William Oughtred

Before Oughtred, Thomas Harriot and William Milbourn are the only Englishmen known to have solved numerical equations of higher degrees. Milbourn published nothing. Harriot slightly modified Vieta’s process by simplifying somewhat the formation of the trial divisor. This method of approximation was the best in existence in Europe until the publication by Wallis in 1685 of Newton’s method of approximation.

It should be stated that, before the time of Newton, the best method of approximation to the roots of numerical equations existed, not in Europe, but in China. As early as the thirteenth century the Chinese possessed a method which is almost identical with what is known today as “Horner’s method.”

LOGARITHMS

Oughtred’s treatment of logarithms is quite in accordance with the more recent practice.49 He explains the finding of the “index” (our “characteristic”); he states that “the sum of two Logarithms is the Logarithm of the Product of their Valors; and their difference is the Logarithm of the Quotient,” that “the Logarithm of the side [436] drawn upon the Index number [2] of dimensions of any Potestas is the logarithm of the same Potestas” [436²], that “the logarithm of any Potestas [436²] divided by the number of its dimensions [2] affordeth the Logarithm of its Root [436].” These statements of Oughtred occur for the first time in the Key of the Mathematicks of 1647; the Clavis of 1631 contains no treatment of logarithms.

If the characteristic of a logarithm is negative, Oughtred indicates this fact by placing the – above the characteristic. He separates the characteristic and mantissa by a comma, but still uses the sign |_ to indicate decimal fractions. He uses the contraction “log.”

INVENTION OF THE SLIDE RULE; CONTROVERSY ON PRIORITY OF INVENTION

Oughtred’s most original line of scientific activity is the one least known to the present generation. Augustus De Morgan, in speaking of Oughtred, who was sometimes called “Oughtred Aetonensis,” remarks: “He is an animal of extinct race, an Eton mathematician. Few Eton men, even of the minority which knows what a sliding rule is, are aware that the inventor was of their own school and college.”50 The invention of the slide rule has, until recently,51 been a matter of dispute; it has been erroneously ascribed to Edmund Gunter, Edmund Wingate, Seth Partridge, and others. We have been able to establish that William Oughtred was the first inventor of slide rules, though not the first to publish thereon. We shall see that Oughtred invented slide rules about 1622, but the descriptions of his instruments were not put into print before 1632 and 1633. Meanwhile one of his own pupils, Richard Delamain, who probably invented the circular slide rule independently, published a description in 1630, at London, in a pamphlet of 32 pages entitled Grammelogia; or the Mathematicall Ring. In editions of this pamphlet which appeared during the following three or four years, various parts were added on, and some parts of the first and second editions eliminated. Thus Delamain antedates Oughtred two years in the publication of a description of a circular slide rule. But Oughtred had invented also a rectilinear slide rule, a description of which appeared in 1633. To the invention of this Oughtred has a clear title. A bitter controversy sprang up between Delamain on one hand, and Oughtred and some of his pupils on the other, on the priority and independence of invention of the circular slide rule. Few inventors and scientific men are so fortunate as to escape contests. The reader needs only to recall the disputes which have arisen, involving the researches of Sir Isaac Newton and Leibniz on the differential and integral calculus, of Thomas Harriot and René Descartes relating to the theory of equations, of Robert Mayer, Hermann von Helmholtz, and Joule on the principle of the conservation of energy, or of Robert Morse, Joseph Henry, Gauss and Weber, and others on the telegraph, to see that questions of priority and independence are not uncommon. The controversy between Oughtred and Delamain embittered Oughtred’s life for many years. He refers to it in print on more than one occasion. We shall confine ourselves at present to the statement that it is by no means clear that Delamain stole the invention from Oughtred; Delamain was probably an independent inventor. Moreover, it is highly probable that the controversy would never have arisen, had not some of Oughtred’s pupils urged and forced him into it. William Forster stated in the preface to the Circles of Proportion of 1632 that while he had been carefully preparing the manuscript for the press, “another to whom the Author [Oughtred] in a louing confidence discouered this intent, using more hast then good speed, went about to preocupate.” It was this passage which started the conflagration. Another pupil, W. Robinson, wrote to Oughtred, when the latter was preparing his Apologeticall Epistle as a reply to Delamain’s countercharges: “Good sir, let me be beholden to you for your Apology whensoever it comes forth, and (if I speak not too late) let me entreat you, whip ignorance well on the blind side, and we may turn him round, and see what part of him is free.”52 As stated previously, Oughtred’s circular slide rule was described by him in his Circles of Proportion, London, 1632, which was translated from Oughtred’s Latin manuscript and then seen through the press by his pupil, William Forster. In 1633 appeared An Addition vnto the Vse of the Instrvment called the Circles of Proportion which contained at the end “The Declaration of the two Rulers for Calculation,” giving a description of Oughtred’s rectilinear slide rule. This Addition was bound with the Circles of Proportion as one volume. About the same time Oughtred described a modified form of the rectilinear slide rule, to be used in London for gauging.53

CHAPTER III

MINOR WORKS

Among the minor works of Oughtred must be ranked his booklet of forty pages to which reference has already been made, entitled, The New Artificial Gauging Line or Rod, London, 1633. His different designs of slide rules and his inventions of sun-dials as well as his exposition of the making of watches show that he displayed unusual interest and talent in the various mathematical instruments. A short tract on watchmaking was brought out in London as an appendix to the Horological Dialogues of a clock- and watchmaker who signed himself “J. S.” (John Smith?). Oughtred’s tract appeared with its own title-page, but with pagination continued from the preceding part, as An Appendix wherein is contained a Method of Calculating all Numbers for Watches. Written originally by that famous Mathematician Mr. William Oughtred, and now made Publick. By J. S. of London, Clock-maker. London, 1675.

“J. S.” says in his preface:

The method following was many years since Compiled by Mr. Oughtred for the use of some Ingenious Gentlemen his friends, who for recreation at the University, studied to find out the reason and Knowledge of Watch-work, which seemed also to be a thing with which Mr. Oughtred himself was much affected, as may in part appear by his putting out of his own Son to the same Trade, for whose use (as I am informed) he did compile a larger tract, but what became of it cannot be known.

Notwithstanding Oughtred’s marked activity in the design of mathematical instruments, and his use of surveying instruments, he always spoke in deprecating terms of their importance and their educational value. In his epistle against Delamain he says:

The Instruments I doe not value or weigh one single penny. If I had been ambitious of praise, or had thought them (or better then they) worthy, at which to have taken my rise, out of my secure and quiet obscuritie, to mount up into glory, and the knowledge of men: I could have done it many yeares before…

Long agoe, when I was a young student of the Mathematicall Sciences, I tryed many wayes and devices to fit my selve with some good Diall or Instrument portable for my pocket, to finde the houre, and try other conclusions by, and accordingly framed for that my purpose both Quadrants, and Rings, and Cylinders, and many other composures. Yet not to my full content and satisfaction; for either they performed but little, or els were patched up with a diversity of lines by an unnaturall and forced contexture. At last I.. found what I had before with much studie and paines in vaine sought for.54

Mention has been made in the previous pages of two of his papers on sun-dials, prepared (as he says) when he was in his twenty-third year. The first was published in the Clavis of 1647. The second paper appeared in his Circles of Proportion.

Both before and after the time of Oughtred much was written on sun-dials. Such instruments were set up against the walls of prominent buildings, much as the faces of clocks in our time. The inscriptions that were put upon sun-dials are often very clever: “I count only the hours of sunshine,” “Alas, how fleeting.” A sun-dial on the grounds of Merchiston Castle, in Edinburgh, where the inventor of logarithms, John Napier, lived for many years, bears the inscription, “Ere time be tint, tak tent of time” (Ere time be lost, take heed of time).

Portable sun-dials were sometimes carried in pockets, as we carry watches. Thus Shakespeare, in As You Like It, Act II, sc. vii:

“And then he drew a diall from his poke.”

Watches were first made for carrying in the pocket about 1658.

Because of this literary, scientific, and practical interest in methods of indicating time it is not surprising that Oughtred devoted himself to the mastery and the advancement of methods of time-measurement.

Besides the accounts previously noted, there came from his pen: The Description and Use of the double Horizontall Dyall: Whereby not onely the hower of the day is shewne; but also the Meridian Line is found: And most Astronomical Questions, which may be done by the Globe, are resolved. Invented and written by W. O., London, 1636.

The “Horizontall Dyall” and “Horologicall Ring” appeared again as appendixes to Oughtred’s translation from the French of a book on mathematical recreations.

The fourth French edition of that work appeared in 1627 at Paris, under the title of Recreations mathematiqve, written by “Henry van Etten,” a pseudonym for the French Jesuit Jean Leurechon (1591-1690). English editions appeared in 1633, 1653, and 1674. The full title of the 1653 edition conveys an idea of the contents of the text: Mathematical Recreations, or, A Collection of many Problemes, extracted out of the Ancient and Modern Philosophers, as Secrets and Experiments in Arithmetick, Geometry, Cosmographie, Horologiographie, Astronomie, Navigation, Musick, Opticks, Architecture, Statick, Mechanicks, Chemistry, Water-works, Fire-works, &c. Not vulgarly manifest till now. Written first in Greek and Latin, lately compil’d in French, by Henry Van Etten, and now in English, with the Examinations and Augmentations of divers Modern Mathematicians. Whereunto is added the Description and Use of the Generall Horologicall Ring. And The Double Horizontall Diall. Invented and written by William Oughtred. London, Printed for William Leake, at the Signe of the Crown in Fleet-street, between the two Temple-Gates. MDCLIII.

The graphic solution of spherical triangles by the accurate drawing of the triangles on a sphere and the measurement of the unknown parts in the drawing was explained by Oughtred in a short tract which was published by his son-in-law, Christopher Brookes, under the following title: The Solution of all Sphaerical Triangles both right and oblique By the Planisphaere: Whereby two of the Sphaerical partes sought, are at one position most easily found out. Published with consent of the Author, By Christopher Brookes, Mathematique Instrument-maker, and Manciple of Wadham Colledge, in Oxford.

Brookes says in the preface:

I have oftentimes seen my Reverend friend Mr. W. O. in his resolution of all sphaericall triangles both right and oblique, to use a planisphaere, without the tedious labour of Trigonometry by the ordinary Canons: which planisphaere he had delineated with his own hands, and used in his calculations more than Forty years before.

Interesting as one of our sources from which Oughtred obtained his knowledge of the conic sections is his study of Mydorge. A tract which he wrote thereon was published by Jonas Moore, in his Arithmetick in two books.. [containing also] the two first books of Mydorgius his conical sections analyzed by that reverend devine Mr. W. Oughtred, Englished and completed with cuts. London, 1660. Another edition bears the date 1688.

To be noted among the minor works of Oughtred are his posthumous papers. He left a considerable number of mathematical papers which his friend Sir Charles Scarborough had revised under his direction and published at Oxford in 1676 in one volume under the title, Gulielmi Oughtredi, Etonensis, quondam Collegii Regalis in Cantabrigia Socii, Opuscula Mathematica hactenus inedita. Its nine tracts are of little interest to a modern reader.

Here we wish to give our reasons for our belief that Oughtred is the author of an anonymous tract on the use of logarithms and on a method of logarithmic interpolation which, as previously noted, appeared as an “Appendix” to Edward Wright’s translation into English of John Napier’s Descriptio, under the title, A Description of the Admirable Table of Logarithmes, London, 1618. The “Appendix” bears the title, “An Appendix to the Logarithmes, showing the practise of the Calculation of Triangles, and also a new and ready way for the exact finding out of such lines and Logarithmes as are not precisely to be found in the Canons.” It is an able tract. A natural guess is that the editor of the book, Samuel Wright, a son of Edward Wright, composed this “Appendix.” More probable is the conjecture which (Dr. J. W. L. Glaisher informs me) was made by Augustus De Morgan, attributing the authorship to Oughtred. Two reasons in support of this are advanced by Dr. Glaisher, the use of x in the “Appendix” as the sign of multiplication (to Oughtred is generally attributed the introduction of the cross × for multiplication in 1631), and the then unusual designation “cathetus” for the vertical leg of a right triangle, a term appearing in Oughtred’s books. We are able to advance a third argument, namely, the occurrence in the “Appendix” of (S*) as the notation for sine complement (cosine), while Seth Ward, an early pupil of Oughtred, in his Idea trigonometriae demonstratae, Oxford, 1654, used a similar notation (S’). It has been stated elsewhere that Oughtred claimed Seth Ward’s exposition of trigonometry as virtually his own. Attention should be called also to the fact that, in his Trigonometria, p. 2, Oughtred uses (’) to designate 180°-angle.

Dr. J. W. L. Glaisher is the first to call attention to other points of interest in this “Appendix.” The interpolations are effected with the aid of a small table containing the logarithms of 72 sines. Except for the omission of the decimal point, these logarithms are natural logarithms – the first of their kind ever published. In this table we find log 10=2302584; in modern notation, this is stated, loge 10=2.302584. The first more extended table of natural logarithms of numbers was published by John Speidell in the 1622 impression of his New Logarithmes, which contains, besides trigonometric tables, the logarithms of the numbers 1-1000.

The “Appendix” contains also the first account of a method of computing logarithms, called the “radix method,” which is usually attributed to Briggs who applied it in his Arithmetica logarithmica, 1624. In general, this method consists in multiplying or dividing a number, whose logarithm is sought, by a suitable factor and resolving the result into factors of the form 1±x/10ⁿ. The logarithm of the number is then obtained by adding the previously calculated logarithms of the factors. The method has been repeatedly rediscovered, by Flower in 1771, Atwood in 1786, Leonelli in 1802, Manning in 1806, Weddle in 1845, Hearn in 1847, and Orchard in 1848.

We conclude with the words of Dr. J. W. L. Glaisher:

The Appendix was an interesting and remarkable contribution to mathematics, for in its sixteen small pages it contains (1) the first use of the sign ×; (2) the first abbreviations, or symbols, for the sine, tangent, cosine, and cotangent; (3) the invention of the radix method of calculating logarithms; (4) the first table of hyperbolic logarithms.55

CHAPTER IV

OUGHTRED’S INFLUENCE UPON MATHEMATICAL PROGRESS AND TEACHING

OUGHTRED AND HARRIOT

Oughtred’s Clavis mathematicae was the most influential mathematical publication in Great Britain which appeared in the interval between John Napier’s Mirifici logarithmorum canonis descriptio, Edinburgh, 1614, and the time, forty years later, when John Wallis began to publish his important researches at Oxford. The year 1631 is of interest as the date of publication, not only of Oughtred’s Clavis, but also of Thomas Harriot’s Artis analyticae praxis. We have no evidence that these two mathematicians ever met. Through their writings they did not influence each other. Harriot died ten years before the appearance of his magnum opus, or ten years before the publication of Oughtred’s Clavis. Strangely, Oughtred, who survived Harriot thirty-nine years, never mentions him. There is no doubt that, of the two, Harriot was the more original mind, more capable of penetrating into new fields of research. But he had the misfortune of having a strong competitor in René Descartes in the development of algebra, so that no single algebraic achievement stands out strongly and conspicuously as Harriot’s own contribution to algebraic science. As a text to serve as an introduction to algebra, Harriot’s Artis analyticae praxis was inferior to Oughtred’s Clavis. The former was a much larger book, not as conveniently portable, compiled after the author’s death by others, and not prepared with the care in the development of the details, nor with the coherence and unity and the profound pedagogic insight which distinguish the work of Oughtred. Nor was Harriot’s position in life such as to be surrounded by so wide a circle of pupils as was Oughtred. To be sure, Harriot had such followers as Torporley, William Lower, and Protheroe in Wales, but this group is small as compared with Oughtred’s.

OUGHTRED’S PUPILS

There was a large number of distinguished men who, in their youth, either visited Oughtred’s home and studied under his roof or else read his Clavis and sought his assistance by correspondence. We permit Aubrey to enumerate some of these pupils in his own gossipy style:

Seth Ward, M.A., a fellow of Sydney Colledge in Cambridge (now bishop of Sarum), came to him, and lived with him halfe a yeare (and he would not take a farthing for his diet), and learned all his mathematiques of him. Sir Jonas More was with him a good while, and learn’t; he was but an ordinary logist before. Sir Charles Scarborough was his scholar; so Dr. John Wallis was his scholar; so was Christopher Wren his scholar, so was Mr… Smethwyck, Regiae Societatis Socius. One Mr. Austin (a most ingeniose man) was his scholar, and studyed so much that he became mad, fell a laughing, and so dyed, to the great griefe of the old gentleman. Mr… Stokes, another scholar, fell mad, and dream’t that the good old gentleman came to him, and gave him good advice, and so he recovered, and is still well. Mr. Thomas Henshawe, Regiae Societatis Socius, was his scholar (then a young gentleman). But he did not so much like any as those that tugged and tooke paines to worke out questions. He taught all free.

He could not endure to see a scholar write an ill hand; he taught them all presently to mend their hands.56

Had Oughtred been the means of guiding the mathematical studies of only John Wallis and Christopher Wren – one the greatest English mathematician between Napier and Newton, the other one of the greatest architects of England – he would have earned profound gratitude. But the foregoing list embraces nine men, most of them distinguished in their day. And yet Aubrey’s list is very incomplete. It is easy to more than double it by adding the names of William Forster, who translated from Latin into English Oughtred’s Circles of Proportion; Arthur Haughton, who brought out the 1660 Oxford edition of the Circles of Proportion; Robert Wood, an educator and politician, who assisted Oughtred in the translation of the Clavis from Latin into English for the edition of 1647; W. Gascoigne, a man of promise, who fell in 1644 at Marston Moor; John Twysden, who was active as a publisher; William Sudell, N. Ewart, Richard Shuttleworth, William Robinson, and William Howard, the son of the Earl of Arundel, for whose instruction Oughtred originally prepared the manuscript treatise that was published in 1631 as the Clavis mathematicae.

Nor must we overlook the names of Lawrence Rooke (who “did admirably well read in Gresham Coll. on the sixth chapt. of the said book,” the Clavis); Christopher Brookes (a maker of mathematical instruments who married a daughter of the famous mathematician); William Leech and William Brearly (who with Robert Wood “have been ready and helpfull incouragers of me [Oughtred] in this labour” of preparing the English Clavis of 1647), and Thomas Wharton, who studied the Clavis and assisted in the editing of the edition of 1647.

The devotion of these pupils offers eloquent testimony, not only of Oughtred’s ability as a mathematician, but also of his power of drawing young men to him – of his personal magnetism. Nor should we omit from the list Richard Delamain, a teacher of mathematics in London, who unfortunately had a bitter controversy with Oughtred on the priority and independence of the invention of the circular slide rule and a form of sun-dial. Delamain became later a tutor in mathematics to King Charles I, and perished in the civil war, before 1645.

OUGHTRED, THE “TODHUNTER OF THE SEVENTEENTHCENTURY”

To afford a clearer view of Oughtred as a teacher and mathematical expositor we quote some passages from various writers and from his correspondence. Anthony Wood57 gives an interesting account of how Seth Ward and Charles Scarborough went from Cambridge University to the obscure home of the country mathematician to be initiated into the mysteries of algebra:

Mr. Cha. Scarborough, then an ingenious young student and fellow of Caius Coll. in the same university, was his [Seth Ward’s] great acquaintance, and both being equally students in that faculty and desirous to perfect themselves, they took a journey to Mr. Will. Oughtred living then at Albury in Surrey, to be informed in many things in his Clavis mathematica which seemed at that time very obscure to them. Mr. Oughtred treated them with great humanity, being very much pleased to see such ingenious young men apply themselves to these studies, and in short time he sent them away well satisfied in their desires. When they returned to Cambridge, they afterwards read the Clav. Math. to their pupils, which was the first time that book was read in the said university. Mr. Laur. Rook, a disciple of Oughtred, I think, and Mr. Ward’s friend, did admirably well read in Gresham Coll. on the sixth chap. of the said book, which obtained him great repute from some and greater from Mr. Ward, who ever after had an especial favour for him.

Anthony Wood makes a similar statement about Thomas Henshaw:

While he remained in that coll. [University College, Oxford] which was five years.. he made an excursion for about 9 months to the famous mathematician Will. Oughtred parson of Aldbury in Surrey, by whom he was initiated in the study of mathematics, and afterwards retiring to his coll. for a time, he at length went to London, was entered a student in the Middle Temple.58

Extracts from letters of W. Gascoigne to Oughtred, of the years 1640 and 1641, throw some light upon mathematical teaching of the time: