How Real Is Euclid’s Elements
Around 530BC, the great philosopher and mathematician Pythagoras went to Magna Graecia (in modern day southern Italy). He built a community to teach and study philosophy and the sacred art of geometry. Pythagoras saw geometry as a part of philosophy that would lead humans to be in touch with the true perfection of the Universe.
The Pythagoreans pursued philosophical enlightenment and shared their knowledge of mathematics. But they were not so interested in making a unified and systematic textbook.
It wasn’t until 300BC that Euclid first compiled a complete book of geometry and mathematics called the Elements. Euclid is considered to be the father of geometry, obviously not because he was the first person who studied geometry. This book wan’t all about his discoveries either. In fact, what he did was that he organized the knowledge of previous mathematicians and the Pythagoreans, and added his own proofs and new discoveries in a systematic and consistent manner.
Needless to say, this book laid the framework for our understanding of geometry and mathematics. But what truly made him the father of geometry was how he put it together. Euclid was the first person using rigorous proof to demonstrate outcomes, and the whole book stood as the height of of logical rigor of mathematics.
Euclid started with definitions (such as “A point is that which has no part”), 5 postulates, and 5 common notions. From there, he deduced other statements called propositions with an unbroken chain of proof. The Elements is essentially a 13 volume textbook of 465 propositions, covering plane geometry, mathematics, and solid geometry.
What’s amazing about these propositions is that everything is built on top of one another. All complex propositions are originated from those initial definitions, postulates, and common notions, showing just how far we can go with a few simple ideas. When you read the Elements in order, nothing you encounter hasn’t been previously proven.
For example, Book 1 Proposition 12
To a given infinite straight line, from a given point which is not on it, to draw a perpendicular straight line.
Let AB be the given infinite straight line, and C the given point which is not on it, thus it is required to draw to the given infinite straight line AB, from the given point C which is not on it, a perpendicular straight line.
For let a point D be taken at random on the other side of the straight line AB, and with centre C and distance CD let the circle EFG be described; [Postulate 3]
Let the straight line EG be bisected at H, [Book 1; Proposition 10] and let the straight line CG, CH, CE be joined. [Postulate 1] I say that CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.
For since GH is equal to HE, and HC is common, the two sides GH, HC are equal to the two sides EH, HC respectively; and the base CG is equal to the base CE; therefore the angle CHG is equal to the angle EHC. [Book 1;Proposition 8]
And they are adjacent angles. But, when a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands. [Definition 10]
Therefore CH has been drawn perpendicular to the given infinite straight line AB from the given point C which is not on it.
Abraham Lincoln was a huge fan of Euclid’s Elements. According to him, it helped him understand what rigorous proof was when he was a self-educated lawyer. He continued reading the Elements to fine tune his mind even after he became a president. There was a conversation with Lincoln about his self education which was published in The Independent in 1864.
In the course of my law reading I constantly came upon the word “demonstrate”. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, “What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?” I consulted Webster’s Dictionary. They told of “certain proof”, “proof beyond the possibility of doubt”, but I could form no idea of what sort of proof that was.
At last I said, “Lincoln, you never can make a lawyer if you do not understand what demonstrate means.” I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.
Despite the huge success of the Elements, there are some dispute about the postulates Euclid first put forward.
1. To draw a straight line from any point to any point.
2. To produce a finite straight line continuously in a straight line.
3. To describe a circle with any centre and distance.
4. That all right angles are equal to one other.
5. 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.
The first four postulates are simple and straightforward. The fifth one essentially means, if you draw a line with two other lines cross it, and their interior angles add up to less than 180 degrees, those two lines will eventually meet at some point. If those two angles add up to 180 degrees, the two lines would never meet. That’s what we call parallel lines. So the Postulate 5 is also known as the parallel postulate.
The problem is that it doesn’t sound so obviously true. It feels more like a proposition rather than a postulate. If it can be proven, all of geometry would be truly consistent and complete. Many mathematicians over the centuries have made attempts to prove the fifth postulate based on the other four, but all failed. They began asking what would happen logically if Postulate 5 were not true.
Mathematicians introduced the case where parallel lines could curve away from each other when the planes they were on were not flat. For example, the lines of longitude on a globe are parallel. But when extended very far, they would meet at the poles. In this concept, all Euclidean definitions, postulates, and propositions break down. Because of the curved surface, you simply can’t draw a straight line, and there are more than one line between two points.
Because flat surface behaves completely different than positively and negatively curves surfaces. This gave rise to entire alternative geometries known as non-Euclidean geometries. The main difference depends on the curvature of the surface upon which the lines are constructed.
In 1915 Albert Einstein proposed General Relativity, where we have to see space and time as one rather than two separate concepts. He described how space-time is curved and warped, and the curvature of space-time is affected by gravity. Accepting this new way of thinking means that we have to accept that we live in a non-Euclidean universe.
Postulate 5 isn’t real. Euclidean geometry, that gave rise to calculus and Newtonian physics, doesn’t represent the world as it actually is. But the reasoning method that Euclid had initiated has remained the central guiding principle of mathematics for over two millennia. It shaped how we see mathematical logic and rigorous proof today. In that sense, it was a huge achievement.
