In 1923, J. D. Bernal, when still an undergraduate, wrote a brilliant Prize Essay for Emmanuel College, Cambridge, in which he applied quaternions to the structure of crystal lattices. It remained unpublished until 1981, when it was published by Birkbeck College under the title 'The analytic theory of point systems', Occasional Paper No. 1, 1981, Department of Crystallography, Birkbeck College, London. Thanks to the efforts of Alan Mackay, the IUCr has acquired a copy which we are happy to make available to a wider audience through this web site. We are grateful to Birkbeck College for allowing us to do so.