## From Euclid to Minkowski

*The history of non-euclidean geometry*

**-0325 Euclid** – “The Elements” was a series of books on mathematics and geometry that established a system of axioms and postulates forming proofs that became a cornerstone of logical reasoning and modern science. In the geometry of planes (2D spaces), the “parallel postulate” establishes that parallel lines on a flat plane will never intersect.

1077 – “Explanations of the Difficulties in the Postulates of Euclid” by **Khayyam (1048)**

1733 – “Euclides Vindicatus” (Euclid Freed of All Blemish) by Saccheri – compared hyperbolic geometry and elliptical geometry to Euclidean geometry.

**1777 – Gauss** began working on non-euclidean geometry at the age of 15 but never published his work.

1794 – “Elements of Geometry” by **Legendre (1752)**

1823 – Janos Bolyai (1801) wrote a letter to his father, outlining the basics of non-euclidean geometry.

**1829 – non-euclidean geometry** was developed independently by Lobachevsky and Bolyai.

1840 – “Geometrical investigations on the theory of parallels” by **Lobachevsky (1792)**

1854 – “On the hypotheses that lie at the foundations of geometry” lecture by **Riemann (1826)**

1868 – “Essay on the interpretation of Non-Euclidean Geometry” by Beltrami

1871 – “On the So-called Non-Euclidean Geometry” by **Klein (1849)**

1908 – **Minkowski (1864)** reformulated Einstein’s 1905 paper on Relativity using non-euclidean geometry as a four dimensional space-time continuum.