Euclid's Fifth Postulate is puzzling for several reasons, among them the uncertainty of its precise subject. Following the Fourth Postulate (“all right angles are equal to one another”), Postulate 5 takes a different tone: “That, if [emphasis added] a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.” Several questions arise: why does the postulate begin (and end) with a conditional statement? Is Euclid working towards what he later proves in Book I, Proposition 32, that the three interior angles of a triangle are equal to two right angles? Or is this a statement about the nature of parallel lines (or apparent parallel lines), as suggested by the use of “produced indefinitely?” Above all, why is this “begged for” in the opening postulates?
A postulate is an opening presupposition in an argument, a premise to be accepted by all parties in the course of a debate or logical proof. Fundamentally, it is “something asked for or demanded,” sharing a common Latin root with the English word petition (Oxford English Dictionary, 3rd Edition, 2006). Euclid does not—or cannot—prove the postulates, but asks us to accept them at the beginning of his geometry as matters of common understanding. Postulate 5 is first cited in support of Proposition 29, concerning the equal interior angles created when a straight line falls across two parallel lines. This is followed by Proposition 30, which depends on Proposition 29 to show that lines parallel to the same line are parallel to one another. Although this would seem obvious from both Definition 23 (“Parallel lines”) and Common Notion 1 (“Things which are equal to the same thing are also equal to one another”), Euclid proves this by comparing the angles created by a straight line falling on three parallel lines. To do so, he relies on Proposition 29, which relies on Postulate 5. Every remaining proposition in Book I, except the construction in Proposition 31, will depend on Proposition 29 and Postulate 5 in some manner.
Clearly Postulate 5 plays an important role in Euclid's system, but is it truly necessary as a postulate? Its complexity compared to the other postulates—and the conditional nature of its claims—make it difficult to accept at face value. Consider Euclid's overall treatment of parallel lines: Definition 23 defines parallel straight lines as “straight lines, which being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.” This definition does not, on the surface, tell us anything about the relationship between two parallel lines beyond the fact that they do not meet. They could be long or short lines, close or very distant from one another, but by definition they will not meet, no matter how far they are extended in either direction.
The use of “indeterminate” confuses the concept of parallel lines further. By Definition 23, parallel lines are straight but indefinite in terms of length, the opposite of having the definite and plainly understood limited lengths of line segments. We might recognize particular line segments AB and CD as being parallel within the terms of a proposition; but cannot project and further extend the segments to determine whether they may or may not intersect at some distant point. Postulate 5 addresses this challenge through a consideration of the angles immediately within our perspective: if two lines (which may or may not be parallel) crossed by a third line form interior angles adding up to less than two right angles, the lines will will intersect at some unknown point on the same side as the angles. Euclid is strangely silent about drawing conclusions from this postulate. He does not conclude that the lines are not parallel, nor does he reach the conclusion of Proposition 29 that the lines are parallel (if they do form two interior angles equally two right angles). This strange conditional formulation of the postulate, and its hesitancy to affirm anything but the intersection of the two lines gives the Postulate a sense of specific purpose that might not conform with broader uses throughout the Elements.
If the conditions of Postulate 5 are met, a triangle is created by the three crossing lines. Euclid leaves this conclusion unstated, but it easily follows that if two lines, both intersected by a third line, having interior angles less than two right angles, form an angle at the spot where they meet one another and so enclose the three sides of a triangle. But Euclid is evasive about this conclusion, keeping to the language he employed to define parallels. “[Lines] produced indefinitely...do not meet” (Definition 23), whereas in Postulate 5 the lines will “meet on that side.” In both cases, the lines are indefinitely produced, even beyond the point of comprehension: lines that never touch no matter how far into a plane they are extended or lines conceivably forming a triangle of great dimensions but minuscule magnitude. Euclid has already defined the basic terms of a triangle, but does not fully deal with the relationship between two right angles and the sum of the three angles of triangle until Book I, Proposition 32, just two propositions after he invoked Postulate 5. The proof of Proposition 32 is simple and direct, relying on the identical angles created by parallel lines set up on a straight line—which in turn is largely supported by Postulate 5.
Euclid does hint at this possibility in Proposition 13, where he establishes that the angles created when one straight line is set up on another straight line equal two right angles. It is possible to deduce from this that if the interior angles established in Postulate 5 are less than two right angles than the line segments may not truly stand in straight lines, but it is not possible to prove that they necessarily meet to form a third angle on the side of the interior angles. Proposition 14 expands on this by establishing that if the angles formed when two line segments meet on a third straight line on one side are equal to two right angles, the line segments are themselves in a straight line with each other. However, this cannot be used to deduce their relationship if they do not equal two right angles. Postulate 5 alone speaks to this situation in the absence of the conclusions of Proposition 32.
Proposition 17 comes closer to reaching the same conclusion of the Fifth Postulate, showing that “in any triangle two angles taken together in any manner are less than two right angles.” This touches on the essence of the Postulate in that it accurately describes the relationship of the angles created by three intersecting lines with the given of an existing triange, but lacks the specificity of which angles will be less than two right angles. Obviously they will be the interior angles of the triangle, but not necessarily the two interior angles of the straight line falling on the other two lines as described in Postulate 5. Moreover, Proposition 17 assumes the existence of a triangle, whereas Postulate 5 suggests that if the sum of the two interior angles are less than the sum of two right angles, a triangle will be formed. This distinction is crucial in the formulation of Proposition 29, a reductio ad absurdum proof that posits the interior angles created when parallel straight lines are crossed by a third line are equal to two right angles, a conclusion rejected on the basis of Postulate 5 and Proposition 27.
A final possibility for avoiding Postulate 5 appears in Propositions 27 & 28, where Euclid finally brings together several points to address the angles created when a third line falls on parallel lines. Proposition 27 argues if the alternate angles created when straight line A falls on straight lines B and C are equal, then B and C parallel to one another. Proposition 28 establishes similar relationships for the exterior and interior angles (via Proposition 15). While neither of these propositions specifically address right angles, Proposition 27 relies on a reductio proof similar to Postulate 5, drawing the parallel lines together to create a hypothetical triangle an opposite interior angle equal to the exterior angle. This conclusion is against both the definition of parallel lines and Proposition 16, and thus rejected. Euclid takes the conclusion of Proposition 27 further in Proposition 29, where he shows via Postulate 5 (and without the use of Proposition 27) the several equal angles formed when parallel lines are crossed by a straight line (and the interior angles on the same side are equal to two right angles). While Euclid can logically make numerous claims about equal angles and parallel lines without the use of Postulate 5, he cannot make specific claims about the magnitude of those angles—especially what two alternate angles less than two right angles will produce when their lines are indefinitely extended.
Much of the difficulty returns to what can be known with certainty about parallel lines. Postulate 5 attempts to establish greater certainty by setting forth measurable circumstances whereby lines may be determined to not be parallel, providing a contrast to the declarative “do not meet one another” of Definition 23. But if the Postulate is really about the nature of parallel lines, why doesn't Euclid say so? He simply says that the lines will meet one the side with the interior angles less than two right angles, without identifying the shape produced as a triangle. The absence of both these conclusions suggests that the Postulate is crafted in such a confined manner so as to limit its direct usage to only that of absolute necessity. And despite his heavy reliance on Proposition 29, Euclid only cites the Postulate directly once more in Book I: Proposition 44, concerning the construction of parallelograms.
This little citation lends credence the suggestion that Postulate 5 must be begged for due to the nature of parallel lines. The majority of Book I following Proposition 29 is concerned with parallel lines, and the proof that the sum of the angles of a triangle equal two right triangles is itself completed using the properties of parallel lines taught in Postulate 5. Despite many difficulties, it may be easier for us to grant Euclid's Postulate 5 than other more absolute--but equally questionable--pronouncements regarding the strange and infinite nature of parallel lines.
Submitted 10 July 2014; corrected 17 July 2014.
