## Synthetic Geometry and Euclid’s Elements

*in*Math

#### Written by Matthew Niemiro

Records on the study of geometry date back thousands of years. Between 2000

and 1600 BC, ancient Babylonians studied the properties of triangles regarding

ratio and proportion, and developed what would become the Pythagorean

Theorem long before Pythagoras (“The Origins of Geometry,” n.d.). Similar

geometric analysis took place independently in other ancient civilizations–the

Egyptians, for example, utilized their relatively advanced understanding of

geometry to study astronomical bodies. Ancient civilizations often documented

their discoveries on clay tablets and papyrus, some of the earliest of which

coming from ancient Egypt. It was not until the 3rd century BC, however, that a truly definitive mathematical text on geometry was written.

Euclid of Alexandria’s Elements (just ‘Elements’, not ‘The Elements’ nor ‘Euclid’s Elements’) is perhaps the most impactful and transformative mathematical text in antiquity. Elements is an index of the extreme, rigorous evaluation of geometry (and, to an extent, elementary number theory) which the ancient Greeks are well known for. The 13 books of Elements are lists of largely synthetic geometrical statements, or geometric ‘axioms’, in order of increasing complexity. These ‘axioms’ are statements on geometric properties which are reasoned to be true, and are then used to support more complex statements.

What is incredibly striking about Elements is that it uses close to no algebraic

equations; it is largely a compilation of synthetic proofs, which by definition do not

use coordinate systems or algebraic equations in their analysis. The text

expands on some otherwise intuitive ideas to articulate astonishingly advanced

proofs where numerical labels and coordinates are completely absent.

It is trivial to say that a line ends at two points–but how can such simple

statements become the basis of advanced spherical geometry? How can it be

used in the analysis of tetrahedrons inscribed in spheres? Such is the level of

rigor to which the ancient Greeks studied geometrical structures, as compiled in

Euclid’s Elements.

Citations:

University of Kentucky (2011). The Origins of Geometry. Retrieved from www.msc.uky.edu/droyster/courses/fall11/MA341/Classnotes/Chapter 01-The Origins of Geometry.pdf

Figure 1. Euclid’s Windmill proof [Image]. Retrieved from https://www.britannica.com/biography/Euclid-Greek-mathematician