A nephew, Charles Gregory was professor of mathematics here to 1739 and was succeeded by his son David Gregory (1712-1765) [Turnbull & Bushnell].
The Regius chair was held by Herbert Westren Turnbull (1921-1950), and E.T. Copson (1950-1969).
Napier (1550-1617) was a student in 1563-1565 but didn't take a degree.
Arbuthnot graduated in medicine in 1696.
In 1753, the University awarded a degree to James Short, the noted telescope maker who had moved from Edinburgh to London in 1738. This was done at his request, though he was already an FRS for 15 years. [Wray].
John Playfair (1748-1819) was a student.
John West (1756-1817) entered as a student in 1769. He later became a teacher of mathematics, with James Ivory and John Leslie being among his students. His Elements of Mathematics of 1784 was popular in Scotland, but he found his income inadequate and emigrated to Jamaica in 1784. [Craik, pp.31-40]
David Brewster (1781-1868), inventor of the kaleidoscope (1814, patented 1817, but not adequately and he didn't make any money from it) and first biographer of Newton, was Principal of the United College of the University in 1838-1859. [FRS, 1813. The kaleidoscope certainly existed in some form before Brewster, but he described it in the Phil. Trans. in 1815, patented it in 1816(?) and popularized it through his Treatise on the Kaleidoscope in 1819. He studied polarization of light and received a Rumford medal for this in 1818.] He also was an inventor of the stereoscope - Wheatstone had a similar device and they are considered the joint inventors, though Huygens had described a similar device in 1659 - and of dioptic lenses for lighthouses. A founder of the BAAS. Kt. in 1831.
John Stuart Mill was Lord Rector in 1866. [Crosland, vol. 2, p.116.]
Chrystal was professor in 1877-1879. D'Arcy Wentworth Thompson (1860-1948), of On Growth and Form, was a professor here from 1917 and there was a bust and a display commemorating him and his work in the foyer of the Physics Building - it was not there in 1992.
J.C. Adams was a Professor here before returning to Cambridge.
The foyer of the Physics Building has an exhibition of scientific instruments, including a great astrolabe made and signed by Humphrey Cole, 1575, a mariner's astrolabe made and signed by Elias Allen in 1616 - the only example definitely known to be made in England - and an orrery by Benjamin Cole, c1750. One of the astrolabes was probably brought by James Gregory, c1673. There is also a wave model by Charles Wheatstone, c1840, of which only one other example is known - at the Royal Institution, London. A microscope of Brewster's is also on display [Wray]. However, Alex Craik ["British Libraries #7: The University of St. Andrews Library", BSHM Newsletter 29 (Summer 1995) 31] reports that the Cole and Allen astrolabes have been removed for safe keeping.
In 1552, Cardan (1501-1576) was summoned to St. Andrews and successfully treated John Hamilton, Archbishop and brother of the Regent of Scotland, for asthma (and general dissipation) so severe that his life was feared for. [Guthrie, pp.162-163] says the consultations took place in Edinburgh and Monimail, Fife.
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An extract from The Mathematical Gazetteer of the British Isles created by David Singmaster