Why was Euclid so Great?
Dr. Kim Johnson at The Lukeion Project
He was a mathematician! Euclid of Alexandria (Εὐκλείδης, which means "renowned, glorious"), 325-265 BC, wrote The Elements. This work not only changed the course of mathematics, but it also changed the way people think of logic and argument forever afterwards.
Who was Euclid, and what were The Elements?
The truth is that we don’t really know much about the personal life of Euclid. He came to work in the library at Alexandria around the time Ptolemy II, Hellenistic monarch of Egypt, was expanding it. We don’t know where he was originally from, although he is possibly one of the few Alexandria natives who came to fame at the library.
We have a few stories written many years after Euclid’s life. When he was teaching Ptolemy geometry, the king asked if there was a shorter way. Euclid responded scornfully, “There is no royal road to mathematics.” When one of his students asked, “When will I ever use this?” (a cry common to modern mathematics students), Euclid responded, “Give the boy some money, for he has to see some profit come from this work.” Mostly, we have Euclid’s masterwork, The Elements.
The Elements is a collection of 13 books (really chapters) collecting most of the Greek knowledge of geometry up to Euclid’s time. Euclid himself did not discover most of the theorems in this book. He simply collected and organized them but the way he organized them was unique.
Previously, Greek mathematicians who wanted to prove something would list the assumptions they were making. They ran into the problem experienced by modern geometry students: what you can assume, and when? After all, you wouldn’t want to prove that the angles in a triangle add up to 180 degrees while inadvertently assuming that very thing. This would be circular reasoning and a fallacy. Euclid’s method avoided such problems by starting with almost nothing and assuming only things that he had previously proven.
The Elements’ starting point included definitions, common notions, and postulates. Definitions are important for any practitioner of logic: Before statements and arguments, one must make the terms precise. Common notions were meant to be clear without proof. For example, things which are equal to the same thing are also equal to each other. Postulates were slightly more complicated but self-evident. After making clear his assumptions, Euclid cataloged much of Greek mathematics. Each theorem was based on the theorems stated before. He included the steps of the proof and diagrams (including labels) to make his reasoning clear. He began with theorems about triangles and parallel lines (precisely where modern geometry students start) and ended proving that there are exactly 5 platonic solids, the primes are infinite, and some numbers are irrational.
This is not to say that Euclid was perfect in 200 BC. Over the years his proofs were corrected, and some of the corrected proofs became part of the canon. His postulates were not enough to prove all his theorems so in the 19th century more postulates and notions were added. The most stunning change came in 1832, when Janos Bolyai and Nicolas Lobachevsky proved that if you change the 5th postulate you get an entirely new geometry. Over all his theories held, and The Elements spread far and wide.
What came before Euclid?
Certainly, people did math before Euclid and without Euclid. The Babylonians had a number system in 2000 BC which would not be surpassed in Europe until after the 1200s when place value and the use of zero started overtaking inefficient Roman numerals. The ancient Egyptians clearly had a solid grasp on geometry. Just look at the pyramids which are still a wonder of the world! Indian mathematics predating Euclid were very advanced and included negative numbers, complicated computations, and zero was considered a number. Chinese mathematicians worked out theories of solving systems of equations, solving polynomials, Pascal’s triangle, and the Pythagorean theorem all without Euclid.
Although these mathematicians made great advancements, Euclid still stands out. These other mathematicians weren’t interested in making their mathematics accessible to the public. Perhaps they didn’t think it was necessary. Perhaps they wanted to keep their secrets to guarantee their future employment or the exclusivity of the scribe and priest class. But most of their work was published with the problem, the solution, and perhaps a method for solving it. Euclid was accessible to anyone willing to put in the work.
The other characteristic of mathematics in other cultures without Euclid was that they were interested in showing solutions to specific problems. Although an intelligent reader could generalize, the writings of mathematicians before and without Euclid would use specific values for the lengths of sides, radii, and other calculations. They did not show the general solution.
What changed after Euclid?
First, most of the old geometry books were destroyed. It’s hard to find any geometry books written before Euclid. Euclid’s work is the only one cited for basic theorems about geometry.
Euclid’s The Elements spread everywhere. For example, there are potsherds with references to his work found at a military outpost found near the Nile plus copies of his book from personal libraries in Babylon and Constantinople. Euclid’s work spread to Baghdad in the 700s, England in the 1000s, and eventually to India and China. Euclid’s work was not a secret but something that anyone could work through and discover for himself. Judging by the computations written in the margins of many ancient copies, some readers were more successful than others.
Euclid’s method was adopted for many other mathematical, scientific, and philosophical efforts. Think about the other variations on The Elements that have been written over 2000 years, including the Elements of Theology by Proclus in the 5th century AD to the Elements of Style by Strunk and White in the 20th century. When Descartes wrote “I think, therefore I am,” he was using Euclid’s method of eliminating all extraneous assumptions to reach the most basic statements we can affirm.
Reading Euclid and learning to think logically came to be one of the cornerstones of a good education. From when it became one of the four liberal arts in the Middle Ages until every schoolchild in England had to pass the “pons asinorum.” Knowing Euclid was essential to training one’s brain to think logically. This is why Lincoln studied Euclid and used his reasoning as an example in a debate against Stephen Douglas. This is why Edna St. Vincent Millay wrote, “Euclid alone has looked on beauty bare.” Anyone who builds or measures, who makes arguments from first principles, or uses science built on any mathematics that came after Euclid, is ultimately referring to Euclid. That’s why Euclid is so great.
The Elements concentrated on building knowledge up from the smallest possible number of assumptions. He wasn’t just interested in knowing what was true, but how you could be sure that you knew it. In addition, Euclid didn’t just solve one problem, he solved them all. One of the previous mathematicians might notice that 3^2+4^2=5^2. Euclid proved that for all right triangles, the square of the hypotenuse was equal to the sum of the squares of the other two sides.
In The Lukeion Project course Counting to Computers: Math from the Dawn of Time to the Digital Age, we investigate the connections between the great mathematicians of the past and the mathematics we learn today. Whether you love, hate, or are indifferent to the standard course of math study, if you are interested in learning more about why we do things the way we do them today, how they were invented in the first place, and the people who were involved, come join us Wednesday afternoons this fall!
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