James Gregory. Portrait of James Gregory (1638- 1675), Scottish mathematician and astronomer. As a mathematician, Gregory proved that a series, the sum of infinite terms, may have a finite sum. He coined the terms 'convergent series' and 'divergent series'. He provided the first proof of the fundamental theorem of calculus. He also made distinctions between different mathematical functions. As an astronomer, Gregory suggested that transits of Mercury or Venus across the solar disk could be used to measure the distance between the Earth and the Sun. In 1663 he published a telescope design (prior to Newton's) which used curved mirrors to focus light rather than lenses.