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James MacCullagh (1809 – 24 October 1847) was an Irish mathematician.

Born in Landahaussy, near Plumbridge, County Tyrone, Ireland, but the family moved to Curly Hill, Strabane when James was about 10. He was a fellow of Trinity College Dublin and a contemporary there of William Rowan Hamilton. Although he worked mostly on optics, he is also remembered for his work on geometry; his most significant work in optics was published in the mid-to-late 1830s; his most significant work on geometry On surfaces of the second order was published in 1843.

In Passages from the Life of a Philosopher, Charles Babbage wrote that MacCullagh was "an excellent friend of mine" and discussed the benefits and drawbacks of the analytical engine with him.

MacCullagh's most important paper on optics, entitled "An essay towards a dynamical theory of crystalline reflection and refraction" was presented to the Royal Irish Academy in December 1839. The paper begins by defining what was then a new concept, subsequently, by Maxwell in 1870, called the curl of a vector field. MacCullagh first showed that the curl is a covariant vector in the sense that its components are transformed in the appropriate manner under coordinate rotation. Taking his cue from George Green, he set out to develop a potential function for a dynamical theory for the transmission of light. MacCullagh found that a conventional potential function proportional to the squared norm of the displacement field was incompatible with known properties of light waves. In order to support only transverse waves, he found that the potential function must be proportional to the squared norm of the curl of the displacement field. It was accepted that his radical choice ruled out any hope for a mechanical model for the ethereal medium. Nevertheless, the field equations stemming from this purely gyrostatic medium were shown to be in accord with all known laws, including those of Snell and Fresnel. At several points, MacCullagh addresses the physical nature of an ethereal medium having such properties. Not surprisingly, he argues against a mechanical interpretation of the ether because he readily admits that no known physical medium could have such a potential function resisting only rotation of its elements. "Concerning the peculiar constitution of the ether, we know nothing and shall suppose nothing, except what is involved in the foregoing assumptions [rectilinear vibrations in a medium of constant density]... Having arrived at the value of [the potential function], we may now take it for the starting point of our theory, and dismiss the assumptions by which we were conducted to it." Despite the success of the theory, physicists and mathematicians were not receptive to the idea of reducing physics to a set of abstract field equations divorced from a mechanical model. The notion of the ether as a compressible fluid or similar physical entity was too deeply ingrained in nineteenth century physical thinking, even for decades after the publication of Maxwell's electromagnetic theory in 1864. MacCullagh's ideas were largely abandoned and forgotten until, in 1880, FitzGerald re-discovered and re-interpreted his findings in the light of Maxwell's work.

MacCullagh died in Dublin at his own hand, perhaps depressed by what he saw as the decline of his mathematical powers.

In May 2009, an Ulster History Circle plaque was unveiled at his family tomb at St Patrick's Church in Upper badoney. The plaque was part of events organised by the Glenelly Historical Society to mark his life.