Euclid's Significance to Mathematics

Euclid

I think Euclid and Euclidean geometry has remained relevant and continually studied throughout history because of the consequential construction of concepts Euclid builds throughout Elements. The first six books, which dealing with plane geometry, attempt to build and demonstrate very basic properties of shapes and geometric elements such as triangles, parallelograms, squares, and rectangles. I believe that Euclid's laying of such fundamental ideas on basic plane geometry is what lures individuals and scholars to constantly refer back to his work. Additionally, as scholars recognized his work as remarkable and respected the vigour, it became the "primary source of geometric reasoning" as stated by van der Waerden; when the mathematics community recognized the clarity of the Elements it created a snowball effect of its popularity.

Referring to Proposition 6 from Heath's translation of Book 1 of the Elements, I believe this is a principle we are taught by our mathematics professors early on in junior high; the idea that a triangle with two equal and opposite angles will have equal sides. However, I feel as though the intuition and moreover, Euclid's demonstration is rarely conveyed to students with thorough engagement. Reading Proposition through Heath's translation creates not only access to the beauty in geometric reasoning, but also logical reasoning where once you challenge a proposition and follow through on the challenge and reach an absurd conclusion you can then validate your original proposition in quite an intuitive way; different from strictly pure geometric proofs.