William Oughtred

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William Oughtred
Wenceslas Hollar - William Oughtred.jpg
William Oughtred engraving by Wenceslaus Hollar
Born5 March 1574
Died30 June 1660(1660-06-30) (aged 86)
EducationEton College
Alma materKing's College, Cambridge
Known forSlide rule
Multiplication "×" sign
Scientific career
InstitutionsKing's College, Cambridge
Notable studentsJohn Wallis
Christopher Wren
Richard Delamain
Seth Ward

William Oughtred (/ˈɔːtərd/ AWT-ed;[1][2] 5 March 1574 – 30 June 1660[2]), also Owtred, Uhtred, etc., was an English mathematician and Anglican clergyman.[3] After John Napier invented logarithms and Edmund Gunter created the logarithmic scales (lines, or rules) upon which slide rules are based, Oughtred was the first to use two such scales sliding by one another to perform direct multiplication and division. He is credited with inventing the slide rule in about 1622.[4] He also introduced the "×" symbol for multiplication and the abbreviations "sin" and "cos" for the sine and cosine functions.[5]

Early life[edit]

The son of Benjamin Oughtred of Eton in Buckinghamshire (now part of Berkshire), William was born there on 5 March 1574/75 and was educated at Eton College, where his father, a writing-master, was one of his teachers.[6] Oughtred had a passion for mathematics, and would often stay awake at nights to learn while others were sleeping.[7] He then attended King's College, Cambridge, where he graduated BA in 1596/97 and MA in 1600, holding a fellowship in the college from 1595 to 1603.[8] He composed a Funeral Ode in Latin for Sir William More of Loseley Park in 1600.[9]

Being admitted to holy orders, he left the University of Cambridge about 1603, when as Master William Oughtred he held the rectoriate of St Mary's Church, Guildford, Surrey.[10] before being instituted as vicar at Shalford in Surrey on 2 July 1605 (at the presentation of the lay patron George Austen, gent.), which he vacated on 18 January 1611.[11] He was presented by Sir (Edward) Randall (lord of the manor) to the rectory of Albury, near Guildford in Surrey and instituted on 16 October 1610,[12] and settled there,[13] serving as rector of Albury for fifty years.[14][15] A primary source for his life is his address "To the English gentrie" in his Just Apologie of c. 1634, which contains autobiographical material.[16]

Marriage and children[edit]

On 20 February 1606, a few months after his institution at Shalford, he married Christsgift Caryll, niece of the Caryll family of Great Tangley Hall at Wonersh[17] in Surrey, of which Lady Elizabeth Aungier (daughter of Sir Francis), wife of Simon Caryll 1607–1619, was matriarch and then dowager until her death in about 1650.[18] The Oughtreds had twelve children, William, Henry, Henry (the first Henry died as a baby), Benjamin, Simon, Margaret, Judith, Edward, Elizabeth, Anne, George, and John. Two of his sons, Benjamin and John, shared Oughtred's interest in instruments and became watchmakers.[19]


Old St Peter and St Paul's Church, Albury, Surrey, where William Oughtred was rector from 1610 to 1660, and where he is buried.

About 1628 he was appointed by the Earl of Arundel to instruct his son in mathematics.[13] He corresponded with some of the most eminent scholars of his time, including William Alabaster, Sir Charles Cavendish, and William Gascoigne.[20][21] He kept up regular contacts with Gresham College, where he knew Henry Briggs and Gunter.[22]

He offered free mathematical tuition to pupils, who included Richard Delamain, and Jonas Moore, making him an influential teacher of a generation of mathematicians. Seth Ward resided with Oughtred for six months to learn contemporary mathematics, and the physician Charles Scarburgh also stayed at Albury; John Wallis, and Christopher Wren corresponded with him.[23] Another Albury pupil was Robert Wood, who helped him get the Clavis through the press.[24]

The invention of the slide rule involved Oughtred in a priority dispute with Delamain. They also disagreed on pedagogy in mathematics, with Oughtred arguing that theory should precede practice.[25][26]

He remained rector until his death in 1660 at Albury, a month after the restoration of Charles II. He was buried in Old St Peter and St Paul's Church, Albury.[27]

Interest in the occult[edit]

According to his contemporaries, Oughtred had an interest in alchemy and astrology.[28] The Hermetic science remained a philosophical touchstone among many reputable scientists of his time, and Oughtred's student Thomas Henshaw copied a Diary and "Practike" given to him by his teacher.[29]

William Lilly, an eminent astrologer, knew Oughtred and claimed in his autobiography to have intervened on his behalf to prevent his ejection by Parliament in 1646;

"About this time, the most famous mathematician of all Europe, Mr. William Oughtred, parson of Aldbury in Surry, was in danger of sequestration by the Committee of or for plundered ministers; (Ambo-dexters they were;) several inconsiderable articles were deposed and sworn against him, material enough to have sequestered him, but that, upon his day of hearing, I applied myself to Sir Bolstrode Whitlock, and all my own old friends, who in such numbers appeared in his behalf, that though the chairman and many other Presbyterian members were stiff against him, yet he was cleared by the major number."[30]

John Aubrey states that (despite their political differences) he was also defended by Sir Richard Onslow. He adds that Oughtred was an astrologer, and successful in the use of natal astrology, saying that he did not know why it should be effective, but believing that some "genius" or "spirit" assisted. According to Aubrey, Elias Ashmole possessed the original copy in Oughtred's handwriting of his rational division of the twelve houses of the zodiac, namely the text which George Wharton inserted in his Almanac for 1659. (The text of the English abstract of The Cabal of the Twelve Houses Astrological by "Morinus" (Jean-Baptiste Morin) appears over Oughtred's name with the date 16 October 1659 in Wharton's publications.[31]) Aubrey suggests that Oughtred was happy to allow the country people to believe that he was capable of conjuring. Aubrey himself had seen a copy of Christopher Cattan's work on Geomancy[32] annotated by Oughtred.[33] He reported that Oughtred had told Bishop Ward and Elias Ashmole that he had received sudden intuitions or solutions to problems when standing in particular places, or leaning against a particular oak or ash tree, "as if infused by a divine genius", after having pondered those problems unsuccessfully for months or years.[34]

Oughtred's name has been mentioned in purported histories of early freemasonry, a suggestion that Oughtred was present at Elias Ashmole's 1646 initiation going back to Thomas De Quincey.[35][36] Oughtred expressed millenarian views to John Evelyn in 1655:

"Came that renowned mathematician, Mr. Oughtred, to see me, I sending my coach to bring him to Wotton, being now very aged. Among other discourse, he told me he thought water to be the philosopher's first matter, and that he was well persuaded of the possibility of their elixir; he believed the sun to be a material fire, the moon a continent, as appears by the late selenographers; he had strong apprehensions of some significant event to happen the following year, from the calculation of difference with the diluvian period; and added that it might possibly be to convert the Jews by our Saviour's visible appearance, or to judge the world; and therefore, his word was, Parate in occursum;[37] he said Original Sin was not met with in the Greek Fathers, yet he believed the thing; this was from some discourse on Dr. Taylor's late book, which I had lent him."[38]


Oughtred's name is remembered in the Oughtred Society, a group formed in the United States in 1991 for collectors of slide rules. It produces the twice-yearly Journal of the Oughtred Society and holds meetings and auctions for its members.[39][40]


Clavis mathematicae, 1652


William Oughtred's most important work is Clavis Mathematicae, The Key to Mathematics, published in 1631 this is a textbook on elementary algebra. Clavis Mathematicae became a classic, reprinted in several editions, this textbook was used by John Wallis and Isaac Newton among others. The book is concise and argued for a less verbose style of mathematics, with a greater dependence on symbols. Drawing on François Viète (though not explicitly), Oughtred also innovated freely with symbols, introducing not only the multiplication sign as now used universally, but also the proportion sign (double colon ::).[41] The book became popular around 15 years later, as mathematics took a greater role in higher education. Wallis wrote the introduction to his 1652 edition, and used it to publicise his skill as cryptographer;[42] in another, Oughtred promoted the talents of Wren.

fgb*Clavis Mathematicae (1631) further Latin editions 1648, 1652, 1667, 169hfv b3; first English edition 1647.

  • Circles of Proportion and the Horizontal Instrument (1632); this was edited by his pupil, William Forster.[43]
  • Trigonometria with Canones sinuum (1657).

Clavis Mathematicae[edit]

The first edition of the Clavis Mathematicae was published in 1631 that consisted of 20 chapters and 88 pages that included algebra and several fundamentals of mathematics.[44] Some changes were then added by Oughtred to the first edition, and a second and third edition were made in 1647 and 1648, having no preface and reducing the book by one chapter.

This book opens up with a discussion of the Hindu-Arabic notation of decimal fractions and later in the book, introduces multiplication and division sign abbreviations of decimal fractions. He also discusses two ways to do long division and introduces the "~" symbol, in terms of mathematics, expressing the difference between two variables.

Circles of Proportion and the Horizontal Instrument[edit]

In the Circles of Proportion and the Horizontal Instrument, Oughtred introduces the abbreviations for trigonometric functions. This book was originally in manuscript before it eventually became published. Also, the slide rule is discussed, an invention that was made by Oughtred which provided a mechanical method of finding logarithmic results.[45]

It is mentioned in this book that John Napier was the first person to ever use to the decimal point and comma, however Bartholomaeus Pitiscus was actually the first to do so.[7]

Trigonometria with Canones sinuum[edit]

Trigonometria contains about 36 pages of writing. In this book, the abbreviations for the trigonometric functions are explained in further detail consisting of mathematical tables.[7]

Slide rules[edit]

Oughtred's invention of the slide rule consisted of taking a single "rule", already known to Gunter, and simplifying the method of employing it. Gunter required the use of a pair of dividers to lay off distances on his rule; Oughtred made the step of sliding two rules past each other to achieve the same ends.[46] His original design of some time in the 1620s was for a circular slide rule; but he was not the first into print with this idea, which was published by Delamain in 1630. The conventional design of a sliding middle section for a linear rule was an invention of the 1650s.[47]

Sun dials[edit]

At the age of 23, Oughtred invented the double horizontal sundial, now named the Oughtred type after him.[48] A short description The description and use of the double Horizontall Dyall (16 pp.) was added to a 1653 edition (in English translation) of the pioneer book on recreational mathematics, Récréations Mathématiques (1624) by Hendrik van Etten, a pseudonym of Jean Leurechon. The translation itself is no longer attributed to Oughtred, but (probably) to Francis Malthus.[49]

Oughtred also invented the Universal equinoctial ring dial.[50]


  1. ^ How to Pronounce: Oughtred Society Retrieved 9 August 2018.
  2. ^ a b Smith, David Eugene (1923). History of Mathematics. 1. p. 392. ISBN 9780486204291.
  3. ^ F. Willmoth, 'Oughtred, William (bap. 1575, d. 1660)', Oxford Dictionary of National Biography (2004).
  4. ^ Smith, David E. (1958). History of Mathematics. Courier Corporation. p. 205. ISBN 9780486204307.
  5. ^ Florian Cajori (1919). A History of Mathematics. Macmillan. p. 157. cajori william-oughtred multiplication.
  6. ^ Wallis, P.J. (1968). "William Oughtred's 'Circles of Proportion' and 'Trigonometries'". Transactions of the Cambridge Bibliographical Society. 4 (5): 372–382. JSTOR 41154471.
  7. ^ a b c Karpinski, Louis C.; Cajori, Florian (1917). "William Oughtred, a Great Seventeenth-Century Teacher of Mathematics". The American Mathematical Monthly. 24 (1): 29–30. doi:10.2307/2972662. hdl:2027/bc.ark:/13960/t6tx40w87. JSTOR 2972662.
  8. ^ "Oughtred, William (OTRT592W)". A Cambridge Alumni Database. University of Cambridge.
  9. ^ 'Funeral ode by William Outhred', Surrey History Centre, ref. 6729/7/129 (Discovery Catalogue).
  10. ^ Church of England Clergy database, Liber Cleri detail, CCEd Record ID: 199392, from British Library Harleian MS 595.
  11. ^ Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Appointment Record ("Owtred") CCEd Record ID: 59030 and Vacancy Evidence Record ("Outhred"), CCEd Record ID: 59115
  12. ^ Church of England Clergy database, Episcopal Register of Thomas Bilson (Winchester), Appointment Record, CCEd Record ID: 59103. The advowson was in the lord of the manor, who was Sir Edward.
  13. ^ a b Chisholm, Hugh, ed. (1911). "Oughtred, William" . Encyclopædia Britannica. 20 (11th ed.). Cambridge University Press. p. 378.
  14. ^ J. and J.A. Venn, Alumni Cantabrigienses Part 1 Vol. III (Cambridge University Press 1924), p. 288 (Internet Archive) (appointment 1610).
  15. ^ "Parishes: Albury", in H.E. Malden (ed.), A History of the County of Surrey, Volume 3 (V.C.H./HMSO, London 1911), pp. 72-77 (British History Online): "was rector from 1610 to 1660".
  16. ^ (W. Oughtred), To the English gentrie, and all others studious of the mathematicks which shall bee readers hereof. The just apologie of Wil: Oughtred, against the slaunderous insimulations of Richard Delamain, in a pamphlet called Grammelogia, or the mathematicall ring, or mirisica logarithmorum projectio circularis (A. Mathewes, London ?1634). Full text at Umich/eebo (Open access). Extracts in F. Cajori (1915) (Further reading).
  17. ^ ODNB, and see Aubrey's Brief Lives, Ed. Oliver Lawson Dick (Ann Arbor, Michigan 1962), pp. 222–224.
  18. ^ 1623 Harleian Visitation of Surrey, Harl Soc. Vol. XLIII (1899), pp. 88–89: cf. C. P. C. wills of John Machell, 1647; Elizabeth Machell, 1650/1656.
  19. ^ "Oughtred biography". www-groups.dcs.st-and.ac.uk.
  20. ^ "Janus: Oughtred, William (? 1574-1660) mathematician". Janus.lib.cam.ac.uk. Retrieved 31 October 2012.
  21. ^ "DSpace at Cambridge: Letter from William Gascoigne to William Oughtred". Dspace.cam.ac.uk. 13 June 2007. Retrieved 31 October 2012. Cite journal requires |journal= (help)
  22. ^ "Loading..." www.compilerpress.atfreeweb.com.[permanent dead link]
  23. ^ Helena Mary Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (1997), p. 42.
  24. ^ Toby Christopher Barnard, Cromwellian Ireland: English Government and Reform in Ireland 1649–1660 (2000), p. 223.
  25. ^ Michelle Selinger, Teaching Mathematics (1994), p. 142.
  26. ^ "The Galileo Project". Galileo.rice.edu. Retrieved 31 October 2012.
  27. ^ "Parishes: Albury", in H.E. Malden (ed.), A History of the County of Surrey, Volume 3 (V.C.H./HMSO, London 1911), pp. 72-77 (British History Online, accessed 6 December 2018).
  28. ^ Keith Thomas, Religion and the Decline of Magic (1973), p. 322 and 452n.
  29. ^ D.R. Dickson, 'Thomas Henshaw and Sir Robert Paston's pursuit of the Red Elixir: an early collaboration between Fellows of the Royal Society', Notes and Records of the Royal Society of London, Vol. 51, No. 1 (Jan., 1997), pp. 57-76, at pp. 67-72.
  30. ^ William Lilly's History of his Life and Times, from the year 1602 to 1681 (Published London 1715), Reprint (Charles Baldwyn, London 1822), pp. 135-37 (Internet Archive).
  31. ^ 'The Cabal of the Twelve Houses Astrological', collected in J. Gadbury (ed.), The Works of that Late Most Excellent Philosopher and Astronomer, Sir George Wharton, bar. collected into one volume (M.H. for John Leigh, London 1683), pp. 189-208. Full text at Oxford/Tcp (open).
  32. ^ La Geomance du Seigneur Christofe de Cattan, Gentilhomme Genevoys. Livre non moins plaisant et recreatif. Avec la roüe de Pythagoras (Gilles Gilles, Paris 1558). Full text (page views) at Internet Archive.
  33. ^ Oughtred may have possessed the English translation by Francis Sparry, The Geomancie of Maister Christopher Catton, a Booke no lesse pleasant and recreative, then of a wittie invention (London 1591).
  34. ^ 'William Oughtred, 1575-1660', in R. Barber (ed.), John Aubrey - Brief Lives: A selection based upon existing contemporary portraits (Folio Society, London 1975), 232-37.
  35. ^ Historico-Critical Inquiry into the Origins of the Rosicrucians and the Free-Masons
  36. ^ E. g. William Wynn Westcott, The Rosicrucians, Past and Present, at Home and Abroad, p. 426.
  37. ^ I.e. "Praeparare in occursum Dei tui, Israel" (Book of Amos, Chapter IV, v. 12): "Prepare to meet thy God, O Israel".
  38. ^ 'Entry for 28 August 1655', in W. Bray (ed.), The Diary of John Evelyn, with a Biographical introduction by the editor, and a special introduction by Richard Garnett, LL.D., 2 vols (M. Walter Dunne, New York and London 1901), I, pp. 305-06 (Internet Archive, Retrieved 5 December 2018).
  39. ^ "The Oughtred Society". The Oughtred Society. Retrieved 18 March 2015.
  40. ^ "Brochure" (PDF). The Oughtred Society. Retrieved 18 March 2015.
  41. ^ Helena Mary Pycior, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra Through the Commentaries on Newton's Universal Arithmetick (1997), p. 48.
  42. ^ "Oxford Figures, Chapter 1: 800 years of mathematical traditions". Mathematical Institute – University of Oxford. 17 September 2007. Archived from the original on 26 October 2012. Retrieved 31 October 2012.
  43. ^ Stephen, Leslie, ed. (1889). "Forster, William (fl.1632)" . Dictionary of National Biography. 20. London: Smith, Elder & Co.
  44. ^ Cajori, Florian (1915). "The Works of William Oughtred". The Monist. 25 (3): 441–466. doi:10.5840/monist191525315. JSTOR 27900548.
  45. ^ Ball, W. W. Rouse (1917). "Review of William Oughtred: a great Seventeenth-century Teacher of Mathematics". Science Progress (1916-1919). 11 (44): 694–695. JSTOR 43426914.
  46. ^ "Slide Rules". Hpmuseum.org. Retrieved 31 October 2012.
  47. ^ "The slide rule – a forgotten tool". Powerhouse Museum Collection. Retrieved 31 October 2012.
  48. ^ "Harvard University – Department of History of Science". Dssmhi1.fas.harvard.edu. Archived from the original on 20 February 2012. Retrieved 31 October 2012.
  49. ^ Heefer, Albrecht. "Récréations Mathématiques (1624) A Study on its Authorship, Sources and Influence" (PDF). logica.ugent.be.
  50. ^ "Royal Museums Greenwich".

Further reading[edit]

External links[edit]