Hamilton did not make one major discovery but two; his characteristic function for the geometry of light, which he extended to dynamics, leading to what is now Hamiltonian mechanics, and the quaternions which played an important part in the emergence of vector analysis. The quaternions themselves having been out of sight for a long time, they can now be found everywhere, from your phone up to the rovers on Mars.
The engraving above, of the steam locomotive Hibernia, shows the state of science a year after Hamilton’s marriage, and two years after the publication of the Third Supplement to an Essay on the Theory of Systems of Rays,* which would lead to Hamiltonian mechanics.
Thus having made his discoveries in a time in which there were no telephones and radios yet, people walked, and travelled by horse, carriage, steam train and steamship, Hamilton’s son reported that “Sir W.” was indifferent to contemporary fame, “arising from his conviction that his belonged to a future age entirely.” He was right, yet little could he know that his work had to wait until quantum mechancis and the dawn of computers before it would start to blossom.
His work being very widely acknowledged now, it is time to brush up his reputation and show him as the happily married genius he was.
* For short descriptions of the original essay and the three supplements see Theory of Systems of Rays.