On the History of Gunter's Scale and the Slide Rule During the Seventeenth Century

In Short

Florian Cajori’s scholarly monograph traces the evolution of mathematical calculating scales in seventeenth-century England. Beginning with Edmund Gunter’s early logarithmic scale, the work examines how various practitioners adapted, reshaped, and expanded these instruments into physical slide rules. The narrative centers on the bitter priority dispute between William Oughtred and his former student Richard Delamain over the invention of the circular slide rule. Through meticulous textual analysis of rare pamphlets, library holdings, and instrument descriptions, Cajori establishes the chronological development of these devices, ending with specialized rectilinear slide rules designed for gauging and timber measurement.

The Story

The intellectual trajectory begins in the early decades of the seventeenth century with Edmund Gunter’s introduction of the logarithmic line of numbers. Rather than relying solely on static tables, practitioners soon realized that projecting logarithms onto physical scales allowed arithmetic and trigonometric operations to be performed mechanically. Early innovators introduced crucial physical modifications: Edmund Wingate carried Gunter’s scale to France, while thinkers like Milbourn and Thomas Brown wrapped logarithmic lines into serpentine spirals and concentric circles to increase precision without making the physical instrument unwieldily large.

The central drama of the book unfolds around 1630, when Richard Delamain published his Grammelogia, describing a circular slide rule featuring a movable inner ring set against a fixed outer circle. Shortly thereafter, William Oughtred published his own account of circular and rectilinear rules, edited by his pupil William Forster. Forster launched a sharp attack against Delamain, accusing him of stealing the concept from Oughtred after receiving private instruction. A fierce public quarrel erupted between the elder master and his former student. Delamain insisted on his priority and independent invention, dedicating expanded editions of his work to King Charles I and introducing advanced designs, including multi-turn spiral projections, concentric cylinder rules, and an early form of the movable runner index.

Cajori systematically analyzes the conflicting testimony, letters, and printed pamphlets exchanged during this controversy. He shows that while Oughtred had devised circular logarithmic scales as early as 1621 and shared them with instrument maker Elias Allen, Delamain was the first to publish a detailed printed description of a circular slide rule in 1630. Oughtred, who prized theoretical rigor and viewed instrument manipulation with a degree of intellectual condescension, favored a design using fixed circles with a pair of opening compass-like legs, whereas Delamain championed the practical mechanical convenience of concentric moving rings.

Following the resolution of the priority debate, the narrative follows the practical evolution of the instrument across the middle and late seventeenth century. Oughtred responded to practical demand by designing a rectilinear brass gauging rod in 1633 for London wine measures. Practical makers like John Brown, Seth Partridge, and Henry Coggeshall adapted logarithmic scales for commercial trades, creating specialised rules for timber measurement, gauging, and trade calculations. The book concludes by surveying surviving brass and boxwood instruments from the period, demonstrating how the circular designs of the early decades gradually gave way to the rectilinear slide rule as the standard mechanical computing tool of the era.

How It Unfolds

The logarithmic scale emerges Innovators adapt Gunter's static logarithmic line into versatile mechanical configurations. Milbourn and Thomas Brown wrap the lines into spirals to maximize scale length on compact brass discs.

Delamain claims the ring Richard Delamain publishes Grammelogia in 1630, providing the earliest printed directions for operating a circular slide rule with movable concentric circles. He demonstrates practical calculations, including simple proportions and financial interest calculations.

The dispute erupts William Forster and William Oughtred accuse Delamain of appropriating Oughtred's unprinted discovery. A heated exchange of pamphlets ensues, with Oughtred dismissing Delamain as an unlettered practitioner corrupting mathematics with mechanical tricks.

Delamain expands his designs In response to the controversy, Delamain publishes enlarged editions of his tract. He proposes sophisticated variations, including multi-turn circular scales, a yard-wide cylindrical slide rule, and an early sliding runner index.

The shift to practical trades Oughtred designs a two-ruler brass gauging rod in 1633 at the request of the Vintners' Company. Subsequent makers like Henry Coggeshall refine two-foot sliding rules specifically tailored for timber measurement and commercial gauging.

The People

William Oughtred A revered English mathematician who invented both circular and rectilinear slide rules around 1621. He seeks theoretical perfection and mathematical demonstration, holding contempt for mere "doers of tricks" on instruments, yet aggressively defends his priority when he feels his original ideas have been stolen.

Richard Delamain A practical teacher of mathematics who claims independent invention of the circular slide rule in 1629. He seeks mechanical efficiency, publishing the first printed instructions for the device and presenting customized silver and brass instruments to King Charles I.

William Forster A loyal pupil of Oughtred who translated his master’s Latin rules into English. He ignites the controversy by publishing a harsh public attack against Delamain in the preface to Oughtred's Circles of Proportion.

Thomas Brown and John Brown Father and son instrument makers who design serpentine spiral scales and composite rules. John Brown promotes easy-to-use instruments, arguing that proper layout allows ordinary workers to solve complex calculations without deep mathematical training.

Elias Allen A prominent London mathematical instrument maker who manufactures brass circular rules and gauging rods for Oughtred, playing a central physical role in turning theoretical designs into real instruments.

In Its Own Voice

"Seeke the first number in the moveable, and bring it to the second number in the fixed, so right against the third number in the moveable, is the answer in the fixed."

Delamain opens his 1630 manual with this direct operational instruction for performing the rule of proportion on a circular slide rule.

"I had the very first moulding . . . but Delamain was already corrupted with doing upon Instruments, and quite lost from ever being made an Artist."

Oughtred contrasts his devoted pupil Forster with Delamain, whom he views as an intellectually shallow tinkerer.

"For what can be more ready and easie, then having set twelve to the length, to see the Content exactly against the Girt or Side of the Square."

Henry Coggeshall highlights the effortless practicality of his specialized timber-measuring rule in his 1677 treatise.

What It's Really About

Beneath its bibliographical and technical details, the book explores the tension between theoretical mathematics and practical craftsmanship in early modern Europe. It contrasts Oughtred's academic view—that instruments are mere secondary aids to true mathematical understanding—with Delamain's belief that accessible mechanical tools democratize computation for non-mathematicians. The work examines how priority claims were established in an era before standard academic journals, relying on private letters, royal dedications, and polemical pamphlets. Ultimately, it illustrates how abstract logarithmic theory was systematically transformed into durable physical hardware, laying the foundations for modern computing tools.

Why Read It Today

Cajori’s study appeals to historians of science, collectors, and anyone fascinated by the origins of computing technology. Reading it feels like doing detective work through rare seventeenth-century archives, tracking down lost pamphlets, analyzing physical brass artifacts, and deciphering bitter marginal notes written by long-dead mathematicians.

The reader must navigate dense bibliographical citations, archaic seventeenth-century spelling, and detailed descriptions of logarithmic graduations. However, what remains with you is a vivid human picture of intellectual rivalry. The book reveals that the early history of technology was driven not just by abstract logic, but by fiery pride, commercial ambition, and personal grudges among the artisans and scholars who first carved logarithmic lines into brass and wood.