John Playfair (1748–1819), professor of mathematics and natural philosophy at the University of Edinburgh, is a relatively obscure figure today, best known as the popularizer of James Hutton's theory of geology. However, Playfair was also involved in mathematics for most of his active career, with his most widely distributed publication, Elements of Geometry (1795), shaping the mathematics education for at least thirteen thousand British students during the nineteenth century. This study focuses on the mathematical context surrounding Elements of Geometry. Specifically, after recounting the background of the text, the paper explores the ways in which Playfair's presentation of elementary geometry reflected three understandings of the terms ‘analysis’ and ‘synthesis’, which were intrinsic components of mathematical culture at the turn of the nineteenth century. In one sense, the words denoted differing styles of mathematical practice in Great Britain and in France. In a second sense, the terms evoked contemporary appeals to ancient methods of proof. Finally, ‘analysis’ and ‘synthesis’ were understood in reference to separate approaches to mathematics education. Playfair's appeals to these understandings help reveal how he viewed himself as a mathematician. Overall, then, this study enriches the standard portrait of a professor who straddled the eighteenth and nineteenth centuries.