Euclid’s fifth postulate

Judith Grabiner has a fine book A historian looks back. The Calculus as Algebra and Selected Writings, MAA 2010, in which she explains how Joseph-Louis Lagrange (actually Italian Giuseppe Lodovico Lagrangia, 1736-1813) developed the derivative as algebra rather than with infinitesimals and limits. His method is more complex than my proposal in Conquest of the Plane (COTP) but his intuition is great.

Grabiner also explains how Lagrange wanted to get rid of Euclid’s fifth postulate. This is equivalent to the property that a point can have only one line parallel to a given other line. The postulate is used to show that the sum of angles in a triangle add up to 180 degrees. With respect to the following diagram: take a line through B that is parallel to AC, and see how the angles α and γ are mirrored, so that α + β + γ = 180 degrees.

Euclid: The sum of angles of a triangle is a half plane.

Euclid: The sum of angles in a triangle is a half plane.

This latter proof is very elegant but also creates a perpetual wonder: Howcome do the angles in a triangle cover a half plane ? And why does this depend upon the fifth postulate that caused so much discussion ?

I always wondered whether it might be explained a bit clearer. Perhaps not as elegant but with faster acceptance and better retention. Let us try to see whether the fifth postulate can be replaced by another one with seemingly less dramatic portent.


Three points not on a line define a circle. Alternatively any triangle can be enclosed by a circle.

Circle and triangle

A circle can be defined by three arbitrary points not on one line. Any triangle can be enclosed by a circle.

We can scale triangle and circle up or down to the angular circle with its center at the origin O = {0, 0} and circumference 1. The unit of measurement of angles now is the plane itself. For example a value of a 1/2 means a half plane or a half turn. The angular circle thus has radius r = 1 / Θ, where Θ = 2 π and is pronounced “Archi” from Archimedes. (Check that its circumference Θ r  = 1.)

The angles of a triangle seem to completely exhaust the angular circle. However, angles are measured from the center of the angular circle. Let us draw the diameters from the corners through the center, which gives AA’, BB’ and CC’. We use the letters α, β and γ now for different angles.

Scaled to the angular circle. Diameters drawn through the corners.

Scaled to the angular circle. Diameters drawn through the corners.

There arise three inner isosceles triangles that use the same radius. The corner at A has angle ∠BAC = α + γ. This angle on the circumference associates with ∠BOC at the center (indicated by a tiny arc-sign) with the proper value ∠BOC = Arc[B, C]. Similarly for the other corners B and C.

For the angles at the center we find ∠AOB + ∠BOC + ∠AOC = Arc[A, B] + Arc[B, C] + Arc[C, A] = 1.

Since this can be done for any triangle, we arrive at the following postulate:

(*) For arbitrary corner A there is a proportion f so that ∠BAC =  α + γ = *∠BOC = f * Arc[B, C]

(1) It follows that the sum of the angles in a triangle equals that proportionality factor f too.

∠BAC + ∠ABC + ∠ACB = f (∠BOC + ∠AOC + ∠AOB) = f.

This also gives ∠BAC + ∠ABC + ∠ACB = 2 (α + β+ γ) = f.

(2) Secondly, we can apply the newly found sum rule to the inner triangles too.

For ΔAOB we find ∠AOB + 2 α = f
For ΔBOC we find ∠BOC + 2 β = f
For ΔAOC we find ∠AOC + 2 γ = f

Adding these we find 1 + 2 (α + β+ γ) = 3 f.
With the above: 1 + f = 3 f, or  f = 1/2.

Combining (1) and (2):

(3) The sum of angles in a triangle is 1/2.

(4) An angle that lies on the circumference of a circle is 1/2 of the associated angle at the center of the circle.

For example: You may check for ΔAOB that we find

∠AOB+ 2 α = Arc[A, B] + f (Arc[B, A’ ] + Arc[A, B’ ]).

Using that Arc[B, A’ ] = Arc[A, B’ ]) we find that

∠AOB+ 2 α = Arc[A, A’ ] = Arc[B, B’ ] = 1/2 (a half plane indeed).


This proof strategy has these advantages:

(i) It emphasizes the measurement of angles, originally by plane sections but replaced by equivalent arcs. It shows that the angular circle is a natural way to measure angles. The 360 degrees came about historically because of the 365 days in the year but the plane itself makes more sense as a unit. (While 360 allows easy calculation: now use percentages: 1/2 = 50%.)

(ii) It shows clearly where the factor f = 1/2 comes from. There is a neat distinction between angles on the circumference and the actual measurement at the center.

(iii) Reversing the equivalences, we now have an elegant proof that a point has only one line parallel to another given line. Euclid’s fifth postulate has become dependent upon (*).

(iv) Non-Euclidean geometry arises from adapting (*). When the axioms are applied to a sphere then a constant f = 1/2 doesn’t make sense. It depends upon the kind of non-Euclidean geometry what the replacement postulate would be.

(v) Lagrange attached value to this discussion because scientists up to Einstein took Euclidean space also as a model for space itself. Grabiner op. cit. p261 suggests that ancient geometry was “the study of geometric figures: triangles, circles, parallelograms, and the like, but by the eighteenth century it had become the study of space (…ref…).” Her reference is to Rosenfeld 1988 ch 5. The implication would be that The Elements wouldn’t be a study of space as well ? I find this hard to believe – though I didn’t read that reference. It would seem to me that Euclid already thought that he axiomised a theory of space. A theory of geometric figures and their properties would not make sense if they were not conceptualised as being in space. That the theory implied ideas about space (like: a finite line might have any length) would be so obvious that it wouldn’t need mention. A fish in water would not speak about it. The Elements clearly isn’t a kosmology like the Timaeus. Given the importance of Plato, Euclid et al. likely regarded their findings limited to known space below the higher spheres, and they didn’t need to speculate like Plato on what lay beyond. This mental set-up still implies a theory of (known) space. The shift in the 18th century likely would be that Plato’s speculative kosmology fell away, so that Euclid’s known space started to apply to the whole universe. Be that as it may be, nevertheless, let me refer to COTP p195-197 for a discussion that Einstein might have been a bit ‘off’ on the issue of measurement error. It may well be that Euclid’s axioms actually define our very notion of space. At least, I find it impossible to think in terms of a “curved space” – i.e. I can imagine a sphere only as an object within Euclidean space.

(vi) Let us return to the selection of postulates. Euclid’s approach might well be better than the alternative given here. The postulate of a single parallel line feels rather natural. The proof on the triangle is so elegant that it may well have highest impact. However, Euclid’s set-up is that the master selects the postulates and that we pupils follow his results. Nowadays we might adopt more daring didactics. (a) Indeed, start with the fifth postulate and use the elegant proof that the angles of a triangle add up to a half plane. Allow for the sense of wonder. (b) Discuss the alternative approach of assuming a constant proportional factor f, as above. (c) Discuss advantages and disadvantages. Then allow pupils their own choice. Indeed, this didactic structure has been used in this weblog entry. (d) Finally, clean up the mess. (1) In the first triangle, the corners were labelled clockwise to get the Greek letters in alphabetical order. The subsequent triangles have been properly labelled counterclockwise. (2) It would have been stricter if the isosceles triangles used indexed labels α1 and α2 but I opted for legibility. (3) Discuss the actual fifth postulate that Euclid used. Perhaps the original discussion about it was caused merely because of its needless complex format.


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