Short answers, pulled from the story.
The Moscow Papyrus dates between 2000 and 1800 BC and contains a formula for calculating the volume of a truncated pyramid known as a frustum. This document served as a practical tool for building rather than abstract theory. Babylonian clay tablets from 1900 BC such as Plimpton 322 reveal astronomers used trapezoid procedures to compute the position and motion of Jupiter.
Euclid wrote his Elements around 300 BC and introduced mathematical rigor through the axiomatic method. This work arranged existing knowledge into a single coherent logical framework of definition, axiom, theorem, and proof. The Elements remained known to all educated people in the West until the middle of the 20th century.
The earliest extant verbal expression of the Pythagorean Theorem appears in the Shatapatha Brahmana from the third century BC. These sutras contain lists of Pythagorean triples such as 3, 4, 5 and 5, 12, 13 which are particular cases of Diophantine equations. The best conjecture is that these arithmetic rules were part of religious ritual requiring three fires burning at three different altars.
Nikolai Ivanovich Lobachevsky, János Bolyai, and Carl Friedrich Gauss discovered non-Euclidean geometries in the 19th century. Gauss's theorem asserts that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied intrinsically as stand-alone spaces and has been expanded into the theory of manifolds and Riemannian geometry.
Alexander Grothendieck introduced scheme theory from the late 1950s through the mid-1970s. This development allowed mathematicians to solve many difficult problems not only in geometry but also in number theory. Wiles's proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory.