## The circles of proportion and the horizontal instrument. The former shewing the manner how to work proportions both simple and compound: and the ready and easy resolving of questions both in arithmetic, geometrie, & astronomie...

London: printed by Augustine Mathewes, 1633.

First edition of one of the most important works in the history of computing, the first printed description of the rectilinear slide rule. Invented by Oughtred about 1621, this was a calculating device based upon Napier’s recently invented logarithms for multiplying and dividing numbers and trigonometric functions. “Slide rules dominated the practice of mathematics from their invention in the reign of King Charles I to their sudden, electronic-calculator driven demise three centuries later” (Jardine). The present work is the second issue (see below) of The circles of proportion, but the first to contain the second part, ‘An addition unto the use of the instrument called the circles of proportion,’ which contains ‘the excellent use of two rulers for calculation’ – this was the rectilinear slide rule. The ‘circle of proportion’ itself was a kind of circular slide rule, but it was much less convenient to use than its rectilinear cousin, and was not much used in practice. The first part of the present work deals with the use of the circle of proportion in general calculation, followed by examples of its application in finding areas and volumes, gauging, military calculations such as determining the number of men it takes to form a square of a certain size, astronomical calculations, the solution of problems in plane and spherical trigonometry, etc. This is followed by a description of another important invention of Oughtred’s, the Horizontal Instrument, which was often engraved on the other side of the circle of proportion. This was based on a stereographic projection and could be used both for finding the time and as a graphical calculator for solving problems concerned with the position of the Sun. The separately-paginated ‘Addition’ to the Circles of Proportion was intended to show how the circles, augmented by other scales, could be used for navigation and also be applied to other problems. It contains a number of interesting examples, including one to find the circumference of the earth. This requires the use of a special level, which is illustrated. But the main importance of the ‘Addition’ is its account of the rectilinear slide rule. Following the publication of Circles of Proportion, Oughtred became embroiled in a priority dispute with Richard Delamain, a former pupil, who published his own account of a circular slide rule in 1630, the point of contention being whether Delamain’s invention was independent. However, Oughtred has undisputed priority, both in invention and publication, for the rectilinear slide-rule, and it was that which revolutionized computational practice. ABPC/RBH record no textually complete copy of any issue of the first edition since Honeyman (this copy).

Provenance: Sir Walter Halsey (1868-1950), Gaddesden Library (bookplate); Robert B. Honeyman (1897-1987), American metallurgical engineer and bibliophile (booklabel) (Sotheby’s, 10 November 1980, lot 2372, £1,050 = \$2,516); Erwin Tomash (1921-2012), American engineer, founder of Dataproducts Corporation, and bibliophile (booklabel).

The principle of the slide rule follows from Napier’s invention of logarithms, announced in his Mirifici logarithmorum canonis descriptio (1614), as a means of simplifying multiplication and division of numbers. Multiplication was performed by adding the logarithms of two numbers together and then finding the number corresponding to the sum of the two logarithms. The logarithms and anti-logarithms had to be looked up in tables. It was soon realised by Edmund Gunter (1581-1626) that logarithms could more rapidly, if less accurately, be read off the logarithmic scale which became known as ‘Gunter’s line,’ first described in his Description and use of the sector (1623), which was a long rule with one side marked with logarithms and the other rulings for navigation. Using a pair of compasses, the logarithms of the two numbers to be multiplied could be added by setting the compasses to the distance from 1 to the first number, then moving the compasses along the scale so that one point was on the second number and the other point indicated the result of the calculation. On a linear scale the result will be the sum of the two numbers, but on a logarithmic scale the result is the product. In Oughtred’s circular slide rule, the logarithmic scales are laid out on a circle and two pointers are attached to the centre to act as a pair of compasses on its side. Finally, Oughtred saw how to do away with the compasses and pointers: 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 compasses, Oughtred created the prototype of the modern slide rule. The rectilinear 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. Its importance was emphasized by James Watt (1736-1819), who made it an essential tool of the Industrial Revolution. In 1850 French army officer Victor Mayer Amdée Mannheim 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.

The way in which the Circles of Proportion came to be published, as well as Oughtred’s attitude to the use of instruments in mathematical education, is described by his pupil William Forster in his ‘Epistle dedicatory’ to Sir Kenelm Digby:

“Being in the time of the long vacation 1630, in the Country, at the house of the Reverend, and my most worthy friend and Teacher, Mr. William Oughtred (to whose instruction I owe both my initiation, and whole progresse in these Sciences), I upon occasion of speech told him of a Ruler of Numbers, Sines, and Tangents, which one had bespoken to be made (such as is usually called Mr. Gunter's Ruler) 6 feet long, to be used with a payre of beame-compasses. He answered that was a poore invention, and the performance very troublesome: But, said he, seeing you are taken with such mechanicall ways of Instruments, I will show you what devises I have had by mee these many yeares. And first, hee brought to mee two Rulers of that sort, to be used by applying one to the other, without any compasses: and after that he shewed mee those lines cast into a circle or Ring, with another moveable circle upon it. I seeing the great expeditenesse of both those wayes, but especially of the latter, wherein it farre excelleth any other Instrument which hath bin knowne; told him, I wondered that he could so many yeares conceale such usefull inventions, not onely from the world, but from my selfe, to whom in other parts and mysteries of Art he had bin so liberall. He answered, That the true way of Art is not by Instruments, but by Demonstration: and that it is a preposterous course of vulgar Teachers, to begin with Instruments and not with the Sciences, and so instead of Artists to make their Schollers only doers of tricks, and as it were Juglers: to the despite of Art, losse of precious time, and betraying of willing and industrious wits unto ignorance, and idlenesse. That the use of Instruments is indeed excellent, if a man be an Artist: but contemptible, being set and opposed to Art. And lastly, that he meant to commend to me the skill of Instruments, but first he would have me well instructed in the Sciences. He also showed me many notes, and Rules for the use of those circles, and of his Horizontall Instrument (which he had projected about 30 yeares before) the most part written in Latine. All which I obtained of him leave to translate into English, and make publique, for the use, and benefit of such as were studious, and lovers of these excellent Sciences.”

Because of his general disdain for mathematical instruments, Oughtred was initially reluctant to allow his invention to be published, but he was persuaded to do so by the appearance of two works by his former pupil Richard Delamain (1600-44), a teacher of mathematics living and working in London. Delamain was in fact the first to publish an account of a circular slide rule in Grammelogia, or the mathematical ring (1630), and of Oughtred’s Horizontal Instrument in his Horizontal Quadrant (1631). There is no doubt that Oughtred had shown Delamain some of his instruments. Moreover, both Delamain and Oughtred were often dealing with the same people, such as the famous instrument-maker Elias Allen, and further information about each other’s findings could of course have been passed via these third-party contacts.

Forster referred to Delamain in his ‘Epistle dedicatory’, though without naming him. He mentioned ‘vulgar Teachers’ in the extract given above, and went on to complain that, while Forster was preparing the book, “another, to whom the Author in a loving confidence discovered this intent, using more haste than good speed, went about to pre-ocupate” the new invention.

After the attack on him in the Circles of Proportion, Delamain re-issued Grammelogia, with additional material stating his own position, both as regards priority and on the question of the use of instruments in mathematical education. Oughtred responded by adding a section to unsold copies of Circles of Proportion headed ‘To the English Gentrie, and all other studious of the Mathematicks, which shall be Readers hereof. The just Apologie of Wil. Oughtred against the slanderous insimulations of Richard Delamain, in a Pamphlet called Grammelogia (etc.)’.

The bibliography of Oughtred’s Circles of Proportion is complex. According to Wallis, there are three issues (though with some variant states). All have the same engraved title page, dated 1632 (though some copies have only a letterpress title, and others only the engraved title):

‘The circles of proportion and the horizontall instrument. Both invented, and the uses of both written in Latine by Mr. W. O. Translated into English and set forth for the publique benefit by William Forster. London. Printed for Elias Allen maker of these and all other mathematical instruments, and are to be sold at his shop over against St Clements church with out Temple-barr, 1632.’

In the first issue, the text on the letterpress title, which is also dated 1632, is almost identical to that on the engraved title (although ‘Circles’ is replaced by ‘Circle’), and the main text contains only the first part, ending at p. 152. In the second issue, offered here, the letterpress title is dated 1633, and new text is added to it: “… newly increased with an additament for navigation. All which rules may also be wrought with the penne by arithmetic, and the canon of triangles … Hereunto is annexed the excellent use of two rulers for calculation …”; this ‘Additament’ is a new 74-page part which follows p. 152 of the first part. It has a separate title page and imprint, but Wallis suggests that it was not available separately. It is this part which contains Oughtred’s description of his rectilinear slide rule. In the third issue, which Wallis dates to 1634, the engraved and letterpress titles are identical to those of the second issue, and a third part, without separate title page, imprint or colophon, is added, headed ‘To the English Gentrie …,” which is largely important today as a source for Oughtred’s biography.