Euclid and Archimedes

Euclid (~ 300 BC)

A parallel figure in the foreground of Raphael’s fresco, on the right is Euclid, from the Greek colony Alexandria in Egypt, by far the most important Greek mathematician, leaning down to demonstrate his mathematical proofs, on what looks like a slate, with calipers, where the students are in discussion, working through the proofs in their heads. Again, this contrast exists between the didactic teaching of a canon and the more learner-centric view of the learner as someone who has to learn by doing and reflecting. 

Elements, in 13 books, is his most famous work, where his theorems and, more importantly, proofs were deduced from axioms. Familiar examples include the proof that the angles of a triangle add up to 180 degrees and Pythagoras’s Theorem. It is this logical rigour that is remarkable, influencing the entire history of mathematics and science. It was used as the main textbook in mathematics for over 2000 years, well into the 20th century and all University students for centuries used this book as part of the quadrivium.

One fascinating feature of Euclid’s Elements, was the first ever algorithm in print, a method to calculate the Greatest Common denominators for any given number, an oft-quoted forerunner for the current age of algorithms.

Beyond this he wrote on the rigour of mathematical proof, conic sections, the geometry of spheres and number theory. In his Phaenomena, Euclid aims at astronomy with a treatment of spherical geometry.

Archimedes (287-212 BC)

Eureka! Is the word most associated with him, where he supposedly submerges a Golden Crown in a bath of water, measuring the displaced volume. The next step, where he divided the mass of the golden crown by its volume, determined whether it was silver or gold. But the story does not appear in any of Archimedes writings.

His reputation rests on his mathematics but also on the practical application of this mathematics. In addition to explanations of levers and fluid mechanics, he is said to have invented the Archimedes Screw for lifting irrigation water, compound pulleys, and many war machines, including an optical device to focus the sun’s rays on invading Roman ships and a crane and claw for sinking ships and improved catapults.

It is his work on circles, spheres and cylinders, parabolas, centres of gravity, law of the lever, curves, conoids, spheroids and floating bodies, along with that famous number ‘pi’, that has ensured his lasting fame. He also appears to have anticipated modern calculus by using a method of exhaustion, increasing the sides of a polygon towards a complete representation of the circle. Archimedes is arguably the greatest of the Greek mathematicians. In the same period, Eraytosthenes (~250BC) used geometry to estimate the circumference of the earth. He noticed that the sun shone down a well in Aswan at midday. On the same day of the year he also measures the shadow of the sun from a column further north in Alexandria, From this he ingeniously calculated the circumference of the earth. 

Influence

Euclid and Archimedes, along with other Greek mathematicians and astronomers put mathematics, mostly geometry, at the heart of the western educational system. It was an indispensable feature for many of the major Greek thinkers who saw it as the foundation for rigorous thinking about the world. They gave mathematics a status in Western thought that has never waned. Its emphasis on geometry, proof and the need for quantitative rigour lies at the heart of later scientific revolutions.

They also made advances in what we would now call engineering, the practical application of science and mathematics into machines and architecture. Beyond this, astronomy also benefited from their mathematical bent.