Gregory was one of the many 17th-century mathematicians who made important contributions to the development of the calculus, although some of his best work remained virtually unknown until long after his death.

Born at Drumoak, Scotland, he studied mathematics at the University of Padua in about 1665 and produced Vera circuli et hyperbolae quadratura (1667; The True Areas of Circles and Hyperbolas). He was particularly interested in expressing functions as series, and he sketched the beginnings of a general theory. It was Gregory who first found series expressions for the trigonometric functions. He introduced the terms ‘convergent’ and ‘divergent’ for series, and was one of the first mathematicians to begin to grasp the difference between the two kinds. Gregory also gave the first proof of the fundamental theorem of calculus.

In addition to his mathematical work Gregory's interests in astronomy led him to do some valuable practical work in optics. He anticipated Newton by recommending a reflecting telescope in his Optica promota (1663; The Advance of Optics). He realized that refracting telescopes would always be limited by aberrations of various kinds. His solution was to use a concave mirror that reflected (rather than a lens that refracted) to minimize these effects. He solved the problem of the observer by having a hole in the primary mirror through which the light could pass to the observer. However, he was unable to find anyone skilled enough actually to construct the telescope.

Gregory held chairs in mathematics at the University of St. Andrews (1669–74) and the University of Edinburgh (1674–75). He died at the age of 37 shortly after going blind.