William Oughtred

William Oughtred
Florian Cajori

Florian Cajori

William Oughtred / A great Seventeenth-Century Teacher of Mathematics

INTRODUCTION

In the year 1660 the Royal Society was founded by royal favor in London, although in reality its inception took place in 1645 when the Philosophical Society (or, as Boyle called it, the “Invisible College”) came into being, which held meetings at Gresham College in London and later in Oxford. It was during the second half of the seventeenth century that Sir Isaac Newton, surrounded by a group of great men – Wallis, Hooke, Barrow, Halley, Cotes – carried on his epoch-making researches in mathematics, astronomy, and physics. But it is not this half-century of science in England, nor any of its great men, that especially engage our attention in this monograph. It is rather the half-century preceding, an epoch of preparation, when in the early times of the House of Stuart the sciences began to flourish in England. Says Dr. A. E. Shipley: “Whatever were the political and moral deficiencies of the Stuart kings, no one of them lacked intelligence in things artistic and scientific.” It was at this time that mathematics, and particularly algebra, began to be cultivated with greater zeal, when elementary algebra with its symbolism as we know it now began to take its shape.

Biographers of Sir Isaac Newton make particular mention of five mathematical books which he read while a young student at Cambridge, namely, Euclid’s Elements, Descartes’s Géométrie, Vieta’s Works, Van Schooten’s Miscellanies, and Oughtred’s Clavis mathematicae. The last of these books has been receiving increasing attention from the historians of algebra in recent years. We have prepared this sketch because we felt that there were points of interest in the life and activity of Oughtred which have not received adequate treatment. Historians have discussed his share in the development of symbolic algebra, but some have fallen into errors, due to inability to examine the original editions of Oughtred’s Clavis mathematicae, which are quite rare and inaccessible to most readers. Moreover, historians have failed utterly to recognize his inventions of mathematical instruments, particularly the slide rule; they have completely overlooked his educational views and his ideas on mathematical teaching. The modern reader may pause with profit to consider briefly the career of this interesting man.

Oughtred was not a professional mathematician. He did not make his livelihood as a teacher of mathematics or as a writer, nor as an engineer who applies mathematics to the control and use of nature’s forces. Oughtred was by profession a minister of the gospel. With him the study of mathematics was a side issue, a pleasure, a recreation. Like the great French algebraist, Vieta, from whom he drew much of his inspiration, he was an amateur mathematician. The word “amateur” must not be taken here in the sense of superficial or unthorough. Great Britain has had many men distinguished in science who pursued science as amateurs. Of such men Oughtred is one of the very earliest.

    F. C.

CHAPTER I

OUGHTRED’S LIFE

AT SCHOOL AND UNIVERSITY

William Oughtred, or, as he sometimes wrote his name, Owtred, was born at Eton, the seat of Eton College, the year of his birth being variously given as 1573, 1574, and 1575. “His father,” says Aubrey, “taught to write at Eaton, and was a scrivener; and understood common arithmetique, and ’twas no small helpe and furtherance to his son to be instructed in it when a schoole-boy.”[1 - Aubrey’s Brief Lives, ed. A. Clark, Vol. II, Oxford, 1898, p. 106.] He was a boy at Eton in the year of the Spanish Armada. At this famous school, which prepared boys for the universities, young Oughtred received thorough training in classical learning.

According to information received from F. L. Clarke, Bursar and Clerk of King’s College, Cambridge, Oughtred was admitted at King’s a scholar from Eton on September 1, 1592, at the age of seventeen. He was made Fellow at King’s on September 1, 1595, while Elizabeth was still on the throne. He received in 1596 the degree of Bachelor of Arts and in 1600 that of Master of Arts. He vacated his fellowship about the beginning of August, 1603. His career at the University of Cambridge we present in his own words. He says:

Next after Eaton schoole, I was bred up in Cambridge in Kings Colledge: of which society I was a member about eleven or twelve yeares: wherein how I behaved my selfe, going hand in hand with the rest of my ranke in the ordinary Academicall studies and exercises, and with what approbation, is well knowne and remembered by many: the time which over and above those usuall studies I employed upon the Mathematicall sciences, I redeemed night by night from my naturall sleep, defrauding my body, and inuring it to watching, cold, and labour, while most others tooke their rest. Neither did I therein seek only my private content, but the benefit of many: and by inciting, assisting, and instructing others, brought many into the love and study of those Arts, not only in our own, but in some other Colledges also: which some at this time (men far better than my selfe in learning, degree, and preferment) will most lovingly acknowledge.[2 - “To the English Gentrie, and all others studious of the Mathematicks, which shall bee Readers hereof. The just Apologie of Wil: Ovghtred, against the slaunderous insimulations of Richard Delamain, in a Pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica logarithmorum projectio circularis” [1633?], p. 8. Hereafter we shall refer to this pamphlet as the Apologeticall Epistle, this name appearing on the page-headings.]

These words describe the struggles which every youth not endowed with the highest genius must make to achieve success. They show, moreover, the kindly feeling toward others and the delight he took throughout life in assisting anyone interested in mathematics. Oughtred’s passion for this study is the more remarkable as neither at Eton nor at Cambridge did it receive emphasis. Even after his time at Cambridge mathematical studies and their applications were neglected there. Jeremiah Horrox was at Cambridge in 1633-35, desiring to make himself an astronomer.

“But many impediments,” says Horrox, “presented themselves: the tedious difficulty of the study itself deterred a mind not yet formed; the want of means oppressed, and still oppresses, the aspirations of my mind: but that which gave me most concern was that there was no one who could instruct me in the art, who could even help my endeavours by joining me in the study; such was the sloth and languor which had seized all… I found that books must be used instead of teachers.”[3 - Companion to the [British] Almanac of 1837, p. 28, in an article by Augustus De Morgan on “Notices of English Mathematical and Astronomical Writers between the Norman Conquest and the Year 1600.”]

Some attention was given to Greek mathematicians, but the works of Italian, German, and French algebraists of the latter part of the sixteenth and beginning of the seventeenth century were quite unknown at Cambridge in Oughtred’s day. It was part of his life-work as a mathematician to make algebra, as it was being developed in his time, accessible to English youths.

At the age of twenty-three Oughtred invented his Easy Way of Delineating Sun-Dials by Geometry, which, though not published until about half a century later, in the first English edition of Oughtred’s Clavis mathematicae in 1647, was in the meantime translated into Latin by Christopher Wren, then a Gentleman Commoner of Wadham College, Oxford, now best known through his architectural creations. In 1600 Oughtred wrote a monograph on the construction of sun-dials upon a plane of any inclination, but that paper was withheld by him from publication until 1632. Sun-dials were interesting objects of study, since watches and pendulum clocks were then still unknown. All sorts of sun-dials, portable and non-portable, were used at that time and long afterward. Several of the college buildings at Oxford and Cambridge have sun-dials even at the present time.

AS RECTOR AND AMATEUR MATHEMATICIAN

It was in 1604 that Oughtred entered upon his professional life-work as a preacher, being instituted to the vicarage of Shalford in Surrey. In 1610 he was made rector of Albury, where he spent the remainder of his long life. Since the era of the Reformation two of the rectors of Albury obtained great celebrity from their varied talents and acquirements – our William Oughtred and Samuel Horsley. Oughtred continued to devote his spare time to mathematics, as he had done in college. A great mathematical invention made by a Scotchman soon commanded his attention – the invention of logarithms. An informant writes as follows:

Lord Napier, in 1614, published at Edinburgh his Mirifici logarithmorum canonis descriptio… It presently fell into the hands of Mr. Briggs, then geometry-reader at Gresham College in London: and that gentleman, forming a design to perfect Lord Napier’s plan, consulted Oughtred upon it; who probably wrote his Treatise of Trigonometry about the same time, since it is evidently formed upon the plan of Lord Napier’s Canon.[4 - New and General Biographical Dictionary (John Nichols), London, 1784, art. “Oughtred.”]

It will be shown later that Oughtred is very probably the author of an “Appendix” which appeared in the 1618 edition of Edward Wright’s translation into English of John Napier’s Descriptio. This “Appendix” relates to logarithms and is an able document, containing several points of historical interest. Mr. Arthur Hutchinson of Pembroke College informs me that in the university library at Cambridge there is a copy of Napier’s Constructio (1619) bound up with a copy of Kepler’s Chilias logarithmorum (1624), that at the beginning of the Constructio is a blank leaf, and before this occurs the title-page only of Napier’s Descriptio (1619), at the top of which appears Oughtred’s autograph. The history of this interesting signature is unknown.

HIS WIFE

In 1606 he married Christ’sgift Caryll, daughter of Caryll, Esq., of Tangley, in an adjoining parish.[5 - Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, p. 132.] We know very little about Oughtred’s family life. The records at King’s College, Cambridge,[6 - Skeleton Collegii Regalis Cantab.: Or A Catalogue of All the Provosts, Fellows and Scholars, of the King’s College.. since the Foundation Thereof, Vol. II, “William Oughtred.”] mention a son, but it is certain that there were more children. A daughter was married to Christopher Brookes. But there is no confirmation of Aubrey’s statements,[7 - Aubrey, op. cit., Vol. II, p. 107.] according to which Oughtred had nine sons and four daughters. Reference to the wife and children is sometimes made in the correspondence with Oughtred. In 1616 J. Hales writes, “I pray let me be remembered, though unknown, to Mistress Oughtred.”[8 - Rigaud, Correspondence of Scientific Men of the Seventeenth Century, Oxford, Vol. I, 1841, p. 5.]

As we shall see later, Oughtred had a great many young men who came to his house and remained there free of charge to receive instruction in mathematics, which was likewise gratuitous. This being the case, certainly great appreciation was due to Mrs. Oughtred, upon whom the burden of hospitality must have fallen. Yet chroniclers are singularly silent in regard to her. Hers was evidently a life of obscurity and service. We greatly doubt the accuracy of the following item handed down by Aubrey; it cannot be a true characterization:

His wife was a penurious woman, and would not allow him to burne candle after supper, by which meanes many a good notion is lost, and many a probleme unsolved; so that Mr. [Thomas] Henshawe, when he was there, bought candle, which was a great comfort to the old man.[9 - Aubrey, op. cit., Vol. II, p. 110.]

IN DANGER OF SEQUESTRATION

Oughtred spent his years in “unremitted attention to his favourite study,” sometimes, it has been whispered, to the neglect of his rectorial duties. Says Aubrey:

I have heard his neighbour ministers say that he was a pittiful preacher; the reason was because he never studyed it, but bent all his thoughts on the mathematiques; but when he was in danger of being sequestred for a royalist, he fell to the study of divinity, and preacht (they sayd) admirably well, even in his old age.[10 - Ibid., p. 111.]

This remark on sequestration brings to mind one of the political and religious struggles of the time, the episcopacy against the independent movements. Says Manning:

In 1646 he was cited before the Committee for Ecclesiastical Affairs, where many articles had been deposed against him; but, by the favour of Sir Bulstrode Whitlock and others, who, at the intercession of William Lilye the Astrologer, appeared in great numbers on his behalf, he had a majority on his side, and so escaped a sequestration.[11 - Op. cit., Vol. II, p. 132.]

Not without interest is the account of this matter given by Lilly himself:

About this Time, the most famous Mathematician of all Europe, (Mr. William Oughtred, Parson of Aldbury in Surrey) was in Danger of Sequestration by the Committee of or for plunder’d Ministers; (Ambo-dexters they were;) several inconsiderable Articles were deposed and sworn against him, material enough to have sequestred him, but that, upon his Day of hearing, I applied my self to Sir Bolstrode Whitlock, and all my own old Friends, who in such Numbers appeared in his Behalf, that though the Chairman and many other Presbyterian Members were stiff against him, yet he was cleared by the major Number. The truth is, he had a considerable Parsonage, and that only was enough to sequester any moderate Judgment: He was also well known to affect his Majesty [Charles I]. In these Times many worthy Ministers lost their Livings or Benefices, for not complying with the Three-penny Directory.[12 - Mr. William Lilly’s History of His Life and Times, From the Year 1602 to 1681, London, 1715, p. 58.]

HIS TEACHING

Oughtred had few personal enemies. His pupils held him in highest esteem and showed deep gratitude; only one pupil must be excepted, Richard Delamain. Against him arose a bitter controversy which saddened the life of Oughtred, then an old man. It involved, as we shall see later, the priority of invention of the circular slide rule and of a horizontal instrument or portable sun-dial. In defense of himself, Oughtred wrote in 1633 or 1634 the Apologeticall Epistle, from which we quoted above. This document contains biographical details, in part as follows:

Ever since my departure from the Vniversity, which is about thirty yeares, I have lived neere to the Towne of Guildford in Surrey: where, whether I have taken so much liberty to the losse of time, and the neglect of my calling the whole Countrey thereabout, both Gentry and others, to whom I am full well knowne, will quickely informe him; my house being not past three and twenty miles from London: and yet I so hid my selve at home, that I seldomly travelled so farre as London once in a yeare. Indeed the life and mind of man cannot endure without some interchangeablenesse of recreation, and pawses from the intensive actions of our severall callings; and every man is drawne with his owne delight. My recreations have been diversity of studies: and as oft as I was toyled with the labour of my owne profession, I have allayed that tediousnesse by walking in the pleasant and more then Elysian fields of the diverse and various parts of humane learning, and not the Mathematics onely.

Even the opponents of Delamain must be grateful to him for having been the means of drawing from Oughtred such interesting biographical details. Oughtred proceeds to tell how, about 1628, he was induced to write his Clavis mathematicae, upon which his reputation as a mathematician largely rests:

About five yeares since, the Earle of Arundell my most honourable Lord in a time of his private retiring to his house in the countrey then at West Horsley, foure small miles from me (though since he hath a house in Aldebury the parish where I live) hearing of me (by what meanes I know not) was pleased to send for me: and afterward at London to appoint mee a Chamber of his owne house: where, at such times, and in such manner as it seemed him good to imploy me, and when I might not inconveniently be spared from my charge, I have been most ready to present my selfe in all humble and affectionate service: I hope also without the offence of God, the transgression of the good Lawes of this Land, neglect of my calling, or the deserved scandall of any good man…

And although I am no mercenary man, nor make profession to teach any one in these arts for gaine and recompence, but as I serve at the Altar, so I live onely of the Altar: yet in those interims that I am at London in my Lords service, I have been still much frequented both by Natives and Strangers, for my resolution and instruction in many difficult poynts of Art; and have most freely and lovingly imparted my selfe and my skill, such as I had, to their contentments, and much honourable acknowledgement of their obligation to my Lord for bringing mee to London, hath beene testifyed by many. Of which my liberallity and unwearyed readinesse to doe good to all, scarce any one can give more ample testimony then R. D. himselfe can: would he be but pleased to allay the shame of this his hot and eager contention, blowne up onely with the full bellowes of intended glory and gaine;.. they [the subjects in which Delamain received assistance from Oughtred] were the first elements of Astronomie concerning the second motions of the fixed starres, and of the Sunne and Moone; they were the first elements of Conics, to delineate those sections: they were the first elements of Optics, Catoptrics, and Dioptrics: of all which you knew nothing at all.

These last passages are instructive as showing what topics were taken up for study with some of his pupils. The chief subject of interest with most of them was algebra, which at that time was just beginning to draw the attention of English lovers of mathematics.

Oughtred carried on an extensive correspondence on mathematical subjects. He was frequently called upon to assist in the solution of knotty problems – sometimes to his annoyance, perhaps, as is shown by the following letter which he wrote in 1642 to a stranger, named Price:

It is true that I have bestowed such vacant time, as I could gain from the study of divinity, (which is my calling,) upon human knowledges, and, amongst other, upon the mathematics, wherein the little skill I have attained, being compared with others of my profession, who for the most part contenting themselves only with their own way, refuse to tread these salebrous and uneasy paths, may peradventure seem the more. But now being in years and mindful of mine end, and having paid dearly for my former delights both in my health and state, besides the prejudice of such, who not considering what incessant labour may produce, reckon so much wanting unto me in my proper calling, as they think I have acquired in other sciences; by which opinion (not of the vulgar only) I have suffered both disrespect, and also hinderance in some small perferments I have aimed at. I have therefore now learned to spare myself, and am not willing to descend again in arenam, and to serve such ungrateful muses. Yet, sir, at your request I have perused your problem… Your problem is easily wrought per Nicomedis conchoidem lineam.[13 - Rigaud, op. cit., Vol. I, p. 60.]

APPEARANCE AND HABITS

Aubrey gives information about the appearance and habits of Oughtred:

He was a little man, had black haire, and blacke eies (with a great deal of spirit). His head was always working. He would drawe lines and diagrams on the dust…

He [his oldest son Benjamin] told me that his father did use to lye a bed till eleaven or twelve a clock, with his doublet on, ever since he can remember. Studyed late at night; went not to bed till 11 a clock; had his tinder box by him; and on the top of his bed-staffe, he had his inke-horne fix’t. He slept but little. Sometimes he went not to bed in two or three nights, and would not come downe to meales till he had found out the quaesitum.

He was more famous abroad for his learning, and more esteemed, then at home. Severall great mathematicians came over into England on purpose to converse with him. His countrey neighbours (though they understood not his worth) knew that there must be extraordinary worth in him, that he was so visited by foreigners…

When learned foreigners came and sawe how privately he lived, they did admire and blesse themselves, that a person of so much worth and learning should not be better provided for…

He has told bishop Ward, and Mr. Elias Ashmole (who was his neighbour), that “on this spott of ground” (or “leaning against this oake” or “that ashe”), “the solution of such or such a probleme came into my head, as if infused by a divine genius, after I had thought on it without successe for a yeare, two, or three.”..

Nicolaus Mercator, Holsatus.. went to see him few yeares before he dyed…