Geometry (which comes from the Greek for ‘measurement of the Earth’) is the branch of mathematics that seeks to address questions of shape, size and space, as well as the relationship between points, lines, curves and surfaces. Ancient humans had long conjured with problems of length, area and volume within the context of everyday life – how could one trade, for instance, olive oil without some concept of its volume? Or how could land be allocated among a tribe without a grasp of its area?
There is evidence that from around 2000 BCE there was a concerted effort among the Egyptians and Babylonians to apply formal mathematical ideas to solving what we now understand to be geometric problems. By the 7th century BCE Thales of Miletus was spearheading the Western tradition, using mathematical theorems to calculate, for instance, the distance of ships from the shore. But it would be a further four centuries or so before the Greek mathematician Euclid laid the foundations of the modern discipline of geometry. His landmark work, the thirteen-volume Elements, took all hitherto established geometric principles and assimilated them into a unified, coherent system. He adopted an axiomatic approach – he first established a small set of axioms and from those deduced a great many more propositions and theorems, which he then sought to prove empirically. In this way, he constructed a comprehensive system of logically deduced geometric knowledge. It set a benchmark of intellectual rigour for mathematicians of all branches for well over two thousand years.
Euclid’s five basic axioms
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.
Euclid’s logical reasoning of geometric truths, as noted by Einstein millennia later, was without precedent, giving human intellect the necessary confidence for its subsequent achievements.