Neither ancient nor modern: Wallis and Barrow on the composition of continua. Part one: Mathematical styles and the composition of continua

John Wallis and Isaac Barrow were key figures in a transitional period in the development of mathematics in early modern England: their work reveals a tension between the emerging algebraic techniques and the more traditional geometric mode of thought. Both men were among the first professional mathematicians in England. Wallis studied at Cambridge, deciphered Royalist codes for Parliament during the Civil War, and was one of the secretaries to the Assembly of Divines at Westminster. He was rewarded for his support of Parliament with the Savilian Professorship of Geometry at Oxford. Barrow was also a student at Cambridge and, in 1660, was appointed Regius Professor of Greek at Trinity College. He subsequently became Professor of Geometry at Gresham College, before finally becoming the Lucasian Professor of Mathematics at Cambridge. The work of both Wallis and Barrow was at the forefront of English mathematics in the second half of the seventeenth century. But even though both enjoyed very similar educations and careers, their mathematical techniques were quite different. Wallis’s style is usually considered algebraic, while Barrow’s is considered geometric. At the same time each man’s work exhibited a similar tension between tradition and innovation - between the mathematical ideas inherited from the Greeks and the demands of the new methods and problems.