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

# William Oughtred

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).