# Definition:Non-Euclidean Geometry

## Definition

**Non-Euclidean geometry** is branch of geometry in which Euclid's fifth postulate does not hold.

## Also see

- Results about
**non-Euclidean geometry**can be found**here**.

## Historical Note

Non-Euclidean geometry was worked by Carl Friedrich Gauss for some years, and by $1820$ had established the main theorems.

However, he kept this all to himself, and it was up to Nikolai Ivanovich Lobachevsky in $1829$ and János Bolyai in $1832$ (independently of each other and Gauss to publish their own work (János Bolyai publishing it as an appendix to *Tentamen iuventutem studiosam in elementa matheosos introducendi* by his father Wolfgang Bolyai).

The reason that Gauss did not publish his own work was because he recognised that the philosophical climate of Germany at the time would have been unable to accept it.

As he wrote to Friedrich Wilhelm Bessel:

*I shall probably not put my very extensive investigations on this subject [ the foundations of geometry ] into publsihable form for a long time, perhaps not in my lifetime, for I dread the shrieks we would hear from the Boeotians if I were to express myself fully on this matter.*

The Boeotians were a tribe of the ancient Greeks, renowned for being of low intelligence.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next):**parallel postulate** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.25$: Gauss ($\text {1777}$ – $\text {1855}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**non-Euclidean geometry** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**non-Euclidean geometry**