His career included the study of geometrical optics, Fourier analysis and quaternions, the last of which made him one of the founders of modern linear algebra.

[6] The emphasis of Hamilton's early education on languages is attributed to the wish of his father to see him employed by the British East India Company.

During this period, he encountered what he believed to be a logical error in Mécanique Céleste, an observation which led Hamilton to be introduced to John Brinkley, then Royal Astronomer of Ireland.

His tutor there was Charles Boyton, a family friend,[3] who brought to his attention the contemporary mathematics published by the group at the École Polytechnique in Paris.

[5]: 209  Despite having his undergraduate career cut short in this way, he earned degrees in both classics and mathematics (BA in 1827, MA in 1837).

[3] The introductory lectures by Hamilton in astronomy were celebrated; in addition to his students, they attracted scholars, poets, and women.

[24][25] Newly appointed to the Observatory, Hamilton set off on a tour in Ireland and England with Alexander Nimmo, who was coaching him on latitude and longitude.

[26] One call was to Sarah Lawrence's school at Gateacre, near Liverpool, where Hamilton had a chance to assess the calculator Master Noakes.

[27] They visited William Wordsworth at Rydal Mount in September of that year, where the writer Caesar Otway was also present.

[30] When Wordsworth visited Dublin in the summer of 1829, in a party with John Marshall and his family, he stayed at Dunsink with Hamilton.

[29]: 411  On a second tour in England with Nimmo in 1831, Hamilton parted from him at Birmingham, to visit the Lawrence sisters and family on his mother's side in the Liverpool area.

[32][33] He was a Christian, described as "a lover of the Bible, an orthodox and attached member of the Established Church", and as having a "profound conviction of the truth of revealed religion".

His first discovery was in an early paper that he communicated in 1823 to John Brinkley, who presented it under the title of Caustics in 1824 to the Royal Irish Academy.

A fuller understanding awaited the microlocal analysis of the middle of the 20th century,[43] The step from optics to dynamics in the application of the method of "Varying Action" was made in 1827, and communicated to the Royal Society, in whose Philosophical Transactions for 1834 and 1835 there are two papers on the subject.

Hamilton's analysis uncovered a deeper mathematical structure than had been previously understood, in particular a symmetry between momentum and position.

In the Dictionary of Irish Biography David Spearman writes:[44] The formulation that he devised for classical mechanics proved to be equally suited to quantum theory, whose development it facilitated.

The Hamiltonian formalism shows no signs of obsolescence; new ideas continue to find this the most natural medium for their description and development, and the function that is now universally known as the Hamiltonian, is the starting point for calculation in almost any area of physics.Many scientists, including Liouville, Jacobi, Darboux, Poincaré, Kolmogorov, Prigogine[45] and Arnold, have extended Hamilton's work, in mechanics, differential equations and symplectic geometry.

[46] Hamilton's mathematical studies seem to have been undertaken and carried to their full development without collaboration, and his writings do not belong to any particular school.

[47] Hamilton was looking for ways of extending complex numbers (which can be viewed as points on a 2-dimensional Argand diagram) to higher spatial dimensions.

[48] Hamilton looked into the solution of the quintic in the theory of equations, examining the results arrived at by Niels Henrik Abel, George Jerrard and others in their respective research.

Of his investigations into the solutions, especially by numerical approximation, of certain classes of physically-important differential equations, only parts were published, at intervals, in the Philosophical Magazine.

[3] Hamilton introduced, as a method of analysis, both quaternions and biquaternions, the extension to eight dimensions by the introduction of complex number coefficients.

His son William Edwin Hamilton brought the Elements of Quaternions, a hefty volume of 762 pages, to publication in 1866.

As copies ran short, a second edition was prepared by Charles Jasper Joly, when the book was split into two volumes, the first appearing in 1899 and the second in 1901.

Later, in 1864, the newly established United States National Academy of Sciences elected its first Foreign Associates, and decided to put Hamilton's name on top of their list.

[51] A plaque under the Broom Bridge, associated with the discovery of quaternions, was unveiled by Éamon de Valera on 13 November 1958.

It is believed by some modern mathematicians that Hamilton's work on quaternions was satirized by Charles Lutwidge Dodgson in Alice in Wonderland.

In particular, the Mad Hatter's tea party was meant to represent the folly of quaternions and the need to revert to Euclidean geometry.