John Pell (1 March 1611 – 12 December 1685) was an English mathematician and political agent abroad. He was made Royal Chair of Mathematics at Orange College by the Prince of Orange, and was under the patronage of Sir Charles Cavendish. He was also a compeer and correspondent of René Descartes and Thomas Hobbes.
Early life He was born at Southwick in West Sussex, England. His father, also named John Pell, was from Southwick, and his mother was Mary Holland, from Halden in Kent. The second of two sons, Pell's older brother was Thomas Pell. By the time he was six, they were orphans, their father dying in 1616 and their mother the following year. John Pell the elder had a fine library, which proved valuable to the young Pell as he grew up. He was educated at Steyning Grammar School and entered Trinity College, Cambridge, at the age of 13. During his university career he became an accomplished linguist; even before taking a B.A. degree in 1629, he corresponded with Henry Briggs and other mathematicians. He was promoted by seniority to M.A. in 1630 and taught in the short-lived Chichester Academy set up by Samuel Hartlib. On 3 July 1632 he married Ithamaria Reginald (also rendered as Ithamara or Ithumaria, with the surname Reginolles), sister of the writer and polymath Bathsua Makin. They had four sons and four daughters. Ithumaria died in 1661. Some time before 1669 Pell remarried. Pell spent much of the 1630s working under Hartlib's influence, on topics in the area of pedagogy, encyclopedism and pansophy, combinatorics, and the legacy of Trithemius. By 1638 he had formulated a proposal for a universal language. In mathematics, he concentrated on expanding the scope of algebra in the theory of equations, and on mathematical tables. As part of a joint lobbying effort with Hartlib to find himself support to continue as a researcher, he had his short Idea of Mathematics printed in October 1638. It brought interested responses from Johann Moriaen and Marin Mersenne.
Academic and diplomat His reputation and the influence of Sir William Boswell, the English resident, with the States-General procured his election in 1644 to the chair of mathematics in Amsterdam, after an earlier attempt immediately after Martin van den Hove left for Leiden had failed. From 1644 he worked on a polemical work, against Longomontanus. For this he put in a large effort soliciting help and testimonials: from Bonaventura Cavalieri, his patron Sir Charles Cavendish, René Descartes, Thomas Hobbes, Mersenne, Claude Mydorge, and Gilles de Roberval. It finally appeared as Controversy with Longomontanus concerning the Quadrature of the Circle (1647). In 1646, on the invitation of Frederick Henry, Prince of Orange, Pell accepted a professorship at the new Orange College at Breda, where he taught until 1652. He realised that war between the English and the Dutch was imminent and that he would be in an extremely difficult position in Breda, so returned to England before the outbreak of the First Anglo-Dutch War in July 1652. After his return, Oliver Cromwell appointed Pell to a post teaching mathematics in London. From 1654 to 1658 Pell acted as Cromwell's political agent in Zürich to the Protestant cantons of Switzerland; he cooperated with Samuel Morland, the English resident at Geneva. Pell was described in Zürich by the English traveller Sir John Reresby in about 1656 as "a strange unknown person, not unsuiting the people he was sent to, nor the master [Cromwell] he came from. They are here so strict in their religion, they suffer not the Venetian ambassador to hear mass in his own house." Cromwell wanted to split the Protestant cantons of Switzerland off to join a Protestant League, with England at its head. However Pell's negotiations were long drawn out and he returned to England to deliver his report only shortly before Cromwell's death. He was unable to report as he waited in vain for an audience with the ailing Cromwell. A mathematical pupil and disciple in Switzerland, from 1657, was Johann Heinrich Rahn, known as Rhonius. Rahn is credited with the invention of the division sign (÷) from one of the classic symbols for Obelus; it has also been attributed to Pell, who taught Rahn a three-column spreadsheet-style technique of tabulation of calculations, and acted as editor for Rahn's 1659 book Teutsche Algebra in which it appeared. This book by Rahn also contained what would become known as the "Pell equation". Diophantine equations was a favourite subject with Pell; he lectured on them at Amsterdam. He is now best remembered, if perhaps erroneously, for the indeterminate equation
a x 2 + 1 = y 2 , {\displaystyle ax^{2}+1=y^{2},}
which is known as Pell's equation. This problem was in fact proposed by Pierre de Fermat first to Bernard Frénicle de Bessy, and in 1657 to all mathematicians. Pell's connection with the problem is through Rahn. It consisted of publication of the solutions of John Wallis and Lord Brouncker in his edition of Thomas Branker's Translation of Rhonius's Algebra (1668); added to his earlier editorial contributions, whatever they were, to the 1659 algebra book written by Rahn (i.e. Rhonius). This new edition by Pell of what was essentially Rahn's work included a great deal of additional material on number theory, amounting to a reply to the 1657 book Exercitationes mathematicae by Frans van Schooten. It is also notable for its inclusion of a Table of Incomposits, an early large factor table.
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