# Definition:Non-Euclidean Geometry/Historical Note

## Historical Note on Non-Euclidean Geometry

Giovanni Girolamo Saccheri was among those attempted to derive Euclid's fifth postulate from the other four, in the process just failing to discover a **non-Euclidean geometry**.

Johann Heinrich Lambert made a number of conjectures regarding non-Euclidean space.

**Non-Euclidean geometry** as a concept in its own right was worked on by Carl Friedrich Gauss for some years, and by $1820$ he had established the main theorems.

However, he kept this all to himself, and it was up to Nikolai Ivanovich Lobachevsky, between $1826$ and $1829$, 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 publishable 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.

In the $1850$s, Bernhard Riemann came up with a different **non-Euclidean geometry** in which no straight line can be drawn through a point parallel to a given straight line.

This sort of **non-Euclidean geometry** is known as Riemannian geometry.

## Sources

- 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):**geometry** - 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):**geometry** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**non-Euclidean geometry**