Originally they were awarded for the most important discoveries in the previous year. The time period was changed to five years then shortened to three years. When Victoria came to the throne in 1837 the conditions for the Royal Medals changed to operate on a three year cycle with Mathematics being one of the subjects for which a Medal could be awarded every third year.
In 1850, however, the conditions were changed again as follows:-
... the Royal Medals in each year should be awarded for the two most important contributions to the advancement of Natural Knowledge, published originally in Her Majesty's dominions within a period of not more than ten years and not less than one year of the date of the award, subject, of course, to Her Majesty's approval. ... in the award of the Royal Medals, one should be given in each of the two great divisions of Natural Knowledge.The list of Royal Medal winners is long, and most, of course, are not mathematicians. The list given below is only a partial list to include those whose biographies are contained in this archive.
1826 James Ivory
... for his Paper on Astronomical Refractions, published in the Philosophical Transactions for the year 1823; and his other valuable Papers on Mathematical Subjects.
1833 John F W Herschel
... for his Paper "on the Investigation of the Orbits of Revolving Double Stars," inserted in the Fifth Volume of the Memoirs of the Royal Astronomical Society.
1835 Sir William R Hamilton
... for the papers published by him in the 16th and 17th volumes of the Transactions of the Royal Irish Academy, entitled Supplement to an Essay on the Theory of Systems of Rays, and more particularly for those investigations at the conclusion of the third ...
1836 John F W Herschel
... for his paper on nebulae and clusters of stars, published in the Philosophical Transactions for 1833.
1838 Henry Fox Talbot
... for his papers entitled Researches in the Integral Calculus, published in the Philosophical Transactions for 1836 and 1837.
1838 Michael Faraday
... for his investigations and discoveries contained in the series of experimental researches in electricity published in the Philosophical Transactions, and more particularly for the seventh series, relating to the definite nature of electrochemical action.
1839 James Ivory
... for his paper on the theory of the astronomical refractions published in the Philosophical Transactions for 1838. Part II.
1840 John F W Herschel
... for his paper entitled On the chemical action of the rays of the solar spectrum on preparations of silver, and other substances, both metallic and non-metallic, and on some photogenic processes, published in the Philosophical Transactions for 1840.
1841 George Ohm
... for his researches into the laws of electric currents contained in various memoirs published in Schweiggers Journal, Poggendorffs Annalen and in a separate work entitled "Die galvanische Kette mathematisch bearbeitet".
1842 James MacCullagh
... for his researches connected with the wave theory of light, contained in the Transactions of the Royal Irish Academy.
1844 George Boole
... for his paper on a general method in analysis, published in the Philosophical Transactions of the present year.
1845 George B Airy
... for his paper on the laws of the tides on the cost of Ireland, as inferred from an extensive series of observations made in connection with the Ordnance Survey of Ireland, published in the Philosophical Transactions for the present year.
1846 Urbain Le Verrier
... for his investigations relative to the disturbances of Uranus by which he proved the existence and predicted the place of the new Planet; the Council considering such prediction confirmed as it was by the immediate discovery of the Planet to be one of the proudest triumphs of modern analysis applied to the Newtonian Theory of Gravitation.
1847 John F W Herschel
... for his work entitled Results of Astronomical Observations made during the years 1834, 1835, 1836, 1837 and 1838, at the Cape of Good Hope; being a completion of a telescopic survey of the whole surface of the visible heavens, commenced in 1825.
1856 William Thomson
... for his various chemical researches relating to electricity, to the motive power of hear, and to other subjects.
1859 Arthur Cayley
... for his mathematical papers published in the Philosophical Transactions, and in various English and foreign journals.
1861 James J Sylvester
... for his various memoirs and researches in mathematical science
1868 George Salmon
... for his researches in analytical geometry and the theory of surfaces, published in the Philosophical Transactions, the Transactions of the Royal Irish Academy, and the Quarterly Journal of Mathematics.
1882 Lord Rayleigh
... for his various papers in mathematical and experimental physics.
1883 Thomas A Hirst
... for his researches in pure mathematics.
1884 George H Darwin
... for his mathematical investigations on the rigidity of the Earth, and on tides.
1886 Peter G Tait
... for his various mathematical and physical researches.
1886 Francis Galton
... for his statistical inquiries into biological phenomena
1888 Osborne Reynolds
... for his investigations in mathematical & experimental physics, and on the application of scientific theory to engineering.
1896 Charles V Boys
... for his invention of Quartz Fibres and investigation of their properties, his improvement of the radio-micrometer and investigations with it, for developments in the art of instantaneous photography, and for his determination of the value of the ...
1897 Andrew R Forsyth
... for his contributions to the progress of pure mathematics, and especially for his work in differential equations and the theory of functions.
1899 George F Fitzgerald
... for his contributions to physical science, especially in the domains of optics and electricity.
1900 Percy A MacMahon
... for the number and range of his contributions to mathematical science
1902 Horace Lamb
... for his investigations in mathematical physics.
1904 William Burnside
... for his researches in mathematics, particularly in the theory of groups.
1906 Alfred G Greenhill
... for his contributions to mathematics, especially the elliptic functions and their applications.
1907 Ernest W Hobson
... on the ground of his investigations in mathematics.
1909 Augustus E H Love
... on the ground of his researches in the theory of elasticity and cognate subjects.
1911 George Chrystal
... on the ground of his work in mathematics and physics.
1914 Ernest W Brown
... for investigations in astronomy, chiefly in the lunar theory
1915 Sir Joseph Larmor
... on the ground of his numerous and important contributions to mathematical and physical science.
1916 Hector M Macdonald
... for his contributions to mathematical physics.
1919 James H Jeans
... on the ground of his researches in applied mathematics
1920 Godfrey H Hardy
... on the ground of his researches in pure mathematics.
1928 Arthur S Eddington
... for his contributions to astrophysics.
1929 John E Littlewood
... for his work on mathematical analysis and the theory of prime numbers.
1934 Sydney Chapman
... for his researches in the kinetic theory of gases, in terrestrial magnetism and in the phenomena of the upper atmosphere.
1938 Ronald A Fisher
... for his important contributions to the theory and practice of statistical methods.
1939 Paul A M Dirac
... for the leading part he had taken in the development of the new quantum mechanics.
1941 Edward A Milne
... for his researches on the atmospheres of the earth and the sun, on the internal constitution of the stars, and on the theory of relativity.
1948 Harold Jeffreys
... for his distinguished work in geophysics and his important contributions to the astronomy of the solar system.
1957 William V D Hodge
... in recognition of his distinguished work on algebraic geometry.
1962 Subrahmanyan Chandrasekhar
... in recognition of his distinguished researches in mathematical physics, particularly those related to the stability of convective motions in fluids with and without magnetic fields.
1964 M J Lighthill
... in recognition of his distinguished contributions to knowledge of the flow of compressible gases, and the mathematical theory of distributions.
1966 Frank Yates
... in recognition of his profound and far-reaching contributions to the statistical methods of experimental biology.
1968 Michael F Atiyah
... in recognition of his distinguished contributions to algebraic geometry and to the study of differential equations by the methods of algebraic topology.
1974 Fred Hoyle
... in recognition of his distinguished contributions to theoretical physics and cosmology.
1983 John F C Kingman
... in recognition of his distinguished researches on queuing theory, on regenerative phenomena, and on mathematical genetics.
1985 Roger Penrose
... for his fundamental contributions to the theory of gravitational collapse and to other geometric aspects of theoretical physics.
1992 Simon Donaldson
... for distinguished for his work which has revolutionized our understanding of four-dimensional geometry.
1996 Andrew Wiles
... in recognition of his achievements in number theory, in particular Fermats Last Theorem and his achievements in algebraic number theory particularly the celebrated main conjecture on cyclotomic fields.
2014 Terence Tao
... for his many deep and varied contributions to mathematics, including harmonic analysis, prime number theory, partial differential equations, combinatorics, computer science, statistics, representation theory, and much more.
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