William Oughtred: A Great Seventeenth-Century Teacher of Mathematics by Florian Cajori — Summary & Review

William Oughtred (c.1573–1660) was a dedicated amateur mathematician and Anglican clergyman whose lifelong passion for mathematics quietly influenced the development of the subject in England and beyond. Educated at Eton and King's College, Cambridge, Oughtred combined his clerical duties with intense mathematical study and teaching. Though he published sparingly until later in life, his works, especially the Clavis mathematicae (1631), became foundational texts that introduced innovative algebraic symbolism and methods.

Oughtred’s Clavis mathematicae was notable for its brevity and symbolic economy, emphasizing the 'analytical art' of algebra and geometry. He introduced or popularized symbols such as the multiplication sign (×), the proportion symbol (::), and the use of a dot (·) for ratio, which influenced later mathematicians like John Wallis. His notation differed from modern conventions—he used colons for grouping instead of parentheses, vowels for unknowns, and consonants for known quantities, following Vieta’s earlier system. Although his work omitted imaginary roots and modern exponential notation, it was praised for clarity and conciseness despite being challenging to read.

Oughtred’s mathematical interests extended beyond algebra. He invented the slide rule around 1622, a practical calculating instrument that sparked a bitter controversy with his pupil Richard Delamain over priority and teaching methods. Oughtred himself was cautious about the early use of instruments in teaching, emphasizing rigorous theoretical understanding before practical application, contrasting with Delamain’s more instrument-focused approach.

His teaching was generous and influential. Oughtred taught many pupils for free, including notable figures such as Seth Ward, John Wallis, Christopher Wren, and Sir Charles Scarborough. His personal magnetism and dedication drew students who often lived with him or corresponded extensively. Despite his generosity, the intensity of his teaching sometimes led to mental strain among pupils, reflecting the demanding nature of his mathematical rigor.

Oughtred’s influence extended into trigonometry, where he used early symbolic abbreviations for trigonometric functions and compiled detailed tables with decimal and centesimal divisions, reflecting the contemporary interest in decimal systems. His work anticipated later notation developments, though he did not adopt the exponential notation introduced by Descartes.

Though Oughtred was more esteemed abroad than at home during his lifetime, his legacy was recognized by later mathematicians such as Isaac Newton, who praised his judgment and teaching. Newton’s 1694 letter recommended Oughtred’s work and highlighted the practical importance of mathematics in navigation, echoing Oughtred’s own exhortations.

The book also explores the historical context of Oughtred’s life, including the challenges of seventeenth-century book trade restrictions, political and religious turmoil, and the slow dissemination of mathematical knowledge. It presents a balanced view of Oughtred’s achievements and limitations, acknowledging uncertainties about some anecdotes, the full extent of his unpublished manuscripts, and the unresolved question of his influence on Descartes.

Overall, the book paints a portrait of a passionate, principled mathematician who valued symbolic clarity and rigorous thought, who taught generously, and who helped bridge the gap between earlier algebraic traditions and the emerging modern mathematics of the seventeenth century.