James began his education at the school in Castledamph and then the family moved to Strabane so that he could receive a good education. His mathematical talents were soon evident, see :-
In Strabane he was, while very young, placed at the only respectable school at that time in the town. Here his genius soon displayed itself. After school hours he was almost constantly employed in solving mathematical problems...To take up a study of the classics he was sent to another school, this time in Lifford. At the age of fifteen MacCullagh entered Trinity College, Dublin. His undergraduate career was one in which he received the highest grade in almost every examination he took. He graduated in 1829 and, after graduating, he entered the Fellowship examinations. In  his attempt in this examination is related:-
MacCullagh entered for the highly competitive fellowship examination, conducted orally in Latin. He was unsuccessful, a hardly surprising result when it is realised how much cramming was required. However, given his youth and inexperience, his performance was nor discreditable, his mark in mathematics being equal to that awarded to the two successful candidates...Not long after failing the Fellowship examination MacCullagh submitted his first papers for publication. One was a paper on giving geometrical results on conic sections, the other was on light, in particular on double refraction in which he gave a clear construction of the Fresnel wave surface. Hamilton wrote a review of these papers in 1830.
Two years later MacCullagh tried again for a Fellowship but this time, after being told that his first answer to the first mathematics question was wrong he refused to answer any further questions. Not surprisingly he failed but immediately he learnt this he sent a letter containing geometrical theorems on the theory of rotations to the examiners. The results were original but, unfortunately for MacCullagh, had been obtained independently by Poinsot who published them in 1834.
MacCullagh did succeed in obtaining a fellowship in 1832 and he was appointed junior assistant to the mathematics professor in Dublin.
In 1833 Hamilton announced his discovery of conical refraction of light. This was a major discovery and, following Hamilton's announcement, MacCullagh published a note on conical refraction in which he claimed at least partial priority for the discovery of conical refraction. He wrote:-
The indeterminate cases of my own theorems, which, optically interpreted, mean conical refraction, of course occurred to me at the time...MacCullagh was particularly cross since he knew Hamilton had studied his 1830 papers containing these theorems since Hamilton had reviewed them. Hamilton was equally cross that his priority for the discovery of conical refraction was being challenged. MacCullagh was forced to admit, what was clearly the truth, that although conical refraction could be deduced from his theorems he had only made that deduction after Hamilton had announced the discovery. MacCullagh had been close to a great discovery but had just failed to make the final step.
In 1835 MacCullagh published on crystalline refraction and reflection. Later he discovered that Franz Neumann had read a paper on the subject to the Berlin Academy in December 1835 which was published in 1837. The two theories were identical except that Franz Neumann developed the theory much further. MacCullagh wrote:-
Franz Neumann's paper is very elaborate, and supersedes, in a great measure, the design which I had formed of treating the subject more fully at my leisure...Although MacCullagh had priority, Franz Neumann had stolen the glory and again MacCullagh had missed out. He produced several other papers on light, the most important being in 1839 when he applied methods used by Green to study reflection and refraction of waves at a surface. Also in 1839 MacCullagh was made an honorary member of the St Andrews Literary and Philosophical Society which had been founded by David Brewster.
In 1842 at a meeting of the British Association in Manchester, there was a discussion on the wave or particle nature of light in which Hamilton, MacCullagh, Bessel, Jacobi, Peacock, David Brewster and others took part. Later in 1842 MacCullagh was awarded the Copley medal of the Royal Society, a particularly great achievement since Bessel was among those considered for the award.
In February 1843 MacCullagh was elected a fellow of the Royal Society. Also in 1843 MacCullagh published his most important work on geometry, namely On surfaces of the second order which described how surfaces such as the ellipsoid could be generated. Although he produced much less work on geometry than on light, it is his work on geometry which has survived and proved in the end the more important. His work on light was of course of less importance after Maxwell published his electro-magnetic theory of light in 1865.
MacCullagh corresponded with many scientists, in particular with John Herschel and with Babbage. He also met with many scientists, some of which have been mentioned above, but during a visit to Turin in 1840 he was invited to Babbage's apartments and there met, among others, Plana and Menabrea.
On Sunday 24 October 1847 MacCullagh committed suicide in his rooms in Dublin. The reasons for his suicide are hard to determine but the best clue may be in a letter which he wrote to Babbage five years before in which he wrote (see ):-
... I have grown very stupid of late, and regularly fail at everything I attempt. What the reason may be I cannot tell. But I begin to be of Newton's opinion, that after a certain age, a man may as well give up mathematics. ...
Article by: J J O'Connor and E F Robertson