet on that side.

The difference between the two formulations lies in the converse of the first formulation:

If a straight line intersects two other straight lines that intersect on some side of first line, the two interior angles on this side add to less than two right angles.

This latter assertion is proved in Euclid's Elements by using the fact that two different lines have at most one intersection point. Conversely this latter assertion implies that two different lines cannot have two intersection points (draw a line passing between the two intersection points and apply the assertion to both sides of this line).

This original formulation of the postulate does not specifically talk about parallel lines; however, its converse and the second formulation imply the existence of parallel lines, since, if the interior angles sum to two right angles, then the two lines do not intersect. Euclid gave the definition of parallel lines in Book I, Definition 23 just before the five postulates.

Euclidean geometry is a geometry that satisfies all of Euclid's axioms, including the parallel postulate and its converse. Non-Euclidean geometries are geometries that do not satisfy the second form of the postulate. A hyperbolic geometry is a geometry that does not satisfy the original postulate. An elliptic geometry is geometry that does not satisfy the converse of the postulate. In particular, in spherical geometry, two lines meet in exactly two points.

The postulate was long considered to be obvious or inevitable, but proofs were elusive. Eventually, it was discovered that inverting the postulate gave valid, albeit different geometries. A geometry where the parallel postulate or its converse does not hold is known as a non-Euclidean geometry. A geometry that is independent of Euclid's fifth postulate and assumes that two different lines have at most one intersection point (i.e., only assumes the modern equivalent of the first four postulates) is known as an absolute geometry (or sometimes "neutral geometry").

Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).

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Between two consecutive states the difference of which is perceptible to our senses, there are always several intermediate states, the difference between which is not perceptible to us; because, in order to be sensuously perceptible, the newly arising state must have reached a certain degree of intensity or of magnitude: it is therefore preceded by degrees of lesser intensity or extension, in passing through which…
Arthur Schopenhauer, On the Fourfold Root of the Principle of Sufficient Reason, and On the Will in Nature
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In Kantian language this is called satisfying equally the law of homogeneity and that of specification. It arises from this peculiar nature of scientific completeness, that the aim of science is not greater certainty—for certainty may be possessed in just as high a degree by the most disconnected particular knowledge—but…
Arthur Schopenhauer, The World as Will and Idea
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For this system, which is called that of _occasional causes_ (because it teaches that God acts on the body at the instance of the soul, and _vice versa_), besides introducing perpetual miracles to establish communication [157] between these two substances, does not obviate the derangement of the natural laws obtaining in each of these same substances, which, in the general opinion, their mutual influence would cause.
Gottfried Wilhelm Leibniz, Theodicy Essays on the Goodness of God, the Freedom of Man and the Origin of Evil
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Parallel postulate

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