# Gauss's anticipation of quaternions and their relation to congruences

Recently i read the article "Hamilton, Rodrigues, Gauss, Quaternions and Rotations: A Historical Reassessment", which can be found freely on the internet. This article is by far the most comprehensive paper i found on the early history of quaternions: it discusses Euler's glimpses of a quaternions algebra, and Gauss's anticipations of it in his posthomously published 1819 fragment "mutations des raumes" (english: "rotations of space").

The article gives a quite complete assessement of this fragment of Gauss: it lists (on p. 11-12) several aspects of Gauss's "algebra of rotations", which correspond almost exactly to the eight subsections of part I of Gauss's fragment. However, it doesn't comment about subsection 5; in it Gauss writes down, as far as i understand it, several congruences involving the elements (i.e, the quaternion coefficients $a,b,c,d$) of a composition of two spatial rotations (two quaternions) modulo the norm of one quaternion.

More specifically, Gauss defines the following:

$$A = a\alpha - b\beta - c\gamma - d\delta$$ $$B = a\beta + b\alpha - c\delta + d\gamma$$ $$C = a\gamma + b\delta + c\alpha - d\beta$$ $$D = a\delta - b\gamma + c\beta + d\alpha$$ and the norms: $m = a^2 + b^2 + c^2 + d^2, \mu = \alpha^2 + \beta^2 + \gamma^2 + \delta^2$, and then observes that, for example:

$$\frac {{C+Di}}{{A + Bi}} \equiv \frac {{c + di}}{{a + bi}} \pmod m$$

One thing that immediately came to my mind is what is "$i$"? he didn't define it previously, and my second question is: what's the meaning of this congruence? in fact Gauss writes six such congruences (anyone interested can find it in Gauss's werke, volume 8, p.359)

• I just want to acknowledge and encourage the groundwork you are doing with tracing lesser known facts through the original sources. I wish I could be of more help but unfortunately I do not have as much time as I used to. – Conifold Aug 8 at 5:15
• Thanks sincerely Conifold! It's very encouraging to know that people here don't only view me as a snooze user. – user2554 Aug 9 at 8:01