Here are historical roots of the
Pentagramma Mirificum. It was discovered by Napier around 1600, and
fully explained by Gauss more than two centuries later, see below.
Section 1.1 of - Fomin and Reading: Root systems and generalized associahedra (Geometric combinatorics, 63-131, IAS/Park City Math. Ser., 13, Amer. Math. Soc., Providence, RI, 2007)
As Fomin points out, it would be a stretch to say that Gauss discovered cluster algebras: the pentagon equation, in its algebraic form, was found by W. Spence in 1809. In the context of Pentagramma Mirificum, the realization that the 5-cycle represents a nontrivial algebraic identity seems to appear first in print by Arthur Cayley; see [A. Cayley, On Gauss's Pentagramma Mirificum, Philosophical Magazine, vol. XLIL (1871), 311-312; The collected mathematical papers of Arthur Cayley. Vol. 7]. In the 20th century, the topic was popularized by H. S. M. Coxeter; see [Kaleidoscopes, pages 85-103] and [Non-Euclidean geometry, Section 12.7]. |
From: Bruce Director: From Plato.s Theaetetus to Gauss.s Pentagramma Mirificum: A Fight for Truth. |

Some further references and sources:

- L. Lewin, Polylogarithms and associated functions, North-Holland Publishing Co., New York-Amsterdam, 1981.
- Problem of the Month November 2006. University of Regina.
- Volumes in hyperbolic 5-space Geometric and Functional Analysis 5 (1995), 640-667.
- Sophie Neustätter: Sphärische Geometrie und Kartographie. Diplomarbeit Wien 2002.
- Böhm, Johannes Ueber Spezialfälle bei der Inhaltsmessung in Räumen konstant Krümmung. Wiss. Z. Friedrich-Schiller-Univ. Jena 5 (1955/56), 157--164.

The pages from the Nachlass of Gauss (Band 3 der Werke):