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Synthetic Geometry and Euclid’s Elements

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.

University of Kentucky (2011). The Origins of Geometry. Retrieved from 01-The Origins of Geometry.pdf

Figure 1. Euclid’s Windmill proof [Image]. Retrieved from

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