Talk:History of Topics in Special Relativity/Lorentz transformation (Quaternions)

section: Euler (1771) – Quaternion rotation parameter

I have no access to Euler's original paper, so I don't know whether the following errors are typing errors in reproducing the reference.

In order to make the given equations work following changes are needed:

A. Regarding "The orthogonal transformation in four dimensions was given by him as"

the term RR must read EE

B. Regarding "and he gave a representation of the sixteen coefficients using eight parameters a,b,c,d,p,q,r,s"

matrix element row 1 col 4: term dp must switch sign to -dp

matrix element row 4 col 3: term cs must switch sign to -cs, term dr must switch sign to -dr

matrix element row 3 col 4: term bp must switch sign to -bp

the resulting matrix must be divided by the scalar sqrt[(d^2+c^2+b^2+a^2)*(s^2+r^2+q^2+p^2)]. Then transpose(B).B is the unit matrix which represents the conditions of A.

Note: Alternatively the scalar may be translated into additional requirements for the eight parameters, e.g.:

sqrt(d^2+c^2+b^2+a^2)=1

sqrt(s^2+r^2+q^2+p^2)=1

which reduces the eight parameters to six independent parameters.

178.165.160.154 (discuss) 07:31, 7 September 2021 (UTC)