On The Puzzling History of Euclid’s Fifth Postulate

At the outset of Euclid’s Elements he offers twenty-three definitions, five postulates, and five common notions (sometimes translated as “axioms”). Of the five postulates, the fifth is the most troubling. It is known as the Parallel Postulate. The word postulate can be roughly translated to mean “request,” “question,” or “hypothesis” (postulat in Latin means “asked”).

The Parallel Postulate is translated from Greek as follows:

“That, if a straight line falling on two straight lines make 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.”

In the picture above, two lines are intersected and their inner angles are less than two right angles, and therefore both lines will meet if extended indefinitely. It defines parallelism. It also deals with concepts of divergence and convergence -the key implication in both concepts is a certain degree of motion.

Why does the Parallel Postulate not occur as a Proposition in Euclid’s Elements? Why is it not, for example, a demonstrated reductio? In some ways the Parallel Postulate begs to have a proof of its claims (note: Euclid’s Proposition 27 in Book I).

The problem with the Parallel Postulate is that a “proof” or demonstration has not been sufficiently made and thus it requires a visual demonstration to understand its claim. The Parallel Postulate requires significant use of the human imagination. We imagine someone drawing two straight and parallel lines and then another line ‘falling’ on the two unparallel lines at an angle that is not perpendicular in a way that both lines will eventually meet. The key term in the Parallel Postulate is indefinitely. In all likelihood, Euclid used the term indefinitely to encourage us to consider a hypothetical exercise in drawing two parallel lines onward without actually doing so -does Euclid’s surface also continue indefinitely? And if so, is the surface indefinitely flat? The crucial distinguishing factor is that Euclid invites us to draw these two parallel lines on a what we might call a mental plane, perhaps not even an existing surface.

Many mathematicians have attempted to prove the Parallel Postulate but to little avail. Ptolemy thought he had proved the Parallel Postulate but Proclus found an error in his proof, and Proclus could not prove the Parallel Postulate either. However, mathematicians have also explored what might happen if the Parallel Postulate was untrue: names like Ibn Al-Haytham, the great poet-mathematician Omar Khayyam, Nasir al-din Al-Tusi, the famous polymath Muhammad ibn Musa al-Khwarizmi, Giovanni Saccheri, Adelard, Descartes, Janos Bolyai, Newton, Leibniz, Carl Gauss, and Nikolai Lobachevsky. The modern shift from the Middle Ages onward was an exploration into the possibilities of geometry. They found that the negation of Euclid’s Parallel Postulate gives rise to new forms of geometry (non-Euclidean geometries). The key distinction relies on the surface upon which the parallel lines are constructed -is it indefinitely flat or is there a curvature to the surface, as is the case with objects in the physical world. In the real world, elliptical geometry better describes shapes that have being, while other forms of geometry are orderly but puzzling, like M.C. Escher’s artwork which displays hyperbolic geometry. Einstein used non-Euclidean geometry to describe the ways in which the space-time continuum becomes warped in the presence of matter in his General Theory of Relativity. This means that the curvature of space implies that straight and parallel lines will, in fact, meet at one point if extended indefinitely.

Thus what began as an ancient quest in search of geometric and Platonic perfection (Pythagoras, Euclid, Proclus etc) was transformed into a project to better understand and map the world around us (Descartes, Newton, Lobachevsky, Einstein etc). The investigation of the true “earth measurement” continues. Defenders of Euclidian geometry argue that Euclid never intended for his Elements to resemble anything existing in the world around us. They say his geometry is pure abstraction in search of perfection. However this poses certain problems because Euclid’s geometry is in fact not pure abstraction. It relies upon a certain understanding and demonstration of postulates in the physical world -i.e. not in some fabled celestial or divine realm. Therefore the question of whether indefinitely parallel lines will ever meet remains a vexing theoretical quandary, and indeed it is worth entertaining the modern position of doubting Euclid and exploring where new geometries take us.

For this reading I used the wonderful translation of Euclid’s Elements by Thomas L. Heath for Green Lion Press. Mr. Heath was a Cambridge scholar who translated Euclid directly from the original Greek in the early 20th century.

On the Definitions, Postulates, and Common Notions of Euclid’s Elements

Euclid’s Elements (“Stoikheîon”) is the foundational text of classical, axiomatic, and deductive geometry (“earth-measurement”). The Elements is composed of thirteen books, each filled with propositions that beautifully unfold a theory of number, shape, proportion, and measurability. The Elements was the essential geomtery textbook for nearly 2,000 years thanks to the preservation efforts of the Byzantines, Arabs, and English. Sadly, the Elements fell out of favor for students in the 20th century and very few, if any, students attempt to summit the extraordinary heights of Euclid in our modern era. The Elements has been cited by every major mathematical and scientific figure including Copernicus, Galileo, Kepler, Newton, Hobbes, Descartes, Spinoza, Whitehead, Russell, Einstein, and so on.

We know almost nothing about Euclid. The only two things we infer about his life, as referenced by ancient sources (primarily Diogenes Laërtius), is that he lived after Plato (died 347 BC) and before Archimedes (287 BC). He worked or perhaps founded a school in Alexandria, Egypt. Thomas L. Heath surmises that Euclid was originally schooled in Athens under the geometric pupils of Plato (in many ways we can see echoes of Plato found in Euclid’s Elements -recall the mathematical instruction of the boy in Plato’s Meno). Take note of a common mistake: Euclid, the author of the Elements, is distinct from Euclid of Megara who appears in Plato’s Theaetetus.

Euclid appears briefly in Archimedes’s On the Sphere and the Cylinder and also in Apollonius’s Conics. There were other “Elements” books circulating in antiquity by Hippocrates, Leo, and Theudius, but Euclid superseded them all and none of the other books have fully survived into the modern day.

Euclid begins his Elements not with a series of “problems” or “equations” like many math modern textbooks but rather with a list of foundational metaphysical claims: Definitions, Postulates, and Common Notions. The Definitions appear first and a general descent occurs. The Postulates follow the Definitions, and lastly we are offered a list of Common Notions. Things that are common occur last in order of importance.

The Definitions are 23 statements (they were later numbered by 16th century editors after the advent of the printing press). The Definitions proceed from small elements to constructions of shapes. They are brief declarations that we can imagine as a response to Socratic questions, “what is…?” The Definitions do not permit a modern conception of the infinite. The first Definition is of a point -an irreducible and indivisible element (“A point is that which has no part”). A point gives us a sense place, perspective, and grounding. A point grants permission to draw a line (“breadthless length”) between two points. Where do we draw these elements? On a surface (“that which has length and breadth only”). A surface is presumed to be flat, unlike modern formulations of elliptical and non-linear geometry (i.e. Lobachevsky). This is evidenced by the final Definition of parallel lines (“straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction”). The assumption is that a) the straight could be produced indefinitely in a hypothetical situation and b) the straight lines are produced on an indefinitely flat plane/surface. This is distinct from modern conceptions of rounded or spherical surfaces upon which to conduct geometric demonstrations. We imagine an ancient geometer demonstrating Euclid’s Definitions in the dirt or on a chalk board.

As the Definitions descend we begin with foundational elements like points and lines (Definitions 1-7), then with Definitions pertaining to proportions between foundational elements like angles (Definitions 8-13), and then Definitions concerning shapes or figures (A figure is defined in Definition 14, Definitions 13-18 concern circles, and Definitions 19-23 concern rectilinear figures). It is worth noting that a plane surface does not appear first in the list of Definitions. Instead human activity (i.e. creating a point and a line) takes precedence over the plane surface. Perhaps Euclid’s Elements was not intended to be translated from the conceptual to the physical world (“earth-measurement”). Perhaps it is meant to be an exploration of the Platonic eidos.

While the Definitions are firm and unquestionable, the Postulates are a series of “requests” or “demands” placed upon the reader. They are a demonstration of the authority or authorship of Euclid. The Postulates do not necessarily deductively follow from the Definitions, rather they are five rules offered by Euclid.

The five Postulates begin with three active requests: first that it is possible to “draw” a straight line between any two points; second that it possible to “produce” a finite straight line; and third that it is possible to “describe” a circle with any center and distance. The descent of the Postulates begins with three active possibilities: ‘drawing’ lines between points in practice and ‘producing’ lines as well as ‘describing’ circles in concept.

The fourth Postulate concerns the equality of all right angles (in other words, there are no modern notions of gradation), and the fifth and final Postulate concerns lines that pass through parallel lines at an angle which will meet if produced indefinitely, and that the intersecting lines will meet at interior angles that are less than two right angles.

Common Notions
The Common Notions are the most democratic of Euclid’s metaphysical claims. They are ideas everyone understands -common to everyone. They are visual, whereas the Definitions and Postulates are more conceptual and analytical. There are five Common Notions: the first four Common Notions concern equality, and the fifth defines the “whole” as greater than the parts (i.e. a triangle is not superseded by its lines or points -it is a whole triangle).

Unlike Aristotle who often begins his books with commonly held opinions and then proceeds into nuanced discussions of greater depth which ultimately yield a higher perspective, Euclid begins his Elements in Platonic fashion -answering Socratic questions as if posed to a geometer -“What is a point?” “What is a line?” “What is a plane surface?” “What is a figure?” Thus, Euclid’s book is as much an examination of the human mind as it is a lesson in mathematics.

For this reading I used the wonderful translation of Euclid’s Elements by Thomas L. Heath for Green Lion Press. Mr. Heath was a Cambridge scholar who translated Euclid directly from the original Greek in the early 20th century.