Tag Archives: euclid

Working Through Book I of Euclid’s Elements

Posted on by
Reply

Recently I re-worked my way through all forty-eight propositions of Book I of Euclid’s Elements. In the book, Euclid offers a captivating introduction to classical geometry, which straddles the world of perfect abstraction on the one hand, yet it also relies upon certain physical principles found in the world around us. For example, in some propositions we experience a kind of imagined motion, or even a gravitational pull that governs Euclid’s visual demonstrations. We use this concept of weight and motion to demonstrate our abstract lines and shapes. In the Elements Euclid serves as a guide while we learn about lines, circles, angles, triangles, and rectilineal figures (all of which rely upon the unproven acceptance of Postulate #5, the “Parallel Postulate”).

As I went through the book this time I was struck by the unfolding plot of the Elements. Insofar as the Elements has a narrative, the plot of Book I begins by introducing us to equilateral triangles and it concludes by proving the equality of all right angles in equilateral figures (triangles and squares). Throughout the book we glean a sense of equality (equilateral triangles and rectilineal figures) while other shapes like circles are merely used to create triangles and rectilineal figures (in other words circles are subordinate to other shapes). Right angles are knowable and equal everywhere, whereas obtuse or acute angles may have varying degrees of distinction -the word mathematics comes from the Greek for ‘the knowable things’ or ‘the art of the knowable things.’ In Book I of Euclid’s Elements, the knowable things share equal attributes.

In Euclid, the concept of spatial area also begins to appear from Proposition #4 onward (“parallellogramic area”). We imagine a sense of transposition that may possible (i.e. that equal triangles may be placed directly on top of one another). What purpose does the concept of spatial area serve for Euclid?

Euclid’s Elements concludes each proposition with the Latin short-script of either Q.E.D. or Q.E.F. Q.E.D. stands for quod erat demonstrandum (or “what was to be shown”) or Q.E.F. or Quod erat faciendum (or “which had to be done”). Euclid used the Greek original of Quod Erat Faciendum (Q.E.F.) to conclude certain propositions that were demonstrations of figures rather than proofs of theorems. For example, Euclid’s first proposition of Book I shows how to construct an equilateral triangle, given one side, and it is concluded with Q.E.F. however most propositions are concluded with Q.E.D.

Lastly, there are by my count 11 reductio ad absurdum proofs (in Latin “reduction to absurdity”) in Book I of the Elements. The first reductio occurs in Proposition #6 (another proposition contains two reductios, as well). The reductios add to the force of the previous propositions by adding a hypothetical converse which, when proven, becomes impossible (“atopon” or literally ‘placeless’ or ‘impossible’ or ‘strange’ or even ‘foreign’). The reductios show us what is truly ‘impossible’ or ‘foreign’ in classical mathematics.

The following images capture my notes on completing the forty-eight propositions of Book I of Euclid’s Elements:

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

On The Puzzling History of Euclid’s Fifth Postulate

Posted on by
Reply

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.