AUTHOR: EUCLID
DATE: C. 300 BC
The Elements is arguably the most significant work of fundamental mathematics ever written, underpinning the teaching of the discipline until well into the twentieth century. Its basic principles went unchallenged until the emergence of Albert Einstein, and while the latter’s General Theory of Relativity demanded a re-evaluation of Euclid’s tenets, Euclid’s principles remain vital components of mathematical understanding, even if their universality has finally been disproven.
We know very little of Euclid the man other than that he was active in Alexandria, Egypt, during the rule of Ptolemy I (c. 367–282 BC) and that he was seemingly known by (and slightly pre-dated) Archimedes. The Elements comprises thirteen books that investigate:
•Geometry (which comes from the Greek for ‘measurement of the earth’) – the mathematical branch that seeks to address questions of shape, size and space, as well as the relationship between points, lines, curves and surfaces
•Number Theory – the branch of mathematics dealing with the properties and relationships of numbers and integers
Euclid provided definitions and laid out assorted postulates and propositions, which he then sought to prove or disprove in a thoroughly modern, logical and scientific way.
As well as introducing his own concepts and theories, he drew on the work of others, including Plato and several of Plato’s followers, perhaps most notably the mathematician and astrologer Eudoxus of Cnidus (whose works are now lost to us). Pythagoras and Hippocrates of Chios were also obvious influences.
Problems of length, area and volume had long impacted on daily human life. How could one trade wine, for instance, if neither the buyer nor seller could be sure how much drink they were dealing with? Or how could a tribe sensibly divide up a slab of land without knowing its area? From the ancient farmer and trader to the modern-day computer scientist or rocket designer, life makes a lot less sense in the absence of geometry.
People had been wrangling with these questions of geometry for thousands of years before Euclid, with the Ancient Egyptians and Babylonians notable for making strides forward in finding mathematical solutions. In the West, meanwhile, Thales of Miletus was using maths to establish the distance of ships from the shore by the seventh century BC. But it was not until Euclid that the mass of geometrical knowledge was assimilated into a unified, coherent system that set the benchmark of intellectual rigour for mathematicians of all branches for well over two thousand years. Where uncertainty had ruled, Euclid helped define mathematics as a discipline of clarity and certainty.
For example, when introducing his work on plane geometry, Euclid outlined five basic axioms (or postulates):
•A straight line may be drawn joining any two points
•Any straight-line segment can be extended indefinitely in a straight line
•With any straight-line segment, a circle can be drawn having the segment as its radius and one endpoint as its centre
•All right angles are equal to one another
•Where two lines intersect a third so that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough
The reach of his work has been vast, and began extending out into the known world very shortly after he finished it. However, in common with other classical works, it became lost to Western Europe for a number of centuries, only being rediscovered in the early twelfth century when an English monk, Adelard of Bath, made a Latin translation from an Arabic translation. (In the medieval Islamic world, such eminent thinkers as Omar Khayyam were well versed in the work even as it was absent in Europe.) First printed in 1482, The Elements has since been published in some one thousand editions and is thought to be the most widely distributed textbook in history. Such scientific and mathematical luminaries as Nicolaus Copernicus, Galileo Galilei, Johannes Kepler and Sir Isaac Newton all to some extent stood upon Euclid’s shoulders, but his approach also influenced philosophers more generally, from Enlightenment figures such as Thomas Hobbes and the archetypal rationalist, Descartes, to later logicians like Bertrand Russell. Euclid’s reliance on formulating testable theories shaped them all. Mathematician Eric Temple Bell has even likened him to a cowboy trussing up a steer: ‘This is the hog-tie, and it is what Euclid did to geometry.’
MAN OF MANY TALENTS
Omar Khayyam (who lived from 1048 to 1131 in Persia) was a giant of the medieval Islamic world, a monumental intellect whose work ranged from mathematics and philosophy to astronomy, history and literature. However, his fame in the West was only secured much later, when a collection of poems attributed to him, the Rubáiyát of Omar Khayyám, was published in English translation. Although his authorship is disputed (experts can reliably assign only a handful of the verses to him), what is more certain is his brilliance as a mathematician. Among his numerous achievements, he calculated the length of the year with remarkable accuracy, putting it at 365.24219858156 days.
For some two thousand years, Euclid was considered the last word in geometry. Then Einstein proved that space can exist in non-Euclidian forms, as around the mouth of a black hole. But even Einstein acknowledged his predecessor’s extraordinary contribution:
Here for the first time the world witnessed the miracle of a logical system which proceeded from step to step with such precision that every single one of its propositions was absolutely indubitable – I refer to Euclid’s Geometry. This admirable triumph of reasoning gave the human intellect the necessary confidence in itself for its subsequent achievements. If Euclid failed to kindle your youthful enthusiasm, then you were not born to be a scientific thinker.