I have seen many times before that Hamilton started off believing he would need a three-dimensional system over the reals in order to describe 3D rotations. He considered numbers of the form $a + bi + cj$, defined summation component-wise and tried to find a suitable multiplication.
My question is: how did he conclude these numbers could not be divided and how did he, from this point, conclude there would need to be four dimensions for the quaternions to work? Any references for further reading are also greatly appreciated!
Thanks in advance!