# William Oughtred

**William Oughtred** ( *AWT-ed*;^{[1]}^{[2]} 5 March 1574 – 30 June 1660^{[2]}) 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]}

Oughtred was born at Eton in Buckinghamshire (now part of Berkshire), and educated at Eton College and King's College, Cambridge of which he became fellow.^{[6]} Being admitted to holy orders, he left the University of Cambridge about 1603, for a living at Shalford in Surrey; he was presented in 1610 to the rectory of Albury, near Guildford in Surrey, where he settled.^{[7]} He was rector of Albury for fifty years.^{[8]} His father was a writing-master at Eton College and taught him as well.^{[9]} Oughtred had a passion for mathematics, as he would stay up most nights and learn, while others were sleeping.^{[10]} A primary source for his life is his address "To the English gentrie" in his *Just Apologie* of c. 1634, which contains autobiographical material.^{[11]}

On 20 February 1606, he married Christsgift Caryll, (niece) of the Caryll family of Tangley Hall at Wonersh^{[12]} 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.^{[13]} 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.^{[14]}

About 1628 he was appointed by the Earl of Arundel to instruct his son in mathematics.^{[7]} He corresponded with some of the most eminent scholars of his time, including William Alabaster, Sir Charles Cavendish, and William Gascoigne.^{[15]}^{[16]} He kept up regular contacts with Gresham College, where he knew Henry Briggs and Gunter.^{[17]}

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.^{[18]} Another Albury pupil was Robert Wood, who helped him get the *Clavis* through the press.^{[19]}

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.^{[20]}^{[21]}

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.^{[22]}

According to his contemporaries, Oughtred had an interest in alchemy and astrology.^{[23]}

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."^{[24]}

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.^{[25]}) Aubrey suggests that Oughtred was happy to allow the country people to believe that he was capable of conjuring. He himself had seen a copy of Christopher Cattan's work on Geomancy^{[26]} annotated by Oughtred.^{[27]} 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.^{[28]}

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.^{[29]}^{[30]} 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*;^{[31]} 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."^{[32]}

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.^{[33]}^{[34]}

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 ::).^{[35]} 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;^{[36]} in another, Oughtred promoted the talents of Wren.

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

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.^{[38]} 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.

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.^{[39]}

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.^{[10]}

*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.^{[10]}

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.^{[40]} 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.^{[41]}

At the age of 23, Oughtred invented the double horizontal sundial, now named the Oughtred type after him.^{[42]} 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.^{[43]}