## William Oughtred

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## William Oughtred

# William Oughtred

Anglican clergyman William Oughtred (1574-1660) is considered one of the world's great mathematiciansdue to his writings on the subject and his invention of the logarithmic slide rule.

Although William Oughtred was by profession an Anglican clergyman, he devoted many years of his life to expanding human understanding in the areas of algebra and calculus as well as to teaching mathematics to gifted students. Oughtred was the author of several books on mathematics and has also been credited by most historians with inventing both the linear and circular slide rules. His innovations extended to the use of many unique mathematical shorthand notations, including the notation "X" for multiplication and "::" for proportion.

## Raised in Academic Environment

Oughtred was born in Eton, Buckinghamshire, England, on March 5, 1574. His father, Benjamin Oughtred, was a scholar who taught writing at Eton School, and through Benjamin's connections the younger Oughtred was educated as a king's scholar at Eton. At age 15 he entered King's College of Cambridge University and became a fellow there in 1595. Oughtred went on to receive his bachelor's degree from King's College, Cambridge, in 1596, followed by a masters of arts degree four years later. Despite the fact that Oughtred's studies at Cambridge consisted predominately of philosophy and theology, as early as age 12 he had demonstrated an extraordinary interest and talent in all things mathematical. As a college student, he had built
on the rudimentary mathematical study provided to him at Eton, studying late into the night after completing his required regular studies. By the time of his graduation from Cambridge, Oughtred had already completed his first work, titled *Easy Method of Mathematical Dialling.*

In 1603 the 29-year-old Oughtred was ordained an Episcopal minister, a common and well-respected career option for an educated man. Applying to the church soon afterward, he gained an appointment as vicar of Shalford in 1604. In 1610 Oughtred was promoted to a position as rector of Albury, near Guilford, Surrey, in which post he served at an annual salary of 100 pounds. During his first years at Albury Oughtred married and set about tending to his parish. Despite the fame he would eventually acquire as a well-known mathematician, he remained dedicated to his flock and held his position as rector of Albury for nearly half a century, until his death in 1660.

Although never formally trained in mathematics, Oughtred clearly had a genius for the subject. Through his writings, he quickly gained renown as a mathematician and soon began to divide the time left to him after his church duties between personal study and the instruction of others. During the 1620s he began to take on as private pupils young men interested in the study of mathematics. These students—among whom were future mathematicians Richard Delamain and John Wallis as well as Christopher Wren, the future architect of St. Paul's Cathedral—shared the home and hospitality of their teacher during their mathematical studies. Eager to impart his mathematical knowledge to these brilliant young minds, Oughtred refused payment, maintaining that he was adequately provided for by his salary as a clergyman. A small man with black hair and a quick, penetrating gaze, he became known for impatiently etching mathematical diagrams in the dust that settled on tables and floors. It was not unusual, in the Oughtred home, to find its owner dressed and awake in the middle of the night while hard at work solving a mathematical problem. On his bed he had permanently affixed an ink-horn, while on the nightstand nearby a candle and tinderbox lay in easy reach, ready for the many nights when a mathematical quandary would demand a solution before Oughtred would allow himself to sleep.

In 1628 Oughtred became math tutor to Lord William Howard, son of the earl of Arundel. Desiring a suitable text to supplement his instruction of the young aristocrat, Oughtred wrote out, in summary form, all that was currently known about arithmetic and algebra. Pleased by the mathematician's efforts on behalf of his son, the earl of Arundel became a patron of Oughtred's and encouraged the rector of Albury to publish his work. The 88-page *Arithmeticae in numeris et speciebus instituto … quasi clavis mathematicae est* —known more commonly as *Clavis mathematicae* —was first published in Latin in 1631. Despite its condensed format, the book quickly drew interest from Oughtred's fellow mathematicians. By the time the second edition of the work was released in 1658, its author's reputation had been cemented in the larger community of European scientists.

In his *Clavis mathematicae* Oughtred describes the Hindu-Arabic system of mathematical notation, sets forth the theory of decimal fractions, and includes a detailed discussion of algebra. Throughout the work he incorporates a number of mathematical shorthand notations he had devised as a way to denote powers, relationships, ratios, and the like. While much of Oughtred's mathematical shorthand was rejected by readers as being too complicated, two of his symbols—"X" for multiplication and "::" for proportion—have gone on to become part of universal mathematical shorthand, along with those of contemporary mathematician and scientist Thomas Harriot (circa 1530-1621). Although Oughtred utilized the notation π as one of his symbols, its use signified only the circumference of a circle, not the ratio of the circumference to the diameter as it has come to denote.

## Developed Logarithmic Slide Rule

The logarithmic slide rule was designed in response to the demands of the scientific renaissance that overtook Europe during Oughtred's lifetime. The astronomical calculus that grew from the work of such men as German astronomer Johann Kepler (1571-1630) and which would appear throughout the work of English scientist Sir Isaac Newton (1624-1727) demanded a means by which the multiplication and division of both extremely small and extremely large numbers could be performed quickly. These scientific and technical calculations were performed with ease using logarithms, which raise or reduce one number to an abbreviated form through the use of exponents.

The invention of logarithms is usually credited to Scottish mathematician and inventor John Napier, baron of Merchiston (1550-1617), who described his invention in 1614 in *Logarithmorum canonis descriptio,* although Swiss watchmaker and mathematician Justus Byrgius (1552-1633) also compiled such a system of mathematical shorthand. Napier's invention was simplified by a colleague at the University of London, professor Henry Briggs (1561-1631), who suggested that the system be designed in base 10 rather than Napier's base "e." Logarithms paved the way for the expanded scientific revolution that followed, allowing that complex operations of products and quotients be completed using simpler additions and subtractions. Their use continued until the advent of the digital calculator and the electronic computer of the twentieth century.

The use of logarithms immediately suggested an instrument that could speed calculations, and that instrument was the slide rule, an analogic calculator that through its mechanism allows for the processing of the variable data represented by logarithms. In 1620 astronomer and mathematician Edmund Gunter (1581-1626) devised "Gunter's Line," a two-foot-long ruler marked with a logarithmic scale. For operations such as the multiplication or division of numbers to several places, lengths along the ruler that are equivalent to the logarithms of the relevant numbers are added and subtracted using a pair of calipers and the result converted back to numeric form through the use of the logarithmic table. Oughtred is believed to have designed the first linear slide rule after less than a year spent wrestling with Gunter's Line and its calipers. Using two rules placed parallel to one another and connected, the position of the numbers relative to each other could now be used to calculate the desired results. By discarding the calipers, Oughtred created the prototype of the modern slide rule.

In its earliest manufactured form slide rules were made of wood, ivory, and even bamboo. They also were designed in several versions: Oughtred's linear and circular versions came first, followed by a cylindrical version, each version adapted for a particular academic discipline. The slide rule quickly gained prominence as a calculating device in every field of science and technology, from astronomy to topography to chemistry to mechanical engineering. However, it was not until the end of the eighteenth century that its importance was made clear by inventor James Watt (1736-1819), who revalued it as a tool of the Industrial Revolution. Demand for slide rules became such that by 1850 they had supplanted the use of Galileo's compass of proportions, an instrument initially intended for military use. In 1850 French army officer Victor Mayer Amdée Mannheim (1831-1906) introduced a transparent slab movable cursor; other modifications and improvements continued to be introduced in the decades that followed, resulting in the slide rule of the twentieth century.

## Later Career Overshadowed by Controversy

The positive reception of his *Clavis mathematicae* within the scientific community prompted Oughtred to write several other books on mathematics. His 1632 work, titled *Circles of Proportion and the Horizontal Instrument,* described both a sundial and a circular form of slide rule that operated like Oughtred's linear slide rule: it was constructed using two concentric rings, one seated inside the other and both of which were inscribed with calibrated logarithmic scales. Ironically, this concentric slide rule, which Oughtred designed for use as a navigational instrument, had been described in a book titled *Grammelogia; or, The Mathematical Ring* published in 1630 by Oughtred's former student, Richard Delamain. Credit for the invention of the circular slide rule was claimed by both teacher and pupil, resulting in an enmity that lasted for the rest of Oughtred's life. Despite the likelihood that Oughtred and Delamain each individually devised the instrument, history has ultimately granted Oughtred credit for the circular slide rule.

During the final decades of his life Oughtred published six more books, among them 1657's *Trigonometria,* which supplements its discussion of two-and three-dimensional triangles with symbolism and tables setting forth the values of trigonometric and logarithmic functions to seven places. His 1651 work, *The Solution of All Spherical Triangles,* discusses the means by which the relative measurements of three-dimensional triangles can be determined; other books by Oughtred cover such subjects as the methods by which the position of the sun can be calculated and a discussion of the art of watchmaking.

Oughtred lived during tumultuous times in England. A staunch supporter of the English crown, he was shocked by the execution of the unpopular King Charles I in January of 1649. Like many who supported the cause of Charles I's son, the Prince of Wales (later Charles II), Oughtred was viewed with suspicion by the Presbyterian-influenced government that desired to take the place of the monarchy through the will of its leader, Oliver Cromwell. During the English Civil War (1642-1646) Oughtred was sequestered and scheduled for trial before Cromwell's puritanical commissioners. Due to the quick action of the astrologer Lilly and the insistence of influential friends, however, the mathematician and teacher was spared. He remained in England throughout Cromwell's reign, despite offers from foreign rulers who had heard of his fame. Oughtred died on June 30, 1660, at the parsonage in Albury. Tradition holds that he died of joy at learning that King Charles II had returned to England from Scotland and been restored to the English throne.

## Books

*Biographical Dictionary of Mathematicians,* Scribner's, 1991.

*Notable Mathematicians,* Gale, 1998.

## Online

*Oughtred Society website,*http://www.oughtred.org (March 15, 2003). □

## Oughtred, William

# OUGHTRED, WILLIAM

(*b.* Eton, Buckinghamshire, England, 5 March 1575; *d.* Albury, near Guildford, Surrey, England, 30 June 1660), *mathematics*

Oughtred’s father was a scrivener who taught writing at Eton and instructed his young son in arithmetic. Oughtred was educated as a king’s scholar at Eton, from which he proceeded to King’s College Cambridge, at the age of fifteen. He became a fellow of his college in 1595, graduated B. A. in 1596, and was awarded the M. A. 1600.

Ordained a priest in 1603, Oughtred at once began his ecclesiastical duties, being presented with the living of Shalford, Surrey. Five years later he became rector of Albury and retained this post until his death. Despite his parochial duties he continued to devote considerable time to mathematics , and in 1628 he was called upon to instruct Lord William Howard, the young son of the earl of Arundel. In carrying out this task he prepared a treatise on arithmetic and algebra. This slight volume, of barely 100 pages, contained almost all that was then known of these two branches of mathematics; it was published in 1631 as *Clavis mathematicae*.

Oughtred’s best-remembered work. the *Clavis* exerted considerable influence in England and on the Continent and immediately established him as a capable mathematician. Both Boyle and Newton held a very high opinion of the work. In a letter to Nathaniel Hawes, treasurer of Christ’s Hospital, dated 25 May 1694 and entitled “A New Scheme of learning for the Mathematical Boys at Christ’s Hospital,” Newton referred to Oughtred as “a man whose judgement (if any man’s) may be relyed on.” In Lord king’s *Life of Locke* we read “ The best Algebra yet extant is Oughtred’s” (I227). John Aubrey, in *Brief Lives*, maintained that Oughtred was more famous abroad for his learning than at home and that several great men came to England for the purpose of meeting him (II, 471).

John Wallis dedicated his *Arithmetica infinitorum* (1655) to Oughtred. A pupil of Oughtred, Wallis never wearied of sounding his praises. In his *Algebra* (1695) he wrote. The *Clavis* doth in as little room delvier as much as the fundamental and useful parts of geometry (as well as of arithmetic and algebra) as any book I know, “ and in its preface he classed Oughtred with the English mathematician Thomas Harriot.

The *Clavis* is not easy reading. The style is very obscure, and rules are so involved as to make them difficult to follow. Oughtred carried symbolism to excess, using signs to denote quantities, their powers, and the fundamental operations in arithmetic and algebra. chief among these were X for multiplication,⊐ for “greater than”;⊏for “less than”; and ~ for difference between. “Ratio was denoted by a dot; proportion, by∷. Thus the proportion A: B=α: β was written A · B ∷α· β. Continued proportion was written ∺ Of the maze of symbols employed by Oughred, only those for multiplication and proportion are till used. Yet, surposingly, there is a complete absence of indices or exponents from his work. Even in later editions of the *Clavis,* Oughtred used Aq, *Ac Aqq, Aqc, Acc, Aqcc, Accc, Aqqcc,*to denote successive powers of *A* up to the tenth. In his *Géométrie* (1637) Descartes had introduced the notation *x*^{n} but restricted its use to cases in which *n* was a positive whole number. Netwon extended this notation to include fractional and negative indices. These first appeared in a letter to Oldenburg for transmission to Leibniz— the famous *Epistola Prior* of June 1676— in which Newton illustrated the newly discovered binomial theorem.

In *La disme,* a short tract published in 1585, Simon Stevin had outlined the principles of decimal fractions. Although a warm admirer of Stevin’s work, Oughtred avoided his clumsy notation and substituted his own, which, although an improvement, was far from satisfactory. He did not use the dot to separate the decimal from the whole number, undoubtedly because he already used it to denote ratio; instead, he wrote a decimal such as 0.56 as 0/5̲6̲.

Oughtred is generally regarded as the inventor of the circular and rectilinear slide rules. Although the former is described in his *Circles of Proportion and the Horizontal Instrument* (1632), a description of the instrument had been published two years earlier by one of his pupils, Richard Delamain, in *Grammelogie, or the Mathematical Ring*. A better quarrle ensued between the two each claiming priority in the invention. There seems to be no very good reason why each should not be credited as an independent inventor. Oughtred’s claim to priority in the invention of the rectilinear slide rule, however, is beyond dispute, since it is known that he had designed the instrument as early as 1621.

In 1657 Oughtred published *Trigonometria*, a work of thirty-six pages dealing with both plane and spherical triangles. Oughtred made free use of the abbreviations *s* for sine *t* for tangent , *se* for secant sco for sine of the complement (or cosine), *tco* for secant cotangent, and *seco* for cosecant. The work also contains tables of sinces, tangents, and secants to seven decimal places as well as tables of logrithms, also to seven places.

It is said that Oughtred, a staunch royalist, died in a transport of joy on hearing the news of the restoration of Charles II.

## BIBLIOGRAPHY

I. Original Works. Oughtred’s chief writing is *Arithmeticae in numeris et speciebus institutio…quasi clavis mathematicae est* (London, 1631); 2nd ed., *Clavis mathematicae* (London, 1648). English translations were made by Robert Wood (1647) and Edmond Halley (1694) Subsequent Latin eds,. appeared at Oxford in 1652, 1667 and 1693.

His other works are *The Circles of Proportion and the Horiziontal Instrument,* W. Forster, trans, (London, 1632), a treatise on navigation; *The Description and Use of the Double Horizional Dial* (London, 1636); *A Most Easy Way for the Delination of plain Sunidals, Only by Gometry*(1647); *The solution of All Spherical Triangles* (Offord, 1651); *Description and use of the General Horological Ring and the Double Horizotal Dial*(London, 1653); *Trignometria* (London, 1657), trans by R. Stokes as *Trignometric* (London, 1675); and *Canones sinuum, tangentium, sectantium et logarithmorum* (London, 1657).

A collection of Oughtred’s papers, mainly on mathematical subjects, was published posthumously under the direction of Charles Scarborough as *Opuscula mathematica hactenus inedita* (Oxford, 1677).

II. Secondary Literature. On Oughtred or his work see John Aubrey, *Briey Lives* Andrew Clark, ed. (Oxford, 1898), II, 106, 113–114, W. W. R. Ball,*A History of the study of Mathematics at Cambridge* (Chicago-London, 1916); *Florian Cajori, William Oughtred, a Great Sevententh Century Teacher of Mathematics* (Chicago-London, 1916); Moritz Cantor,*volesungne über Geschite der Mathematick*, 2nd, ed., II (Liepzig, 1913), 720–721; charles Hutton, *Philosophical and Mathematical Dictionary, new* (London, 1918), II, 141–142; and S. J. Riguaud, ed., *Correspondence of scientific Men of the seventeenth Century*, I(Oxford, 1841), 11, 16,66.

J. F. Scott

## William Oughtred

# William Oughtred

**1574-1660**

**English Mathematician**

An Anglican priest who tutored students in mathematics, William Oughtred invented both the linear and the circular slide rules. He also introduced the symbol x for multiplication, and :: for proportion.

Oughtred was born in Eton on March 5, 1574. His father, Benjamin, taught writing at Eton School, which Oughtred attended as a king's scholar. At age 15, he entered King's College at Cambridge, where he earned his B.A. degree seven years later, as well as an M.A. in 1600. Though mathematics was not his area of study, it was during this time that Oughtred developed a keen interest in the subject such that even after he was ordained as a priest, he tutored math students without receiving any pay. His salary as a clergyman, he said, was sufficient to cover his needs.

Following his ordination in 1603, Oughtred received an assignment to serve as vicar of Shalford in 1604. Six years later, he was appointed rector of Albury, where he would remain until his death 50 years later. During the 1620s, he began tutoring, and in 1628 the Earl of Arundel offered to become Oughtred's patron in return for his teaching the earl's son, Lord William Howard.

Oughtred's *Clavis mathematicae,* published in 1631, was probably intended as a textbook for young Lord William, but with its summation of all significant arithmetic and algebraic knowledge up to that time, it proved to have much wider appeal. The author used the book to introduce a number of proposed mathematical symbols, but most were cumbersome and failed to catch on; yet x and :: became permanent fixtures of the landscape.

Ultimately Oughtred, encouraged by the warm reception *Clavis mathematicae* received, would publish numerous other books. In *The Circles of Proportion and the Horizontal Instrument* (1632), he discussed the idea of a slide rule as an instrument to aid in navigation. Though one of his students later claimed that *he* had invented it, Oughtred is generally credited with both the circular and the linear slide rule, which he may have invented as early as 1621. The slide rule would remain an important tool of computation for more than 350 years, until the advent of handheld calculators.

In the English Civil War (1642-60), Oughtred supported the Crown against Oliver Cromwell's radical Puritans, and with the triumph of Cromwell was forced to keep a low profile. By that point he was already up in years, and he died on June 30, 1660, not long after King Charles II had been restored to the throne.

**JUDSON KNIGHT**