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diff --git a/old/46815-0.txt b/old/46815-0.txt new file mode 100644 index 0000000..7688a57 --- /dev/null +++ b/old/46815-0.txt @@ -0,0 +1,3929 @@ +The Project Gutenberg EBook of William Oughtred, by Florian Cajori + +This eBook is for the use of anyone anywhere in the United States and most +other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms of +the Project Gutenberg License included with this eBook or online at +www.gutenberg.org. If you are not located in the United States, you'll have +to check the laws of the country where you are located before using this ebook. + +Title: William Oughtred + A great Seventeenth-Century Teacher of Mathematics + +Author: Florian Cajori + +Release Date: September 9, 2014 [EBook #46815] + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK WILLIAM OUGHTRED *** + + + + +Produced by Brenda Lewis, Stephen Hutcheson, and the Online +Distributed Proofreading Canada Team at +http://www.pgdpcanada.net + + + + + + + WILLIAM OUGHTRED + + + + + WILLIAM OUGHTRED + A GREAT SEVENTEENTH-CENTURY + TEACHER OF + MATHEMATICS + + + BY + FLORIAN CAJORI, Ph.D. + Professor of Mathematics + Colorado College + + CHICAGO LONDON + THE OPEN COURT PUBLISHING COMPANY + 1916 + + Copyright 1916 By + The Open Court Publishing Co. + + All Rights Reserved + + Published September 1916 + + + Composed and Printed By + The University of Chicago Press + Chicago, Illinois, U.S.A. + + + + + TABLE OF CONTENTS + + + PAGE + Introduction 1 + CHAPTER + I. Oughtred’s Life 3 + At School and University 3 + As Rector and Amateur Mathematician 6 + His Wife 7 + In Danger of Sequestration 8 + His Teaching 9 + Appearance and Habits 12 + Alleged Travel Abroad 14 + His Death 15 + II. Principal Works 17 + Clavis mathematicae 17 + Circles of Proportion and Trigonometrie 35 + Solution of Numerical Equations 39 + Logarithms 46 + Invention of the Slide Rule; Controversy on Priority of Invention 46 + III. Minor Works 50 + IV. Oughtred’s Influence upon Mathematical Progress and Teaching 57 + Oughtred and Harriot 57 + Oughtred’s Pupils 58 + Oughtred, the “Todhunter of the Seventeenth Century” 60 + Was Descartes Indebted to Oughtred? 69 + The Spread of Oughtred’s Notations 73 + V. Oughtred’s Ideas on the Teaching of Mathematics 84 + General Statement 84 + Mathematics, “a Science of the Eye” 85 + Rigorous Thinking and the Use of Instruments 87 + Newton’s Comments on Oughtred 94 + Index 97 + + + + + 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] 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] + +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] + +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] + +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] We know very little about Oughtred’s +family life. The records at King’s College, Cambridge,[6] 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] 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] + +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] + + + 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] + +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] + +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] + + + 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] + + + 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. . . . . + + The right hon^ble Thomas Howard, earle of Arundel and Surrey, Lord High + Marshall of England, was his great patron, and loved him intirely. One + time they were like to have been killed together by the fall at Albury + of a grott, which fell downe but just as they were come out.[14] + +Oughtred’s friends convey the impression that, in the main, Oughtred +enjoyed a comfortable living at Albury. Only once appear indications of +financial embarrassment. About 1634 one of his pupils, W. Robinson, +writes as follows: + + I protest unto you sincerely, were I as able as some, at whose hands + you have merited exceedingly, or (to speak more absolutely) as able as + willing, I would as freely give you 500 l. per ann. as 500 pence; and I + cannot but be astonished at this our age, wherein pelf and dross is + made their summum bonum, and the best part of man, with the true + ornaments thereof, science and knowledge, are so slighted. . . . .[15] + +In his letters Oughtred complains several times of the limitations for +work and the infirmities due to his advancing old age. The impression he +made upon others was quite different. Says one biographer: + + He sometimes amused himself with archery, and sometimes practised as a + surveyor of land. . . . . He was sprightly and active, when more than + eighty years of age.[16] + +Another informant says that Oughtred was + + as facetious in Greek and Latine as solid in Arithmetique, Astronomy, + and the sphere of all Measures, Musick, etc.; exact in his style as in + his judgment; handling his Cube, and other Instruments at eighty, as + steadily, as others did at thirty; owing this, he said, to temperance + and Archery; principling his people with plain and solid truths, as he + did the world with great and useful Arts; advancing new Inventions in + all things but Religion. Which in its old order and decency he + maintained secure in his privacy, prudence, meekness, simplicity, + resolution, patience, and contentment.[17] + + + ALLEGED TRAVEL ABROAD + +According to certain sources of information, Oughtred traveled on the +European Continent and was invited to change his abode to the Continent. +We have seen no statement from Oughtred himself on this matter. He seldom +referred to himself in his books and letters. The autobiography contained +in his Apologeticall Epistle was written a quarter of a century before +his death. Aubrey gives the following: + + In the time of the civill warres the duke of Florence invited him over, + and offered him 500 li. per annum; but he would not accept it, because + of his religion.[18] + +A portrait of Oughtred, painted in 1646 by Hollar and inserted in the +English edition of the Clavis of 1647, contains underneath the following +lines: + + “Haec est Oughtredi senio labantis imago + Itala quam cupiit, Terra Britanna tulit.” + +In the sketch of Oughtred by Owen Manning it is confessed that “it is not +known to what this alludes; but possibly he might have been in Italy with +his patron, the Earl of Arundel.”[19] It would seem quite certain either +that Oughtred traveled in Europe or that he received some sort of an +offer to settle in Italy. In view of Aubrey’s explicit statement and of +Oughtred’s well-known habit of confining himself to his duties and +studies in his own parish, seldom going even as far as London, we +strongly incline to the opinion that he did not travel on the Continent, +but that he received an offer from some patron of the sciences—possibly +some distinguished visitor—to settle in Italy. + + + HIS DEATH + +He died at Albury, June 30, 1660, aged about eighty-six years. Of his +last days and death, Aubrey speaks as follows: + + Before he dyed he burned a world of papers, and sayd that the world was + not worthy of them; he was so superb. He burned also severall printed + bookes, and would not stirre, till they were consumed. . . . . I + myselfe have his Pitiscus, imbelished with his excellent marginall + notes, which I esteeme as a great rarity. I wish I could also have got + his Bilingsley’s Euclid, which John Collins sayes was full of his + annotations. . . . . + + Ralph Greatrex, his great friend, the mathematicall instrument-maker, + sayed he conceived he dyed with joy for the comeing-in of the king, + which was the 29th of May before. “And are yee sure he is + restored?”—“Then give me a glasse of sack to drinke his sacred + majestie’s health.” His spirits were then quite upon the wing to fly + away. . . . .[20] + +In this passage, as in others, due allowance must be made for Aubrey’s +lack of discrimination. He was not in the habit of sifting facts from +mere gossip. That Oughtred should have declared that the world was not +worthy of his papers or manuscripts is not in consonance with the +sweetness of disposition ordinarily attributed to him. More probable was +the feeling that the papers he burned—possibly old sermons—were of no +particular value to the world. That he did not destroy a large mass of +mathematical manuscripts is evident from the fact that a considerable +number of them came after his death into the hands of Sir Charles +Scarborough, M.D., under whose supervision some of them were carefully +revised and published at Oxford in 1677 under the title of Opuscula +mathematica hactenus inedita. + +Aubrey’s story of Oughtred’s mode of death has been as widely circulated +in every modern biographical sketch as has his slander of Mrs. Oughtred +by claiming that she was so penurious that she would deny him the use of +candles to read by. Oughtred died on June 30; the Restoration occurred on +May 29. No doubt Oughtred rejoiced over the Restoration, but the story of +his drinking “a glass of sack” to his Majesty’s health, and then dying of +joy is surely apocryphal. De Morgan humorously remarks, “It should be +added, by way of excuse, that he was eighty-six years old.”[21] + + + + + CHAPTER II + PRINCIPAL WORKS + + + “CLAVIS MATHEMATICAE” + +Passing to the consideration of Oughtred’s mathematical books, we begin +with the observation that he showed a marked disinclination to give his +writings to the press. His first paper on sun-dials was written at the +age of twenty-three, but we are not aware that more than one brief +mathematical manuscript was printed before his fifty-seventh year. In +every instance, publication in printed form seems to have been due to +pressure exerted by one or more of his patrons, pupils, or friends. Some +of his manuscripts were lent out to his pupils, who prepared copies for +their own use. In some instances they urged upon him the desirability of +publication and assisted in preparing copy for the printer. The earliest +and best-known book of Oughtred was his Clavis mathematicae, to which +repeated allusion has already been made. As he himself informs us, he was +employed by the Earl of Arundel about 1628 to instruct the Earl’s son, +Lord William Howard (afterward Viscount Stafford) in the mathematics. For +the use of this young man Oughtred composed a treatise on algebra which +was published in Latin in the year 1631 at the urgent request of a +kinsman of the young man, Charles Cavendish, a patron of learning. + +The Clavis mathematicae,[22] in its first edition of 1631, was a booklet +of only 88 small pages. Yet it contained in very condensed form the +essentials of arithmetic and algebra as known at that time. + +Aside from the addition of four tracts, the 1631 edition underwent some +changes in the editions of 1647 and 1648, which two are much alike. The +twenty chapters of 1631 are reduced to nineteen in 1647 and in all the +later editions. Numerous minute alterations from the 1631 edition occur +in all parts of the books of 1647 and 1648. The material of the last +three chapters of the 1631 edition is rearranged, with some slight +additions here and there. The 1648 edition has no preface. In the print +of 1652 there are only slight alterations from the 1648 edition; after +that the book underwent hardly any changes, except for the number of +tracts appended, and brief explanatory notes added at the close of the +chapters in the English editions of 1694 and 1702. The 1652 and 1667 +editions were seen through the press by John Wallis; the 1698 impression +contains on the title-page the words: Ex Recognitione D. Johannis Wallis, +S.T.D. Geometriae Professoris Saviliani. + +The cost of publishing may be a matter of some interest. When arranging +for the printing of the 1667 edition of the Clavis, Wallis wrote Collins: +“I told you in my last what price she [Mrs. Lichfield] expects for it, as +I have formerly understood from her, viz., £ 40 for the impression, which +is about 9½d. a book.”[23] + +As compared with other contemporary works on algebra, Oughtred’s +distinguishes itself for the amount of symbolism used, particularly in +the treatment of geometric problems. Extraordinary emphasis was placed +upon what he called in the Clavis the “analytical art.”[24] By that term +he did not mean our modern analysis or analytical geometry, but the art +“in which by taking the thing sought as knowne, we finde out that we +seeke.”[25] He meant to express by it condensed processes of rigid, +logical deduction expressed by appropriate symbols, as contrasted with +mere description or elucidation by passages fraught with verbosity. In +the preface to the first edition (1631) he says: + + In this little book I make known . . . . the rules relating to + fundamentals, collected together, just like a bundle, and adapted to + the explanation of as many problems as possible. + +As stated in this preface, one of his reasons for publishing the book, is + + . . . . that like Ariadne I might offer a thread to mathematical study + by which the mysteries of this science might be revealed, and direction + given to the best authors of antiquity, Euclid, Archimedes, the great + geometrician Apollonius of Perga, and others, so as to be easily and + thoroughly understood, their theorems being added, not only because to + many they are the height and depth of mathematical science (I ignore + the would-be mathematicians who occupy themselves only with the + so-called practice, which is in reality mere juggler’s tricks with + instruments, the surface so to speak, pursued with a disregard of the + great art, a contemptible picture), but also to show with what keenness + they have penetrated, with what mass of equations, comparisons, + reductions, conversions and disquisitions these heroes have ornamented, + increased and invented this most beautiful science. + +The Clavis opens with an explanation of the Hindu-Arabic notation and of +decimal fractions. Noteworthy is the absence of the words “million,” +“billion,” etc. Though used on the Continent by certain mathematical +writers long before this, these words did not become current in English +mathematical books until the eighteenth century. The author was a great +admirer of decimal fractions, but failed to introduce the notation which +in later centuries came to be universally adopted. Oughtred wrote 0.56 in +this manner 0|56; the point he used to designate ratio. Thus 3:4 was +written by him 3·4. The decimal point (or comma) was first used by the +inventor of logarithms, John Napier, as early as 1616 and 1617. Although +Oughtred had mastered the theory of logarithms soon after their +publication in 1614 and was a great admirer of Napier, he preferred to +use the dot for the designation of ratio. This notation of ratio is used +in all his mathematical books, except in two instances. The two dots (:) +occur as symbols of ratio in some parts of Oughtred’s posthumous work, +Opuscula mathematica hactenus inedita, Oxford, 1677, but may have been +due to the editors and not to Oughtred himself. Then again the two dots +(:) are used to designate ratio on the last two pages of the tables of +the Latin edition of Oughtred’s Trigonometria of 1657. In all other parts +of that book the dot (·) is used. Probably someone who supervised the +printing of the tables introduced the (:) on the last two pages, +following the logarithmic tables, where methods of interpolation are +explained. The probability of this conjecture is the stronger, because in +the English edition of the Trigonometrie, brought out the same year +(1657) but after the Latin edition, the notation (:) at the end of the +book is replaced by the usual (·), except that in some copies of the +English edition the explanations at the end are omitted altogether. + +Oughtred introduces an interesting, and at the same time new, feature of +an abbreviated multiplication and an abbreviated division of decimal +fractions. On this point he took a position far in advance of his time. +The part on abbreviated multiplication was rewritten in slightly enlarged +form and with some unimportant alterations in the later edition of the +Clavis. We give it as it occurs in the revision. Four cases are given. In +finding the product of 246|914 and 35|27, “if you would have the Product +without any Parts” (without any decimal part), “set the place of Unity of +the lesser under the place of Unity in the greater: as in the Example,” +writing the figures of the lesser number in inverse order. From the +example it will be seen that he begins by multiplying by 3, the +right-hand digit of the multiplier. In the first edition of the Clavis he +began with 7, the left digit. Observe also that he “carries” the nearest +tens in the product of each lower digit and the upper digit one place to +its right. For instance, he takes 7×4=28 and carries 3, then he finds +7×2+3=17 and writes down 17. + + 2 4 6|9 1 4 + 7 2|5 3 + ------- + 7 4 0 7 + 1 2 3 5 + 4 9 + 1 7 + ------- + 8 7 0 8 + +The second case supposes that “you would have the Product with some +places of parts” (decimals), say 4: “Set the place of Unity of the lesser +Number under the Fourth place of the Parts of the greater.” The +multiplication of 246|914 by 35|27 is now performed thus: + + 2 4 6|9 1 4 + 7 2|5 3 + --------------- + 7 4 0 7 4 2 0 0 + 1 2 3 4 5 7 0 0 + 4 9 3 8 2 8 + 1 7 2 8 4 0 + --------------- + 8 7 0 8|6 5 6 8 + +In the third and fourth cases are considered factors which appear as +integers, but are in reality decimals; for instance, the sine of 54° is +given in the tables as 80902 when in reality it is .80902. + +Of interest as regards the use of the word “parabola” is the following: +“The Number found by Division is called the Quotient, or also Parabola, +because it arises out of the Application of a plain Number to a given +Longitude, that a congruous Latitude may be found.”[26] This is in +harmony with etymological dictionaries which speak of a parabola as the +application of a given area to a given straight line. The dividend or +product is the area; the divisor or factor is the line. + +Oughtred gives two processes of long division. The first is identical +with the modern process, except that the divisor is written below every +remainder, each digit of the divisor being crossed out as soon as it has +been used in the partial multiplication. The second method of long +division is one of the several types of the old “scratch method.” This +antiquated process held its place by the side of the modern method in all +editions of the Clavis. The author divides 467023 by 357|0926425, giving +the following instructions: “Take as many of the first Figures of the +Divisor as are necessary, for the first Divisor, and then in every +following particular Division drop one of the Figures of the Divisor +towards the Left Hand, till you have got a competent Quotient.” He does +not explain abbreviated division as thoroughly as abbreviated +multiplication. + + 17 + 3̸0̸3̸ + 2̸8̸0̸3̸ + 1̸0̸9̸9̸3̸0̸ + 3̣5̣7̣|0̣9̣2̣6425) 4̸6̸7̸0̸2̸3̸ (1307|80 + 3̸5̸7̸0̸9̸3̸ + 1̸0̸7̸1̸2̸7̸ + 2̸5̸0̸0̸ + 2̸8̸6̸ + + +Oughtred does not examine the degree of reliability or accuracy of his +processes of abbreviated multiplication and division. Here as in other +places he gives in condensed statement the mode of procedure, without +further discussion. + +He does not attempt to establish the rules for the addition, subtraction, +multiplication, and division of positive and negative numbers. “If the +Signs are both alike, the Product will be affirmative, if unlike, +negative”; then he proceeds to applications. This attitude is superior to +that of many writers of the eighteenth and nineteenth centuries, on +pedagogical as well as logical grounds: pedagogically, because the +beginner in the study of algebra is not in a position to appreciate an +abstract train of thought, as every teacher well knows, and derives +better intellectual exercise from the applications of the rules to +problems; logically, because the rule of signs in multiplication does not +admit of rigorous proof, unless some other assumption is first made which +is no less arbitrary than the rule itself. It is well known that the +proofs of the rule of signs given by eighteenth-century writers are +invalid. Somewhere they involve some surreptitious assumption. This +criticism applies even to the proof given by Laplace, which tacitly +assumes the distributive law in multiplication. + +A word should be said on Oughtred’s definition of + and -. He recognizes +their double function in algebra by saying (Clavis, 1631, p. 2): “Signum +additionis, sive affirmationis, est + plus” and “Signum subductionis, +sive negationis est - minus.” They are symbols which indicate the quality +of numbers in some instances and operations of addition or subtraction in +other instances. In the 1694 edition of the Clavis, thirty-four years +after the death of Oughtred, these symbols are defined as signifying +operations only, but are actually used to signify the quality of numbers +as well. In this respect the 1694 edition marks a recrudescence. + +The characteristic in the Clavis that is most striking to a modern reader +is the total absence of indexes or exponents. There is much discussion in +the leading treatises of the latter part of the sixteenth and the early +part of the seventeenth century on the theory of indexes, but the modern +exponential notation, aⁿ, is of later date. The modern notation, for +positive integral exponents, first appears in Descartes’ Géométrie, 1637; +fractional and negative exponents were first used in the modern form by +Sir Isaac Newton, in his announcement of the binomial formula, in a +letter written in 1676. This total absence of our modern exponential +notation in Oughtred’s Clavis gives it a strange aspect. Like Vieta, +Oughtred uses ordinarily the capital letters, A, B, C, . . . . to +designate given numbers; A² is written Aq, A³ is written Ac; for A⁴, A⁵, +A⁶ he has, respectively, Aqq, Aqc, Acc. Only on rare occasions, usually +when some parallelism in notation is aimed at, does he use small +letters[27] to represent numbers or magnitudes. Powers of binomials or +polynomials are marked by prefixing the capital letters Q (for square), C +(for cube), QQ (for the fourth power), QC (for the fifth power), etc. + +Oughtred does not express aggregation by (). Parentheses had been used by +Girard, and by Clavius as early as 1609,[28] but did not come into +general use in mathematical language until the time of Leibniz and the +Bernoullis. Oughtred indicates aggregation by writing a colon (:) at both +ends. Thus, Q:A-E: means with him (A-E)². Similarly, √q:A+E: means +√(A+E). The two dots at the end are frequently omitted when the part +affected includes all the terms of the polynomial to the end. Thus, +C:A+B-E=.. means (A+B-E)³=.. There are still further departures from this +notation, but they occur so seldom that we incline to the interpretation +that they are simply printer’s errors. For proportion Oughtred uses the +symbol (::). The proportion a:b=c:d appears in his notation a·b::c·d. +Apparently, a proportion was not fully recognized in this day as being +the expression of an equality of ratios. That probably explains why he +did not use = here as in the notation of ordinary equations. Yet Oughtred +must have been very close to the interpretation of a proportion as an +equality; for he says in his Elementi decimi Euclidis declaratio, +“proportio, sive ratio aequalis ::” That he introduced this extra symbol +when the one for equality was sufficient is a misfortune. Simplicity +demands that no unnecessary symbols be introduced. However, Oughtred’s +symbolism is certainly superior to those which preceded. Consider the +notation of Clavius.[29] He wrote 20:60=4:x, x=12, thus: “20·60·4? fiunt +12.” The insufficiency of such a notation in the more involved +expressions frequently arising in algebra is readily seen. Hence +Oughtred’s notation (::) was early adopted by English mathematicians. It +was used by John Wallis at Oxford, by Samuel Foster at Gresham College, +by James Gregory of Edinburgh, by the translators into English of Rahn’s +algebra, and by many other early writers. Oughtred has been credited +generally with the introduction of St. Andrew’s cross × as the symbol for +multiplication in the Clavis of 1631. We have discovered that this +symbol, or rather the letter x which closely resembles it, occurs as the +sign of multiplication thirteen years earlier in an anonymous “Appendix +to the Logarithmes, shewing the practise of the Calculation of Triangles +etc.” to Edward Wright’s translation of John Napier’s Descriptio, +published in 1618.[30] Later we shall give our reasons for believing that +Oughtred is the author of that “Appendix.” The × has survived as a symbol +of multiplication. + +Another symbol introduced by Oughtred and found in modern books is ~, +expressing difference; thus C~D signifies the difference between C and D, +even when D is the larger number.[31] This symbol was used by John Wallis +in 1657.[32] + +Oughtred represented in symbols also certain composite expressions, as +for instance A+E=Z, A-E=X, where A is greater than E. He represented by a +symbol also each of the following: A²+E², A³+E³, A²-E², A³-E³. + +Oughtred practically translated the tenth book of Euclid from its +ponderous rhetorical form into that of brief symbolism. An appeal to the +eye was a passion with Oughtred. The present writer has collected the +different mathematical symbols used by Oughtred and has found more than +one hundred and fifty of them. + +The differences between the seven different editions of the Clavis lie +mainly in the special parts appended to some editions and dropped in the +latest editions. The part which originally constituted the Clavis was not +materially altered, except in two or three of the original twenty +chapters. These changes were made in the editions of 1647 and 1648. After +the first edition, great stress was laid upon the theory of indices upon +the very first page, as also in passages farther on. Of course, Oughtred +did not have our modern notation of indices or exponents, but their +theory had been a part of algebra and arithmetic for some time. Oughtred +incorporated this theory in his brief exposition of the Hindu-Arabic +notation and in his explanation of logarithms. As previously pointed out, +the last three chapters of the 1631 edition were considerably rearranged +in the later editions and combined into two chapters, so that the Clavis +proper had nineteen chapters instead of twenty in the additions after the +first. These chapters consisted of applications of algebra to geometry +and were so framed as to constitute a severe test of the student’s grip +of the subject. The very last problem deals with the division of angles +into equal parts. He derives the cubic equation upon which the trisection +depends algebraically, also the equations of the fifth degree and seventh +degree upon which the divisions of the angle into 5 and 7 equal parts +depend, respectively. The exposition was severely brief, yet accurate. He +did not believe in conducting the reader along level paths or along +slight inclines. He was a guide for mountain-climbers, and woe unto him +who lacked nerve. + +Oughtred lays great stress upon expansions of powers of a binomial. He +makes use of these expansions in the solution of numerical equations. To +one who does not specialize in the history of mathematics such expansions +may create surprise, for did not Newton invent the binomial theorem after +the death of Oughtred? As a matter of fact, the expansions of positive +integral powers of a binomial were known long before Newton, not only to +seventeenth-century but even to eleventh-century mathematicians. +Oughtred’s Clavis of 1631 gave the binomial coefficients for all powers +up to and including the tenth. What Newton really accomplished was the +generalization of the binomial expansion which makes it applicable to +negative and fractional exponents and converts it into an infinite +series. + +As a specimen of Oughtred’s style of writing we quote his solution of +quadratic equations, accompanied by a translation into English and into +modern mathematical symbols. + +As a preliminary step[33] he lets + + Z=A+E and A>E; + +he lets also X=A-E. From these relations he obtains identities which, in +modern notation, are ¼Z²-AE=(½Z-E)²=¼X². Now, if we know Z and AE, we can +find ½X. Then ½(Z+X)=A, and ½(Z-X)=E, and + + A=½Z+√(¼Z²-AE). + +Having established these preliminaries, he proceeds thus: + + Datis igitur linea inaequaliter secta Z (10), & rectangulo sub + segmentis AE (21) qui gnomon est: datur semidifferentia segmentorum ½X: + & per consequens ipsa segmenta. Nam ponatur alterutrum segmentum A: + alterum erit Z-A: Rectangulum auctem est ZA-A_q=AE. Et quia dantur Z & + AE: estque ¼Z_q-AE=¼X_q: & per 5c. 18, ½Z+½X=A: & ½Z-½X=E: Aequatio sic + resoluetur: ½Z±√_q:¼Z_q-AE:=A {maius segment/minus segment. + + Itaque proposita equatione, in qua sunt tres species aequaliter in + ordine tabellae adscendentes, altissima autem species ponitur negata: + Magnitudo data coefficiens mediam speciem est linea bisecanda: & + magnitudo absoluta data, ad quam sit aequatio, est rectangulum sub + segmentis inaequalibus, sine gnomon: vt ZA-A_q=AE: in numeris autem + 10l-l_q=21: Estque A, vel 1l, alterutrum segmentum inaequale. Inuenitur + autem sic: + + Dimidiata coefficiens median speciem est Z/2 (5); cuius quadratum est + Z_q/4 (25): ex hoc tolle AE (21) absolutum: eritque Z_q/4-AE (4) + quadratum semidifferentiae segmentorum: latus huius quadratum (2) est + semidifferentia: quam si addas ad Z/2 (5) semissem coefficientis, sive + lineae bisecandae, erit maius segment.; sin detrahas, erit minus + segment: Dico Z/2±√_q:Z_q/4-AE:=A {maius segmentum/minus segmentum. + +We translate the Latin passage, using the modern exponential notation and +parentheses, as follows: + + Given therefore an unequally divided line Z (10), and a rectangle + beneath the segments AE (21) which is a gnomon. Half the difference of + the segments ½X is given, and consequently the segment itself. For, if + one of the two segments is placed equal to A, the other will be Z-A. + Moreover, the rectangle is ZA-A²=AE. And because Z and AE are given, + and there is ¼Z²-AE=¼X², and by 5c.18, ½Z+½X=A, and ½Z-½X=E, the + equation will be solved thus: ½Z±√(¼Z²-AE)=A {major segment/minor + segment. + + And so an equation having been proposed in which three species (terms) + are in equally ascending powers, the highest species, moreover, being + negative, the given magnitude which constitutes the middle species is + the line to be bisected. And the given absolute magnitude to which it + is equal is the rectangle beneath the unequal segments, without gnomon. + As ZA-A²=AE, or in numbers, 10x-x²=21. And A or x is one of the two + unequal segments. It may be found thus: + + The half of the middle species is Z/2 (5), its square is Z²/4 (25). + From it subtract the absolute term AE (21), and Z²/4-AE (4) will be the + square of half the difference of the segments. The square root of this, + √[(Z²/2)²-AE] (2), is half the difference. If you add it to half the + coefficient Z/2 (5), or half the line to be bisected, the longer + segment is obtained; if you subtract it, the smaller segment is + obtained. I say: Z/2±√(Z²/4-AE)=A {major segment/minor segment. + +The quadratic equation Aq+ZA=AE receives similar treatment. This and the +preceding equation, ZA-Aq=AE, constitute together a solution of the +general quadratic equation, x²+ax=b, provided that E or Z are not +restricted to positive values, but admit of being either positive or +negative, a case not adequately treated by Oughtred. Imaginary numbers +and imaginary roots receive no consideration whatever. + +A notation suggested by Vieta and favored by Girard made vowels stand for +unknowns and consonants for knowns. This conventionality was adopted by +Oughtred in parts of his algebra, but not throughout. Near the beginning +he used Q to designate the unknown, though usually this letter stood with +him for the “square” of the expression after it.[34] + +It is of some interest that Oughtred used π/δ to signify the ratio of the +circumference to the diameter of a circle. Very probably this notation is +the forerunner of the π=3.14159 . . . . used in 1706 by William Jones. +Oughtred first used π/δ in the 1647 edition of the Clavis mathematicae. +In the 1652 edition he says, “Si in circulo sit 7.22::δ·π::113.355:erit +δ·π::2 R.P: periph.” This notation was adopted by Isaac Barrow, who used +it extensively. David Gregory[35] used π/ρ in 1697, and De Moivre[36] +used c/r about 1697, to designate the ratio of the circumference to the +radius. + +We quote the description of the Clavis that was given by Oughtred’s +greatest pupil, John Wallis. It contains additional information of +interest to us. Wallis devotes chap. xv of his Treatise of Algebra, +London, 1685, pp. 67-69, to Mr. Oughtred and his Clavis, saying: + + Mr. William Oughtred (our Country-man) in his Clavis Mathematicae, (or + Key of Mathematicks,) first published in the Year 1631, follows Vieta + (as he did Diophantus) in the use of the Cossick Denominations; + omitting (as he had done) the names of Sursolids, and contenting + himself with those of Square and Cube, and the Compounds of these. + + But he doth abridge Vieta’s Characters or Species, using only the + letters q, c, &c. which in Vieta are expressed (at length) by Quadrate, + Cube, &c. For though when Vieta first introduced this way of Specious + Arithmetick, it was more necessary (the thing being new,) to express it + in words at length: Yet when the thing was once received in practise, + Mr. Oughtred (who affected brevity, and to deliver what he taught as + briefly as might be, and reduce all to a short view,) contented himself + with single Letters instead of those words. + + Thus what Vieta would have written + + A Quadrate, into B Cube, + ------------------------ Equal to FG Plane, + CDE Solid, + + would with him be thus expressed + + A_q B_c + ------- = FG. + C D E + + And the better to distinguish upon the first view, what quantities were + Known, and what Unknown, he doth (usually) denote the Known to + Consonants, and the Unknown by Vowels; as Vieta (for the same reason) + had done before him. + + He doth also (to very great advantage) make use of several Ligatures, + or Compendious Notes, to signify Summs, Differences, and Rectangles of + several Quantities. As for instance, Of two Quantities A (the Greater), + and E (the Lesser), the Sum he calls Z, the Difference X, the Rectangle + AE. . . . . + + Which being of (almost) a constant signification with him throughout, + do save a great circumlocution of words, (each Letter serving instead + of a Definition;) and are also made use of (with very great advantage) + to discover the true nature of divers intricate Operations, arising + from the various compositions of such Parts, Sums, Differences, and + Rectangles; (of which there is great plenty in his Clavis, Cap. 11, 16, + 18, 19. and elsewhere,) which without such Ligatures, or Compendious + Notes, would not be easily discovered or apprehended. . . . . + + I know there are who find fault with his Clavis, as too obscure, + because so short, but without cause; for his words be always full, but + not Redundant, and need only a little attention in the Reader to weigh + the force of every word, and the Syntax of it; . . . . And this, when + once apprehended, is much more easily retained, than if it were + expressed with the prolixity of some other Writers; where a Reader must + first be at the pains to weed out a great deal of superfluous Language, + that he may have a short prospect of what is material; which is here + contracted for him in a short Synopsis. . . . . + + Mr. Oughtred in his Clavis, contents himself (for the most part) with + the solution of Quadratick Equations, without proceeding (or very + sparingly) to Cubick Equations, and those of Higher Powers; having + designed that Work for an Introduction into Algebra so far, leaving the + Discussion of Superior Equations for another work. . . . . He contents + himself likewise in Resolving Equations, to take notice of the + Affirmative or Positive Roots; omitting the Negative or Ablative Roots, + and such as are called Imaginary or Impossible Roots. And of those + which, he calls Ambiguous Equations, (as having more Affirmative Roots + than one,) he doth not (that I remember) any where take notice of more + than Two Affirmative Roots: (Because in Quadratick Equations, which are + those he handleth, there are indeed no more.) Whereas yet in Cubick + Equations, there may be Three, and in those of Higher Powers, yet more. + Which Vieta was well aware of, and mentioneth in some of his Writings; + and of which Mr. Oughtred could not be ignorant. + + + “CIRCLES OF PROPORTION” AND “TRIGONOMETRIE” + +Oughtred wrote and had published three important mathematical books, the +Clavis, the Circles of Proportion,[37] and a Trigonometrie.[38] This last +appeared in the year 1657 at London, in both Latin and English. + +It is claimed that the trigonometry was “neither finished nor published +by himself, but collected out of his scattered papers; and though he +connived at the printing it, yet imperfectly done, as appears by his +MSS.; and one of the printed Books, corrected by his own Hand.”[39] +Doubtless more accurate on this point is a letter of Richard Stokes who +saw the book through the press: + + I have procured your Trigonometry to be written over in a fair hand, + which when finished I will send to you, to know if it be according to + your mind; for I intend (since you were pleased to give your assent) to + endeavour to print it with Mr. Briggs his Tables, and so soon as I can + get the Prutenic Tables I will turn those of the sun and moon, and send + them to you.[40] + +In the preface to the Latin edition Stokes writes: + + Since this trigonometry was written for private use without the + intention of having it published, it pleased the Reverend Author, + before allowing it to go to press, to expunge some things, to change + other things and even to make some additions and insert more lucid + methods of exposition. + +This much is certain, the Trigonometry bears the impress characteristic +of Oughtred. Like all his mathematical writings, the book was very +condensed. Aside from the tables, the text covered only 36 pages. Plane +and spherical triangles were taken up together. The treatise is known in +the history of trigonometry as among the very earliest works to adopt a +condensed symbolism so that equations involving trigonometric functions +could be easily taken in by the eye. In the work of 1657, contractions +are given as follows: s=sine, t=tangent, se=secant, s co=cosine (sine +complement), t co=cotangent, se co=cosecant, log=logarithm, Z cru=sum of +the sides of a rectangle or right angle, X cru=difference of these sides. +It has been generally overlooked by historians that Oughtred used the +abbreviations of trigonometric functions, named above, a quarter of a +century earlier, in his Circles of Proportion, 1632, 1633. Moreover, he +used sometimes also the abbreviations which are current at the present +time, namely sin=sine, tan=tangent, sec=secant. We know that the Circles +of Proportion existed in manuscript many years before they were +published. The symbol sv for sinus versus occurs in the Clavis of 1631. +The great importance of well-chosen symbols needs no emphasis to readers +of the present day. With reference to Oughtred’s trigonometric symbols. +Augustus De Morgan said: + + This is so very important a step, simple as it is, that Euler is justly + held to have greatly advanced trigonometry by its introduction. Nobody + that we know of has noticed that Oughtred was master of the + improvement, and willing to have taught it, if people would have + learnt.[41] + +We find, however, that even Oughtred cannot be given the whole credit in +this matter. By or before 1631 several other writers used abbreviations +of the trigonometric functions. As early as 1624 the contractions sin for +sine and tan for tangent appear on the drawing representing Gunter’s +scale, but Gunter did not use them in his books, except in the drawing of +his scale.[42] A closer competitor for the honor of first using these +trigonometric abbreviations is Richard Norwood in his Trigonometrie, +London, 1631, where s stands for sine, t for tangent, sc for sine +complement (cosine), tc for tangent complement (cotangent), and sec for +secant. Norwood was a teacher of mathematics in London and a well-known +writer of books on navigation. Aside from the abbreviations just cited +Norwood did not use nearly as much symbolism in his mathematics as did +Oughtred. + +Mention should be made of trigonometric symbols used even earlier than +any of the preceding, in “An Appendix to the Logarithmes, shewing the +practise of the Calculation of Triangles, etc.,” printed in Edward +Wright’s edition of Napier’s A Description of the Admirable Table of +Logarithmes, London, 1618. We referred to this “Appendix” in tracing the +origin of the sign ×. It contains, on p. 4, the following passage: “For +the Logarithme of an arch or an angle I set before (s), for the +antilogarithme or compliment thereof (s*) and for the Differential (t).” +In further explanation of this rather unsatisfactory passage, the author +(Oughtred?) says, “As for example: sB+BC=CA. that is, the Logarithme of +an angle B. at the Base of a plane right-angled triangle, increased by +the addition of the Logarithm of BC, the hypothenuse thereof, is equall +to the Logarithme of CA the cathetus.” + +Here “logarithme of an angle B” evidently means “log sin B,” just as with +Napier, “Logarithms of the arcs” signifies really “Logarithms of the +sines of the angles.” In Napier’s table, the numbers in the column marked +“Differentiae” signify log sine minus log cosine of an angle; that is, +the logarithms of the tangents. This explains the contraction (t) in the +“Appendix.” The conclusion of all this is that as early as 1618 the signs +s, s*, t were used for sine, cosine, and tangent, respectively. + +John Speidell, in his Breefe Treatise of Sphaericall Triangles, London, +1627, uses Si. for sine, T. and Tan for tangent, Se. for secant, Si. Co. +for cosine, Se. Co. for cosecant, T. Co. for cotangent. + +The innovation of designating the sides and angles of a triangle by A, B, +C, and a, b, c, so that A was opposite a, B opposite b, and C opposite c, +is attributed to Leonard Euler (1753), but was first used by Richard +Rawlinson of Queen’s College, Oxford, sometimes after 1655 and before +1668. Oughtred did not use Rawlinson’s notation.[43] + +In trigonometry English writers of the first half of the seventeenth +century used contractions more freely than their continental +contemporaries; even more freely, indeed, than English writers of a later +period. Von Braunmühl, the great historian of trigonometry, gives +Oughtred much praise for his trigonometry, and points out that half a +century later the army of writers on trigonometry had hardly yet reached +the standard set by Oughtred’s analysis.[44] Oughtred must be credited +also with the first complete proof that was given to the first two of +“Napier’s analogies.” His trigonometry contains seven-place tables of +sines, tangents, and secants, and six-place tables of logarithmic sines +and tangents; also seven-place logarithmic tables of numbers. At the time +of Oughtred there was some agitation in favor of a wider introduction of +decimal systems. This movement is reflected in those tables which contain +the centesimal division of the degree, a practice which is urged for +general adoption in our own day, particularly by the French. + + + SOLUTION OF NUMERICAL EQUATIONS + +In the solution of numerical equations Oughtred does not mention the +sources from which he drew, but the method is substantially that of the +great French algebraist Vieta, as explained in a publication which +appeared in 1600 in Paris under the title, De numerosa potestatum purarum +atque adfectarum ad exegesin resolutione tractatus. In view of the fact +that Vieta’s process has been described inaccurately by leading modern +historians including H. Hankel[45] and M. Cantor,[46] it may be worth +while to go into some detail.[47] By them it is made to appear as +identical with the procedure given later by Newton. The two are not the +same. The difference lies in the divisor used. What is now called +“Newton’s method” is Newton’s method as modified by Joseph Raphson.[48] +The Newton-Raphson method of approximation to the roots of an equation +f(x)=0 is usually given the form a-[f(a)/f´(a)], where a is an +approximate value of the required root. It will be seen that the divisor +is f´(a). Vieta’s divisor is different; it is + + |f(a+s₁)-f(a)|-s₁ⁿ, + +where f(x) is the left of the equation f(x)=k, n is the degree of +equation, and s₁ is a unit of the denomination of the digit next to be +found. Thus in x³+420000x=247651713, it can be shown that 417 is +approximately a root; suppose that a has been taken to be 400, then +s₁=10; but if, at the next step of approximation, a is taken to be 410, +then s₁=1. In this example, taking a=400, Vieta’s divisor would have been +9120000; Newton’s divisor would have been 900000. + +A comparison of Vieta’s method with the Newton-Raphson method reveals the +fact that Vieta’s divisor is more reliable, but labors under the very +great disadvantage of requiring a much larger amount of computation. The +latter divisor is accurate enough and easier to compute. Altogether the +Newton-Raphson process marks a decided advance over that of Vieta. + +As already stated, it is the method of Vieta that Oughtred explains. The +Englishman’s exposition is an improvement on that of Vieta, printed forty +years earlier. Nevertheless, Oughtred’s explanation is far from easy to +follow. The theory of equations was at that time still in its primitive +stage of development. Algebraic notation was not sufficiently developed +to enable the argument to be condensed into a form easily surveyed. So +complicated does Vieta’s process of approximation appear that M. Cantor +failed to recognize that Vieta possessed a uniform mode of procedure. But +when one has in mind the general expression for Vieta’s divisor which we +gave above, one will recognize that there was marked uniformity in +Vieta’s approximations. + +Oughtred allows himself twenty-eight sections in which to explain the +process and at the close cannot forbear remarking that 28 is a “perfect” +number (being equal to the sum of its divisors, 1, 2, 4, 7, 14). + +The early part of his exposition shows how an equation may be transformed +so as to make its roots 10, 100, 1000, or 10^m times smaller. This +simplifies the task of “locating a root”; that is, of finding between +what integers the root lies. + +Taking one of Oughtred’s equations, x⁴-72x³+238600x=8725815, upon +dividing 72x³ by 10, 238600x by 1000, and 8725815 by 10,000, we obtain +x⁴-7·2x³+238·6x=872·5. Dividing both sides by x, we obtain +x³+238·6-7·2x²=x)872·5. Letting x=4, we have 64+238·6-115·2=187·4. + +But 4)872·5(218·1; 4 is too small. Next let x=5, we have +125+238·6-180=183·6. + +But 5)872·5(174·5; 5 is too large. We take the lesser value, x=4, or in +the original equation, x=40. This method may be used to find the second +digit in the root. Oughtred divides both sides of the equation by x², and +obtains x²+x)238600-72x=x²)8725815. He tries x=47 and x=48, and finds +that x=47. + +He explains also how the last computation may be done by logarithms. +Thereby he established for himself the record of being the first to use +logarithms in the solution of affected equations. + +As an illustration of Oughtred’s method of approximation after the root +sought has been located, we have chosen for brevity a cubic in preference +to a quartic. We selected the equation x³+420000x=247651713. By the +process explained above a root is found to lie between x=400 and x=500. +From this point on, the approximation as given by Oughtred is as shown on +p. 43. + +In further explanation of this process, observe that the given equation +is of the form L_c+C_qL=D_c, where L_c is our x, C_q=420000, +D_c=247651713. In the first step of approximation, let L=A+E, where A=400 +and E is, as yet, undetermined. We have + + L_c=(A+E)³=A³+3A²E+3AE²+E³ + +and + + C_qL=420000(A+E). + +Subtract from 247651713 the sum of the known terms A³ (his A_c) and +420000 A (his C_qA). This sum is 232000000 the remainder is 15651713. + + “Exemplum II + + 1c+42̣00̣00̣l=247̇651̇7̣1̣3̣̇ + + Hoc est, L_c+C_qL=D_c + + 2 4 7̇ | 6 5 1̇ | 7̣ 1̣ 3̣̇ | ( 4 1 7 + ------+-------+-------+------------ + 4 2 | 0 0 0 | 0 | C_q + ------+-------+-------+------------ + 6 4 | | | A_c + 1 6 8 | 0 0 0 | 0 | C_q A + ------+-------+-------+------------ + 2 3 2 | 0 0 0 | 0 | Ablatit. + =================================== +R 1 5 | 6 5 1̇ | 7 1 3̣ | + ------+-------+-------+------------ + 4 | 8 | | 3 A_q + | 1 2 | | 3 A + 4 | 2 0 0 | 0 0 | C_q + ------+-------+-------+------------ + 9 | 1 2 0 | 0 0 | Divisor. + ------+-------+-------+------------ + 4 | 8 | | 3 A_q E + | 1 2 | | 3 A E_q + | 1 | | E_c + 4 | 2 0 0 | 0 0 | C_q E + ------+-------+-------+------------ + 9 | 1 2 1 | 0 0 | Ablatit. + =================================== +R 6 | 5 3 0 | 7 1 3̣̇ | 4 | 1 | + ------+-------+-------+------------ ----+-----+--- + | 5 0 4 | 3 | 3 A_q | | + | 1 | 2 3 | 3 A 1 6 | 8 | + | 4 2 0 | 0 0 0 | C_q | 1 | + ------+-------+-------+------------ ----+-----+--- + | 9 2 5 | 5 3 0 | Divisor. 1 6 8 1 + ------+-------+-------+------------ + 3 | 5 3 0 | 1 | 3 A_q E + | 6 0 | 2 7 | 3 A E_q + | | 3 4 3 | E_c + 2 | 9 4 0 | 0 0 0 | C_q E + ------+-------+-------+------------ + 6 | 5 3 0 | 7 1 3 | Ablatit.” + +Next, he evaluates the coefficients of E in 3A²E and 420000E, also 3A, +the coefficient of E². He obtains 3A²=480000, 3A=1200, C_q=420000. He +interprets 3A² and C_q as tens, 3A as hundreds. Accordingly, he obtains +as their sum 9120000, which is the divisor for finding the second digit +in the approximation. Observe that this divisor is the value of +|f(a+s₁)-f(a)|-s₁ⁿ in our general expression, where a=400, s₁=10, n=3, +f(x)=x³+420000x. + +Dividing the remainder 15651713 by 9120000, he obtains the integer 1 in +ten’s place; thus E=10, approximately. He now computes the terms 3A²E, +3AE² and E³ to be, respectively, 4800000, 120000, 1000. Their sum is +9121000. Subtracting it from the previous remainder, 15651713, leaves the +new remainder, 6530713. + +From here on each step is a repetition of the preceding step. The new A +is 410, the new E is to be determined. We have now in closer +approximation, L=A+E. This time we do not subtract A³ and C_qA, because +this subtraction is already affected by the preceding work. + +We find the second trial divisor by computing the sum of 3A², 3A and C_q; +that is, the sum of 504300, 1230, 420000, which is 925530. Again, this +divisor can be computed by our general expression for divisors, by taking +a=410, s₁=1, n=3. + +Dividing 6530713 by 925530 yields the integer 7. Thus E=7. Computing +3A²E, 3AE², E³ and subtracting their sum, the remainder is 0. Hence 417 +is an exact root of the given equation. + +Since the extraction of a cube root is merely the solution of a pure +cubic equation, x³=n, the process given above may be utilized in finding +cube roots. This is precisely what Oughtred does in chap. xiv of his +Clavis. If the foregoing computation is modified by putting C_q=0, the +process will yield the approximate cube root of 247651713. + +Oughtred solves 16 examples by the process of approximation here +explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In +all cases he finds only one or two real roots. Of the roots sought, five +are irrational, the remaining are rational and are computed to their +exact values. Three of the computed roots have 2 figures each, 9 roots +have 3 figures each, 4 roots have 4 figures each. While no attempt is +made to secure all the roots—methods of computing complex roots were +invented much later—he computes roots of equations which involve large +coefficients and some of them are of a degree as high as the fifth. In +view of the fact that many editions of the Clavis were issued, one +impression as late as 1702, it contributed probably more than any other +book to the popularization of Vieta’s method in England. + +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, log_e 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 SEVENTEENTH + CENTURY” + +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 Wood[57] 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: + + Amongst the mathematical rarities these times have afforded, there are + none of that small number I (a late intruder into these studies) have + yet viewed, which so fully demonstrates their authors’ great abilities + as your Clavis, not richer in augmentations, than valuable for + contraction; . . . . + + Your belief that there is in all inventions aliquid divinum, an + infusion beyond human cogitations, I am confident will appear notably + strengthened, if you please to afford this truth belief, that I entered + upon these studies accidentally after I betook myself to the country, + having never had so much aid as to be taught addition, nor the + discourse of an artist (having left both Oxford and London before I + knew what any proposition in geometry meant) to inform me what were the + best authors.[59] + +The following extracts from two letters by W. Robinson, written before +the appearance of the 1647 English edition of the Clavis, express the +feeling of many readers of the Clavis on its extreme conciseness and +brevity of explanation: + + I shall long exceedingly till I see your Clavis turned into a + pick-lock; and I beseech you enlarge it, and explain it what you can, + for we shall not need to fear either tautology or superfluity; you are + naturally concise, and your clear judgment makes you both methodical + and pithy; and your analytical way is indeed the only way. . . . . + + I will once again earnestly entreat you, that you be rather diffuse in + the setting forth of your English mathematical Clavis, than concise, + considering that the wisest of men noted of old, and said stultorum + infinitus est numerus, these arts cannot be made too easy, they are so + abstruse of themselves, and men either so lazy or dull, that their + fastidious wits take a loathing at the very entrance of these studies, + unless it be sweetened on with plainness and facility. Brevity may well + argue a learned author, that without any excess or redundance, either + of matter or words, can give the very substance and essence of the + thing treated of; but it seldom makes a learned scholar; and if one be + capable, twenty are not; and if the master sum up in brief the pith of + his own long labours and travails, it is not easy to imagine that + scholars can with less labour than it cost their masters dive into the + depths thereof.[60] + +Here is the judgment of another of Oughtred’s friends: + + . . . . with the character I received from your and my noble friend Sir + Charles Cavendish, then at Paris, of your second edition of the same + piece, made me at my return into England speedily to get, and + diligently peruse the same. Neither truly did I find my expectation + deceived; having with admiration often considered how it was possible + (even in the hardest things of geometry) to deliver so much matter in + so few words, yet with such demonstrative clearness and perspicuity: + and hath often put me in mind of learned Mersennus his judgment (since + dead) of it, that there was more matter comprehended in that little + book than in Diophantus, and all the ancients. . . . .[61] + +Oughtred’s own feeling was against diffuseness in textbook writing. In +his revisions of his Clavis the original character of that book was not +altered. In his reply to W. Robinson, Oughtred said: + + . . . . But my art for all such mathematical inventions I have set down + in my Clavis Mathematica, which therefore in my title I say is tum + logisticae cum analyticae adeoque totius mathematicae quasi clavis, + which if any one of a mathematical genius will carefully study, (and + indeed it must be carefully studied,) he will not admire others, but + himself do wonders. But I (such is my tenuity) have enough fungi vice + cotis, acutum reddere quae ferrum valet, exsors ipsa secandi, or like + the touchstone, which being but a stone, base and little worth, can + shew the excellence and riches of gold.[62] + +John Wallis held Oughtred’s Clavis in high regard. When in correspondence +with John Collins concerning plans for a new edition, Wallis wrote in +1666-67, six years after the death of Oughtred: + + . . . . But for the goodness of the book in itself, it is that (I + confess) which I look upon as a very good book, and which doth in as + little room deliver as much of the fundamental and useful part of + geometry (as well as of arithmetic and algebra) as any book I know; and + why it should not be now acceptable I do not see. It is true, that as + in other things so in mathematics, fashions will daily alter, and that + which Mr. Oughtred designed by great letters may be now by others be + designed by small; but a mathematician will, with the same ease and + advantage, understand A_c, and a³ or aaa. . . . . And the like I judge + of Mr. Oughtred’s Clavis, which I look upon (as those pieces of Vieta + who first went in that way) as lasting books and classic authors in + this kind; to which, notwithstanding, every day may make new additions. + . . . . + + But I confess, as to my own judgment, I am not for making the book + bigger, because it is contrary to the design of it, being intended for + a manual or contract; whereas comments, by enlarging it, do rather + destroy it. . . . . But it was by him intended, in a small epitome, to + give the substance of what is by others delivered in larger volumes. . + . . .[63] + +That there continued to be a group of students and teachers who desired a +fuller exposition than is given by Oughtred is evident from the +appearance, over fifty years after the first publication of the Clavis, +of a booklet by Gilbert Clark, entitled Oughtredus Explicatus, London, +1682. A review of this appeared in the Acta Eruditorum (Leipzig, 1684), +on p. 168, wherein Oughtred is named “clarissimus Angliae mathematicus.” +John Collins wrote Wallis in 1666-67 that Clark, “who lives with Sir +Justinian Isham, within seven miles of Northampton, . . . . intimates he +wrote a comment on the Clavis, which lay long in the hands of a printer, +by whom he was abused, meaning Leybourne.”[64] + +We shall have occasion below to refer to Oughtred’s inability to secure a +copy of a noted Italian mathematical work published a few years before. +In those days the condition of the book trade in England must have been +somewhat extraordinary. Dr. J. W. L. Glaisher throws some light upon this +subject.[65] He found in the Calendar of State Papers, Domestic Series, +1637, a petition to Archbishop Laud in which it is set forth that when +Hooganhuysen, a Dutchman, “heretofore complained of in the High +Commission for importing books printed beyond the seas,” had been bound +“not to bring in any more,” one Vlacq (the computer and publisher of +logarithmic tables) “kept up the same agency and sold books in his stead. +. . . . Vlacq is now preparing to go beyond the seas to avoid answering +his late bringing over nine bales of books contrary to the decree of the +Star Chamber.” Judgment was passed that, “Considering the ill-consequence +and scandal that would arise by strangers importing and venting in this +kingdom books printed beyond the seas,” certain importations be +prohibited, and seized if brought over. + +This want of easy intercommunication of results of scientific research in +Oughtred’s time is revealed in the following letter, written by Oughtred +to Robert Keylway, in 1645: + + I speak this the rather, and am induced to a better confidence of your + performance, by reason of a geometric-analytical art or practice found + out by one Cavalieri, an Italian, of which about three years since I + received information by a letter from Paris, wherein was praelibated + only a small taste thereof, yet so that I divine great enlargement of + the bounds of the mathematical empire will ensue. I was then very + desirous to see the author’s own book while my spirits were more free + and lightsome, but I could not get it in France. Since, being more + stept into years, daunted and broken with the sufferings of these + disastrous times, I must content myself to keep home, and not put out + to any foreign discoveries.[66] + +It was in 1655, when Oughtred was about eighty years old, that John +Wallis, the great forerunner of Newton in Great Britain, began to publish +his great researches on the arithmetic of infinites. Oughtred rejoiced +over the achievements of his former pupil. In 1655, Oughtred wrote John +Wallis as follows: + + I have with unspeakable delight, so far as my necessary businesses, the + infirmness of my health, and the greatness of my age (approaching now + to an end) would permit, perused your most learned papers, of several + choice arguments, which you sent me: wherein I do first with + thankfulness acknowledge to God, the Father of lights, the great light + he hath given you; and next I congratulate you, even with admiration, + the clearness and perspicacity of your understanding and genius, who + have not only gone, but also opened a way into these profoundest + mysteries of art, unknown and not thought of by the ancients. With + which your mysterious inventions I am the more affected, because full + twenty years ago, the learned patron of learning, Sir Charles + Cavendish, shewed me a paper written, wherein were some few excellent + new theorems, wrought by the way, as I suppose, of Cavalieri, which I + wrought over again more agreeably to my way. The paper, wherein I + wrought it, I shewed to many, whereof some took copies, but my own I + cannot find. I mention it for this, because I saw therein a light + breaking out for the discovery of wonders to be revealed to mankind, in + this last age of the world: which light I did salute as afar off, and + now at a nearer distance embrace in your prosperous beginnings. Sir, + that you are pleased to mention my name in your never dying papers, + that is your noble favour to me, who can add nothing to your glory, but + only my applause. . . . .[67] + +The last sentence has reference to Wallis’ appreciative and eulogistic +reference to Oughtred in the preface. It is of interest to secure the +opinion of later English writers who knew Oughtred only through his +books. John Locke wrote in his journal under the date, June 24, 1681, +“the best algebra yet extant is Outred’s.”[68] John Collins, who is known +in the history of mathematics chiefly through his very extensive +correspondence with nearly all mathematicians of his day, was inclined to +be more critical. He wrote Wallis about 1667: + + It was not my intent to disparage the author, though I know many that + did lightly esteem him when living, some whereof are at rest, as Mr. + Foster and Mr. Gibson. . . . . You grant the author is brief, and + therefore obscure, and I say it is but a collection, which, if himself + knew, he had done well to have quoted his authors, whereto the reader + might have repaired. You do not like those words of Vieta in his + theorems, ex adjunctione plano solidi, plus quadrato quadrati, etc., + and think Mr. Oughtred the first that abridged those expressions by + symbols; but I dissent, and tell you ’twas done before by Cataldus, + Geysius, and Camillus Gloriosus,[69] who in his first decade of + exercises, (not the first tract,) printed at Naples in 1627, which was + four years before the first edition of the Clavis, proposeth this + equation just as I here give it you, viz. + 1ccc+16qcc+41qqc-2304cc-18364qc-133000qq-54505c+3728q+8064 N aequatur + 4608, finds N or a root of it to be 24, and composeth the whole out of + it for proof, just in Mr. Oughtred’s symbols and method. Cataldus on + Vieta came out fifteen years before, and I cannot quote that, as not + having it by me. + + . . . . And as for Mr. Oughtred’s method of symbols, this I say to it; + it may be proper for you as a commentator to follow it, but divers I + know, men of inferior rank that have good skill in algebra, that + neither use nor approve it. . . . . Is not A⁵ sooner wrote than A_qc? + Let A be 2, the cube of 2 is 8, which squared is 64: one of the + questions between Maghet Grisio and Gloriosus is whether 64=A_cc or + A_qc. The Cartesian method tells you it is A⁶, and decides the doubt. . + . . .[70] + +There is some ground for the criticisms passed by Collins. To be sure, +the first edition of the Clavis is dated 1631—six years before Descartes +suggested the exponential notation which came to be adopted as the +symbolism in our modern algebra. But the second edition of the Clavis, +1647, appeared ten years after Descartes’ innovation. Had Oughtred seen +fit to adopt the new exponential notation in 1647, the step would have +been epoch-making in the teaching of algebra in England. We have seen no +indication that Oughtred was familiar with Descartes’ Géométrie of 1637. + +The year preceding Oughtred’s death Mr. John Twysden expressed himself as +follows in the preface to his Miscellanies: + + It remains that I should adde something touching the beginning, and use + of these Sciences. . . . . I shall only, to their honours, name some of + our own Nation yet living, who have happily laboured upon both stages. + That succeeding ages may understand that in this of ours, there yet + remained some who were neither ignorant of these Arts, as if they had + held them vain, nor condemn them as superfluous. Amongst them all let + Mr. William Oughtred, of Aeton, be named in the first place, a Person + of venerable grey haires, and exemplary piety, who indeed exceeds all + praise we can bestow upon him. Who by an easie method, and admirable + Key, hath unlocked the hidden things of geometry. Who by an accurate + Trigonometry and furniture of Instruments, hath inriched, as well + geometry, as Astronomy. Let D. John Wallis, and D. Seth Ward, succeed + in the next place, both famous Persons, and Doctors in Divinity, the + one of geometry, the other of astronomy, Savilian Professors in the + University of Oxford.[71] + +The astronomer Edmund Halley, in his preface to the 1694 English edition +of the Clavis, speaks of this book as one of “so established a +reputation, that it were needless to say anything thereof,” though “the +concise Brevity of the author is such, as in many places to need +Explication, to render it Intelligible to the less knowing Mathematical +matters.” + +In closing this part of our monograph, we quote the testimony of Robert +Boyle, the experimental physicist, as given May 8, 1647, in a letter to +Mr. Hartlib: + + The Englishing of, and additions to Oughtred’s Clavis mathematica does + much content me, I having formerly spent much study on the original of + that algebra, which I have long since esteemed a much more instructive + way of logic, than that of Aristotle.[72] + + + WAS DESCARTES INDEBTED TO OUGHTRED? + +This question first arose in the seventeenth century, when John Wallis, +of Oxford, in his Algebra (the English edition of 1685, and more +particularly the Latin edition of 1693), raised the issue of Descartes’ +indebtedness to the English scientists, Thomas Harriot and William +Oughtred. In discussing matters of priority between Harriot and +Descartes, relating to the theory of equations, Wallis is generally held +to have shown marked partiality to Harriot. Less attention has been given +by historians of mathematics to Descartes’ indebtedness to Oughtred. Yet +this question is of importance in tracing Oughtred’s influence upon his +time. + +On January 8, 1688-89, Samuel Morland addressed a letter of inquiry to +John Wallis, containing a passage which we translate from the Latin: + + Some time ago I read in the elegant and truly precious book that you + have written on Algebra, about Descartes, this philosopher so extolled + above all for having arrived at a very perfect system by his own + powers, without the aid of others, this Descartes, I say, who has + received in geometry very great light from our Oughtred and our + Harriot, and has followed their track though he carefully suppressed + their names. I stated this in a conversation with a professor in + Utrecht (where I reside at present). He requested me to indicate to him + the page-numbers in the two authors which justified this accusation. I + admitted that I could not do so. The Géométrie of Descartes is not + sufficiently familiar to me, although with Oughtred I am fairly + familiar. I pray you therefore that you will assume this burden. Give + me at least those references to passages of the two authors from the + comparison of which the plagiarism by Descartes is the most + striking.[73] + +Following Morland’s letter in the De algebra tractatus, is printed +Wallis’ reply, dated March 12, 1688 (“Stilo Angliae”), which is, in part, +as follows: + + I nowhere give him the name of a plagiarist; I would not appear so + impolite. However this I say, the major part of his algebra (if not + all) is found before him in other authors (notably in our Harriot) whom + he does not designate by name. That algebra may be applied to geometry, + and that it is in fact so applied, is nothing new. Passing the ancients + in silence, we state that this has been done by Vieta, Ghetaldi, + Oughtred and others, before Descartes. They have resolved by algebra + and specious arithmetic [literal arithmetic] many geometrical problems. + . . . . But the question is not as to application of algebra to + geometry (a thing quite old), but of the Cartesian algebra considered + by itself. + +Wallis then indicates in the 1659 edition of Descartes’ Géométrie where +the subjects treated on the first six pages are found in the writings of +earlier algebraists, particularly of Harriot and Oughtred. For example, +what is found on the first page of Descartes, relating to addition, +subtraction, multiplication, division, and root extraction, is declared +by Wallis to be drawn from Vieta, Ghetaldi, and Oughtred. + +It is true that Descartes makes no mention of modern writers, except once +of Cardan. But it was not the purpose of Descartes to write a history of +algebra. To be sure, references to such of his immediate predecessors as +he had read would not have been out of place. Nevertheless, Wallis fails +to show that Descartes made illegitimate use of anything he may have seen +in Harriot or Oughtred. + +The first inquiry to be made is, Did Descartes possess copies of the +books of Harriot and Oughtred? It is only in recent time that this +question has been answered as to Harriot. As to Oughtred, it is still +unanswered. It is now known that Descartes had seen Harriot’s Artis +analyticae praxis (1631). Descartes wrote a letter to Constantin Huygens +in which he states that he is sending Harriot’s book.[74] + +An able discussion of the question, what effect, if any, Oughtred’s +Clavis mathematicae of 1631 had upon Descartes’[75] Géométrie of 1637, is +given by H. Bosmans in a recent article. According to Bosmans no evidence +has been found that Descartes possessed a copy of Oughtred’s book, or +that he had examined it. Bosmans believes nevertheless that Descartes was +influenced by the Clavis, either directly or indirectly. He says: + + If Descartes did not read it carefully, which is not proved, he was + none the less well informed with regard to it. No one denies his + intimate knowledge of the intellectual movement of his time. The Clavis + mathematica enjoyed a rapid success. It is impossible that, at least + indirectly, he did not know the more original ideas which it contained. + Far from belittling Descartes, as I much desire to repeat, this rather + makes him the greater.[76] + +We ourselves would hardly go as far as does Bosmans. Unless Descartes +actually examined a copy of Oughtred it is not likely that he was +influenced by Oughtred in appreciable degree. Book reviews were quite +unknown in those days. No evidence has yet been adduced to show that +Descartes obtained a knowledge of Oughtred by correspondence. A most +striking feature about Oughtred’s Clavis is its notation. No trace of the +Englishman’s symbolism has been pointed out in Descartes’ Géométrie of +1637. Only six years intervened between the publication of the Clavis and +the Géométrie. It took longer than this period for the Clavis to show +evidence of its influence upon mathematical books published in England; +it is not probable that abroad the contact was more immediate than at +home. Our study of seventeenth-century algebra has led us to the +conviction that Oughtred deserves a higher place in the development of +this science than is usually accorded to him; but that it took several +decennia for his influence fully to develop. + + + THE SPREAD OF OUGHTRED’S NOTATIONS + +An idea of Oughtred’s influence upon mathematical thought and teaching +can be obtained from the spread of his symbolism. This study indicates +that the adoption was not immediate. The earliest use that we have been +able to find of Oughtred’s notation for proportion, A.B::C.D, occurs +nineteen years after the Clavis mathematicae of 1631. In 1650 John Kersey +brought out in London an edition of Edmund Wingates’ Arithmetique made +easie, in which this notation is used. After this date publications +employing it became frequent, some of them being the productions of +pupils of Oughtred. We have seen it in Vincent Wing (1651),[77] Seth Ward +(1653),[78] John Wallis (1655),[79] in “R. B.,” a schoolmaster in +Suffolk,[80] Samuel Foster (1659),[81] Jonas Moore (1660),[82] and Isaac +Barrow (1657).[83] In the latter part of the seventeenth century +Oughtred’s notation, A.B::C.D, became the prevalent, though not +universal, notation in Great Britain. A tremendous impetus to their +adoption was given by Seth Ward, Isaac Barrow, and particularly by John +Wallis, who was rising to international eminence as a mathematician. + +In France we have noticed Oughtred’s notation for proportion in +Franciscus Dulaurens (1667),[84] J. Prestet (1675),[85] R. P. Bernard +Lamy (1684),[86] Ozanam (1691),[87] De l’Hospital (1696),[88] R. P. Petro +Nicolas (1697).[89] + +In the Netherlands we have noticed it in R. P. Bernard Lamy (1680),[90] +and in an anonymous work of 1690.[91] In German and Italian works of the +seventeenth century we have not seen Oughtred’s notation for proportion. + +In England a modified notation soon sprang up in which ratio was +indicated by two dots instead of a single dot, thus A:B::C:D. The reason +for the change lies probably in the inclination to use the single dot to +designate decimal fractions. W. W. Beman pointed out that this modified +symbolism (:) for ratio is found as early as 1657 in the end of the +trigonometric and logarithmic tables that were bound with Oughtred’s +Trigonometria.[92] It is not probable, however, that this notation was +used by Oughtred himself. The Trigonometria proper has Oughtred’s +A.B::C.D throughout. Moreover, in the English edition of this +trigonometry, which appeared the same year, 1657, but subsequent to the +Latin edition, the passages which contained the colon as the symbol for +ratio, when not omitted, are recast, and the regular Oughtredian notation +is introduced. In Oughtred’s posthumous work, Opuscula mathematica +hactenus inedita, 1677, the colon appears quite often but is most likely +due to the editor of the book. + +We have noticed that the notation A:B::C:D antedates the year 1657. +Vincent Wing, the astronomer, published in 1651 in London the Harmonicon +coeleste, in which is found not only Oughtred’s notation A.B::C.D but +also the modified form of it given above. The two are used +interchangeably. His later works, the Logistica astronomica (1656), +Doctrina spherica (1655), and Doctrina theorica, published in one volume +in London, all use the symbols A:B::C:D exclusively. The author of a book +entitled, An Idea of Arithmetick at first designed for the use of the +Free Schoole at Thurlow in Suffolk . . . . by R. B., Schoolmaster there, +London, 1655, writes A:a::C:c, though part of the time he uses Oughtred’s +unmodified notation. + +We can best indicate the trend in England by indicating the authors of +the seventeenth century whom we have found using the notation A:B::C:D +and the authors of the eighteenth century whom we have found using +A.B::C.D. The former notation was the less common during the seventeenth +but the more common during the eighteenth century. We have observed the +symbols A:B::C:D (besides the authors already named) in John Collins +(1659),[93] James Gregory (1663),[94] Christopher Wren (1668-69),[95] +William Leybourn (1673),[96] William Sanders (1686),[97] John Hawkins +(1684),[98] Joseph Raphson (1697),[99] E. Wells (1698),[100] and John +Ward (1698).[101] + +Of English eighteenth-century authors the following still clung to the +notation A.B::C.D: John Harris’ translation of F. Ignatius Gaston Pardies +(1701),[102] George Shelley (1704),[103] Sam Cobb (1709),[104] J. Collins +in Commercium Epistolicum (1712), John Craig (1718),[105] Jo. Wilson +(1724).[106] The latest use of A.B::C.D which has come to our notice is +in the translation of the Analytical Institutions of Maria G. Agnesi, +made by John Colson sometime before 1760, but which was not published +until 1801. During the seventeenth century the notation A:B::C:D acquired +almost complete ascendancy in England. + +In France Oughtred’s unmodified notation A.B::C.D, having been adopted +later, was also discarded later than in England. An approximate idea of +the situation appears from the following data. The notation A.B::C.D was +used by M. Carré (1700),[107] M. Guisnée (1705),[108] M. de Fontenelle +(1727),[109] M. Varignon (1725),[110] M. Robillard (1753),[111] M. +Sebastien le Clerc (1764),[112] Clairaut (1731),[113] M. L’Hospital +(1781).[114] + +In Italy Oughtred’s modified notation a, b::c, d was used by Maria G. +Agnesi in her Instituzioni analitiche, Milano, 1748. The notation +a:b::c:d found entrance the latter part of the eighteenth century. In +Germany the symbolism a:b=c:d, suggested by Leibniz, found wider +acceptance.[115] + +It is evident from the data presented that Oughtred proposed his notation +for ratio and proportion at a time when the need of a specific notation +began to be generally felt, that his symbol for ratio a.b was temporarily +adopted in England and France but gave way in the eighteenth century to +the symbol a:b, that Oughtred’s symbol for proportion :: found almost +universal adoption in England and France and was widely used in Italy, +the Netherlands, the United States, and to some extent in Germany; it has +survived to the present time but is now being gradually displaced by the +sign of equality =. + +Oughtred’s notation to express aggregation of terms has received little +attention from historians but is nevertheless interesting. His books, as +well as those of John Wallis, are full of parentheses but they are not +used as symbols of aggregation in algebra; they are simply marks of +punctuation for parenthetical clauses. We have seen that Oughtred writes +(a+b)² and √(a+b) thus, Q:a+b:, √:a+b:, or Q:a+b, √:a+b, using on rarer +occasions a single dot in place of the colon. This notation did not +originate with Oughtred, but, in slightly modified form, occurs in +writings from the Netherlands. In 1603 C. Dibvadii in geometriam Evclidis +demonstratio numeralis, Leyden, contains many expressions of this sort, +√·136+√2048, signifying √(136+√2048). The dot is used to indicate that +the root of the binomial (not of 136 alone) is called for. This notation +is used extensively in Ludolphi à Cevlen de circulo, Leyden, 1619, and in +Willebrordi Snellii De circuli dimensione, Leyden, 1621. In place of the +single dot Oughtred used the colon (:), probably to avoid confusion with +his notation for ratio. To avoid further possibility of uncertainty he +usually placed the colon both before and after the algebraic expression +under aggregation. This notation was adopted by John Wallis and Isaac +Barrow. It is found in the writings of Descartes. Together with Vieta’s +horizontal bar, placed over two or more terms, it constituted the means +used almost universally for denoting aggregation of terms in algebra. +Before Oughtred the use of parentheses had been suggested by Clavius[116] +and Girard.[117] The latter wrote, for instance, √(2+√3). While +parentheses never became popular in algebra before the time of Leibniz +and the Bernoullis they were by no means lost sight of. We are able to +point to the following authors who made use of them: I. Errard de +Bar-le-Duc (1619),[118] Jacobo de Billy (1643),[119] one of whose books +containing this notation was translated into English, and also the +posthumous works of Samuel Foster.[120] J. W. L. Glaisher points out that +parentheses were used by Norwood in his Trigonometrie (1631), p. 30.[121] + +The symbol for the arithmetical difference between two numbers, ~, is +usually attributed to John Wallis, but it occurs in Oughtred’s Clavis +mathematicae of 1652, in the tract on Elementi decimi Euclidis +declaratio, at an earlier date than in any of Wallis’ books. As Wallis +assisted in putting this edition through the press it is possible, though +not probable, that the symbol was inserted by him. Were the symbol +Wallis’, Oughtred would doubtless have referred to its origin in the +preface. During the eighteenth century the symbol found its way into +foreign texts even in far-off Italy.[122] It is one of three symbols +presumably invented by Oughtred and which are still used at the present +time. The others are × and ::. + +The curious and ill-chosen symbols, |̲̅ ̅ for “greater than,” and ̲ ̲̅| +for “less than,” were certain to succumb in their struggle for existence +against Harriot’s admirably chosen > and <. Yet such was the reputation +of Oughtred that his symbols were used in England quite extensively +during the seventeenth and the beginning of the eighteenth century. +Considerable confusion has existed among algebraists and also among +historians as to what Oughtred’s symbols really were. Particularly is +this true of the sign for “less than” which is frequently written ̅ ̲̅|. +Oughtred’s symbols, or these symbols turned about in some way, have been +used by Seth Ward,[123] John Wallis,[124] Isaac Barrow,[125] John +Kersey,[126] E. Wells,[127] John Hawkins,[128] Tho. Baker,[129] Richard +Sault,[130] Richard Rawlinson,[131] Franciscus Dulaurens,[132] James +Milnes,[133] George Cheyne,[134] John Craig,[135] Jo. Wilson,[136] and J. +Collins.[137] + +General acceptance has been accorded to Oughtred’s symbol ×. The first +printed appearance of this symbol for multiplication in 1618 in the form +of the letter x hardly explains its real origin. The author of the +“Appendix” (be he Oughtred or someone else) may not have used the letter +x at all, but may have written the cross ×, called the St. Andrew’s +cross, while the printer, in the absence of any type accurately +representing that cross, may have substituted the letter x in its place. +The hypothesis that the symbol × of multiplication owes its origin to the +old habit of using directed bars to indicate that two numbers are to be +combined, as for instance in the multiplication of 23 and 34, thus, + + 2 3 + |\ /| + | x | + |/ \| + 3 4 + ------- + 7 8 2 + +has been advanced by two writers, C. Le Paige[138] and Gravelaar.[139] +Bosmans is more inclined to the belief that Oughtred adopted the symbol +somewhat arbitrarily, much as he did the numerous symbols in his Elementi +decimi Euclidis declaratio.[140] + +Le Paige’s and Gravelaar’s theory finds some support in the fact that the +cross ×, without the two additional vertical lines shown above, occurs in +a commentary published by Oswald Schreshensuchs[141] in 1551, where the +sign is written between two factors placed one above the other. + + + + + CHAPTER V + OUGHTRED’S IDEAS ON THE TEACHING OF MATHEMATICS + + + GENERAL STATEMENT + +Nowhere has Oughtred given a full and systematic exposition of his views +on mathematical teaching. Nevertheless, he had very pronounced and +clear-cut ideas on the subject. That a man who was not a teacher by +profession should have mature views on teaching is most interesting. We +gather his ideas from the quality of the books he published, from his +prefaces, and from passages in his controversial writing against +Delamain. As we proceed to give quotations unfolding Oughtred’s views, we +shall observe that three points receive special emphasis: (1) an appeal +to the eye through suitable symbolism; (2) emphasis upon rigorous +thinking; (3) the postponement of the use of mathematical instruments +until after the logical foundations of a subject have been thoroughly +mastered. + +The importance of these tenets is immensely reinforced by the conditions +of the hour. This voice from the past speaks wisdom to specialists of +today. Recent methods of determining educational values and the modern +cult of utilitarianism have led some experts to extraordinary +conclusions. Laboratory methods of testing, by the narrowness of their +range, often mislead. Thus far they have been inferior to the word of a +man of experience, insight, and conviction. + + + MATHEMATICS, “A SCIENCE OF THE EYE” + +Oughtred was a great admirer of the Greek mathematicians—Euclid, +Archimedes, Apollonius of Perga, Diophantus. But in reading their works +he experienced keenly what many modern readers have felt, namely, that +the almost total absence of mathematical symbols renders their writings +unnecessarily difficult to read. Statements that can be compressed into a +few well-chosen symbols which the eye is able to survey as a whole are +expressed in long-drawn-out sentences. A striking illustration of the +importance of symbolism is afforded by the history of the formula + + ix=log(cos x+i sin x). + +It was given in Roger Cotes’ Harmonia mensurarum, 1722, not in symbols, +but expressed in rhetorical form, destitute of special aids to the eye. +The result was that the theorem remained in the book undetected for 185 +years and was meanwhile rediscovered by others. Owing to the prominence +of Cotes as a mathematician it is very improbable that such a thing could +have happened had the theorem been thrust into view by the aid of +mathematical symbols. + +In studying the ancient authors Oughtred is reported to have written down +on the margin of the printed page some of the theorems and their proofs, +expressed in the symbolic language of algebra. + +In the preface of his Clavis of 1631 and of 1647 he says: + + Wherefore, that I might more clearly behold the things themselves, I + uncasing the Propositions and Demonstrations out of their covert of + words, designed them in notes and species appearing to the very eye. + After that by comparing the divers affections of Theorems, inequality, + proportion, affinity, and dependence, I tryed to educe new out of them. + +It was this motive which led him to introduce the many abbreviations in +algebra and trigonometry to which reference has been made in previous +pages. The pedagogical experience of recent centuries has indorsed +Oughtred’s view, provided of course that the pupil is carefully taught +the exact meaning of the symbols. There have been and there still are +those who oppose the intensive use of symbolism. In our day the new +symbolism for all mathematics, suggested by the school of Peano in Italy, +can hardly be said to be received with enthusiasm. In Oughtred’s day +symbolism was not yet the fashion. To be convinced of this fact one need +only open a book of Edmund Gunter, with whom Oughtred came in contact in +his youth, or consult the Principia of Sir Isaac Newton, who flourished +after Oughtred. The mathematical works of Gunter and Newton, particularly +the former, are surprisingly destitute of mathematical symbols. The +philosopher Hobbes, in a controversy with John Wallis, criticized the +latter for that “Scab of Symbols,” whereupon Wallis replied: + + I wonder how you durst touch M. Oughtred for fear of catching the Scab. + For, doubtlesse, his book is as much covered over with the Scab of + Symbols, as any of mine. . . . . As for my Treatise of Conick Sections, + you say, it is covered over with the Scab of Symbols, that you had not + the patience to examine whether it is well or ill demonstrated.[142] + +Oughtred maintained his view of the importance of symbols on many +different occasions. Thus, in his Circles of Proportion, 1632, p. 20: + + This manner of setting downe Theoremes, whether they be Proportions, or + Equations, by Symboles or notes of words, is most excellent, + artificiall, and doctrinall. Wherefore I earnestly exhort every one, + that desireth though but to looke into these noble Sciences + Mathematicall, to accustome themselves unto it: and indeede it is + easie, being most agreeable to reason, yea even to sence. And out of + this working may many singular consectaries be drawne: which without + this would, it may be, for ever lye hid. + + + RIGOROUS THINKING AND THE USE OF INSTRUMENTS + +The author’s elevated concept of mathematical study as conducive to +rigorous thinking shines through the following extract from his preface +to the 1647 Clavis: + + . . . . Which Treatise being not written in the usuall synthetical + manner, nor with verbous expressions, but in the inventive way of + Analitice, and with symboles or notes of things instead of words, + seemed unto many very hard; though indeed it was but their owne + diffidence, being scared by the newnesse of the delivery; and not any + difficulty in the thing it selfe. For this specious and symbolicall + manner, neither racketh the memory with multiplicity of words, nor + chargeth the phantasie with comparing and laying things together; but + plainly presenteth to the eye the whole course and processe of every + operation and argumentation. + + Now my scope and intent in the first Edition of that my Key was, and in + this New Filing, or rather forging of it, is, to reach out to the + ingenious lovers of these Sciences, as it were Ariadnes thread, to + guide them through the intricate Labyrinth of these studies, and to + direct them for the more easie and full understanding of the best and + antientest Authors. . . . . That they may not only learn their + propositions, which is the highest point of Art that most Students aime + at; but also may perceive with what solertiousnesse, by what engines of + aequations, Interpretations, Comparations, Reductions, and + Disquisitions, those antient Worthies have beautified, enlarged, and + first found out this most excellent Science. . . . . Lastly, by framing + like questions problematically, and in a way of Analysis, as if they + were already done, resolving them into their principles, I sought out + reasons and means whereby they might be effected. And by this course of + practice, not without long time, and much industry, I found out this + way for the helpe and facilitation of Art. + +Still greater emphasis upon rigorous thinking in mathematics is laid in +the preface to the Circles of Proportion and in some parts of his +Apologeticall Epistle against Delamain. In that preface William Forster +quotes the reply of Oughtred to the question how he (Oughtred) had for so +many years concealed his invention of the slide rule from himself +(Forster) whom he had taught so many other things. The reply was: + + 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 in-stead of Artists, to + make their Scholers only doers of tricks, and as it were Iuglers: to + the despite of Art, losse of previous time, and betraying of willing + and industrious wits, vnto ignorance, and idlenesse. That the vse 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 haue me + well instructed in the Sciences.” + +Delamain took a different view, arguing that instruments might very well +be placed in the hands of pupils from the start. At the time of this +controversy Delamain supported himself by teaching mathematics in London +and he advertised his ability to give instruction in mathematics, +including the use of instruments. Delamain brought the charge against +Oughtred of unjustly calling “many of the [British] Nobility and Gentry +doers of trickes and juglers.” To this Oughtred replies: + + As I did to Delamain and to some others, so I did to William Forster: I + freely gave him my helpe and instruction in these faculties: only this + was the difference, I had the very first moulding (as I may say) of + this latter: But Delamain was already corrupted with doring upon + Instruments, and quite lost from ever being made an Artist: I suffered + not William Forster for some time so much as speake of any Instrument, + except only the Globe it selfe; and to explicate, and worke the + questions of the Sphaere, by the way of the Analemma: which also + himselfe did describe for the present occasion. And this my restraint + from such pleasing avocations, and holding him to the strictnesse of + percept, brought forth this fruit, that in short time, even by his owne + skill, he could not onely use any Instrument he should see, but also + was able to delineate the like, and devise others.[143] + +As representing Delamain’s views, we make the following selection from +his Grammelogia (London, about 1633), the part near the end of the book +and bearing the title, “In the behalfe of vulgar Teachers and others,” +where Delamain refers to Oughtred’s charge that the scholars of “vulgar” +teachers are “doers of tricks, as it were iuglers.” Delamain says: + + . . . . Which words are neither cautelous, nor subterfugious, but are + as downe right in their plainnesse, as they are touching, and + pernitious, by two much derogating from many, and glancing upon many + noble personages, with too grosse, if not too base an attribute, in + tearming them doers of tricks, as it were to iuggle: because they + perhaps make use of a necessitie in the furnishing of themselves with + such knowledge by Practicall Instrumentall operation, when their more + weighty negotiations will not permit them for Theoreticall figurative + demonstration; those that are guilty of the aspertion, and are touched + therewith may answer for themselves, and studie to be more + Theoreticall, than Practicall: for the Theory, is as the Mother that + produceth the daughter, the very sinewes and life of Practise, the + excellencie and highest degree of true Mathematicall Knowledge: but for + those that would make but a step as it were into that kind of Learning, + whose onely desire is expedition, and facilitie, both which by the + generall consent of all are best effected with Instrument, rather then + with tedious regular demonstrations, it was ill to checke them so + grosly, not onely in what they have Practised, but abridging them also + of their liberties with what they may Practise, which aspertion may not + easily be slighted off by any glosse or Apologie, without an Ingenuous + confession, or some mentall reservation: To which vilification, + howsoever, in the behalfe of my selfe, and others, I answer; That + Instrumentall operation is not only the Compendiating, and facilitating + of Art, but even the glory of it, whole demonstration both of the + making, and operation is soly in the science, and to an Artist or + disputant proper to be knowne, and so to all, who would truly know the + cause of the Mathematicall operations in their originall; But, for none + to know the use of a Mathematicall Instrumen[t], except he knowes the + cause of its operation, is somewhat too strict, which would keepe many + from affecting the Art, which of themselves are ready enough every + where, to conceive more harshly of the difficultie, and impossibilitie + of attayning any skill therein, then it deserves, because they see + nothing but obscure propositions, and perplex and intricate + demonstrations before their eyes, whose unsavoury tartnes, to an + unexperienced palate like bitter pills is sweetned over, and made + pleasant with an Instrumentall compendious facilitie, and made to goe + downe the more readily, and yet to retaine the same vertue, and + working; And me thinkes in this queasy age, all helpes may bee used to + procure a stomacke, all bates and invitations to the declining studie + of so noble a Science, rather then by rigid Method and generall Lawes + to scarre men away. All are not of like disposition, neither all (as + was sayd before) propose the same end, some resolve to wade, others to + put a finger in onely, or wet a hand: now thus to tye them to an + obscure and Theoricall forme of teaching, is to crop their hope, even + in the very bud. . . . . The beginning of a mans knowledge even in the + use of an Instrument, is first founded on doctrinal precepts, and these + precepts may be conceived all along in its use: and are so farre from + being excluded, that they doe necessarily concomitate and are contained + therein: the practicke being better understood by the doctrinall part, + and this later explained by the Instrumentall, making precepts obvious + unto sense, and the Theory going along with the Instrument, better + informing and inlightning the understanding, etc. vis vnita fortior, so + as if that in Phylosophy bee true, Nihil est [in] intellectu quod non + prius fuit in sensu. + +The difference between Oughtred and Delamain as to the use of +mathematical instruments raises important questions. Should the slide +rule be placed in the hands of a boy before, or after, he has mastered +the theory of logarithms? Should logarithmic tables be withheld from him +until the theoretical foundation is laid in the mind of the pupil? Is it +a good thing to let a boy use a surveying instrument unless he first +learns trigonometry? Is it advisable to permit a boy to familiarize +himself with the running of a dynamo before he has mastered the +underlying principles of electricity? Does the use of instruments +ordinarily discourage a boy from mastery of the theory? Or does such +manipulation constitute a natural and pleasing approach to the abstract? +On this particular point, who showed the profounder psychological +insight, Oughtred or Delamain? + +In July, 1914, there was held in Edinburgh a celebration of the +three-hundredth anniversary of the invention of logarithms. On that +occasion there was collected at Edinburgh university one of the largest +exhibits ever seen of modern instruments of calculation. The opinion was +expressed by an experienced teacher that “weapons as those exhibited +there are for men and not for boys, and such danger as there may be in +them is of the same character as any form of too early specialization.” + +It is somewhat of a paradox that Oughtred, who in his student days and +during his active years felt himself impelled to invent sun-dials, +planispheres, and various types of slide rules—instruments which +represent the most original contributions which he handed down to +posterity—should discourage the use of such instruments in teaching +mathematics to beginners. That without the aid of instruments he himself +should have succeeded so well in attracting and inspiring young men +constitutes the strongest evidence of his transcendent teaching ability. +It may be argued that his pedagogic dogma, otherwise so excellent, here +goes contrary to the course he himself followed instinctively in his +self-education along mathematical lines. We read that Sir Isaac Newton, +as a child, constructed sun-dials, windmills, kites, paper lanterns, and +a wooden clock. Should these activities have been suppressed? Ordinary +children are simply Isaac Newtons on a smaller intellectual scale. Should +their activities along these lines be encouraged or checked? + +On the other hand, it may be argued that the paradox alluded to above +admits of explanation, like all paradoxes, and that there is no +inconsistency between Oughtred’s pedagogic views and his own course of +development. If he invented sun-dials, he must have had a comprehension +of the cosmic motions involved; if he solved spherical triangles +graphically by the aid of the planisphere, he must have understood the +geometry of the sphere, so far as it relates to such triangles; if he +invented slide rules, he had beforehand a thorough grasp of logarithms. +The question at issue does not involve so much the invention of +instruments, as the use by the pupil of instruments already constructed, +before he fully understands the theory which is involved. Nor does Sir +Isaac Newton’s activity as a child establish Delamain’s contention. Of +course, a child should not be discouraged from manual activity along the +line of producing interesting toys in imitation of structures and +machines that he sees, but to introduce him to the realm of abstract +thought by the aid of instruments is a different proposition, fraught +with danger. A boy may learn to use a slide rule mechanically and, +because of his ability to obtain practical results, feel justified in +foregoing the mastery of underlying theory; or he may consider the +ability of manipulating a surveying instrument quite sufficient, even +though he be ignorant of geometry and trigonometry; or he may learn how +to operate a dynamo and an electric switchboard and be altogether +satisfied, though having no grasp of electrical science. Thus instruments +draw a youth aside from the path leading to real intellectual attainments +and real efficiency; they allure him into lanes which are often blind +alleys. Such were the views of Oughtred. + +Who was right, Oughtred or Delamain? It may be claimed that there is a +middle ground which more nearly represents the ideal procedure in +teaching. Shall the slide rule be placed in the student’s hands at the +time when he is engaged in the mastery of principles? Shall there be an +alternate study of the theory of logarithms and of the slide rule—on the +idea of one hand washing the other—until a mastery of both the theory and +the use of the instrument has been attained? Does this method not produce +the best and most lasting results? Is not this Delamain’s actual +contention? We leave it to the reader to settle these matters from his +own observation, knowledge, and experience. + + + NEWTON’S COMMENTS ON OUGHTRED + +Oughtred is an author who has been found to be of increasing interest to +modern historians of mathematics. But no modern writer has, to our +knowledge, pointed out his importance in the history of the teaching of +mathematics. Yet his importance as a teacher did receive recognition in +the seventeenth century by no less distinguished a scientist than Sir +Isaac Newton. On May 25, 1694, Sir Isaac Newton wrote a long letter in +reply to a request for his recommendation on a proposed new course of +study in mathematics at Christ’s Hospital. Toward the close of his +letter, Newton says: + + And now I have told you my opinion in these things, I will give you Mr. + Oughtred’s, a Man whose judgment (if any man’s) may be safely relyed + upon. For he in his book of the circles of proposition, in the end of + what he writes about Navigation (page 184) has this exhortation to + Seamen. “And if,” saith he, “the Masters of Ships and Pilots will take + the pains in the Journals of their Voyages diligently and faithfully to + set down in severall columns, not onely the Rumb they goe on and the + measure of the Ships way in degrees, and the observation of Latitude + and variation of their compass; but alsoe their conjectures and reason + of their correction they make of the aberrations they shall find, and + the qualities and condition of their ship, and the diversities and + seasons of the winds, and the secret motions or agitations of the Seas, + when they begin, and how long they continue, how farr they extend and + with what inequality; and what else they shall observe at Sea worthy + consideration, and will be pleased freely to communicate the same with + Artists, such as are indeed skilfull in the Mathematicks and lovers and + enquirers of the truth: I doubt not but that there shall be in + convenient time, brought to light many necessary precepts which may + tend to y^e perfecting of Navigation, and the help and safety of such + whose Vocations doe inforce them to commit their lives and estates in + the vast Ocean to the providence of God.” Thus farr that very good and + judicious man Mr. Oughtred. I will add, that if instead of sending the + Observations of Seamen to able Mathematicians at Land, the Land would + send able Mathematicians to Sea, it would signify much more to the + improvem^t of Navigation and safety of Mens lives and estates on that + element.[144] + +May Oughtred prove as instructive to the modern reader as he did to +Newton! + + + + + Footnotes + + +[1]Aubrey’s Brief Lives, ed. A. Clark, Vol. II, Oxford, 1898, p. 106. + +[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. + +[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.” + +[4]New and General Biographical Dictionary (John Nichols), London, 1784, + art. “Oughtred.” + +[5]Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, p. 132. + +[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.” + +[7]Aubrey, op. cit., Vol. II, p. 107. + +[8]Rigaud, Correspondence of Scientific Men of the Seventeenth Century, + Oxford, Vol. I, 1841, p. 5. + +[9]Aubrey, op. cit., Vol. II, p. 110. + +[10]Ibid., p. 111. + +[11]Op. cit., Vol. II, p. 132. + +[12]Mr. William Lilly’s History of His Life and Times, From the Year 1602 + to 1681, London, 1715, p. 58. + +[13]Rigaud, op. cit., Vol. I, p. 60. + +[14]Aubrey, op. cit., Vol. II, p. 107. + +[15]Rigaud, op. cit., Vol. I, p. 16. + +[16]Owen Manning, op. cit., p. 132. + +[17]New and General Biographical Dictionary (John Nichols), London, 1784, + art. “Oughtred.” + +[18]Op. cit., Vol. II, p. 110. + +[19]Rev. Owen Manning, The History and Antiquities of Surrey, Vol. II, + London, 1809, p. 132. + +[20]Op. cit., Vol. II, 1898, p. 111. + +[21]Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago and + London, 1915, Vol. II, p. 303. + +[22]The full title of the Clavis of 1631 is as follows: Arithmeticae in + numeris et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae, + atqve adeo totivs mathematicae, qvasi clavis est.—Ad nobilissimvm + spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur, + Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae & + Svrriae, Comitis Mareschalli Angliae, &c filium.—Londini, Apud Thomam + Harpervm. M.DC.XXXI. + + In all there appeared five Latin editions, the second in 1648 at + London, the third in 1652 at Oxford, the fourth in 1667 at Oxford, the + fifth in 1693 and 1698 at Oxford. There were two independent English + editions: the first in 1647 at London, translated in greater part by + Robert Wood of Lincoln College, Oxford, as is stated in the preface to + the 1652 Latin edition; the second in 1694 and 1702 is a new + translation, the preface being written and the book recommended by the + astronomer Edmund Halley. The 1694 and 1702 impressions labored under + the defect of many sense-disturbing errors due to careless reading of + the proofs. All the editions of the Clavis, after the first edition, + had one or more of the following tracts added on: + + Eq.=De Aequationum affectarvm resolvtione in numeris. + Eu.=Elementi decimi Euclidis declaratio. + So.=De Solidis regularibus, tractatus. + An.=De Anatocismo, sive usura composita. + Fa.=Regula falsae positionis. + Ar.=Theorematum in libris Archimedis de Sphaera & cylindro declaratio. + Ho.=Horologia scioterica in plano, geometricè delineandi modus. + + The abbreviated titles given here are, of course, our own. The lists + of tracts added to the Clavis mathematicae of 1631 in its later + editions, given in the order in which the tracts appear in each + edition, are as follows: Clavis of 1647, Eq., An., Fa., Ho.; Clavis of + 1648, Eq., An., Fa., Eu., So.; Clavis of 1652, Eq., Eu., So., An., + Fa., Ar., Ho.; Clavis of 1667, Eq., Eu., So., An., Fa., Ar., Ho.; + Clavis of 1693 and 1698, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of + 1694 and 1702, Eq. + + The title-page of the Clavis was considerably modified after the first + edition. Thus, the 1652 Latin edition has this title-page: Guilelmi + Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia Socii, + Clavis mathematicae denvo limata, sive potius fabricata. Cum aliis + quibusdam ejusdem commentationibus, quae in sequenti pagina + recensentur. Editio tertia auctior & emendatior. Oxoniae, Excudebat + Leon. Lichfield, Veneunt apud Tho. Robinson. 1652. + +[23]Rigaud, op. cit., Vol. II, p. 476. + +[24]See, for instance, the Clavis mathematicae of 1652, where he + expresses himself thus (p. 4): “Speciosa haec Arithmetica arti + Analyticae (per quam ex sumptione quaesiti, tanquam noti, investigatur + quaesitum) multo accommodatior est, quam illa numerosa.” + +[25]Oughtred, The Key of the Mathematicks, London, 1647, p. 4. + +[26]Clavis 1694, p. 19, and the Clavis of 1631, p. 8. + +[27]See for instance, Oughtred’s Elementi decimi Euclidis declaratio, + 1652, p. 1, where he uses A and E, and also a and e. + +[28]See Christophori Clavii Bambergensis Operum mathematicorum, tomus + secundus, Moguntiae, M.DC.XI, algebra, p. 39. + +[29]Christophori Clavii operum mathematicorum Tomus Secundus, Moguntiae, + M.DC.XI, Epitome arithmeticae, p. 36. + +[30]See F. Cajori, “The Cross × as a Symbol of Multiplication,” in + Nature, Vol. XCIV (1914), p. 363. + +[31]See Elementi decimi Euclidis declaratio, 1652, p. 2. + +[32]See Johannis Wallisii Operum mathematicorum pars prima, Oxonii, 1657, + p. 247. + +[33]Clavis of 1631, chap. xix, sec. 5, p. 50. + +[34]We have noticed the representation of known quantities by consonants + and the unknown by vowels in Wingate’s Arithmetick made easie, edited + by John Kersey, London, 1650, algebra, p. 382; and in the second part, + section 19, of Jonas Moore’s Arithmetick in two parts, London, 1660, + Moore suggests as an alternative the use of z, y, x, etc., for the + unknowns. The practice of representing unknowns by vowels did not + spread widely in England. + +[35]Philosophical Transactions, Vol. XIX, No. 231, London, p. 652. + +[36]Ibid., Vol. XIX, p. 56. + +[37]There are two title-pages to the edition of 1632. The first + title-page is as follows: The Circles of Proportion and The + Horizontall Instrument. Both invented, and the vses 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. T. Cecill Sculp. + + In 1633 there was added the following, with a separate title-page: An + addition vnto the Vse of the Instrvment called the Circles of + Proportion. . . . . London, 1633, this being followed by Oughtred’s To + the English Gentrie etc. In the British Museum there is a copy of + another impression of the Circles of Proportion, dated 1639, with the + Addition vnto the Vse of the Instrument etc., bearing the original + date, 1633, and with the epistle, To the English Gentrie, etc., + inserted immediately after Forster’s dedication, instead of at the end + of the volume. + +[38]The complete title of the English edition is as follows: + Trigonometrie, or, The manner of calculating the Sides and Angles of + Triangles, by the Mathematical Canon, demonstrated. By William + Oughtred Etonens. And published by Richard Stokes Fellow of Kings + Colledge in Cambridge, and Arthur Haughton Gentleman. London, Printed + by R. and W. Leybourn, for Thomas Johnson at the Golden Key in St. + Pauls Church-yard. M.DC.LVII. + +[39]Jer. Collier, The Great Historical, Geographical, Genealogical and + Poetical Dictionary, Vol. II, London, 1701, art. “Oughtred.” + +[40]Rigaud op. cit., Vol. I, p. 82. + +[41]A. De Morgan, Budget of Paradoxes, London, 1872, p. 451; 2d ed., + Chicago, 1915, Vol. II, p. 303. + +[42]E. Gunter, Description and Use of the Sector, the Crosse-staffe and + other Instruments, London, 1624, second book, p. 31. + +[43]F. Cajori, “On the History of a Notation in Trigonometry,” Nature, + Vol. XCIV, 1915, pp. 642, 643. + +[44]A. von Braunmühl, Geschichte der Trigonometrie, 2. Teil, Leipzig, + 1903, pp. 42, 91. + +[45]H. Hankel, Geschichte der Mathematik in Alterthum und Mittelalter, + Leipzig, 1874, pp. 369, 370. + +[46]M. Cantor, Vorlesungen über Geschichte der Mathematik, II, 1900, pp. + 640, 641. + +[47]This matter has been discussed in a paper by F. Cajori, “A History of + the Arithmetical Methods of Approximation, etc., Colorado College + Publication, General Series No. 51, 1910, pp. 182-84. Later this + subject was again treated by G. Eneström in Bibliotheca mathematica, + 3. Folge, Vol. XI, 1911, pp. 234, 235. + +[48]See F. Cajori, op. cit., p. 193. + +[49]See William Oughtred’s Key of the Mathematicks, London, 1694, pp. + 173-75, tract, “Of the Resolution of the Affected Equations,” or any + edition of the Clavis after the first. + +[50]A. De Morgan, op. cit., p. 451; 2d ed., Vol. II, p. 303. + +[51]See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909, + pp. 7-14, Addenda, p. ii. + +[52]Rigaud, op. cit., Vol. I, p. 12. + +[53]The New Artificial Gauging Line or Rod: together with rules + concerning the use thereof: Invented and written by William Oughtred, + London, 1633. + +[54]W. Oughtred, Apologeticall Epistle, p. 13. + +[55]Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, (1915), + p. 169. In this article Glaisher republishes the “Appendix” in full. + +[56]Aubrey, op. cit., Vol. II, 1898, p. 108. + +[57]Wood’s Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247. + +[58]Wood, op. cit., Vol. II, p. 445. + +[59]Rigaud, op. cit., Vol. I, pp. 33, 35. + +[60]Rigaud, op. cit., Vol. I, pp. 16, 26. + +[61]Rigaud, op. cit., Vol. I, p. 66. + +[62]Ibid., Vol. I, p. 9. + +[63]Rigaud, op. cit., Vol. II, p. 475. + +[64]Ibid., Vol. II, p. 471. + +[65]J. W. L. Glaisher, “On Early Logarithmic Tables, and Their + Calculators,” Philosophical Magazine, 4th Ser., Vol. XLV (1873), pp. + 378, 379. + +[66]Rigaud, op. cit., Vol. I, p. 65. + +[67]Rigaud, op. cit., Vol. I, p. 87. + +[68]King’s Life of John Locke, Vol. I, London, 1830, p. 227. + +[69]Exercitationum Mathematicarum Decas prima, Naples, 1627, and probably + Cataldus’ Transformatio Geometrica, Bonon., 1612. + +[70]Rigaud, op. cit., Vol. II, pp. 477-80. + +[71]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, + Sometimes publike Professor of Astronomie in Gresham Colledge in + London, by John Twysden, London, 1659. + +[72]The Works of the Honourable Robert Boyle in five volumes, to which is + prefixed the Life of the Author, Vol. I, London, 1744, p. 24. + +[73]The letter is printed in John Wallis’ De algebra tractatus, 1693, p. + 206. + +[74]See La Correspondance de Descartes, published by Charles Adam and + Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457. + +[75]H. Bosmans, S.J., “La première édition de la Clavis Mathematica + d’Oughtred. Son influence sur la Géométrie de Descartes,” Annales de + la société scientifique de Bruxelles, 35th year, 1910-11, Part II, pp. + 24-78. + +[76]Ibid., p. 78. + +[77]Vincent Wing, Harmonicon coeleste, London, 1651, p. 5. + +[78]Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenta + inquisitio brevis, Oxford, 1653, p. 7. + +[79]John Wallis, Elenchus geometriae Hobbianae, Oxford, 1655, p. 48. + +[80]An Idea of Arithmetick, at first designed for the use of the Free + Schoole at Thurlow in Suffolk. . . . . By R. B., Schoolmaster there, + London, 1655, p. 6. + +[81]The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster + . . . . by John Twysden, London, 1659, p. 1. + +[82]Moor’s Arithmetick in two Books, London, 1660, p. 89. + +[83]Isaac Barrow, Euclidis data, Cambridge, 1657, p. 2. + +[84]Francisci Dulaurens Specima mathematica, Paris, 1667, p. 1. + +[85]Elémens des mathématiques, Paris, 1675, Preface signed “J. P.” + +[86]Nouveaux élémens de géométrie, Paris, 1692 (permission to print + 1684). + +[87]Ozanam, Dictionnaire mathématique, Paris, 1691, p. 12. + +[88]Analyse des infiniment petits, Paris, 1696, p. 11. + +[89]Petro Nicolas, De conchoidibus et cissoidibus exercitationes + geometricae, Toulouse, 1697, p. 17. + +[90]R. P. Bernard Lamy, Elémens des mathématiques, Amsterdam, 1692 + (permission to print 1680). + +[91]Nouveaux élémens de géométrie, 2d ed., The Hague, 1690, p. 304. + +[92]W. W. Beman in L’intermédiaire des mathématiciens, Paris, Vol. IX, + 1902, p. 229, question 2424. + +[93]John Collins, The Mariner’s Plain Scale New Plain’d, London, 1659, p. + 25. + +[94]James Gregory, Optica promota, London, 1663, pp. 19, 48. + +[95]Philosophical Transactions, Vol. III, London, p. 868. + +[96]William Leybourn, The Line of Proportion, London, 1673, p. 14. + +[97]Elementa geometriae . . . . a Gulielmo Sanders, Glasgow, 1686, p. 3. + +[98]Cocker’s Decimal Arithmetick, . . . . perused by John Hawkins, + London, 1695 (preface dated 1684), p. 41. + +[99]Joseph Raphson, Analysis Aequationum universalis, London, 1697, p. + 26. + +[100]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, + 1698, p. 107. + +[101]John Ward, A Compendium of Algebra, 2d ed., London, 1698, p. 62. + +[102]Plain Elements of Geometry and Plain Trigonometry, London, 1701, p. + 63. + +[103]George Shelley, Wingate’s Arithmetick, London, 1704, p. 343. + +[104]A Synopsis of Algebra, Being a posthumous work of John Alexander of + Bern, Swisserland. . . . . Done from the Latin by Sam. Cobb, London, + 1709, p. 16. + +[105]John Craig, De Calculo fluentium, London, 1718, p. 35. The notation + A:B::C:D is given also. + +[106]Trigonometry, 2d ed., Edinburgh, 1724, p. 11. + +[107]Méthode pour la mésure des surfaces, la dimension des solides . . . + . par M. Carré de l’académie r. des sciences, 1700, p. 59. + +[108]Application de l’algèbre à géométrie . . . . Paris, 1705. + +[109]Elémens de la géométrie de l’infini, by M. de Fontenelle, Paris, + 1727, p. 110. + +[110]Eclaircissemens sur l’analyse des infiniment petits, by M. Varignon, + Paris, 1725, p. 87. + +[111]Application de la géométrie ordinaire et des calculs différentiel et + intégral, by M. Robillard, Paris, 1753. + +[112]Traité de géométrie théorique et pratique, new ed., Paris, 1764, p. + 15. + +[113]Recherches sur les courbes à double courbure, Paris, 1731, p. 13. + +[114]Analyse des infiniment petits, by the Marquis de L’Hospital. New ed. + by M. Le Fèvre, Paris, 1781, p. 41. In this volume passages in fine + print, probably supplied by the editor, contain the notation a:b::c:d; + the parts in large type give Oughtred’s original notation. + +[115]The tendency during the eighteenth century is shown in part by the + following data: Jacobi Bernoulli Opera, Tomus primus, Geneva, 1744, + gives B.A::D.C on p. 368, the paper having been first published in + 1688; on p. 419 is given GE:AG=LA:ML, the paper having been first + published in 1689. Bernhardi Nieuwentiit, Considerationes circa + analyseos ad quantitates infinitè parvas applicatae principia, + Amsterdam, 1694, p. 20, and Analysis infinitorum, Amsterdam, 1695, on + p. 276, have x:c::s:r. Paul Halcken’s Deliciae mathematicae, Hamburg, + 1719, gives a:b::c:d. Johannis Baptistae Caraccioli, Geometria + algebraica universa, Rome, 1759, p. 79, has a.b::c.d. Delle corde + ouverto fibre elastiche schediasmi fisico-matematici del conte + Giordano Riccati, Bologna, 1767, p. 65, gives P:b::r:ds. “Produzioni + mathematiche” del Conte Giulio Carlo de Fagnano, Vol. I, Pesario, + 1750, p. 193, has a.b::c.d. L. Mascheroni, Géométrie du compas, + translated by A. M. Carette, Paris, 1798, p. 188, gives + √(3):2::√(2):Lp. Danielis Melandri and Paulli Frisi, De theoria lunae + commentarii, Parma, 1769, p. 13, has a:b::c:d. Vicentio Riccato and + Hieronymo Saladino, Institutiones analyticae, Vol. I, Bologna, 1765, + p. 47, gives x:a::m:n+m. R. G. Boscovich, Opera pertinentia ad opticam + et astronomiam, Bassani, 1785, p. 409, uses a:b::c:d. Jacob Bernoulli, + Ars Conjectandi, Basel, 1713, has n-r.n-1::c.d. Pavlini Chelvicii, + Institutiones analyticae, editio post tertiam Romanam prima in + Germania, Vienna, 1761, p. 2, a.b::c.d. Christiani Wolfii, Elementa + matheseos universae, Vol. III, Geneva, 1735, p. 63, has AB:AE=1:q. + Johann Bernoulli, Opera omnia, Vol. I, Lausanne and Geneva, 1742, p. + 43, has a:b=c:d. D. C. Walmesley, Analyse des mesures des rapports et + des angles, Paris, 1749, uses extensively a.b::c.d, later a:b::c:d. G. + W. Krafft, Institutiones geometriae sublimoris, Tübingen, 1753, p. + 194, has a:b=c:d. J. H. Lambert, Photometria, 1760, p. 104, has C:π + =BC²:MH². Meccanica sublime del Dott. Domenico Bartaloni, Naples, + 1765, has a:b::c:d. Occasionally ratio is not designated by a.b, nor + by a:b, but by a, b, as for instance in A. de Moivre’s Doctrine of + Chance, London, 1756, p. 34, where he writes a, b::1, q. A further + variation in the designation of ratio is found in James Atkinson’s + Epitome of the Art of Navigation, London, 1718, p. 24, namely, + 3..2::72..48. Curious notations are given in Rich. Balam’s Algebra, + London, 1653. + +[116]Chr. Clavii Operum mathematicorum tomus secundus, Mayence, 1611, + Algebra, p. 39. + +[117]Invention nouvelle en l’algèbre, by Albert Girard, Amsterdam, 1629, + p. 17. + +[118]La géométrie et pratique générale d’icelle, par I. Errard de + Bar-le-Duc, Ingénieur ordinaire de sa Majesté, 3d ed., revised by D. + H. P. E. M., Paris, 1619, p. 216. + +[119]Novae geometriae clavis algebra, authore P. Jacobo de Billy, Paris, + 1643, p. 157; also an Abridgement of the Precepts of Algebra. Written + in French by James de Billy, London, 1659, p. 346. + +[120]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, + Sometime publike Professor of Astronomie in Gresham Colledge in + London, London, 1659, p. 7. + +[121]Quarterly Jour. of Pure and Applied Math., Vol. XLVI (London, 1915), + p. 191. + +[122]Pietro Cossali, Origine, trasporto in Italia primi progressi in essa + dell’ algebra, Vol. I, Parmense, 1797, p. 52. + +[123]In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio + brevis, Auctore Setho Wardo, Oxford, 1653, p. 1. + +[124]John Wallis, Algebra, London, 1685, p. 321, and in some of his other + works. He makes greater use of Harriot’s symbols. + +[125]Euclidis data, 1657, p. 1; also Euclidis elementorum libris XV, + London, 1659, p. 1. + +[126]John Kersey, Algebra, London, 1673, p. 321. + +[127]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, + 1698, p. 142. + +[128]Cocker’s Decimal Arithmetick, perused by John Hawkins, London, 1695 + (preface dated 1684), p. 278. + +[129]Th. Baker, The Geometrical Key, London, 1684, p. 15. + +[130]Richard Sault, A New Treatise of Algebra, London (no date). + +[131]Richard Rawlinson in a pamphlet without date, issued sometime + between 1655 and 1668, containing trigonometric formulas. There is a + copy in the British Museum. + +[132]F. Dulaurens, Specima mathematica, Paris, 1667, p. 1. + +[133]J. Milnes, Sectionum conicarum elementa, Oxford, 1702, p. 42. + +[134]Cheyne, Philosophical Principles of Natural Religion, London, 1705, + p. 55. + +[135]J. Craig, De calculo fluentium, London, 1718, p. 86. + +[136]Jo. Wilson, Trigonometry, 2d ed., Edinburgh, 1724, p. v. + +[137]Commercium Epistolicum, 1712, p. 20. + +[138]C. Le Paige, “Sur l’origine de certains signes d’opération,” Annales + de la société scientifique de Bruxelles, 16th year, 1891-92, Part II, + pp. 79-82. + +[139]Gravelaar, “Over den oorsprong van ons maalteeken (×),” Wiskundig + Tijdschrift, 6th year. We have not had access to this article. + +[140]H. Bosmans, op. cit., p. 40. + +[141]Claudii Ptolemaei . . . . annotationes, Bâle, 1551. This reference + is taken from the Encyclopédie des sciences mathématiques, Tome I, + Vol. I, Fasc. 1, p. 40. + +[142]Due Correction for Mr. Hobbes. Or Schoole Discipline, for not saying + his Lessons right. In answer to his Six Lessons, directed to the + Professors of Mathematicks. By the Professor of Geometry. Oxford, + 1656, pp. 7, 47, 50. + +[143]Oughtred, Apologeticall Epistle, p. 27. + +[144]J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes, + London, 1850, pp. 279-92. + + + + + INDEX + + + Adam, Charles, 71 + Agnesi, Maria G., 77 + Alexander, J., 76 + Allen, E., 35 + Analysis, 19, 20 + Apollonius of Perga, 20, 85 + Archimedes, 18, 20, 85 + Aristotle, 69 + Ashmole, E., 13 + Atkinson, J., 79 + Atwood, 56 + Aubrey, 3, 7, 8, 12-16, 58, 59 + Austin, 58 + + Baker, T., 82 + Balam, R., 79 + Bar-le-Duc, de, 80 + Barrow, S., 1, 32, 73, 74, 80, 81 + Bartaloni, D., 79 + Beman, W. W., 74, 75 + Bernoulli, Jakob, 78-80 + Bernoulli, John, 79, 80 + Billingsley’s Euclid, 15 + Billion, 20 + Billy, Jacobo de, 80 + Binomial formula, 25, 29 + Bliss, P., 60 + Boscovich, R. G., 78 + Bosmans, H., 72, 83 + Boyle, R., 1, 69 + Braunmühl, von, 39 + Brearly, W., 59 + Briggs, 6, 36, 55 + Brookes, Christopher, 7, 53, 59 + + Cajori, F., 27, 39, 40, 47 + Cantor, M., 40, 41 + Caraccioli, J. B., 78 + Cardan, 71 + Carré, 77 + Carrete, N. M., 78 + Caryll, C., 7 + Cataldi, 67 + Cavalieri, 65, 66 + Cavendish, Charles, 17, 62, 66 + Charles I, 9, 60 + Chelvicius, P., 79 + Cheyne, G., 82 + _Circles of Proportion_, 35, 37, 48, 49, 51, 59, 87, 88 + Clairaut, 77 + Clark, A., 3 + Clark, G., 63 + Clarke, F. L., 3 + _Clavis mathematicae_, 1, 5, 10, 14, 17-35, 45, 46, 51, 57-63, + 68-73, 81, 85, 87 + Clavius, 26, 80 + Clerc, le, 77 + Cobb, S., 76 + Cocker, 76, 82 + Collins, John, 15, 19, 63, 64, 67, 68, 76, 82 + Colson, J., 77 + Conchoid, 12 + Conic sections, 11, 53 + Cossali, P., 81 + Cotes, R., 1, 85 + Craig, J., 76, 82 + Cross, symbol of multiplication, 27, 38, 55, 56, 82, 83 + Cubic equations, 28, 34, 42, 45 + + Decimal fractions, notation of, 21 + Degree, centesimal division, 39 + Delamain, R., 4, 9, 10, 11, 47, 48, 51, 60, 84, 88, 89, 91, 93, 94 + De Moivre, 32, 79 + De Morgan, A., 5, 16, 37, 46, 47, 54 + Descartes, R., 1, 25, 47, 57, 68-72, 80 + Dibuadius, 79 + Difference, symbol for, 27, 81 + Diophantus, 63, 85 + Division, abbreviated, 21, 23, 24 + Dulaurens, F., 74, 82 + + Earl of Arundel, 10, 13, 15, 17 + Edleston, J., 95 + Eneström, G., 40 + Equations, solution of, 18, 28, 29, 31, 34, 39-45, 87 + Errard de Bar-le-Duc, 80 + Eton College, 3, 4 + Euclid, 1, 15, 18, 20, 25, 27, 28, 79, 81, 83, 85 + Euler, L., 37, 39 + Ewart, 59 + Exponents, 25, 28, 29 + + Fagnano, de, 78 + Flower, 56 + Fontenelle, de, 77 + Forster, W., 35, 48, 59, 88 + Foster, S., 27, 67, 69, 73, 80, 89 + Frisi, P., 78 + + Gascoigne, 59, 61 + Gauss, C. F., 48 + Geysius, 67 + Ghetaldi, 70, 71 + Gibson, 67 + Girard, A., 32, 80 + Glaisher, J. W. L., 54-56, 64, 80 + Glorioso, 67, 68 + _Grammelogia_, 4, 47, 89 + Gravelaar, 83 + Greater than, symbol for, 81 + Greatrex, R., 15 + Gregory, D., 32 + Gregory, J., 27, 76 + Gresham College, 1, 6, 27, 59, 61, 80 + Guisnée, 77 + Gunter, E., 37, 47, 86 + Gunter’s scale, 37 + + Halcken, P., 78 + Hales, J., 7 + Halley, E., 1, 18, 69 + Hankel, H., 40 + Harper, T., 18 + Harriot, T., 45, 47, 57, 58, 69-71, 81 + Harris, J., 76 + Hartlib, 69 + Haughton, A., 35, 59 + Hawkins, J., 76, 82 + Hearn, 56 + Helmholtz, 48 + Henry, J., 48 + Henry van Etten, 52, 53 + Henshaw, T., 8, 58, 61 + Hobbes, 73, 86 + Hollar, 14 + Holsatus, 13 + Hooganhuysen, 64 + Hooke, Rb., 1 + Horner’s method, 45 + Horology, 18, 50 + Horrox, J., 4 + Hospital, de l’, 74, 77 + Howard, Th. _See_ Earl of Arundel. + Howard, W., 17, 18, 59 + Hutchinson, A., 6 + + Invisible college, 1 + + Joule, 48 + + Kepler, J., 6 + Kersey, J., 32, 73, 82 + Keylway, R., 65 + King, 67 + Kings College, Cambridge, 3, 35 + Krafft, G. W., 79 + + Lambert, J. H., 79 + Lamy, R. P. B., 74 + Laud, Archbishop, 65 + Leake, W., 53 + Le Clerc, 77 + Leech, W., 59 + Le Fèvre, 77 + Leibniz, 47, 78, 80 + Leonelli, 56 + Le Paige, de, 83 + Less than, symbol for, 81 + Leurechon, 52 + Leybourn, 35, 64, 76 + Lichfield, Mrs., 19 + Lilly, W., 8, 9 + Locke, J., 67 + Logarithms, 6, 21, 27, 28, 38, 39, 42, 46, 54-56, 65, 92, 93; + natural, 55; + radix method of computing, 55, 56 + Lower, W., 58 + Ludolph à Ceulen, 79 + + Manning, 56 + Manning, O., 7, 8, 13-15 + Mascheroni, L., 78 + Mayer, R., 47 + Melandri, D., 78 + Mercator, N., 13 + Mersenne, 63 + Milbourn, W., 45 + Million, 20 + Milnes, J., 82 + Moivre, de, 32, 79 + Moore, Jonas, 32, 54, 58, 73 + Moreland, S., 70 + Morse, R., 48 + Multiplication, abbreviated, 21, 22, 24; + symbol for, 27, 82, 83 + Mydorge, 54 + + Napier, J., 6, 7, 21, 27, 38, 39, 52, 54, 57, 59 + Napier’s analogies, 39 + Newton, Sir Isaac, 1, 25, 29, 40, 41, 45, 47, 59, 65, 86, 92-95 + Nichols, J., 6, 14 + Nicolas, R. P. P., 74 + Nieuwentiit, B., 78 + Norwood, R., 37, 38, 80 + + _Opuscula mathematica hactenus inedita_, 16, 21, 75 + Orchard, 56 + _Oughtredus explicatus_, 64 + Ozanam, 74 + + π, symbol for, 32 + Paige, C. de, 83 + Pardies, 76 + Parentheses, 26, 79, 80 + Partridge, S., 47 + Peano, 86 + Perfect number, 41 + Pitiscus, 15 + Planisphere, 53, 92, 93 + Prestet, J., 74 + Price, 11 + Proportion, notation for, 26, 27, 73-79 + Protheroe, 58 + Ptolemy, 83 + + Quadratic equation, 29, 31, 34 + + Radix method, 55, 56 + Rahn, 27 + Raphson, J., 40, 41, 76 + Ratio, notation of, 21, 73-80 + Rawlinson, R., 39, 82 + Regula falsa, 18 + Regular solids, 18 + Riccati, G., 78 + Riccati, V., 78 + Rigaud, 7, 12, 13, 19, 48, 61-66, 68 + Robillard, 77 + Robinson, W., 13, 48, 59, 62, 63 + Rooke, L., 59, 61 + + Saladini, H., 78 + Sanders, W., 76 + Sault, R., 82 + Scarborough, Charles, 16, 54, 58, 60 + Schooten, Van, 1 + Schreshensuchs, O., 83 + Scratch method, 23 + Shakespeare, 52 + Shelley, G., 76 + Shipley, A. E., 1 + Shuttleworth, 59 + Slide rule, 9, 46-49, 50, 60, 88, 93 + Smethwyck, 58 + Smith, J., 50 + Snellius, W., 79 + Solids, regular, 18 + Speidell, John, 38, 55 + Spherical triangles, 53, 54, 93 + Stokes, R., 35, 36, 58 + Sudell, 59 + Sun dials, 5, 9, 50, 51, 52, 60, 92 + + Tannery, P., 71 + Todhunter, 60 + Torporley, 58 + Triangles, spherical, 53, 54, 93 + _Trigonometria_, 21, 36, 55, 75 + Trigonometric functions, symbols for, 36, 37, 55, 56 + _Trigonometrie_, 21, 35, 39 + Trisection of angles, 28 + Twysden, 59, 68, 69, 73 + + Varignon, 77 + Vieta, 1, 2, 25, 32, 33, 35, 39-41, 45, 63, 67, 70, 71 + Vlack, 65 + Von Braunmühl, 39 + + Wadham College, 5, 53 + Wallis, John, 1, 19, 27, 33, 45, 57-59, 63, 64, 66-74, 79-81, 86 + Walmesley, D. C., 79 + Ward, Bishop, 13 + Ward, John, 76 + Ward, Seth, 55, 58, 60, 68, 73, 74, 81 + Watch-making, 18, 50 + Weber, W. E., 48 + Weddle, 56 + Wells, E., 76, 82 + Wharton, 60 + Whitlock, B., 8, 9 + Wilson, J., 77, 82 + Wing, V., 73, 75 + Wingate, E., 32, 47, 73 + Wolf, Christian, 79 + Wood, A., 60, 61 + Wood, R., 18, 59 + Wren, Christopher, 5, 58, 59, 76 + Wright, E., 6, 27, 38, 54 + Wright, S., 54 + + + + + Transcriber’s Notes + + +A handful of typos, mostly misplaced punctuation, were silently +corrected. + +HTML and UTF text versions make heavy use of mathematical symbols: +particularly superscripts, subscripts, and combining characters. Some +viewers may require user assistance to find fonts containing these +characters. + +The text versions miss much of the formatting, especially in mathematical +formulas: + + +--Several arithmetic examples must be viewed in a monospaced font (which + recognizes combining characters) to be legible. + +--Formulas under the horizontal line of a square root symbol are + parenthesized. + +--Subscripts are preceded by “_”. + +--Superscripts are preceded by “^”. + +--Italics, used primarily in formulas and bibliographical entries, are + not indicated in the text. + +--Italics in the index are delimited by “_”. + +--Underlines are not indicated (in particular, in the fractional part of + a decimal number in Oughtred's notation). + +--The idiosyncratic “greater than” and “less than” symbols are indicated + as {symbol} in the ASCII version. + +--Overdots, underdots, and slashmarks around digits (in the long division + example) are not indicated in the ASCII version. + +--Greek letters are spelled out within {curly brackets} in the ASCII + version. + + + + + + + +End of the Project Gutenberg EBook of William Oughtred, by Florian Cajori + +*** END OF THIS PROJECT GUTENBERG EBOOK WILLIAM OUGHTRED *** + +***** This file should be named 46815-0.txt or 46815-0.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/4/6/8/1/46815/ + +Produced by Brenda Lewis, Stephen Hutcheson, and the Online +Distributed Proofreading Canada Team at +http://www.pgdpcanada.net + +Updated editions will replace the previous one--the old editions will +be renamed. + +Creating the works from print editions not protected by U.S. copyright +law means that no one owns a United States copyright in these works, +so the Foundation (and you!) can copy and distribute it in the United +States without permission and without paying copyright +royalties. 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You may copy it, give it away or re-use it under the terms of +the Project Gutenberg License included with this eBook or online at +www.gutenberg.org. If you are not located in the United States, you'll have +to check the laws of the country where you are located before using this ebook. + +Title: William Oughtred + A great Seventeenth-Century Teacher of Mathematics + +Author: Florian Cajori + +Release Date: September 9, 2014 [EBook #46815] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK WILLIAM OUGHTRED *** + + + + +Produced by Brenda Lewis, Stephen Hutcheson, and the Online +Distributed Proofreading Canada Team at +http://www.pgdpcanada.net + + + + + + + WILLIAM OUGHTRED + + + + + WILLIAM OUGHTRED + A GREAT SEVENTEENTH-CENTURY + TEACHER OF + MATHEMATICS + + + BY + FLORIAN CAJORI, Ph.D. + Professor of Mathematics + Colorado College + + CHICAGO LONDON + THE OPEN COURT PUBLISHING COMPANY + 1916 + + Copyright 1916 By + The Open Court Publishing Co. + + All Rights Reserved + + Published September 1916 + + + Composed and Printed By + The University of Chicago Press + Chicago, Illinois, U.S.A. + + + + + TABLE OF CONTENTS + + + PAGE + Introduction 1 + CHAPTER + I. Oughtred's Life 3 + At School and University 3 + As Rector and Amateur Mathematician 6 + His Wife 7 + In Danger of Sequestration 8 + His Teaching 9 + Appearance and Habits 12 + Alleged Travel Abroad 14 + His Death 15 + II. Principal Works 17 + Clavis mathematicae 17 + Circles of Proportion and Trigonometrie 35 + Solution of Numerical Equations 39 + Logarithms 46 + Invention of the Slide Rule; Controversy on Priority of Invention 46 + III. Minor Works 50 + IV. Oughtred's Influence upon Mathematical Progress and Teaching 57 + Oughtred and Harriot 57 + Oughtred's Pupils 58 + Oughtred, the "Todhunter of the Seventeenth Century" 60 + Was Descartes Indebted to Oughtred? 69 + The Spread of Oughtred's Notations 73 + V. Oughtred's Ideas on the Teaching of Mathematics 84 + General Statement 84 + Mathematics, "a Science of the Eye" 85 + Rigorous Thinking and the Use of Instruments 87 + Newton's Comments on Oughtred 94 + Index 97 + + + + + 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 Gomtrie, 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] 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] + +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] + +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] + +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] We know very little about Oughtred's +family life. The records at King's College, Cambridge,[6] 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] 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] + +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] + + + 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] + +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] + +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] + + + 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] + + + 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. . . . . + + The right hon^ble Thomas Howard, earle of Arundel and Surrey, Lord High + Marshall of England, was his great patron, and loved him intirely. One + time they were like to have been killed together by the fall at Albury + of a grott, which fell downe but just as they were come out.[14] + +Oughtred's friends convey the impression that, in the main, Oughtred +enjoyed a comfortable living at Albury. Only once appear indications of +financial embarrassment. About 1634 one of his pupils, W. Robinson, +writes as follows: + + I protest unto you sincerely, were I as able as some, at whose hands + you have merited exceedingly, or (to speak more absolutely) as able as + willing, I would as freely give you 500 l. per ann. as 500 pence; and I + cannot but be astonished at this our age, wherein pelf and dross is + made their summum bonum, and the best part of man, with the true + ornaments thereof, science and knowledge, are so slighted. . . . .[15] + +In his letters Oughtred complains several times of the limitations for +work and the infirmities due to his advancing old age. The impression he +made upon others was quite different. Says one biographer: + + He sometimes amused himself with archery, and sometimes practised as a + surveyor of land. . . . . He was sprightly and active, when more than + eighty years of age.[16] + +Another informant says that Oughtred was + + as facetious in Greek and Latine as solid in Arithmetique, Astronomy, + and the sphere of all Measures, Musick, etc.; exact in his style as in + his judgment; handling his Cube, and other Instruments at eighty, as + steadily, as others did at thirty; owing this, he said, to temperance + and Archery; principling his people with plain and solid truths, as he + did the world with great and useful Arts; advancing new Inventions in + all things but Religion. Which in its old order and decency he + maintained secure in his privacy, prudence, meekness, simplicity, + resolution, patience, and contentment.[17] + + + ALLEGED TRAVEL ABROAD + +According to certain sources of information, Oughtred traveled on the +European Continent and was invited to change his abode to the Continent. +We have seen no statement from Oughtred himself on this matter. He seldom +referred to himself in his books and letters. The autobiography contained +in his Apologeticall Epistle was written a quarter of a century before +his death. Aubrey gives the following: + + In the time of the civill warres the duke of Florence invited him over, + and offered him 500 li. per annum; but he would not accept it, because + of his religion.[18] + +A portrait of Oughtred, painted in 1646 by Hollar and inserted in the +English edition of the Clavis of 1647, contains underneath the following +lines: + + "Haec est Oughtredi senio labantis imago + Itala quam cupiit, Terra Britanna tulit." + +In the sketch of Oughtred by Owen Manning it is confessed that "it is not +known to what this alludes; but possibly he might have been in Italy with +his patron, the Earl of Arundel."[19] It would seem quite certain either +that Oughtred traveled in Europe or that he received some sort of an +offer to settle in Italy. In view of Aubrey's explicit statement and of +Oughtred's well-known habit of confining himself to his duties and +studies in his own parish, seldom going even as far as London, we +strongly incline to the opinion that he did not travel on the Continent, +but that he received an offer from some patron of the sciences--possibly +some distinguished visitor--to settle in Italy. + + + HIS DEATH + +He died at Albury, June 30, 1660, aged about eighty-six years. Of his +last days and death, Aubrey speaks as follows: + + Before he dyed he burned a world of papers, and sayd that the world was + not worthy of them; he was so superb. He burned also severall printed + bookes, and would not stirre, till they were consumed. . . . . I + myselfe have his Pitiscus, imbelished with his excellent marginall + notes, which I esteeme as a great rarity. I wish I could also have got + his Bilingsley's Euclid, which John Collins sayes was full of his + annotations. . . . . + + Ralph Greatrex, his great friend, the mathematicall instrument-maker, + sayed he conceived he dyed with joy for the comeing-in of the king, + which was the 29th of May before. "And are yee sure he is + restored?"--"Then give me a glasse of sack to drinke his sacred + majestie's health." His spirits were then quite upon the wing to fly + away. . . . .[20] + +In this passage, as in others, due allowance must be made for Aubrey's +lack of discrimination. He was not in the habit of sifting facts from +mere gossip. That Oughtred should have declared that the world was not +worthy of his papers or manuscripts is not in consonance with the +sweetness of disposition ordinarily attributed to him. More probable was +the feeling that the papers he burned--possibly old sermons--were of no +particular value to the world. That he did not destroy a large mass of +mathematical manuscripts is evident from the fact that a considerable +number of them came after his death into the hands of Sir Charles +Scarborough, M.D., under whose supervision some of them were carefully +revised and published at Oxford in 1677 under the title of Opuscula +mathematica hactenus inedita. + +Aubrey's story of Oughtred's mode of death has been as widely circulated +in every modern biographical sketch as has his slander of Mrs. Oughtred +by claiming that she was so penurious that she would deny him the use of +candles to read by. Oughtred died on June 30; the Restoration occurred on +May 29. No doubt Oughtred rejoiced over the Restoration, but the story of +his drinking "a glass of sack" to his Majesty's health, and then dying of +joy is surely apocryphal. De Morgan humorously remarks, "It should be +added, by way of excuse, that he was eighty-six years old."[21] + + + + + CHAPTER II + PRINCIPAL WORKS + + + "CLAVIS MATHEMATICAE" + +Passing to the consideration of Oughtred's mathematical books, we begin +with the observation that he showed a marked disinclination to give his +writings to the press. His first paper on sun-dials was written at the +age of twenty-three, but we are not aware that more than one brief +mathematical manuscript was printed before his fifty-seventh year. In +every instance, publication in printed form seems to have been due to +pressure exerted by one or more of his patrons, pupils, or friends. Some +of his manuscripts were lent out to his pupils, who prepared copies for +their own use. In some instances they urged upon him the desirability of +publication and assisted in preparing copy for the printer. The earliest +and best-known book of Oughtred was his Clavis mathematicae, to which +repeated allusion has already been made. As he himself informs us, he was +employed by the Earl of Arundel about 1628 to instruct the Earl's son, +Lord William Howard (afterward Viscount Stafford) in the mathematics. For +the use of this young man Oughtred composed a treatise on algebra which +was published in Latin in the year 1631 at the urgent request of a +kinsman of the young man, Charles Cavendish, a patron of learning. + +The Clavis mathematicae,[22] in its first edition of 1631, was a booklet +of only 88 small pages. Yet it contained in very condensed form the +essentials of arithmetic and algebra as known at that time. + +Aside from the addition of four tracts, the 1631 edition underwent some +changes in the editions of 1647 and 1648, which two are much alike. The +twenty chapters of 1631 are reduced to nineteen in 1647 and in all the +later editions. Numerous minute alterations from the 1631 edition occur +in all parts of the books of 1647 and 1648. The material of the last +three chapters of the 1631 edition is rearranged, with some slight +additions here and there. The 1648 edition has no preface. In the print +of 1652 there are only slight alterations from the 1648 edition; after +that the book underwent hardly any changes, except for the number of +tracts appended, and brief explanatory notes added at the close of the +chapters in the English editions of 1694 and 1702. The 1652 and 1667 +editions were seen through the press by John Wallis; the 1698 impression +contains on the title-page the words: Ex Recognitione D. Johannis Wallis, +S.T.D. Geometriae Professoris Saviliani. + +The cost of publishing may be a matter of some interest. When arranging +for the printing of the 1667 edition of the Clavis, Wallis wrote Collins: +"I told you in my last what price she [Mrs. Lichfield] expects for it, as +I have formerly understood from her, viz., 40 for the impression, which +is about 91/2d. a book."[23] + +As compared with other contemporary works on algebra, Oughtred's +distinguishes itself for the amount of symbolism used, particularly in +the treatment of geometric problems. Extraordinary emphasis was placed +upon what he called in the Clavis the "analytical art."[24] By that term +he did not mean our modern analysis or analytical geometry, but the art +"in which by taking the thing sought as knowne, we finde out that we +seeke."[25] He meant to express by it condensed processes of rigid, +logical deduction expressed by appropriate symbols, as contrasted with +mere description or elucidation by passages fraught with verbosity. In +the preface to the first edition (1631) he says: + + In this little book I make known . . . . the rules relating to + fundamentals, collected together, just like a bundle, and adapted to + the explanation of as many problems as possible. + +As stated in this preface, one of his reasons for publishing the book, is + + . . . . that like Ariadne I might offer a thread to mathematical study + by which the mysteries of this science might be revealed, and direction + given to the best authors of antiquity, Euclid, Archimedes, the great + geometrician Apollonius of Perga, and others, so as to be easily and + thoroughly understood, their theorems being added, not only because to + many they are the height and depth of mathematical science (I ignore + the would-be mathematicians who occupy themselves only with the + so-called practice, which is in reality mere juggler's tricks with + instruments, the surface so to speak, pursued with a disregard of the + great art, a contemptible picture), but also to show with what keenness + they have penetrated, with what mass of equations, comparisons, + reductions, conversions and disquisitions these heroes have ornamented, + increased and invented this most beautiful science. + +The Clavis opens with an explanation of the Hindu-Arabic notation and of +decimal fractions. Noteworthy is the absence of the words "million," +"billion," etc. Though used on the Continent by certain mathematical +writers long before this, these words did not become current in English +mathematical books until the eighteenth century. The author was a great +admirer of decimal fractions, but failed to introduce the notation which +in later centuries came to be universally adopted. Oughtred wrote 0.56 in +this manner 0|56; the point he used to designate ratio. Thus 3:4 was +written by him 34. The decimal point (or comma) was first used by the +inventor of logarithms, John Napier, as early as 1616 and 1617. Although +Oughtred had mastered the theory of logarithms soon after their +publication in 1614 and was a great admirer of Napier, he preferred to +use the dot for the designation of ratio. This notation of ratio is used +in all his mathematical books, except in two instances. The two dots (:) +occur as symbols of ratio in some parts of Oughtred's posthumous work, +Opuscula mathematica hactenus inedita, Oxford, 1677, but may have been +due to the editors and not to Oughtred himself. Then again the two dots +(:) are used to designate ratio on the last two pages of the tables of +the Latin edition of Oughtred's Trigonometria of 1657. In all other parts +of that book the dot () is used. Probably someone who supervised the +printing of the tables introduced the (:) on the last two pages, +following the logarithmic tables, where methods of interpolation are +explained. The probability of this conjecture is the stronger, because in +the English edition of the Trigonometrie, brought out the same year +(1657) but after the Latin edition, the notation (:) at the end of the +book is replaced by the usual (), except that in some copies of the +English edition the explanations at the end are omitted altogether. + +Oughtred introduces an interesting, and at the same time new, feature of +an abbreviated multiplication and an abbreviated division of decimal +fractions. On this point he took a position far in advance of his time. +The part on abbreviated multiplication was rewritten in slightly enlarged +form and with some unimportant alterations in the later edition of the +Clavis. We give it as it occurs in the revision. Four cases are given. In +finding the product of 246|914 and 35|27, "if you would have the Product +without any Parts" (without any decimal part), "set the place of Unity of +the lesser under the place of Unity in the greater: as in the Example," +writing the figures of the lesser number in inverse order. From the +example it will be seen that he begins by multiplying by 3, the +right-hand digit of the multiplier. In the first edition of the Clavis he +began with 7, the left digit. Observe also that he "carries" the nearest +tens in the product of each lower digit and the upper digit one place to +its right. For instance, he takes 7x4=28 and carries 3, then he finds +7x2+3=17 and writes down 17. + + 2 4 6|9 1 4 + 7 2|5 3 + ------- + 7 4 0 7 + 1 2 3 5 + 4 9 + 1 7 + ------- + 8 7 0 8 + +The second case supposes that "you would have the Product with some +places of parts" (decimals), say 4: "Set the place of Unity of the lesser +Number under the Fourth place of the Parts of the greater." The +multiplication of 246|914 by 35|27 is now performed thus: + + 2 4 6|9 1 4 + 7 2|5 3 + --------------- + 7 4 0 7 4 2 0 0 + 1 2 3 4 5 7 0 0 + 4 9 3 8 2 8 + 1 7 2 8 4 0 + --------------- + 8 7 0 8|6 5 6 8 + +In the third and fourth cases are considered factors which appear as +integers, but are in reality decimals; for instance, the sine of 54^o is +given in the tables as 80902 when in reality it is .80902. + +Of interest as regards the use of the word "parabola" is the following: +"The Number found by Division is called the Quotient, or also Parabola, +because it arises out of the Application of a plain Number to a given +Longitude, that a congruous Latitude may be found."[26] This is in +harmony with etymological dictionaries which speak of a parabola as the +application of a given area to a given straight line. The dividend or +product is the area; the divisor or factor is the line. + +Oughtred gives two processes of long division. The first is identical +with the modern process, except that the divisor is written below every +remainder, each digit of the divisor being crossed out as soon as it has +been used in the partial multiplication. The second method of long +division is one of the several types of the old "scratch method." This +antiquated process held its place by the side of the modern method in all +editions of the Clavis. The author divides 467023 by 357|0926425, giving +the following instructions: "Take as many of the first Figures of the +Divisor as are necessary, for the first Divisor, and then in every +following particular Division drop one of the Figures of the Divisor +towards the Left Hand, till you have got a competent Quotient." He does +not explain abbreviated division as thoroughly as abbreviated +multiplication. + + 17 + 303 + 2803 + 109930 + 357|0926425) 467023 (1307|80 + 357093 + 107127 + 2500 + 286 + + +Oughtred does not examine the degree of reliability or accuracy of his +processes of abbreviated multiplication and division. Here as in other +places he gives in condensed statement the mode of procedure, without +further discussion. + +He does not attempt to establish the rules for the addition, subtraction, +multiplication, and division of positive and negative numbers. "If the +Signs are both alike, the Product will be affirmative, if unlike, +negative"; then he proceeds to applications. This attitude is superior to +that of many writers of the eighteenth and nineteenth centuries, on +pedagogical as well as logical grounds: pedagogically, because the +beginner in the study of algebra is not in a position to appreciate an +abstract train of thought, as every teacher well knows, and derives +better intellectual exercise from the applications of the rules to +problems; logically, because the rule of signs in multiplication does not +admit of rigorous proof, unless some other assumption is first made which +is no less arbitrary than the rule itself. It is well known that the +proofs of the rule of signs given by eighteenth-century writers are +invalid. Somewhere they involve some surreptitious assumption. This +criticism applies even to the proof given by Laplace, which tacitly +assumes the distributive law in multiplication. + +A word should be said on Oughtred's definition of + and -. He recognizes +their double function in algebra by saying (Clavis, 1631, p. 2): "Signum +additionis, sive affirmationis, est + plus" and "Signum subductionis, +sive negationis est - minus." They are symbols which indicate the quality +of numbers in some instances and operations of addition or subtraction in +other instances. In the 1694 edition of the Clavis, thirty-four years +after the death of Oughtred, these symbols are defined as signifying +operations only, but are actually used to signify the quality of numbers +as well. In this respect the 1694 edition marks a recrudescence. + +The characteristic in the Clavis that is most striking to a modern reader +is the total absence of indexes or exponents. There is much discussion in +the leading treatises of the latter part of the sixteenth and the early +part of the seventeenth century on the theory of indexes, but the modern +exponential notation, a^n, is of later date. The modern notation, for +positive integral exponents, first appears in Descartes' Gomtrie, 1637; +fractional and negative exponents were first used in the modern form by +Sir Isaac Newton, in his announcement of the binomial formula, in a +letter written in 1676. This total absence of our modern exponential +notation in Oughtred's Clavis gives it a strange aspect. Like Vieta, +Oughtred uses ordinarily the capital letters, A, B, C, . . . . to +designate given numbers; A^2 is written Aq, A^3 is written Ac; for A^4, +A^5, A^6 he has, respectively, Aqq, Aqc, Acc. Only on rare occasions, +usually when some parallelism in notation is aimed at, does he use small +letters[27] to represent numbers or magnitudes. Powers of binomials or +polynomials are marked by prefixing the capital letters Q (for square), C +(for cube), QQ (for the fourth power), QC (for the fifth power), etc. + +Oughtred does not express aggregation by (). Parentheses had been used by +Girard, and by Clavius as early as 1609,[28] but did not come into +general use in mathematical language until the time of Leibniz and the +Bernoullis. Oughtred indicates aggregation by writing a colon (:) at both +ends. Thus, Q:A-E: means with him (A-E)^2. Similarly, {root}q:A+E: means +{root}(A+E). The two dots at the end are frequently omitted when the part +affected includes all the terms of the polynomial to the end. Thus, +C:A+B-E=.. means (A+B-E)^3=.. There are still further departures from +this notation, but they occur so seldom that we incline to the +interpretation that they are simply printer's errors. For proportion +Oughtred uses the symbol (::). The proportion a:b=c:d appears in his +notation ab::cd. Apparently, a proportion was not fully recognized in +this day as being the expression of an equality of ratios. That probably +explains why he did not use = here as in the notation of ordinary +equations. Yet Oughtred must have been very close to the interpretation +of a proportion as an equality; for he says in his Elementi decimi +Euclidis declaratio, "proportio, sive ratio aequalis ::" That he +introduced this extra symbol when the one for equality was sufficient is +a misfortune. Simplicity demands that no unnecessary symbols be +introduced. However, Oughtred's symbolism is certainly superior to those +which preceded. Consider the notation of Clavius.[29] He wrote 20:60=4:x, +x=12, thus: "20604? fiunt 12." The insufficiency of such a notation in +the more involved expressions frequently arising in algebra is readily +seen. Hence Oughtred's notation (::) was early adopted by English +mathematicians. It was used by John Wallis at Oxford, by Samuel Foster at +Gresham College, by James Gregory of Edinburgh, by the translators into +English of Rahn's algebra, and by many other early writers. Oughtred has +been credited generally with the introduction of St. Andrew's cross x as +the symbol for multiplication in the Clavis of 1631. We have discovered +that this symbol, or rather the letter x which closely resembles it, +occurs as the sign of multiplication thirteen years earlier in an +anonymous "Appendix to the Logarithmes, shewing the practise of the +Calculation of Triangles etc." to Edward Wright's translation of John +Napier's Descriptio, published in 1618.[30] Later we shall give our +reasons for believing that Oughtred is the author of that "Appendix." The +x has survived as a symbol of multiplication. + +Another symbol introduced by Oughtred and found in modern books is ~, +expressing difference; thus C~D signifies the difference between C and D, +even when D is the larger number.[31] This symbol was used by John Wallis +in 1657.[32] + +Oughtred represented in symbols also certain composite expressions, as +for instance A+E=Z, A-E=X, where A is greater than E. He represented by a +symbol also each of the following: A^2+E^2, A^3+E^3, A^2-E^2, A^3-E^3. + +Oughtred practically translated the tenth book of Euclid from its +ponderous rhetorical form into that of brief symbolism. An appeal to the +eye was a passion with Oughtred. The present writer has collected the +different mathematical symbols used by Oughtred and has found more than +one hundred and fifty of them. + +The differences between the seven different editions of the Clavis lie +mainly in the special parts appended to some editions and dropped in the +latest editions. The part which originally constituted the Clavis was not +materially altered, except in two or three of the original twenty +chapters. These changes were made in the editions of 1647 and 1648. After +the first edition, great stress was laid upon the theory of indices upon +the very first page, as also in passages farther on. Of course, Oughtred +did not have our modern notation of indices or exponents, but their +theory had been a part of algebra and arithmetic for some time. Oughtred +incorporated this theory in his brief exposition of the Hindu-Arabic +notation and in his explanation of logarithms. As previously pointed out, +the last three chapters of the 1631 edition were considerably rearranged +in the later editions and combined into two chapters, so that the Clavis +proper had nineteen chapters instead of twenty in the additions after the +first. These chapters consisted of applications of algebra to geometry +and were so framed as to constitute a severe test of the student's grip +of the subject. The very last problem deals with the division of angles +into equal parts. He derives the cubic equation upon which the trisection +depends algebraically, also the equations of the fifth degree and seventh +degree upon which the divisions of the angle into 5 and 7 equal parts +depend, respectively. The exposition was severely brief, yet accurate. He +did not believe in conducting the reader along level paths or along +slight inclines. He was a guide for mountain-climbers, and woe unto him +who lacked nerve. + +Oughtred lays great stress upon expansions of powers of a binomial. He +makes use of these expansions in the solution of numerical equations. To +one who does not specialize in the history of mathematics such expansions +may create surprise, for did not Newton invent the binomial theorem after +the death of Oughtred? As a matter of fact, the expansions of positive +integral powers of a binomial were known long before Newton, not only to +seventeenth-century but even to eleventh-century mathematicians. +Oughtred's Clavis of 1631 gave the binomial coefficients for all powers +up to and including the tenth. What Newton really accomplished was the +generalization of the binomial expansion which makes it applicable to +negative and fractional exponents and converts it into an infinite +series. + +As a specimen of Oughtred's style of writing we quote his solution of +quadratic equations, accompanied by a translation into English and into +modern mathematical symbols. + +As a preliminary step[33] he lets + + Z=A+E and A>E; + +he lets also X=A-E. From these relations he obtains identities which, in +modern notation, are 1/4Z^2-AE=(1/2Z-E)^2=1/4X^2. Now, if we know Z and +AE, we can find 1/2X. Then 1/2(Z+X)=A, and 1/2(Z-X)=E, and + + A=1/2Z+{root}(1/4Z^2-AE). + +Having established these preliminaries, he proceeds thus: + + Datis igitur linea inaequaliter secta Z (10), & rectangulo sub + segmentis AE (21) qui gnomon est: datur semidifferentia segmentorum + 1/2X: & per consequens ipsa segmenta. Nam ponatur alterutrum segmentum + A: alterum erit Z-A: Rectangulum auctem est ZA-A_q=AE. Et quia dantur Z + & AE: estque 1/4Z_q-AE=1/4X_q: & per 5c. 18, 1/2Z+1/2X=A: & + 1/2Z-1/2X=E: Aequatio sic resoluetur: 1/2Z+/-{root}_q:1/4Z_q-AE:=A + {maius segment/minus segment. + + Itaque proposita equatione, in qua sunt tres species aequaliter in + ordine tabellae adscendentes, altissima autem species ponitur negata: + Magnitudo data coefficiens mediam speciem est linea bisecanda: & + magnitudo absoluta data, ad quam sit aequatio, est rectangulum sub + segmentis inaequalibus, sine gnomon: vt ZA-A_q=AE: in numeris autem + 10l-l_q=21: Estque A, vel 1l, alterutrum segmentum inaequale. Inuenitur + autem sic: + + Dimidiata coefficiens median speciem est Z/2 (5); cuius quadratum est + Z_q/4 (25): ex hoc tolle AE (21) absolutum: eritque Z_q/4-AE (4) + quadratum semidifferentiae segmentorum: latus huius quadratum (2) est + semidifferentia: quam si addas ad Z/2 (5) semissem coefficientis, sive + lineae bisecandae, erit maius segment.; sin detrahas, erit minus + segment: Dico Z/2+/-{root}_q:Z_q/4-AE:=A {maius segmentum/minus + segmentum. + +We translate the Latin passage, using the modern exponential notation and +parentheses, as follows: + + Given therefore an unequally divided line Z (10), and a rectangle + beneath the segments AE (21) which is a gnomon. Half the difference of + the segments 1/2X is given, and consequently the segment itself. For, + if one of the two segments is placed equal to A, the other will be Z-A. + Moreover, the rectangle is ZA-A^2=AE. And because Z and AE are given, + and there is 1/4Z^2-AE=1/4X^2, and by 5c.18, 1/2Z+1/2X=A, and + 1/2Z-1/2X=E, the equation will be solved thus: + 1/2Z+/-{root}(1/4Z^2-AE)=A {major segment/minor segment. + + And so an equation having been proposed in which three species (terms) + are in equally ascending powers, the highest species, moreover, being + negative, the given magnitude which constitutes the middle species is + the line to be bisected. And the given absolute magnitude to which it + is equal is the rectangle beneath the unequal segments, without gnomon. + As ZA-A^2=AE, or in numbers, 10x-x^2=21. And A or x is one of the two + unequal segments. It may be found thus: + + The half of the middle species is Z/2 (5), its square is Z^2/4 (25). + From it subtract the absolute term AE (21), and Z^2/4-AE (4) will be + the square of half the difference of the segments. The square root of + this, {root}[(Z^2/2)^2-AE] (2), is half the difference. If you add it + to half the coefficient Z/2 (5), or half the line to be bisected, the + longer segment is obtained; if you subtract it, the smaller segment is + obtained. I say: Z/2+/-{root}(Z^2/4-AE)=A {major segment/minor segment. + +The quadratic equation Aq+ZA=AE receives similar treatment. This and the +preceding equation, ZA-Aq=AE, constitute together a solution of the +general quadratic equation, x^2+ax=b, provided that E or Z are not +restricted to positive values, but admit of being either positive or +negative, a case not adequately treated by Oughtred. Imaginary numbers +and imaginary roots receive no consideration whatever. + +A notation suggested by Vieta and favored by Girard made vowels stand for +unknowns and consonants for knowns. This conventionality was adopted by +Oughtred in parts of his algebra, but not throughout. Near the beginning +he used Q to designate the unknown, though usually this letter stood with +him for the "square" of the expression after it.[34] + +It is of some interest that Oughtred used {pi/delta} to signify the ratio +of the circumference to the diameter of a circle. Very probably this +notation is the forerunner of the {pi}=3.14159 . . . . used in 1706 by +William Jones. Oughtred first used {pi/delta} in the 1647 edition of the +Clavis mathematicae. In the 1652 edition he says, "Si in circulo sit +7.22::{deltapi}::113.355:erit {deltapi}::2 R.P: periph." This notation +was adopted by Isaac Barrow, who used it extensively. David Gregory[35] +used {pi/rho} in 1697, and De Moivre[36] used c/r about 1697, to +designate the ratio of the circumference to the radius. + +We quote the description of the Clavis that was given by Oughtred's +greatest pupil, John Wallis. It contains additional information of +interest to us. Wallis devotes chap. xv of his Treatise of Algebra, +London, 1685, pp. 67-69, to Mr. Oughtred and his Clavis, saying: + + Mr. William Oughtred (our Country-man) in his Clavis Mathematicae, (or + Key of Mathematicks,) first published in the Year 1631, follows Vieta + (as he did Diophantus) in the use of the Cossick Denominations; + omitting (as he had done) the names of Sursolids, and contenting + himself with those of Square and Cube, and the Compounds of these. + + But he doth abridge Vieta's Characters or Species, using only the + letters q, c, &c. which in Vieta are expressed (at length) by Quadrate, + Cube, &c. For though when Vieta first introduced this way of Specious + Arithmetick, it was more necessary (the thing being new,) to express it + in words at length: Yet when the thing was once received in practise, + Mr. Oughtred (who affected brevity, and to deliver what he taught as + briefly as might be, and reduce all to a short view,) contented himself + with single Letters instead of those words. + + Thus what Vieta would have written + + A Quadrate, into B Cube, + ------------------------ Equal to FG Plane, + CDE Solid, + + would with him be thus expressed + + A_q B_c + ------- = FG. + C D E + + And the better to distinguish upon the first view, what quantities were + Known, and what Unknown, he doth (usually) denote the Known to + Consonants, and the Unknown by Vowels; as Vieta (for the same reason) + had done before him. + + He doth also (to very great advantage) make use of several Ligatures, + or Compendious Notes, to signify Summs, Differences, and Rectangles of + several Quantities. As for instance, Of two Quantities A (the Greater), + and E (the Lesser), the Sum he calls Z, the Difference X, the Rectangle + AE. . . . . + + Which being of (almost) a constant signification with him throughout, + do save a great circumlocution of words, (each Letter serving instead + of a Definition;) and are also made use of (with very great advantage) + to discover the true nature of divers intricate Operations, arising + from the various compositions of such Parts, Sums, Differences, and + Rectangles; (of which there is great plenty in his Clavis, Cap. 11, 16, + 18, 19. and elsewhere,) which without such Ligatures, or Compendious + Notes, would not be easily discovered or apprehended. . . . . + + I know there are who find fault with his Clavis, as too obscure, + because so short, but without cause; for his words be always full, but + not Redundant, and need only a little attention in the Reader to weigh + the force of every word, and the Syntax of it; . . . . And this, when + once apprehended, is much more easily retained, than if it were + expressed with the prolixity of some other Writers; where a Reader must + first be at the pains to weed out a great deal of superfluous Language, + that he may have a short prospect of what is material; which is here + contracted for him in a short Synopsis. . . . . + + Mr. Oughtred in his Clavis, contents himself (for the most part) with + the solution of Quadratick Equations, without proceeding (or very + sparingly) to Cubick Equations, and those of Higher Powers; having + designed that Work for an Introduction into Algebra so far, leaving the + Discussion of Superior Equations for another work. . . . . He contents + himself likewise in Resolving Equations, to take notice of the + Affirmative or Positive Roots; omitting the Negative or Ablative Roots, + and such as are called Imaginary or Impossible Roots. And of those + which, he calls Ambiguous Equations, (as having more Affirmative Roots + than one,) he doth not (that I remember) any where take notice of more + than Two Affirmative Roots: (Because in Quadratick Equations, which are + those he handleth, there are indeed no more.) Whereas yet in Cubick + Equations, there may be Three, and in those of Higher Powers, yet more. + Which Vieta was well aware of, and mentioneth in some of his Writings; + and of which Mr. Oughtred could not be ignorant. + + + "CIRCLES OF PROPORTION" AND "TRIGONOMETRIE" + +Oughtred wrote and had published three important mathematical books, the +Clavis, the Circles of Proportion,[37] and a Trigonometrie.[38] This last +appeared in the year 1657 at London, in both Latin and English. + +It is claimed that the trigonometry was "neither finished nor published +by himself, but collected out of his scattered papers; and though he +connived at the printing it, yet imperfectly done, as appears by his +MSS.; and one of the printed Books, corrected by his own Hand."[39] +Doubtless more accurate on this point is a letter of Richard Stokes who +saw the book through the press: + + I have procured your Trigonometry to be written over in a fair hand, + which when finished I will send to you, to know if it be according to + your mind; for I intend (since you were pleased to give your assent) to + endeavour to print it with Mr. Briggs his Tables, and so soon as I can + get the Prutenic Tables I will turn those of the sun and moon, and send + them to you.[40] + +In the preface to the Latin edition Stokes writes: + + Since this trigonometry was written for private use without the + intention of having it published, it pleased the Reverend Author, + before allowing it to go to press, to expunge some things, to change + other things and even to make some additions and insert more lucid + methods of exposition. + +This much is certain, the Trigonometry bears the impress characteristic +of Oughtred. Like all his mathematical writings, the book was very +condensed. Aside from the tables, the text covered only 36 pages. Plane +and spherical triangles were taken up together. The treatise is known in +the history of trigonometry as among the very earliest works to adopt a +condensed symbolism so that equations involving trigonometric functions +could be easily taken in by the eye. In the work of 1657, contractions +are given as follows: s=sine, t=tangent, se=secant, s co=cosine (sine +complement), t co=cotangent, se co=cosecant, log=logarithm, Z cru=sum of +the sides of a rectangle or right angle, X cru=difference of these sides. +It has been generally overlooked by historians that Oughtred used the +abbreviations of trigonometric functions, named above, a quarter of a +century earlier, in his Circles of Proportion, 1632, 1633. Moreover, he +used sometimes also the abbreviations which are current at the present +time, namely sin=sine, tan=tangent, sec=secant. We know that the Circles +of Proportion existed in manuscript many years before they were +published. The symbol sv for sinus versus occurs in the Clavis of 1631. +The great importance of well-chosen symbols needs no emphasis to readers +of the present day. With reference to Oughtred's trigonometric symbols. +Augustus De Morgan said: + + This is so very important a step, simple as it is, that Euler is justly + held to have greatly advanced trigonometry by its introduction. Nobody + that we know of has noticed that Oughtred was master of the + improvement, and willing to have taught it, if people would have + learnt.[41] + +We find, however, that even Oughtred cannot be given the whole credit in +this matter. By or before 1631 several other writers used abbreviations +of the trigonometric functions. As early as 1624 the contractions sin for +sine and tan for tangent appear on the drawing representing Gunter's +scale, but Gunter did not use them in his books, except in the drawing of +his scale.[42] A closer competitor for the honor of first using these +trigonometric abbreviations is Richard Norwood in his Trigonometrie, +London, 1631, where s stands for sine, t for tangent, sc for sine +complement (cosine), tc for tangent complement (cotangent), and sec for +secant. Norwood was a teacher of mathematics in London and a well-known +writer of books on navigation. Aside from the abbreviations just cited +Norwood did not use nearly as much symbolism in his mathematics as did +Oughtred. + +Mention should be made of trigonometric symbols used even earlier than +any of the preceding, in "An Appendix to the Logarithmes, shewing the +practise of the Calculation of Triangles, etc.," printed in Edward +Wright's edition of Napier's A Description of the Admirable Table of +Logarithmes, London, 1618. We referred to this "Appendix" in tracing the +origin of the sign x. It contains, on p. 4, the following passage: "For +the Logarithme of an arch or an angle I set before (s), for the +antilogarithme or compliment thereof (s*) and for the Differential (t)." +In further explanation of this rather unsatisfactory passage, the author +(Oughtred?) says, "As for example: sB+BC=CA. that is, the Logarithme of +an angle B. at the Base of a plane right-angled triangle, increased by +the addition of the Logarithm of BC, the hypothenuse thereof, is equall +to the Logarithme of CA the cathetus." + +Here "logarithme of an angle B" evidently means "log sin B," just as with +Napier, "Logarithms of the arcs" signifies really "Logarithms of the +sines of the angles." In Napier's table, the numbers in the column marked +"Differentiae" signify log sine minus log cosine of an angle; that is, +the logarithms of the tangents. This explains the contraction (t) in the +"Appendix." The conclusion of all this is that as early as 1618 the signs +s, s*, t were used for sine, cosine, and tangent, respectively. + +John Speidell, in his Breefe Treatise of Sphaericall Triangles, London, +1627, uses Si. for sine, T. and Tan for tangent, Se. for secant, Si. Co. +for cosine, Se. Co. for cosecant, T. Co. for cotangent. + +The innovation of designating the sides and angles of a triangle by A, B, +C, and a, b, c, so that A was opposite a, B opposite b, and C opposite c, +is attributed to Leonard Euler (1753), but was first used by Richard +Rawlinson of Queen's College, Oxford, sometimes after 1655 and before +1668. Oughtred did not use Rawlinson's notation.[43] + +In trigonometry English writers of the first half of the seventeenth +century used contractions more freely than their continental +contemporaries; even more freely, indeed, than English writers of a later +period. Von Braunmhl, the great historian of trigonometry, gives +Oughtred much praise for his trigonometry, and points out that half a +century later the army of writers on trigonometry had hardly yet reached +the standard set by Oughtred's analysis.[44] Oughtred must be credited +also with the first complete proof that was given to the first two of +"Napier's analogies." His trigonometry contains seven-place tables of +sines, tangents, and secants, and six-place tables of logarithmic sines +and tangents; also seven-place logarithmic tables of numbers. At the time +of Oughtred there was some agitation in favor of a wider introduction of +decimal systems. This movement is reflected in those tables which contain +the centesimal division of the degree, a practice which is urged for +general adoption in our own day, particularly by the French. + + + SOLUTION OF NUMERICAL EQUATIONS + +In the solution of numerical equations Oughtred does not mention the +sources from which he drew, but the method is substantially that of the +great French algebraist Vieta, as explained in a publication which +appeared in 1600 in Paris under the title, De numerosa potestatum purarum +atque adfectarum ad exegesin resolutione tractatus. In view of the fact +that Vieta's process has been described inaccurately by leading modern +historians including H. Hankel[45] and M. Cantor,[46] it may be worth +while to go into some detail.[47] By them it is made to appear as +identical with the procedure given later by Newton. The two are not the +same. The difference lies in the divisor used. What is now called +"Newton's method" is Newton's method as modified by Joseph Raphson.[48] +The Newton-Raphson method of approximation to the roots of an equation +f(x)=0 is usually given the form a-[f(a)/f'(a)], where a is an +approximate value of the required root. It will be seen that the divisor +is f'(a). Vieta's divisor is different; it is + + |f(a+s_1)-f(a)|-s_1^n, + +where f(x) is the left of the equation f(x)=k, n is the degree of +equation, and s_1 is a unit of the denomination of the digit next to be +found. Thus in x^3+420000x=247651713, it can be shown that 417 is +approximately a root; suppose that a has been taken to be 400, then +s_1=10; but if, at the next step of approximation, a is taken to be 410, +then s_1=1. In this example, taking a=400, Vieta's divisor would have +been 9120000; Newton's divisor would have been 900000. + +A comparison of Vieta's method with the Newton-Raphson method reveals the +fact that Vieta's divisor is more reliable, but labors under the very +great disadvantage of requiring a much larger amount of computation. The +latter divisor is accurate enough and easier to compute. Altogether the +Newton-Raphson process marks a decided advance over that of Vieta. + +As already stated, it is the method of Vieta that Oughtred explains. The +Englishman's exposition is an improvement on that of Vieta, printed forty +years earlier. Nevertheless, Oughtred's explanation is far from easy to +follow. The theory of equations was at that time still in its primitive +stage of development. Algebraic notation was not sufficiently developed +to enable the argument to be condensed into a form easily surveyed. So +complicated does Vieta's process of approximation appear that M. Cantor +failed to recognize that Vieta possessed a uniform mode of procedure. But +when one has in mind the general expression for Vieta's divisor which we +gave above, one will recognize that there was marked uniformity in +Vieta's approximations. + +Oughtred allows himself twenty-eight sections in which to explain the +process and at the close cannot forbear remarking that 28 is a "perfect" +number (being equal to the sum of its divisors, 1, 2, 4, 7, 14). + +The early part of his exposition shows how an equation may be transformed +so as to make its roots 10, 100, 1000, or 10^m times smaller. This +simplifies the task of "locating a root"; that is, of finding between +what integers the root lies. + +Taking one of Oughtred's equations, x^4-72x^3+238600x=8725815, upon +dividing 72x^3 by 10, 238600x by 1000, and 8725815 by 10,000, we obtain +x^4-72x^3+2386x=8725. Dividing both sides by x, we obtain +x^3+2386-72x^2=x)8725. Letting x=4, we have 64+2386-1152=1874. + +But 4)8725(2181; 4 is too small. Next let x=5, we have +125+2386-180=1836. + +But 5)8725(1745; 5 is too large. We take the lesser value, x=4, or in +the original equation, x=40. This method may be used to find the second +digit in the root. Oughtred divides both sides of the equation by x^2, +and obtains x^2+x)238600-72x=x^2)8725815. He tries x=47 and x=48, and +finds that x=47. + +He explains also how the last computation may be done by logarithms. +Thereby he established for himself the record of being the first to use +logarithms in the solution of affected equations. + +As an illustration of Oughtred's method of approximation after the root +sought has been located, we have chosen for brevity a cubic in preference +to a quartic. We selected the equation x^3+420000x=247651713. By the +process explained above a root is found to lie between x=400 and x=500. +From this point on, the approximation as given by Oughtred is as shown on +p. 43. + +In further explanation of this process, observe that the given equation +is of the form L_c+C_qL=D_c, where L_c is our x, C_q=420000, +D_c=247651713. In the first step of approximation, let L=A+E, where A=400 +and E is, as yet, undetermined. We have + + L_c=(A+E)^3=A^3+3A^2E+3AE^2+E^3 + +and + + C_qL=420000(A+E). + +Subtract from 247651713 the sum of the known terms A^3 (his A_c) and +420000 A (his C_qA). This sum is 232000000 the remainder is 15651713. + + "Exemplum II + + 1c+420000l=247651713 + + Hoc est, L_c+C_qL=D_c + + 2 4 7 | 6 5 1 | 7 1 3 | ( 4 1 7 + ------+-------+-------+------------ + 4 2 | 0 0 0 | 0 | C_q + ------+-------+-------+------------ + 6 4 | | | A_c + 1 6 8 | 0 0 0 | 0 | C_q A + ------+-------+-------+------------ + 2 3 2 | 0 0 0 | 0 | Ablatit. + =================================== +R 1 5 | 6 5 1 | 7 1 3 | + ------+-------+-------+------------ + 4 | 8 | | 3 A_q + | 1 2 | | 3 A + 4 | 2 0 0 | 0 0 | C_q + ------+-------+-------+------------ + 9 | 1 2 0 | 0 0 | Divisor. + ------+-------+-------+------------ + 4 | 8 | | 3 A_q E + | 1 2 | | 3 A E_q + | 1 | | E_c + 4 | 2 0 0 | 0 0 | C_q E + ------+-------+-------+------------ + 9 | 1 2 1 | 0 0 | Ablatit. + =================================== +R 6 | 5 3 0 | 7 1 3 | 4 | 1 | + ------+-------+-------+------------ ----+-----+--- + | 5 0 4 | 3 | 3 A_q | | + | 1 | 2 3 | 3 A 1 6 | 8 | + | 4 2 0 | 0 0 0 | C_q | 1 | + ------+-------+-------+------------ ----+-----+--- + | 9 2 5 | 5 3 0 | Divisor. 1 6 8 1 + ------+-------+-------+------------ + 3 | 5 3 0 | 1 | 3 A_q E + | 6 0 | 2 7 | 3 A E_q + | | 3 4 3 | E_c + 2 | 9 4 0 | 0 0 0 | C_q E + ------+-------+-------+------------ + 6 | 5 3 0 | 7 1 3 | Ablatit." + +Next, he evaluates the coefficients of E in 3A^2E and 420000E, also 3A, +the coefficient of E^2. He obtains 3A^2=480000, 3A=1200, C_q=420000. He +interprets 3A^2 and C_q as tens, 3A as hundreds. Accordingly, he obtains +as their sum 9120000, which is the divisor for finding the second digit +in the approximation. Observe that this divisor is the value of +|f(a+s_1)-f(a)|-s_1^n in our general expression, where a=400, s_1=10, +n=3, f(x)=x^3+420000x. + +Dividing the remainder 15651713 by 9120000, he obtains the integer 1 in +ten's place; thus E=10, approximately. He now computes the terms 3A^2E, +3AE^2 and E^3 to be, respectively, 4800000, 120000, 1000. Their sum is +9121000. Subtracting it from the previous remainder, 15651713, leaves the +new remainder, 6530713. + +From here on each step is a repetition of the preceding step. The new A +is 410, the new E is to be determined. We have now in closer +approximation, L=A+E. This time we do not subtract A^3 and C_qA, because +this subtraction is already affected by the preceding work. + +We find the second trial divisor by computing the sum of 3A^2, 3A and +C_q; that is, the sum of 504300, 1230, 420000, which is 925530. Again, +this divisor can be computed by our general expression for divisors, by +taking a=410, s_1=1, n=3. + +Dividing 6530713 by 925530 yields the integer 7. Thus E=7. Computing +3A^2E, 3AE^2, E^3 and subtracting their sum, the remainder is 0. Hence +417 is an exact root of the given equation. + +Since the extraction of a cube root is merely the solution of a pure +cubic equation, x^3=n, the process given above may be utilized in finding +cube roots. This is precisely what Oughtred does in chap. xiv of his +Clavis. If the foregoing computation is modified by putting C_q=0, the +process will yield the approximate cube root of 247651713. + +Oughtred solves 16 examples by the process of approximation here +explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In +all cases he finds only one or two real roots. Of the roots sought, five +are irrational, the remaining are rational and are computed to their +exact values. Three of the computed roots have 2 figures each, 9 roots +have 3 figures each, 4 roots have 4 figures each. While no attempt is +made to secure all the roots--methods of computing complex roots were +invented much later--he computes roots of equations which involve large +coefficients and some of them are of a degree as high as the fifth. In +view of the fact that many editions of the Clavis were issued, one +impression as late as 1702, it contributed probably more than any other +book to the popularization of Vieta's method in England. + +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" [436^2], that "the logarithm of any Potestas +[436^2] 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 Ren 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 x 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^o-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, log_e 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/10^n. 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 x; (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 Ren 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 SEVENTEENTH + CENTURY" + +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 Wood[57] 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: + + Amongst the mathematical rarities these times have afforded, there are + none of that small number I (a late intruder into these studies) have + yet viewed, which so fully demonstrates their authors' great abilities + as your Clavis, not richer in augmentations, than valuable for + contraction; . . . . + + Your belief that there is in all inventions aliquid divinum, an + infusion beyond human cogitations, I am confident will appear notably + strengthened, if you please to afford this truth belief, that I entered + upon these studies accidentally after I betook myself to the country, + having never had so much aid as to be taught addition, nor the + discourse of an artist (having left both Oxford and London before I + knew what any proposition in geometry meant) to inform me what were the + best authors.[59] + +The following extracts from two letters by W. Robinson, written before +the appearance of the 1647 English edition of the Clavis, express the +feeling of many readers of the Clavis on its extreme conciseness and +brevity of explanation: + + I shall long exceedingly till I see your Clavis turned into a + pick-lock; and I beseech you enlarge it, and explain it what you can, + for we shall not need to fear either tautology or superfluity; you are + naturally concise, and your clear judgment makes you both methodical + and pithy; and your analytical way is indeed the only way. . . . . + + I will once again earnestly entreat you, that you be rather diffuse in + the setting forth of your English mathematical Clavis, than concise, + considering that the wisest of men noted of old, and said stultorum + infinitus est numerus, these arts cannot be made too easy, they are so + abstruse of themselves, and men either so lazy or dull, that their + fastidious wits take a loathing at the very entrance of these studies, + unless it be sweetened on with plainness and facility. Brevity may well + argue a learned author, that without any excess or redundance, either + of matter or words, can give the very substance and essence of the + thing treated of; but it seldom makes a learned scholar; and if one be + capable, twenty are not; and if the master sum up in brief the pith of + his own long labours and travails, it is not easy to imagine that + scholars can with less labour than it cost their masters dive into the + depths thereof.[60] + +Here is the judgment of another of Oughtred's friends: + + . . . . with the character I received from your and my noble friend Sir + Charles Cavendish, then at Paris, of your second edition of the same + piece, made me at my return into England speedily to get, and + diligently peruse the same. Neither truly did I find my expectation + deceived; having with admiration often considered how it was possible + (even in the hardest things of geometry) to deliver so much matter in + so few words, yet with such demonstrative clearness and perspicuity: + and hath often put me in mind of learned Mersennus his judgment (since + dead) of it, that there was more matter comprehended in that little + book than in Diophantus, and all the ancients. . . . .[61] + +Oughtred's own feeling was against diffuseness in textbook writing. In +his revisions of his Clavis the original character of that book was not +altered. In his reply to W. Robinson, Oughtred said: + + . . . . But my art for all such mathematical inventions I have set down + in my Clavis Mathematica, which therefore in my title I say is tum + logisticae cum analyticae adeoque totius mathematicae quasi clavis, + which if any one of a mathematical genius will carefully study, (and + indeed it must be carefully studied,) he will not admire others, but + himself do wonders. But I (such is my tenuity) have enough fungi vice + cotis, acutum reddere quae ferrum valet, exsors ipsa secandi, or like + the touchstone, which being but a stone, base and little worth, can + shew the excellence and riches of gold.[62] + +John Wallis held Oughtred's Clavis in high regard. When in correspondence +with John Collins concerning plans for a new edition, Wallis wrote in +1666-67, six years after the death of Oughtred: + + . . . . But for the goodness of the book in itself, it is that (I + confess) which I look upon as a very good book, and which doth in as + little room deliver as much of the fundamental and useful part of + geometry (as well as of arithmetic and algebra) as any book I know; and + why it should not be now acceptable I do not see. It is true, that as + in other things so in mathematics, fashions will daily alter, and that + which Mr. Oughtred designed by great letters may be now by others be + designed by small; but a mathematician will, with the same ease and + advantage, understand A_c, and a^3 or aaa. . . . . And the like I judge + of Mr. Oughtred's Clavis, which I look upon (as those pieces of Vieta + who first went in that way) as lasting books and classic authors in + this kind; to which, notwithstanding, every day may make new additions. + . . . . + + But I confess, as to my own judgment, I am not for making the book + bigger, because it is contrary to the design of it, being intended for + a manual or contract; whereas comments, by enlarging it, do rather + destroy it. . . . . But it was by him intended, in a small epitome, to + give the substance of what is by others delivered in larger volumes. . + . . .[63] + +That there continued to be a group of students and teachers who desired a +fuller exposition than is given by Oughtred is evident from the +appearance, over fifty years after the first publication of the Clavis, +of a booklet by Gilbert Clark, entitled Oughtredus Explicatus, London, +1682. A review of this appeared in the Acta Eruditorum (Leipzig, 1684), +on p. 168, wherein Oughtred is named "clarissimus Angliae mathematicus." +John Collins wrote Wallis in 1666-67 that Clark, "who lives with Sir +Justinian Isham, within seven miles of Northampton, . . . . intimates he +wrote a comment on the Clavis, which lay long in the hands of a printer, +by whom he was abused, meaning Leybourne."[64] + +We shall have occasion below to refer to Oughtred's inability to secure a +copy of a noted Italian mathematical work published a few years before. +In those days the condition of the book trade in England must have been +somewhat extraordinary. Dr. J. W. L. Glaisher throws some light upon this +subject.[65] He found in the Calendar of State Papers, Domestic Series, +1637, a petition to Archbishop Laud in which it is set forth that when +Hooganhuysen, a Dutchman, "heretofore complained of in the High +Commission for importing books printed beyond the seas," had been bound +"not to bring in any more," one Vlacq (the computer and publisher of +logarithmic tables) "kept up the same agency and sold books in his stead. +. . . . Vlacq is now preparing to go beyond the seas to avoid answering +his late bringing over nine bales of books contrary to the decree of the +Star Chamber." Judgment was passed that, "Considering the ill-consequence +and scandal that would arise by strangers importing and venting in this +kingdom books printed beyond the seas," certain importations be +prohibited, and seized if brought over. + +This want of easy intercommunication of results of scientific research in +Oughtred's time is revealed in the following letter, written by Oughtred +to Robert Keylway, in 1645: + + I speak this the rather, and am induced to a better confidence of your + performance, by reason of a geometric-analytical art or practice found + out by one Cavalieri, an Italian, of which about three years since I + received information by a letter from Paris, wherein was praelibated + only a small taste thereof, yet so that I divine great enlargement of + the bounds of the mathematical empire will ensue. I was then very + desirous to see the author's own book while my spirits were more free + and lightsome, but I could not get it in France. Since, being more + stept into years, daunted and broken with the sufferings of these + disastrous times, I must content myself to keep home, and not put out + to any foreign discoveries.[66] + +It was in 1655, when Oughtred was about eighty years old, that John +Wallis, the great forerunner of Newton in Great Britain, began to publish +his great researches on the arithmetic of infinites. Oughtred rejoiced +over the achievements of his former pupil. In 1655, Oughtred wrote John +Wallis as follows: + + I have with unspeakable delight, so far as my necessary businesses, the + infirmness of my health, and the greatness of my age (approaching now + to an end) would permit, perused your most learned papers, of several + choice arguments, which you sent me: wherein I do first with + thankfulness acknowledge to God, the Father of lights, the great light + he hath given you; and next I congratulate you, even with admiration, + the clearness and perspicacity of your understanding and genius, who + have not only gone, but also opened a way into these profoundest + mysteries of art, unknown and not thought of by the ancients. With + which your mysterious inventions I am the more affected, because full + twenty years ago, the learned patron of learning, Sir Charles + Cavendish, shewed me a paper written, wherein were some few excellent + new theorems, wrought by the way, as I suppose, of Cavalieri, which I + wrought over again more agreeably to my way. The paper, wherein I + wrought it, I shewed to many, whereof some took copies, but my own I + cannot find. I mention it for this, because I saw therein a light + breaking out for the discovery of wonders to be revealed to mankind, in + this last age of the world: which light I did salute as afar off, and + now at a nearer distance embrace in your prosperous beginnings. Sir, + that you are pleased to mention my name in your never dying papers, + that is your noble favour to me, who can add nothing to your glory, but + only my applause. . . . .[67] + +The last sentence has reference to Wallis' appreciative and eulogistic +reference to Oughtred in the preface. It is of interest to secure the +opinion of later English writers who knew Oughtred only through his +books. John Locke wrote in his journal under the date, June 24, 1681, +"the best algebra yet extant is Outred's."[68] John Collins, who is known +in the history of mathematics chiefly through his very extensive +correspondence with nearly all mathematicians of his day, was inclined to +be more critical. He wrote Wallis about 1667: + + It was not my intent to disparage the author, though I know many that + did lightly esteem him when living, some whereof are at rest, as Mr. + Foster and Mr. Gibson. . . . . You grant the author is brief, and + therefore obscure, and I say it is but a collection, which, if himself + knew, he had done well to have quoted his authors, whereto the reader + might have repaired. You do not like those words of Vieta in his + theorems, ex adjunctione plano solidi, plus quadrato quadrati, etc., + and think Mr. Oughtred the first that abridged those expressions by + symbols; but I dissent, and tell you 'twas done before by Cataldus, + Geysius, and Camillus Gloriosus,[69] who in his first decade of + exercises, (not the first tract,) printed at Naples in 1627, which was + four years before the first edition of the Clavis, proposeth this + equation just as I here give it you, viz. + 1ccc+16qcc+41qqc-2304cc-18364qc-133000qq-54505c+3728q+8064 N aequatur + 4608, finds N or a root of it to be 24, and composeth the whole out of + it for proof, just in Mr. Oughtred's symbols and method. Cataldus on + Vieta came out fifteen years before, and I cannot quote that, as not + having it by me. + + . . . . And as for Mr. Oughtred's method of symbols, this I say to it; + it may be proper for you as a commentator to follow it, but divers I + know, men of inferior rank that have good skill in algebra, that + neither use nor approve it. . . . . Is not A^5 sooner wrote than A_qc? + Let A be 2, the cube of 2 is 8, which squared is 64: one of the + questions between Maghet Grisio and Gloriosus is whether 64=A_cc or + A_qc. The Cartesian method tells you it is A^6, and decides the doubt. + . . . .[70] + +There is some ground for the criticisms passed by Collins. To be sure, +the first edition of the Clavis is dated 1631--six years before Descartes +suggested the exponential notation which came to be adopted as the +symbolism in our modern algebra. But the second edition of the Clavis, +1647, appeared ten years after Descartes' innovation. Had Oughtred seen +fit to adopt the new exponential notation in 1647, the step would have +been epoch-making in the teaching of algebra in England. We have seen no +indication that Oughtred was familiar with Descartes' Gomtrie of 1637. + +The year preceding Oughtred's death Mr. John Twysden expressed himself as +follows in the preface to his Miscellanies: + + It remains that I should adde something touching the beginning, and use + of these Sciences. . . . . I shall only, to their honours, name some of + our own Nation yet living, who have happily laboured upon both stages. + That succeeding ages may understand that in this of ours, there yet + remained some who were neither ignorant of these Arts, as if they had + held them vain, nor condemn them as superfluous. Amongst them all let + Mr. William Oughtred, of Aeton, be named in the first place, a Person + of venerable grey haires, and exemplary piety, who indeed exceeds all + praise we can bestow upon him. Who by an easie method, and admirable + Key, hath unlocked the hidden things of geometry. Who by an accurate + Trigonometry and furniture of Instruments, hath inriched, as well + geometry, as Astronomy. Let D. John Wallis, and D. Seth Ward, succeed + in the next place, both famous Persons, and Doctors in Divinity, the + one of geometry, the other of astronomy, Savilian Professors in the + University of Oxford.[71] + +The astronomer Edmund Halley, in his preface to the 1694 English edition +of the Clavis, speaks of this book as one of "so established a +reputation, that it were needless to say anything thereof," though "the +concise Brevity of the author is such, as in many places to need +Explication, to render it Intelligible to the less knowing Mathematical +matters." + +In closing this part of our monograph, we quote the testimony of Robert +Boyle, the experimental physicist, as given May 8, 1647, in a letter to +Mr. Hartlib: + + The Englishing of, and additions to Oughtred's Clavis mathematica does + much content me, I having formerly spent much study on the original of + that algebra, which I have long since esteemed a much more instructive + way of logic, than that of Aristotle.[72] + + + WAS DESCARTES INDEBTED TO OUGHTRED? + +This question first arose in the seventeenth century, when John Wallis, +of Oxford, in his Algebra (the English edition of 1685, and more +particularly the Latin edition of 1693), raised the issue of Descartes' +indebtedness to the English scientists, Thomas Harriot and William +Oughtred. In discussing matters of priority between Harriot and +Descartes, relating to the theory of equations, Wallis is generally held +to have shown marked partiality to Harriot. Less attention has been given +by historians of mathematics to Descartes' indebtedness to Oughtred. Yet +this question is of importance in tracing Oughtred's influence upon his +time. + +On January 8, 1688-89, Samuel Morland addressed a letter of inquiry to +John Wallis, containing a passage which we translate from the Latin: + + Some time ago I read in the elegant and truly precious book that you + have written on Algebra, about Descartes, this philosopher so extolled + above all for having arrived at a very perfect system by his own + powers, without the aid of others, this Descartes, I say, who has + received in geometry very great light from our Oughtred and our + Harriot, and has followed their track though he carefully suppressed + their names. I stated this in a conversation with a professor in + Utrecht (where I reside at present). He requested me to indicate to him + the page-numbers in the two authors which justified this accusation. I + admitted that I could not do so. The Gomtrie of Descartes is not + sufficiently familiar to me, although with Oughtred I am fairly + familiar. I pray you therefore that you will assume this burden. Give + me at least those references to passages of the two authors from the + comparison of which the plagiarism by Descartes is the most + striking.[73] + +Following Morland's letter in the De algebra tractatus, is printed +Wallis' reply, dated March 12, 1688 ("Stilo Angliae"), which is, in part, +as follows: + + I nowhere give him the name of a plagiarist; I would not appear so + impolite. However this I say, the major part of his algebra (if not + all) is found before him in other authors (notably in our Harriot) whom + he does not designate by name. That algebra may be applied to geometry, + and that it is in fact so applied, is nothing new. Passing the ancients + in silence, we state that this has been done by Vieta, Ghetaldi, + Oughtred and others, before Descartes. They have resolved by algebra + and specious arithmetic [literal arithmetic] many geometrical problems. + . . . . But the question is not as to application of algebra to + geometry (a thing quite old), but of the Cartesian algebra considered + by itself. + +Wallis then indicates in the 1659 edition of Descartes' Gomtrie where +the subjects treated on the first six pages are found in the writings of +earlier algebraists, particularly of Harriot and Oughtred. For example, +what is found on the first page of Descartes, relating to addition, +subtraction, multiplication, division, and root extraction, is declared +by Wallis to be drawn from Vieta, Ghetaldi, and Oughtred. + +It is true that Descartes makes no mention of modern writers, except once +of Cardan. But it was not the purpose of Descartes to write a history of +algebra. To be sure, references to such of his immediate predecessors as +he had read would not have been out of place. Nevertheless, Wallis fails +to show that Descartes made illegitimate use of anything he may have seen +in Harriot or Oughtred. + +The first inquiry to be made is, Did Descartes possess copies of the +books of Harriot and Oughtred? It is only in recent time that this +question has been answered as to Harriot. As to Oughtred, it is still +unanswered. It is now known that Descartes had seen Harriot's Artis +analyticae praxis (1631). Descartes wrote a letter to Constantin Huygens +in which he states that he is sending Harriot's book.[74] + +An able discussion of the question, what effect, if any, Oughtred's +Clavis mathematicae of 1631 had upon Descartes'[75] Gomtrie of 1637, is +given by H. Bosmans in a recent article. According to Bosmans no evidence +has been found that Descartes possessed a copy of Oughtred's book, or +that he had examined it. Bosmans believes nevertheless that Descartes was +influenced by the Clavis, either directly or indirectly. He says: + + If Descartes did not read it carefully, which is not proved, he was + none the less well informed with regard to it. No one denies his + intimate knowledge of the intellectual movement of his time. The Clavis + mathematica enjoyed a rapid success. It is impossible that, at least + indirectly, he did not know the more original ideas which it contained. + Far from belittling Descartes, as I much desire to repeat, this rather + makes him the greater.[76] + +We ourselves would hardly go as far as does Bosmans. Unless Descartes +actually examined a copy of Oughtred it is not likely that he was +influenced by Oughtred in appreciable degree. Book reviews were quite +unknown in those days. No evidence has yet been adduced to show that +Descartes obtained a knowledge of Oughtred by correspondence. A most +striking feature about Oughtred's Clavis is its notation. No trace of the +Englishman's symbolism has been pointed out in Descartes' Gomtrie of +1637. Only six years intervened between the publication of the Clavis and +the Gomtrie. It took longer than this period for the Clavis to show +evidence of its influence upon mathematical books published in England; +it is not probable that abroad the contact was more immediate than at +home. Our study of seventeenth-century algebra has led us to the +conviction that Oughtred deserves a higher place in the development of +this science than is usually accorded to him; but that it took several +decennia for his influence fully to develop. + + + THE SPREAD OF OUGHTRED'S NOTATIONS + +An idea of Oughtred's influence upon mathematical thought and teaching +can be obtained from the spread of his symbolism. This study indicates +that the adoption was not immediate. The earliest use that we have been +able to find of Oughtred's notation for proportion, A.B::C.D, occurs +nineteen years after the Clavis mathematicae of 1631. In 1650 John Kersey +brought out in London an edition of Edmund Wingates' Arithmetique made +easie, in which this notation is used. After this date publications +employing it became frequent, some of them being the productions of +pupils of Oughtred. We have seen it in Vincent Wing (1651),[77] Seth Ward +(1653),[78] John Wallis (1655),[79] in "R. B.," a schoolmaster in +Suffolk,[80] Samuel Foster (1659),[81] Jonas Moore (1660),[82] and Isaac +Barrow (1657).[83] In the latter part of the seventeenth century +Oughtred's notation, A.B::C.D, became the prevalent, though not +universal, notation in Great Britain. A tremendous impetus to their +adoption was given by Seth Ward, Isaac Barrow, and particularly by John +Wallis, who was rising to international eminence as a mathematician. + +In France we have noticed Oughtred's notation for proportion in +Franciscus Dulaurens (1667),[84] J. Prestet (1675),[85] R. P. Bernard +Lamy (1684),[86] Ozanam (1691),[87] De l'Hospital (1696),[88] R. P. Petro +Nicolas (1697).[89] + +In the Netherlands we have noticed it in R. P. Bernard Lamy (1680),[90] +and in an anonymous work of 1690.[91] In German and Italian works of the +seventeenth century we have not seen Oughtred's notation for proportion. + +In England a modified notation soon sprang up in which ratio was +indicated by two dots instead of a single dot, thus A:B::C:D. The reason +for the change lies probably in the inclination to use the single dot to +designate decimal fractions. W. W. Beman pointed out that this modified +symbolism (:) for ratio is found as early as 1657 in the end of the +trigonometric and logarithmic tables that were bound with Oughtred's +Trigonometria.[92] It is not probable, however, that this notation was +used by Oughtred himself. The Trigonometria proper has Oughtred's +A.B::C.D throughout. Moreover, in the English edition of this +trigonometry, which appeared the same year, 1657, but subsequent to the +Latin edition, the passages which contained the colon as the symbol for +ratio, when not omitted, are recast, and the regular Oughtredian notation +is introduced. In Oughtred's posthumous work, Opuscula mathematica +hactenus inedita, 1677, the colon appears quite often but is most likely +due to the editor of the book. + +We have noticed that the notation A:B::C:D antedates the year 1657. +Vincent Wing, the astronomer, published in 1651 in London the Harmonicon +coeleste, in which is found not only Oughtred's notation A.B::C.D but +also the modified form of it given above. The two are used +interchangeably. His later works, the Logistica astronomica (1656), +Doctrina spherica (1655), and Doctrina theorica, published in one volume +in London, all use the symbols A:B::C:D exclusively. The author of a book +entitled, An Idea of Arithmetick at first designed for the use of the +Free Schoole at Thurlow in Suffolk . . . . by R. B., Schoolmaster there, +London, 1655, writes A:a::C:c, though part of the time he uses Oughtred's +unmodified notation. + +We can best indicate the trend in England by indicating the authors of +the seventeenth century whom we have found using the notation A:B::C:D +and the authors of the eighteenth century whom we have found using +A.B::C.D. The former notation was the less common during the seventeenth +but the more common during the eighteenth century. We have observed the +symbols A:B::C:D (besides the authors already named) in John Collins +(1659),[93] James Gregory (1663),[94] Christopher Wren (1668-69),[95] +William Leybourn (1673),[96] William Sanders (1686),[97] John Hawkins +(1684),[98] Joseph Raphson (1697),[99] E. Wells (1698),[100] and John +Ward (1698).[101] + +Of English eighteenth-century authors the following still clung to the +notation A.B::C.D: John Harris' translation of F. Ignatius Gaston Pardies +(1701),[102] George Shelley (1704),[103] Sam Cobb (1709),[104] J. Collins +in Commercium Epistolicum (1712), John Craig (1718),[105] Jo. Wilson +(1724).[106] The latest use of A.B::C.D which has come to our notice is +in the translation of the Analytical Institutions of Maria G. Agnesi, +made by John Colson sometime before 1760, but which was not published +until 1801. During the seventeenth century the notation A:B::C:D acquired +almost complete ascendancy in England. + +In France Oughtred's unmodified notation A.B::C.D, having been adopted +later, was also discarded later than in England. An approximate idea of +the situation appears from the following data. The notation A.B::C.D was +used by M. Carr (1700),[107] M. Guisne (1705),[108] M. de Fontenelle +(1727),[109] M. Varignon (1725),[110] M. Robillard (1753),[111] M. +Sebastien le Clerc (1764),[112] Clairaut (1731),[113] M. L'Hospital +(1781).[114] + +In Italy Oughtred's modified notation a, b::c, d was used by Maria G. +Agnesi in her Instituzioni analitiche, Milano, 1748. The notation +a:b::c:d found entrance the latter part of the eighteenth century. In +Germany the symbolism a:b=c:d, suggested by Leibniz, found wider +acceptance.[115] + +It is evident from the data presented that Oughtred proposed his notation +for ratio and proportion at a time when the need of a specific notation +began to be generally felt, that his symbol for ratio a.b was temporarily +adopted in England and France but gave way in the eighteenth century to +the symbol a:b, that Oughtred's symbol for proportion :: found almost +universal adoption in England and France and was widely used in Italy, +the Netherlands, the United States, and to some extent in Germany; it has +survived to the present time but is now being gradually displaced by the +sign of equality =. + +Oughtred's notation to express aggregation of terms has received little +attention from historians but is nevertheless interesting. His books, as +well as those of John Wallis, are full of parentheses but they are not +used as symbols of aggregation in algebra; they are simply marks of +punctuation for parenthetical clauses. We have seen that Oughtred writes +(a+b)^2 and {root}(a+b) thus, Q:a+b:, {root}:a+b:, or Q:a+b, {root}:a+b, +using on rarer occasions a single dot in place of the colon. This +notation did not originate with Oughtred, but, in slightly modified form, +occurs in writings from the Netherlands. In 1603 C. Dibvadii in +geometriam Evclidis demonstratio numeralis, Leyden, contains many +expressions of this sort, {root}136+{root}2048, signifying +{root}(136+{root}2048). The dot is used to indicate that the root of the +binomial (not of 136 alone) is called for. This notation is used +extensively in Ludolphi Cevlen de circulo, Leyden, 1619, and in +Willebrordi Snellii De circuli dimensione, Leyden, 1621. In place of the +single dot Oughtred used the colon (:), probably to avoid confusion with +his notation for ratio. To avoid further possibility of uncertainty he +usually placed the colon both before and after the algebraic expression +under aggregation. This notation was adopted by John Wallis and Isaac +Barrow. It is found in the writings of Descartes. Together with Vieta's +horizontal bar, placed over two or more terms, it constituted the means +used almost universally for denoting aggregation of terms in algebra. +Before Oughtred the use of parentheses had been suggested by Clavius[116] +and Girard.[117] The latter wrote, for instance, {root}(2+{root}3). While +parentheses never became popular in algebra before the time of Leibniz +and the Bernoullis they were by no means lost sight of. We are able to +point to the following authors who made use of them: I. Errard de +Bar-le-Duc (1619),[118] Jacobo de Billy (1643),[119] one of whose books +containing this notation was translated into English, and also the +posthumous works of Samuel Foster.[120] J. W. L. Glaisher points out that +parentheses were used by Norwood in his Trigonometrie (1631), p. 30.[121] + +The symbol for the arithmetical difference between two numbers, ~, is +usually attributed to John Wallis, but it occurs in Oughtred's Clavis +mathematicae of 1652, in the tract on Elementi decimi Euclidis +declaratio, at an earlier date than in any of Wallis' books. As Wallis +assisted in putting this edition through the press it is possible, though +not probable, that the symbol was inserted by him. Were the symbol +Wallis', Oughtred would doubtless have referred to its origin in the +preface. During the eighteenth century the symbol found its way into +foreign texts even in far-off Italy.[122] It is one of three symbols +presumably invented by Oughtred and which are still used at the present +time. The others are x and ::. + +The curious and ill-chosen symbols, {symbol} for "greater than," and +{symbol} for "less than," were certain to succumb in their struggle for +existence against Harriot's admirably chosen > and <. Yet such was the +reputation of Oughtred that his symbols were used in England quite +extensively during the seventeenth and the beginning of the eighteenth +century. Considerable confusion has existed among algebraists and also +among historians as to what Oughtred's symbols really were. Particularly +is this true of the sign for "less than" which is frequently written +{symbol}. Oughtred's symbols, or these symbols turned about in some way, +have been used by Seth Ward,[123] John Wallis,[124] Isaac Barrow,[125] +John Kersey,[126] E. Wells,[127] John Hawkins,[128] Tho. Baker,[129] +Richard Sault,[130] Richard Rawlinson,[131] Franciscus Dulaurens,[132] +James Milnes,[133] George Cheyne,[134] John Craig,[135] Jo. Wilson,[136] +and J. Collins.[137] + +General acceptance has been accorded to Oughtred's symbol x. The first +printed appearance of this symbol for multiplication in 1618 in the form +of the letter x hardly explains its real origin. The author of the +"Appendix" (be he Oughtred or someone else) may not have used the letter +x at all, but may have written the cross x, called the St. Andrew's +cross, while the printer, in the absence of any type accurately +representing that cross, may have substituted the letter x in its place. +The hypothesis that the symbol x of multiplication owes its origin to the +old habit of using directed bars to indicate that two numbers are to be +combined, as for instance in the multiplication of 23 and 34, thus, + + 2 3 + |\ /| + | x | + |/ \| + 3 4 + ------- + 7 8 2 + +has been advanced by two writers, C. Le Paige[138] and Gravelaar.[139] +Bosmans is more inclined to the belief that Oughtred adopted the symbol +somewhat arbitrarily, much as he did the numerous symbols in his Elementi +decimi Euclidis declaratio.[140] + +Le Paige's and Gravelaar's theory finds some support in the fact that the +cross x, without the two additional vertical lines shown above, occurs in +a commentary published by Oswald Schreshensuchs[141] in 1551, where the +sign is written between two factors placed one above the other. + + + + + CHAPTER V + OUGHTRED'S IDEAS ON THE TEACHING OF MATHEMATICS + + + GENERAL STATEMENT + +Nowhere has Oughtred given a full and systematic exposition of his views +on mathematical teaching. Nevertheless, he had very pronounced and +clear-cut ideas on the subject. That a man who was not a teacher by +profession should have mature views on teaching is most interesting. We +gather his ideas from the quality of the books he published, from his +prefaces, and from passages in his controversial writing against +Delamain. As we proceed to give quotations unfolding Oughtred's views, we +shall observe that three points receive special emphasis: (1) an appeal +to the eye through suitable symbolism; (2) emphasis upon rigorous +thinking; (3) the postponement of the use of mathematical instruments +until after the logical foundations of a subject have been thoroughly +mastered. + +The importance of these tenets is immensely reinforced by the conditions +of the hour. This voice from the past speaks wisdom to specialists of +today. Recent methods of determining educational values and the modern +cult of utilitarianism have led some experts to extraordinary +conclusions. Laboratory methods of testing, by the narrowness of their +range, often mislead. Thus far they have been inferior to the word of a +man of experience, insight, and conviction. + + + MATHEMATICS, "A SCIENCE OF THE EYE" + +Oughtred was a great admirer of the Greek mathematicians--Euclid, +Archimedes, Apollonius of Perga, Diophantus. But in reading their works +he experienced keenly what many modern readers have felt, namely, that +the almost total absence of mathematical symbols renders their writings +unnecessarily difficult to read. Statements that can be compressed into a +few well-chosen symbols which the eye is able to survey as a whole are +expressed in long-drawn-out sentences. A striking illustration of the +importance of symbolism is afforded by the history of the formula + + ix=log(cos x+i sin x). + +It was given in Roger Cotes' Harmonia mensurarum, 1722, not in symbols, +but expressed in rhetorical form, destitute of special aids to the eye. +The result was that the theorem remained in the book undetected for 185 +years and was meanwhile rediscovered by others. Owing to the prominence +of Cotes as a mathematician it is very improbable that such a thing could +have happened had the theorem been thrust into view by the aid of +mathematical symbols. + +In studying the ancient authors Oughtred is reported to have written down +on the margin of the printed page some of the theorems and their proofs, +expressed in the symbolic language of algebra. + +In the preface of his Clavis of 1631 and of 1647 he says: + + Wherefore, that I might more clearly behold the things themselves, I + uncasing the Propositions and Demonstrations out of their covert of + words, designed them in notes and species appearing to the very eye. + After that by comparing the divers affections of Theorems, inequality, + proportion, affinity, and dependence, I tryed to educe new out of them. + +It was this motive which led him to introduce the many abbreviations in +algebra and trigonometry to which reference has been made in previous +pages. The pedagogical experience of recent centuries has indorsed +Oughtred's view, provided of course that the pupil is carefully taught +the exact meaning of the symbols. There have been and there still are +those who oppose the intensive use of symbolism. In our day the new +symbolism for all mathematics, suggested by the school of Peano in Italy, +can hardly be said to be received with enthusiasm. In Oughtred's day +symbolism was not yet the fashion. To be convinced of this fact one need +only open a book of Edmund Gunter, with whom Oughtred came in contact in +his youth, or consult the Principia of Sir Isaac Newton, who flourished +after Oughtred. The mathematical works of Gunter and Newton, particularly +the former, are surprisingly destitute of mathematical symbols. The +philosopher Hobbes, in a controversy with John Wallis, criticized the +latter for that "Scab of Symbols," whereupon Wallis replied: + + I wonder how you durst touch M. Oughtred for fear of catching the Scab. + For, doubtlesse, his book is as much covered over with the Scab of + Symbols, as any of mine. . . . . As for my Treatise of Conick Sections, + you say, it is covered over with the Scab of Symbols, that you had not + the patience to examine whether it is well or ill demonstrated.[142] + +Oughtred maintained his view of the importance of symbols on many +different occasions. Thus, in his Circles of Proportion, 1632, p. 20: + + This manner of setting downe Theoremes, whether they be Proportions, or + Equations, by Symboles or notes of words, is most excellent, + artificiall, and doctrinall. Wherefore I earnestly exhort every one, + that desireth though but to looke into these noble Sciences + Mathematicall, to accustome themselves unto it: and indeede it is + easie, being most agreeable to reason, yea even to sence. And out of + this working may many singular consectaries be drawne: which without + this would, it may be, for ever lye hid. + + + RIGOROUS THINKING AND THE USE OF INSTRUMENTS + +The author's elevated concept of mathematical study as conducive to +rigorous thinking shines through the following extract from his preface +to the 1647 Clavis: + + . . . . Which Treatise being not written in the usuall synthetical + manner, nor with verbous expressions, but in the inventive way of + Analitice, and with symboles or notes of things instead of words, + seemed unto many very hard; though indeed it was but their owne + diffidence, being scared by the newnesse of the delivery; and not any + difficulty in the thing it selfe. For this specious and symbolicall + manner, neither racketh the memory with multiplicity of words, nor + chargeth the phantasie with comparing and laying things together; but + plainly presenteth to the eye the whole course and processe of every + operation and argumentation. + + Now my scope and intent in the first Edition of that my Key was, and in + this New Filing, or rather forging of it, is, to reach out to the + ingenious lovers of these Sciences, as it were Ariadnes thread, to + guide them through the intricate Labyrinth of these studies, and to + direct them for the more easie and full understanding of the best and + antientest Authors. . . . . That they may not only learn their + propositions, which is the highest point of Art that most Students aime + at; but also may perceive with what solertiousnesse, by what engines of + aequations, Interpretations, Comparations, Reductions, and + Disquisitions, those antient Worthies have beautified, enlarged, and + first found out this most excellent Science. . . . . Lastly, by framing + like questions problematically, and in a way of Analysis, as if they + were already done, resolving them into their principles, I sought out + reasons and means whereby they might be effected. And by this course of + practice, not without long time, and much industry, I found out this + way for the helpe and facilitation of Art. + +Still greater emphasis upon rigorous thinking in mathematics is laid in +the preface to the Circles of Proportion and in some parts of his +Apologeticall Epistle against Delamain. In that preface William Forster +quotes the reply of Oughtred to the question how he (Oughtred) had for so +many years concealed his invention of the slide rule from himself +(Forster) whom he had taught so many other things. The reply was: + + 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 in-stead of Artists, to + make their Scholers only doers of tricks, and as it were Iuglers: to + the despite of Art, losse of previous time, and betraying of willing + and industrious wits, vnto ignorance, and idlenesse. That the vse 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 haue me + well instructed in the Sciences." + +Delamain took a different view, arguing that instruments might very well +be placed in the hands of pupils from the start. At the time of this +controversy Delamain supported himself by teaching mathematics in London +and he advertised his ability to give instruction in mathematics, +including the use of instruments. Delamain brought the charge against +Oughtred of unjustly calling "many of the [British] Nobility and Gentry +doers of trickes and juglers." To this Oughtred replies: + + As I did to Delamain and to some others, so I did to William Forster: I + freely gave him my helpe and instruction in these faculties: only this + was the difference, I had the very first moulding (as I may say) of + this latter: But Delamain was already corrupted with doring upon + Instruments, and quite lost from ever being made an Artist: I suffered + not William Forster for some time so much as speake of any Instrument, + except only the Globe it selfe; and to explicate, and worke the + questions of the Sphaere, by the way of the Analemma: which also + himselfe did describe for the present occasion. And this my restraint + from such pleasing avocations, and holding him to the strictnesse of + percept, brought forth this fruit, that in short time, even by his owne + skill, he could not onely use any Instrument he should see, but also + was able to delineate the like, and devise others.[143] + +As representing Delamain's views, we make the following selection from +his Grammelogia (London, about 1633), the part near the end of the book +and bearing the title, "In the behalfe of vulgar Teachers and others," +where Delamain refers to Oughtred's charge that the scholars of "vulgar" +teachers are "doers of tricks, as it were iuglers." Delamain says: + + . . . . Which words are neither cautelous, nor subterfugious, but are + as downe right in their plainnesse, as they are touching, and + pernitious, by two much derogating from many, and glancing upon many + noble personages, with too grosse, if not too base an attribute, in + tearming them doers of tricks, as it were to iuggle: because they + perhaps make use of a necessitie in the furnishing of themselves with + such knowledge by Practicall Instrumentall operation, when their more + weighty negotiations will not permit them for Theoreticall figurative + demonstration; those that are guilty of the aspertion, and are touched + therewith may answer for themselves, and studie to be more + Theoreticall, than Practicall: for the Theory, is as the Mother that + produceth the daughter, the very sinewes and life of Practise, the + excellencie and highest degree of true Mathematicall Knowledge: but for + those that would make but a step as it were into that kind of Learning, + whose onely desire is expedition, and facilitie, both which by the + generall consent of all are best effected with Instrument, rather then + with tedious regular demonstrations, it was ill to checke them so + grosly, not onely in what they have Practised, but abridging them also + of their liberties with what they may Practise, which aspertion may not + easily be slighted off by any glosse or Apologie, without an Ingenuous + confession, or some mentall reservation: To which vilification, + howsoever, in the behalfe of my selfe, and others, I answer; That + Instrumentall operation is not only the Compendiating, and facilitating + of Art, but even the glory of it, whole demonstration both of the + making, and operation is soly in the science, and to an Artist or + disputant proper to be knowne, and so to all, who would truly know the + cause of the Mathematicall operations in their originall; But, for none + to know the use of a Mathematicall Instrumen[t], except he knowes the + cause of its operation, is somewhat too strict, which would keepe many + from affecting the Art, which of themselves are ready enough every + where, to conceive more harshly of the difficultie, and impossibilitie + of attayning any skill therein, then it deserves, because they see + nothing but obscure propositions, and perplex and intricate + demonstrations before their eyes, whose unsavoury tartnes, to an + unexperienced palate like bitter pills is sweetned over, and made + pleasant with an Instrumentall compendious facilitie, and made to goe + downe the more readily, and yet to retaine the same vertue, and + working; And me thinkes in this queasy age, all helpes may bee used to + procure a stomacke, all bates and invitations to the declining studie + of so noble a Science, rather then by rigid Method and generall Lawes + to scarre men away. All are not of like disposition, neither all (as + was sayd before) propose the same end, some resolve to wade, others to + put a finger in onely, or wet a hand: now thus to tye them to an + obscure and Theoricall forme of teaching, is to crop their hope, even + in the very bud. . . . . The beginning of a mans knowledge even in the + use of an Instrument, is first founded on doctrinal precepts, and these + precepts may be conceived all along in its use: and are so farre from + being excluded, that they doe necessarily concomitate and are contained + therein: the practicke being better understood by the doctrinall part, + and this later explained by the Instrumentall, making precepts obvious + unto sense, and the Theory going along with the Instrument, better + informing and inlightning the understanding, etc. vis vnita fortior, so + as if that in Phylosophy bee true, Nihil est [in] intellectu quod non + prius fuit in sensu. + +The difference between Oughtred and Delamain as to the use of +mathematical instruments raises important questions. Should the slide +rule be placed in the hands of a boy before, or after, he has mastered +the theory of logarithms? Should logarithmic tables be withheld from him +until the theoretical foundation is laid in the mind of the pupil? Is it +a good thing to let a boy use a surveying instrument unless he first +learns trigonometry? Is it advisable to permit a boy to familiarize +himself with the running of a dynamo before he has mastered the +underlying principles of electricity? Does the use of instruments +ordinarily discourage a boy from mastery of the theory? Or does such +manipulation constitute a natural and pleasing approach to the abstract? +On this particular point, who showed the profounder psychological +insight, Oughtred or Delamain? + +In July, 1914, there was held in Edinburgh a celebration of the +three-hundredth anniversary of the invention of logarithms. On that +occasion there was collected at Edinburgh university one of the largest +exhibits ever seen of modern instruments of calculation. The opinion was +expressed by an experienced teacher that "weapons as those exhibited +there are for men and not for boys, and such danger as there may be in +them is of the same character as any form of too early specialization." + +It is somewhat of a paradox that Oughtred, who in his student days and +during his active years felt himself impelled to invent sun-dials, +planispheres, and various types of slide rules--instruments which +represent the most original contributions which he handed down to +posterity--should discourage the use of such instruments in teaching +mathematics to beginners. That without the aid of instruments he himself +should have succeeded so well in attracting and inspiring young men +constitutes the strongest evidence of his transcendent teaching ability. +It may be argued that his pedagogic dogma, otherwise so excellent, here +goes contrary to the course he himself followed instinctively in his +self-education along mathematical lines. We read that Sir Isaac Newton, +as a child, constructed sun-dials, windmills, kites, paper lanterns, and +a wooden clock. Should these activities have been suppressed? Ordinary +children are simply Isaac Newtons on a smaller intellectual scale. Should +their activities along these lines be encouraged or checked? + +On the other hand, it may be argued that the paradox alluded to above +admits of explanation, like all paradoxes, and that there is no +inconsistency between Oughtred's pedagogic views and his own course of +development. If he invented sun-dials, he must have had a comprehension +of the cosmic motions involved; if he solved spherical triangles +graphically by the aid of the planisphere, he must have understood the +geometry of the sphere, so far as it relates to such triangles; if he +invented slide rules, he had beforehand a thorough grasp of logarithms. +The question at issue does not involve so much the invention of +instruments, as the use by the pupil of instruments already constructed, +before he fully understands the theory which is involved. Nor does Sir +Isaac Newton's activity as a child establish Delamain's contention. Of +course, a child should not be discouraged from manual activity along the +line of producing interesting toys in imitation of structures and +machines that he sees, but to introduce him to the realm of abstract +thought by the aid of instruments is a different proposition, fraught +with danger. A boy may learn to use a slide rule mechanically and, +because of his ability to obtain practical results, feel justified in +foregoing the mastery of underlying theory; or he may consider the +ability of manipulating a surveying instrument quite sufficient, even +though he be ignorant of geometry and trigonometry; or he may learn how +to operate a dynamo and an electric switchboard and be altogether +satisfied, though having no grasp of electrical science. Thus instruments +draw a youth aside from the path leading to real intellectual attainments +and real efficiency; they allure him into lanes which are often blind +alleys. Such were the views of Oughtred. + +Who was right, Oughtred or Delamain? It may be claimed that there is a +middle ground which more nearly represents the ideal procedure in +teaching. Shall the slide rule be placed in the student's hands at the +time when he is engaged in the mastery of principles? Shall there be an +alternate study of the theory of logarithms and of the slide rule--on the +idea of one hand washing the other--until a mastery of both the theory +and the use of the instrument has been attained? Does this method not +produce the best and most lasting results? Is not this Delamain's actual +contention? We leave it to the reader to settle these matters from his +own observation, knowledge, and experience. + + + NEWTON'S COMMENTS ON OUGHTRED + +Oughtred is an author who has been found to be of increasing interest to +modern historians of mathematics. But no modern writer has, to our +knowledge, pointed out his importance in the history of the teaching of +mathematics. Yet his importance as a teacher did receive recognition in +the seventeenth century by no less distinguished a scientist than Sir +Isaac Newton. On May 25, 1694, Sir Isaac Newton wrote a long letter in +reply to a request for his recommendation on a proposed new course of +study in mathematics at Christ's Hospital. Toward the close of his +letter, Newton says: + + And now I have told you my opinion in these things, I will give you Mr. + Oughtred's, a Man whose judgment (if any man's) may be safely relyed + upon. For he in his book of the circles of proposition, in the end of + what he writes about Navigation (page 184) has this exhortation to + Seamen. "And if," saith he, "the Masters of Ships and Pilots will take + the pains in the Journals of their Voyages diligently and faithfully to + set down in severall columns, not onely the Rumb they goe on and the + measure of the Ships way in degrees, and the observation of Latitude + and variation of their compass; but alsoe their conjectures and reason + of their correction they make of the aberrations they shall find, and + the qualities and condition of their ship, and the diversities and + seasons of the winds, and the secret motions or agitations of the Seas, + when they begin, and how long they continue, how farr they extend and + with what inequality; and what else they shall observe at Sea worthy + consideration, and will be pleased freely to communicate the same with + Artists, such as are indeed skilfull in the Mathematicks and lovers and + enquirers of the truth: I doubt not but that there shall be in + convenient time, brought to light many necessary precepts which may + tend to y^e perfecting of Navigation, and the help and safety of such + whose Vocations doe inforce them to commit their lives and estates in + the vast Ocean to the providence of God." Thus farr that very good and + judicious man Mr. Oughtred. I will add, that if instead of sending the + Observations of Seamen to able Mathematicians at Land, the Land would + send able Mathematicians to Sea, it would signify much more to the + improvem^t of Navigation and safety of Mens lives and estates on that + element.[144] + +May Oughtred prove as instructive to the modern reader as he did to +Newton! + + + + + Footnotes + + +[1]Aubrey's Brief Lives, ed. A. Clark, Vol. II, Oxford, 1898, p. 106. + +[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. + +[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." + +[4]New and General Biographical Dictionary (John Nichols), London, 1784, + art. "Oughtred." + +[5]Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, p. 132. + +[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." + +[7]Aubrey, op. cit., Vol. II, p. 107. + +[8]Rigaud, Correspondence of Scientific Men of the Seventeenth Century, + Oxford, Vol. I, 1841, p. 5. + +[9]Aubrey, op. cit., Vol. II, p. 110. + +[10]Ibid., p. 111. + +[11]Op. cit., Vol. II, p. 132. + +[12]Mr. William Lilly's History of His Life and Times, From the Year 1602 + to 1681, London, 1715, p. 58. + +[13]Rigaud, op. cit., Vol. I, p. 60. + +[14]Aubrey, op. cit., Vol. II, p. 107. + +[15]Rigaud, op. cit., Vol. I, p. 16. + +[16]Owen Manning, op. cit., p. 132. + +[17]New and General Biographical Dictionary (John Nichols), London, 1784, + art. "Oughtred." + +[18]Op. cit., Vol. II, p. 110. + +[19]Rev. Owen Manning, The History and Antiquities of Surrey, Vol. II, + London, 1809, p. 132. + +[20]Op. cit., Vol. II, 1898, p. 111. + +[21]Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago and + London, 1915, Vol. II, p. 303. + +[22]The full title of the Clavis of 1631 is as follows: Arithmeticae in + numeris et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae, + atqve adeo totivs mathematicae, qvasi clavis est.--Ad nobilissimvm + spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur, + Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae & + Svrriae, Comitis Mareschalli Angliae, &c filium.--Londini, Apud Thomam + Harpervm. M.DC.XXXI. + + In all there appeared five Latin editions, the second in 1648 at + London, the third in 1652 at Oxford, the fourth in 1667 at Oxford, the + fifth in 1693 and 1698 at Oxford. There were two independent English + editions: the first in 1647 at London, translated in greater part by + Robert Wood of Lincoln College, Oxford, as is stated in the preface to + the 1652 Latin edition; the second in 1694 and 1702 is a new + translation, the preface being written and the book recommended by the + astronomer Edmund Halley. The 1694 and 1702 impressions labored under + the defect of many sense-disturbing errors due to careless reading of + the proofs. All the editions of the Clavis, after the first edition, + had one or more of the following tracts added on: + + Eq.=De Aequationum affectarvm resolvtione in numeris. + Eu.=Elementi decimi Euclidis declaratio. + So.=De Solidis regularibus, tractatus. + An.=De Anatocismo, sive usura composita. + Fa.=Regula falsae positionis. + Ar.=Theorematum in libris Archimedis de Sphaera & cylindro declaratio. + Ho.=Horologia scioterica in plano, geometric delineandi modus. + + The abbreviated titles given here are, of course, our own. The lists + of tracts added to the Clavis mathematicae of 1631 in its later + editions, given in the order in which the tracts appear in each + edition, are as follows: Clavis of 1647, Eq., An., Fa., Ho.; Clavis of + 1648, Eq., An., Fa., Eu., So.; Clavis of 1652, Eq., Eu., So., An., + Fa., Ar., Ho.; Clavis of 1667, Eq., Eu., So., An., Fa., Ar., Ho.; + Clavis of 1693 and 1698, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of + 1694 and 1702, Eq. + + The title-page of the Clavis was considerably modified after the first + edition. Thus, the 1652 Latin edition has this title-page: Guilelmi + Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia Socii, + Clavis mathematicae denvo limata, sive potius fabricata. Cum aliis + quibusdam ejusdem commentationibus, quae in sequenti pagina + recensentur. Editio tertia auctior & emendatior. Oxoniae, Excudebat + Leon. Lichfield, Veneunt apud Tho. Robinson. 1652. + +[23]Rigaud, op. cit., Vol. II, p. 476. + +[24]See, for instance, the Clavis mathematicae of 1652, where he + expresses himself thus (p. 4): "Speciosa haec Arithmetica arti + Analyticae (per quam ex sumptione quaesiti, tanquam noti, investigatur + quaesitum) multo accommodatior est, quam illa numerosa." + +[25]Oughtred, The Key of the Mathematicks, London, 1647, p. 4. + +[26]Clavis 1694, p. 19, and the Clavis of 1631, p. 8. + +[27]See for instance, Oughtred's Elementi decimi Euclidis declaratio, + 1652, p. 1, where he uses A and E, and also a and e. + +[28]See Christophori Clavii Bambergensis Operum mathematicorum, tomus + secundus, Moguntiae, M.DC.XI, algebra, p. 39. + +[29]Christophori Clavii operum mathematicorum Tomus Secundus, Moguntiae, + M.DC.XI, Epitome arithmeticae, p. 36. + +[30]See F. Cajori, "The Cross x as a Symbol of Multiplication," in + Nature, Vol. XCIV (1914), p. 363. + +[31]See Elementi decimi Euclidis declaratio, 1652, p. 2. + +[32]See Johannis Wallisii Operum mathematicorum pars prima, Oxonii, 1657, + p. 247. + +[33]Clavis of 1631, chap. xix, sec. 5, p. 50. + +[34]We have noticed the representation of known quantities by consonants + and the unknown by vowels in Wingate's Arithmetick made easie, edited + by John Kersey, London, 1650, algebra, p. 382; and in the second part, + section 19, of Jonas Moore's Arithmetick in two parts, London, 1660, + Moore suggests as an alternative the use of z, y, x, etc., for the + unknowns. The practice of representing unknowns by vowels did not + spread widely in England. + +[35]Philosophical Transactions, Vol. XIX, No. 231, London, p. 652. + +[36]Ibid., Vol. XIX, p. 56. + +[37]There are two title-pages to the edition of 1632. The first + title-page is as follows: The Circles of Proportion and The + Horizontall Instrument. Both invented, and the vses 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. T. Cecill Sculp. + + In 1633 there was added the following, with a separate title-page: An + addition vnto the Vse of the Instrvment called the Circles of + Proportion. . . . . London, 1633, this being followed by Oughtred's To + the English Gentrie etc. In the British Museum there is a copy of + another impression of the Circles of Proportion, dated 1639, with the + Addition vnto the Vse of the Instrument etc., bearing the original + date, 1633, and with the epistle, To the English Gentrie, etc., + inserted immediately after Forster's dedication, instead of at the end + of the volume. + +[38]The complete title of the English edition is as follows: + Trigonometrie, or, The manner of calculating the Sides and Angles of + Triangles, by the Mathematical Canon, demonstrated. By William + Oughtred Etonens. And published by Richard Stokes Fellow of Kings + Colledge in Cambridge, and Arthur Haughton Gentleman. London, Printed + by R. and W. Leybourn, for Thomas Johnson at the Golden Key in St. + Pauls Church-yard. M.DC.LVII. + +[39]Jer. Collier, The Great Historical, Geographical, Genealogical and + Poetical Dictionary, Vol. II, London, 1701, art. "Oughtred." + +[40]Rigaud op. cit., Vol. I, p. 82. + +[41]A. De Morgan, Budget of Paradoxes, London, 1872, p. 451; 2d ed., + Chicago, 1915, Vol. II, p. 303. + +[42]E. Gunter, Description and Use of the Sector, the Crosse-staffe and + other Instruments, London, 1624, second book, p. 31. + +[43]F. Cajori, "On the History of a Notation in Trigonometry," Nature, + Vol. XCIV, 1915, pp. 642, 643. + +[44]A. von Braunmhl, Geschichte der Trigonometrie, 2. Teil, Leipzig, + 1903, pp. 42, 91. + +[45]H. Hankel, Geschichte der Mathematik in Alterthum und Mittelalter, + Leipzig, 1874, pp. 369, 370. + +[46]M. Cantor, Vorlesungen ber Geschichte der Mathematik, II, 1900, pp. + 640, 641. + +[47]This matter has been discussed in a paper by F. Cajori, "A History of + the Arithmetical Methods of Approximation, etc., Colorado College + Publication, General Series No. 51, 1910, pp. 182-84. Later this + subject was again treated by G. Enestrm in Bibliotheca mathematica, + 3. Folge, Vol. XI, 1911, pp. 234, 235. + +[48]See F. Cajori, op. cit., p. 193. + +[49]See William Oughtred's Key of the Mathematicks, London, 1694, pp. + 173-75, tract, "Of the Resolution of the Affected Equations," or any + edition of the Clavis after the first. + +[50]A. De Morgan, op. cit., p. 451; 2d ed., Vol. II, p. 303. + +[51]See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909, + pp. 7-14, Addenda, p. ii. + +[52]Rigaud, op. cit., Vol. I, p. 12. + +[53]The New Artificial Gauging Line or Rod: together with rules + concerning the use thereof: Invented and written by William Oughtred, + London, 1633. + +[54]W. Oughtred, Apologeticall Epistle, p. 13. + +[55]Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, (1915), + p. 169. In this article Glaisher republishes the "Appendix" in full. + +[56]Aubrey, op. cit., Vol. II, 1898, p. 108. + +[57]Wood's Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247. + +[58]Wood, op. cit., Vol. II, p. 445. + +[59]Rigaud, op. cit., Vol. I, pp. 33, 35. + +[60]Rigaud, op. cit., Vol. I, pp. 16, 26. + +[61]Rigaud, op. cit., Vol. I, p. 66. + +[62]Ibid., Vol. I, p. 9. + +[63]Rigaud, op. cit., Vol. II, p. 475. + +[64]Ibid., Vol. II, p. 471. + +[65]J. W. L. Glaisher, "On Early Logarithmic Tables, and Their + Calculators," Philosophical Magazine, 4th Ser., Vol. XLV (1873), pp. + 378, 379. + +[66]Rigaud, op. cit., Vol. I, p. 65. + +[67]Rigaud, op. cit., Vol. I, p. 87. + +[68]King's Life of John Locke, Vol. I, London, 1830, p. 227. + +[69]Exercitationum Mathematicarum Decas prima, Naples, 1627, and probably + Cataldus' Transformatio Geometrica, Bonon., 1612. + +[70]Rigaud, op. cit., Vol. II, pp. 477-80. + +[71]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, + Sometimes publike Professor of Astronomie in Gresham Colledge in + London, by John Twysden, London, 1659. + +[72]The Works of the Honourable Robert Boyle in five volumes, to which is + prefixed the Life of the Author, Vol. I, London, 1744, p. 24. + +[73]The letter is printed in John Wallis' De algebra tractatus, 1693, p. + 206. + +[74]See La Correspondance de Descartes, published by Charles Adam and + Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457. + +[75]H. Bosmans, S.J., "La premire dition de la Clavis Mathematica + d'Oughtred. Son influence sur la Gomtrie de Descartes," Annales de + la socit scientifique de Bruxelles, 35th year, 1910-11, Part II, pp. + 24-78. + +[76]Ibid., p. 78. + +[77]Vincent Wing, Harmonicon coeleste, London, 1651, p. 5. + +[78]Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenta + inquisitio brevis, Oxford, 1653, p. 7. + +[79]John Wallis, Elenchus geometriae Hobbianae, Oxford, 1655, p. 48. + +[80]An Idea of Arithmetick, at first designed for the use of the Free + Schoole at Thurlow in Suffolk. . . . . By R. B., Schoolmaster there, + London, 1655, p. 6. + +[81]The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster + . . . . by John Twysden, London, 1659, p. 1. + +[82]Moor's Arithmetick in two Books, London, 1660, p. 89. + +[83]Isaac Barrow, Euclidis data, Cambridge, 1657, p. 2. + +[84]Francisci Dulaurens Specima mathematica, Paris, 1667, p. 1. + +[85]Elmens des mathmatiques, Paris, 1675, Preface signed "J. P." + +[86]Nouveaux lmens de gomtrie, Paris, 1692 (permission to print + 1684). + +[87]Ozanam, Dictionnaire mathmatique, Paris, 1691, p. 12. + +[88]Analyse des infiniment petits, Paris, 1696, p. 11. + +[89]Petro Nicolas, De conchoidibus et cissoidibus exercitationes + geometricae, Toulouse, 1697, p. 17. + +[90]R. P. Bernard Lamy, Elmens des mathmatiques, Amsterdam, 1692 + (permission to print 1680). + +[91]Nouveaux lmens de gomtrie, 2d ed., The Hague, 1690, p. 304. + +[92]W. W. Beman in L'intermdiaire des mathmaticiens, Paris, Vol. IX, + 1902, p. 229, question 2424. + +[93]John Collins, The Mariner's Plain Scale New Plain'd, London, 1659, p. + 25. + +[94]James Gregory, Optica promota, London, 1663, pp. 19, 48. + +[95]Philosophical Transactions, Vol. III, London, p. 868. + +[96]William Leybourn, The Line of Proportion, London, 1673, p. 14. + +[97]Elementa geometriae . . . . a Gulielmo Sanders, Glasgow, 1686, p. 3. + +[98]Cocker's Decimal Arithmetick, . . . . perused by John Hawkins, + London, 1695 (preface dated 1684), p. 41. + +[99]Joseph Raphson, Analysis Aequationum universalis, London, 1697, p. + 26. + +[100]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, + 1698, p. 107. + +[101]John Ward, A Compendium of Algebra, 2d ed., London, 1698, p. 62. + +[102]Plain Elements of Geometry and Plain Trigonometry, London, 1701, p. + 63. + +[103]George Shelley, Wingate's Arithmetick, London, 1704, p. 343. + +[104]A Synopsis of Algebra, Being a posthumous work of John Alexander of + Bern, Swisserland. . . . . Done from the Latin by Sam. Cobb, London, + 1709, p. 16. + +[105]John Craig, De Calculo fluentium, London, 1718, p. 35. The notation + A:B::C:D is given also. + +[106]Trigonometry, 2d ed., Edinburgh, 1724, p. 11. + +[107]Mthode pour la msure des surfaces, la dimension des solides . . . + . par M. Carr de l'acadmie r. des sciences, 1700, p. 59. + +[108]Application de l'algbre gomtrie . . . . Paris, 1705. + +[109]Elmens de la gomtrie de l'infini, by M. de Fontenelle, Paris, + 1727, p. 110. + +[110]Eclaircissemens sur l'analyse des infiniment petits, by M. Varignon, + Paris, 1725, p. 87. + +[111]Application de la gomtrie ordinaire et des calculs diffrentiel et + intgral, by M. Robillard, Paris, 1753. + +[112]Trait de gomtrie thorique et pratique, new ed., Paris, 1764, p. + 15. + +[113]Recherches sur les courbes double courbure, Paris, 1731, p. 13. + +[114]Analyse des infiniment petits, by the Marquis de L'Hospital. New ed. + by M. Le Fvre, Paris, 1781, p. 41. In this volume passages in fine + print, probably supplied by the editor, contain the notation a:b::c:d; + the parts in large type give Oughtred's original notation. + +[115]The tendency during the eighteenth century is shown in part by the + following data: Jacobi Bernoulli Opera, Tomus primus, Geneva, 1744, + gives B.A::D.C on p. 368, the paper having been first published in + 1688; on p. 419 is given GE:AG=LA:ML, the paper having been first + published in 1689. Bernhardi Nieuwentiit, Considerationes circa + analyseos ad quantitates infinit parvas applicatae principia, + Amsterdam, 1694, p. 20, and Analysis infinitorum, Amsterdam, 1695, on + p. 276, have x:c::s:r. Paul Halcken's Deliciae mathematicae, Hamburg, + 1719, gives a:b::c:d. Johannis Baptistae Caraccioli, Geometria + algebraica universa, Rome, 1759, p. 79, has a.b::c.d. Delle corde + ouverto fibre elastiche schediasmi fisico-matematici del conte + Giordano Riccati, Bologna, 1767, p. 65, gives P:b::r:ds. "Produzioni + mathematiche" del Conte Giulio Carlo de Fagnano, Vol. I, Pesario, + 1750, p. 193, has a.b::c.d. L. Mascheroni, Gomtrie du compas, + translated by A. M. Carette, Paris, 1798, p. 188, gives + {root}(3):2::{root}(2):Lp. Danielis Melandri and Paulli Frisi, De + theoria lunae commentarii, Parma, 1769, p. 13, has a:b::c:d. Vicentio + Riccato and Hieronymo Saladino, Institutiones analyticae, Vol. I, + Bologna, 1765, p. 47, gives x:a::m:n+m. R. G. Boscovich, Opera + pertinentia ad opticam et astronomiam, Bassani, 1785, p. 409, uses + a:b::c:d. Jacob Bernoulli, Ars Conjectandi, Basel, 1713, has + n-r.n-1::c.d. Pavlini Chelvicii, Institutiones analyticae, editio post + tertiam Romanam prima in Germania, Vienna, 1761, p. 2, a.b::c.d. + Christiani Wolfii, Elementa matheseos universae, Vol. III, Geneva, + 1735, p. 63, has AB:AE=1:q. Johann Bernoulli, Opera omnia, Vol. I, + Lausanne and Geneva, 1742, p. 43, has a:b=c:d. D. C. Walmesley, + Analyse des mesures des rapports et des angles, Paris, 1749, uses + extensively a.b::c.d, later a:b::c:d. G. W. Krafft, Institutiones + geometriae sublimoris, Tbingen, 1753, p. 194, has a:b=c:d. J. H. + Lambert, Photometria, 1760, p. 104, has C:{pi}=BC^2:MH^2. Meccanica + sublime del Dott. Domenico Bartaloni, Naples, 1765, has a:b::c:d. + Occasionally ratio is not designated by a.b, nor by a:b, but by a, b, + as for instance in A. de Moivre's Doctrine of Chance, London, 1756, p. + 34, where he writes a, b::1, q. A further variation in the designation + of ratio is found in James Atkinson's Epitome of the Art of + Navigation, London, 1718, p. 24, namely, 3..2::72..48. Curious + notations are given in Rich. Balam's Algebra, London, 1653. + +[116]Chr. Clavii Operum mathematicorum tomus secundus, Mayence, 1611, + Algebra, p. 39. + +[117]Invention nouvelle en l'algbre, by Albert Girard, Amsterdam, 1629, + p. 17. + +[118]La gomtrie et pratique gnrale d'icelle, par I. Errard de + Bar-le-Duc, Ingnieur ordinaire de sa Majest, 3d ed., revised by D. + H. P. E. M., Paris, 1619, p. 216. + +[119]Novae geometriae clavis algebra, authore P. Jacobo de Billy, Paris, + 1643, p. 157; also an Abridgement of the Precepts of Algebra. Written + in French by James de Billy, London, 1659, p. 346. + +[120]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, + Sometime publike Professor of Astronomie in Gresham Colledge in + London, London, 1659, p. 7. + +[121]Quarterly Jour. of Pure and Applied Math., Vol. XLVI (London, 1915), + p. 191. + +[122]Pietro Cossali, Origine, trasporto in Italia primi progressi in essa + dell' algebra, Vol. I, Parmense, 1797, p. 52. + +[123]In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio + brevis, Auctore Setho Wardo, Oxford, 1653, p. 1. + +[124]John Wallis, Algebra, London, 1685, p. 321, and in some of his other + works. He makes greater use of Harriot's symbols. + +[125]Euclidis data, 1657, p. 1; also Euclidis elementorum libris XV, + London, 1659, p. 1. + +[126]John Kersey, Algebra, London, 1673, p. 321. + +[127]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, + 1698, p. 142. + +[128]Cocker's Decimal Arithmetick, perused by John Hawkins, London, 1695 + (preface dated 1684), p. 278. + +[129]Th. Baker, The Geometrical Key, London, 1684, p. 15. + +[130]Richard Sault, A New Treatise of Algebra, London (no date). + +[131]Richard Rawlinson in a pamphlet without date, issued sometime + between 1655 and 1668, containing trigonometric formulas. There is a + copy in the British Museum. + +[132]F. Dulaurens, Specima mathematica, Paris, 1667, p. 1. + +[133]J. Milnes, Sectionum conicarum elementa, Oxford, 1702, p. 42. + +[134]Cheyne, Philosophical Principles of Natural Religion, London, 1705, + p. 55. + +[135]J. Craig, De calculo fluentium, London, 1718, p. 86. + +[136]Jo. Wilson, Trigonometry, 2d ed., Edinburgh, 1724, p. v. + +[137]Commercium Epistolicum, 1712, p. 20. + +[138]C. Le Paige, "Sur l'origine de certains signes d'opration," Annales + de la socit scientifique de Bruxelles, 16th year, 1891-92, Part II, + pp. 79-82. + +[139]Gravelaar, "Over den oorsprong van ons maalteeken (x)," Wiskundig + Tijdschrift, 6th year. We have not had access to this article. + +[140]H. Bosmans, op. cit., p. 40. + +[141]Claudii Ptolemaei . . . . annotationes, Ble, 1551. This reference + is taken from the Encyclopdie des sciences mathmatiques, Tome I, + Vol. I, Fasc. 1, p. 40. + +[142]Due Correction for Mr. Hobbes. Or Schoole Discipline, for not saying + his Lessons right. In answer to his Six Lessons, directed to the + Professors of Mathematicks. By the Professor of Geometry. Oxford, + 1656, pp. 7, 47, 50. + +[143]Oughtred, Apologeticall Epistle, p. 27. + +[144]J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes, + London, 1850, pp. 279-92. + + + + + INDEX + + + Adam, Charles, 71 + Agnesi, Maria G., 77 + Alexander, J., 76 + Allen, E., 35 + Analysis, 19, 20 + Apollonius of Perga, 20, 85 + Archimedes, 18, 20, 85 + Aristotle, 69 + Ashmole, E., 13 + Atkinson, J., 79 + Atwood, 56 + Aubrey, 3, 7, 8, 12-16, 58, 59 + Austin, 58 + + Baker, T., 82 + Balam, R., 79 + Bar-le-Duc, de, 80 + Barrow, S., 1, 32, 73, 74, 80, 81 + Bartaloni, D., 79 + Beman, W. W., 74, 75 + Bernoulli, Jakob, 78-80 + Bernoulli, John, 79, 80 + Billingsley's Euclid, 15 + Billion, 20 + Billy, Jacobo de, 80 + Binomial formula, 25, 29 + Bliss, P., 60 + Boscovich, R. G., 78 + Bosmans, H., 72, 83 + Boyle, R., 1, 69 + Braunmhl, von, 39 + Brearly, W., 59 + Briggs, 6, 36, 55 + Brookes, Christopher, 7, 53, 59 + + Cajori, F., 27, 39, 40, 47 + Cantor, M., 40, 41 + Caraccioli, J. B., 78 + Cardan, 71 + Carr, 77 + Carrete, N. M., 78 + Caryll, C., 7 + Cataldi, 67 + Cavalieri, 65, 66 + Cavendish, Charles, 17, 62, 66 + Charles I, 9, 60 + Chelvicius, P., 79 + Cheyne, G., 82 + _Circles of Proportion_, 35, 37, 48, 49, 51, 59, 87, 88 + Clairaut, 77 + Clark, A., 3 + Clark, G., 63 + Clarke, F. L., 3 + _Clavis mathematicae_, 1, 5, 10, 14, 17-35, 45, 46, 51, 57-63, + 68-73, 81, 85, 87 + Clavius, 26, 80 + Clerc, le, 77 + Cobb, S., 76 + Cocker, 76, 82 + Collins, John, 15, 19, 63, 64, 67, 68, 76, 82 + Colson, J., 77 + Conchoid, 12 + Conic sections, 11, 53 + Cossali, P., 81 + Cotes, R., 1, 85 + Craig, J., 76, 82 + Cross, symbol of multiplication, 27, 38, 55, 56, 82, 83 + Cubic equations, 28, 34, 42, 45 + + Decimal fractions, notation of, 21 + Degree, centesimal division, 39 + Delamain, R., 4, 9, 10, 11, 47, 48, 51, 60, 84, 88, 89, 91, 93, 94 + De Moivre, 32, 79 + De Morgan, A., 5, 16, 37, 46, 47, 54 + Descartes, R., 1, 25, 47, 57, 68-72, 80 + Dibuadius, 79 + Difference, symbol for, 27, 81 + Diophantus, 63, 85 + Division, abbreviated, 21, 23, 24 + Dulaurens, F., 74, 82 + + Earl of Arundel, 10, 13, 15, 17 + Edleston, J., 95 + Enestrm, G., 40 + Equations, solution of, 18, 28, 29, 31, 34, 39-45, 87 + Errard de Bar-le-Duc, 80 + Eton College, 3, 4 + Euclid, 1, 15, 18, 20, 25, 27, 28, 79, 81, 83, 85 + Euler, L., 37, 39 + Ewart, 59 + Exponents, 25, 28, 29 + + Fagnano, de, 78 + Flower, 56 + Fontenelle, de, 77 + Forster, W., 35, 48, 59, 88 + Foster, S., 27, 67, 69, 73, 80, 89 + Frisi, P., 78 + + Gascoigne, 59, 61 + Gauss, C. F., 48 + Geysius, 67 + Ghetaldi, 70, 71 + Gibson, 67 + Girard, A., 32, 80 + Glaisher, J. W. L., 54-56, 64, 80 + Glorioso, 67, 68 + _Grammelogia_, 4, 47, 89 + Gravelaar, 83 + Greater than, symbol for, 81 + Greatrex, R., 15 + Gregory, D., 32 + Gregory, J., 27, 76 + Gresham College, 1, 6, 27, 59, 61, 80 + Guisne, 77 + Gunter, E., 37, 47, 86 + Gunter's scale, 37 + + Halcken, P., 78 + Hales, J., 7 + Halley, E., 1, 18, 69 + Hankel, H., 40 + Harper, T., 18 + Harriot, T., 45, 47, 57, 58, 69-71, 81 + Harris, J., 76 + Hartlib, 69 + Haughton, A., 35, 59 + Hawkins, J., 76, 82 + Hearn, 56 + Helmholtz, 48 + Henry, J., 48 + Henry van Etten, 52, 53 + Henshaw, T., 8, 58, 61 + Hobbes, 73, 86 + Hollar, 14 + Holsatus, 13 + Hooganhuysen, 64 + Hooke, Rb., 1 + Horner's method, 45 + Horology, 18, 50 + Horrox, J., 4 + Hospital, de l', 74, 77 + Howard, Th. _See_ Earl of Arundel. + Howard, W., 17, 18, 59 + Hutchinson, A., 6 + + Invisible college, 1 + + Joule, 48 + + Kepler, J., 6 + Kersey, J., 32, 73, 82 + Keylway, R., 65 + King, 67 + Kings College, Cambridge, 3, 35 + Krafft, G. W., 79 + + Lambert, J. H., 79 + Lamy, R. P. B., 74 + Laud, Archbishop, 65 + Leake, W., 53 + Le Clerc, 77 + Leech, W., 59 + Le Fvre, 77 + Leibniz, 47, 78, 80 + Leonelli, 56 + Le Paige, de, 83 + Less than, symbol for, 81 + Leurechon, 52 + Leybourn, 35, 64, 76 + Lichfield, Mrs., 19 + Lilly, W., 8, 9 + Locke, J., 67 + Logarithms, 6, 21, 27, 28, 38, 39, 42, 46, 54-56, 65, 92, 93; + natural, 55; + radix method of computing, 55, 56 + Lower, W., 58 + Ludolph Ceulen, 79 + + Manning, 56 + Manning, O., 7, 8, 13-15 + Mascheroni, L., 78 + Mayer, R., 47 + Melandri, D., 78 + Mercator, N., 13 + Mersenne, 63 + Milbourn, W., 45 + Million, 20 + Milnes, J., 82 + Moivre, de, 32, 79 + Moore, Jonas, 32, 54, 58, 73 + Moreland, S., 70 + Morse, R., 48 + Multiplication, abbreviated, 21, 22, 24; + symbol for, 27, 82, 83 + Mydorge, 54 + + Napier, J., 6, 7, 21, 27, 38, 39, 52, 54, 57, 59 + Napier's analogies, 39 + Newton, Sir Isaac, 1, 25, 29, 40, 41, 45, 47, 59, 65, 86, 92-95 + Nichols, J., 6, 14 + Nicolas, R. P. P., 74 + Nieuwentiit, B., 78 + Norwood, R., 37, 38, 80 + + _Opuscula mathematica hactenus inedita_, 16, 21, 75 + Orchard, 56 + _Oughtredus explicatus_, 64 + Ozanam, 74 + + {pi}, symbol for, 32 + Paige, C. de, 83 + Pardies, 76 + Parentheses, 26, 79, 80 + Partridge, S., 47 + Peano, 86 + Perfect number, 41 + Pitiscus, 15 + Planisphere, 53, 92, 93 + Prestet, J., 74 + Price, 11 + Proportion, notation for, 26, 27, 73-79 + Protheroe, 58 + Ptolemy, 83 + + Quadratic equation, 29, 31, 34 + + Radix method, 55, 56 + Rahn, 27 + Raphson, J., 40, 41, 76 + Ratio, notation of, 21, 73-80 + Rawlinson, R., 39, 82 + Regula falsa, 18 + Regular solids, 18 + Riccati, G., 78 + Riccati, V., 78 + Rigaud, 7, 12, 13, 19, 48, 61-66, 68 + Robillard, 77 + Robinson, W., 13, 48, 59, 62, 63 + Rooke, L., 59, 61 + + Saladini, H., 78 + Sanders, W., 76 + Sault, R., 82 + Scarborough, Charles, 16, 54, 58, 60 + Schooten, Van, 1 + Schreshensuchs, O., 83 + Scratch method, 23 + Shakespeare, 52 + Shelley, G., 76 + Shipley, A. E., 1 + Shuttleworth, 59 + Slide rule, 9, 46-49, 50, 60, 88, 93 + Smethwyck, 58 + Smith, J., 50 + Snellius, W., 79 + Solids, regular, 18 + Speidell, John, 38, 55 + Spherical triangles, 53, 54, 93 + Stokes, R., 35, 36, 58 + Sudell, 59 + Sun dials, 5, 9, 50, 51, 52, 60, 92 + + Tannery, P., 71 + Todhunter, 60 + Torporley, 58 + Triangles, spherical, 53, 54, 93 + _Trigonometria_, 21, 36, 55, 75 + Trigonometric functions, symbols for, 36, 37, 55, 56 + _Trigonometrie_, 21, 35, 39 + Trisection of angles, 28 + Twysden, 59, 68, 69, 73 + + Varignon, 77 + Vieta, 1, 2, 25, 32, 33, 35, 39-41, 45, 63, 67, 70, 71 + Vlack, 65 + Von Braunmhl, 39 + + Wadham College, 5, 53 + Wallis, John, 1, 19, 27, 33, 45, 57-59, 63, 64, 66-74, 79-81, 86 + Walmesley, D. C., 79 + Ward, Bishop, 13 + Ward, John, 76 + Ward, Seth, 55, 58, 60, 68, 73, 74, 81 + Watch-making, 18, 50 + Weber, W. E., 48 + Weddle, 56 + Wells, E., 76, 82 + Wharton, 60 + Whitlock, B., 8, 9 + Wilson, J., 77, 82 + Wing, V., 73, 75 + Wingate, E., 32, 47, 73 + Wolf, Christian, 79 + Wood, A., 60, 61 + Wood, R., 18, 59 + Wren, Christopher, 5, 58, 59, 76 + Wright, E., 6, 27, 38, 54 + Wright, S., 54 + + + + + Transcriber's Notes + + +A handful of typos, mostly misplaced punctuation, were silently +corrected. + +HTML and UTF text versions make heavy use of mathematical symbols: +particularly superscripts, subscripts, and combining characters. Some +viewers may require user assistance to find fonts containing these +characters. + +The text versions miss much of the formatting, especially in mathematical +formulas: + + +--Several arithmetic examples must be viewed in a monospaced font (which + recognizes combining characters) to be legible. + +--Formulas under the horizontal line of a square root symbol are + parenthesized. + +--Subscripts are preceded by "_". + +--Superscripts are preceded by "^". + +--Italics, used primarily in formulas and bibliographical entries, are + not indicated in the text. + +--Italics in the index are delimited by "_". + +--Underlines are not indicated (in particular, in the fractional part of + a decimal number in Oughtred's notation). + +--The idiosyncratic "greater than" and "less than" symbols are indicated + as {symbol} in the ASCII version. + +--Overdots, underdots, and slashmarks around digits (in the long division + example) are not indicated in the ASCII version. + +--Greek letters are spelled out within {curly brackets} in the ASCII + version. + + + + + + + +End of the Project Gutenberg EBook of William Oughtred, by Florian Cajori + +*** END OF THIS PROJECT GUTENBERG EBOOK WILLIAM OUGHTRED *** + +***** This file should be named 46815-8.txt or 46815-8.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/4/6/8/1/46815/ + +Produced by Brenda Lewis, Stephen Hutcheson, and the Online +Distributed Proofreading Canada Team at +http://www.pgdpcanada.net + +Updated editions will replace the previous one--the old editions will +be renamed. + +Creating the works from print editions not protected by U.S. copyright +law means that no one owns a United States copyright in these works, +so the Foundation (and you!) can copy and distribute it in the United +States without permission and without paying copyright +royalties. 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text-indent:-1em; margin-top:0; margin-bottom:0; text-align:left; } +p.t, div.t, .t { margin-left:5em; text-indent:-3em; margin-top:0; margin-bottom:0; text-align:left; } +p.lr, div.lr, .lr { display:block; margin-left:0em; margin-right:1em; margin-top:0; margin-bottom:0; text-align:right; } +.fnblock { margin-top:2em; } +.fndef { text-align:justify; margin-top:1.5em; margin-left:1.5em; text-indent:-1.5em; } +.fndef p.fncont, .fndef dl { margin-left:1.5em; text-indent:0em; } +dl.catalog dd { font-style:italic; } +dl.catalog dt { margin-top:1em; } +.clear { clear:both; } +</style> +</head> +<body> + + +<pre> + +The Project Gutenberg EBook of William Oughtred, by Florian Cajori + +This eBook is for the use of anyone anywhere in the United States and most +other parts of the world at no cost and with almost no restrictions +whatsoever. You may copy it, give it away or re-use it under the terms of +the Project Gutenberg License included with this eBook or online at +www.gutenberg.org. If you are not located in the United States, you'll have +to check the laws of the country where you are located before using this ebook. + +Title: William Oughtred + A great Seventeenth-Century Teacher of Mathematics + +Author: Florian Cajori + +Release Date: September 9, 2014 [EBook #46815] + +Language: English + +Character set encoding: UTF-8 + +*** START OF THIS PROJECT GUTENBERG EBOOK WILLIAM OUGHTRED *** + + + + +Produced by Brenda Lewis, Stephen Hutcheson, and the Online +Distributed Proofreading Canada Team at +http://www.pgdpcanada.net + + + + + + +</pre> + +<p class="center"><b><span class="large">WILLIAM OUGHTRED</span></b></p> +<div class="titlepg"> +<h1>WILLIAM OUGHTRED +<br /><span class="smaller">A GREAT SEVENTEENTH-CENTURY +<br />TEACHER OF</span> +<br /><span class="small">MATHEMATICS</span></h1> +<p class="center"><span class="small">BY</span> +<br />FLORIAN CAJORI, <span class="sc">Ph.D.</span> +<br /><span class="small"><i>Professor of Mathematics</i> +<br /><i>Colorado College</i></span></p> +<p class="center"><span class="smaller">CHICAGO LONDON</span> +<br /><span class="small">THE OPEN COURT PUBLISHING COMPANY +<br />1916</span></p> +<p class="center"><span class="smaller"><span class="sc">Copyright 1916 By</span> +<br /><span class="sc">The Open Court Publishing Co.</span></span></p> +<p class="center"><span class="smaller">All Rights Reserved</span></p> +<p class="center"><span class="smaller">Published September 1916</span></p> +<p class="tbcenter"><span class="smaller">Composed and Printed By +<br />The University of Chicago Press +<br />Chicago, Illinois, U.S.A.</span></p> +</div> +<div class="pb" id="Page_v">[v]</div> +<h2>TABLE OF CONTENTS</h2> +<dl class="toc"> +<dt class="jr"><span class="small">PAGE</span></dt> +<dt><a href="#c1"><span class="sc">Introduction</span></a> 1</dt> +<dt class="jl"><span class="small">CHAPTER</span></dt> +<dt><a href="#c2"><span class="chn">I.</span> <span class="sc">Oughtred’s Life</span></a> 3</dt> +<dd><a href="#c3">At School and University</a> 3</dd> +<dd><a href="#c4">As Rector and Amateur Mathematician</a> 6</dd> +<dd><a href="#c5">His Wife</a> 7</dd> +<dd><a href="#c6">In Danger of Sequestration</a> 8</dd> +<dd><a href="#c7">His Teaching</a> 9</dd> +<dd><a href="#c8">Appearance and Habits</a> 12</dd> +<dd><a href="#c9">Alleged Travel Abroad</a> 14</dd> +<dd><a href="#c10">His Death</a> 15</dd> +<dt><a href="#c11"><span class="chn">II.</span> <span class="sc">Principal Works</span></a> 17</dt> +<dd><a href="#c12"><i>Clavis mathematicae</i></a> 17</dd> +<dd><a href="#c13"><i>Circles of Proportion</i> and <i>Trigonometrie</i></a> 35</dd> +<dd><a href="#c14">Solution of Numerical Equations</a> 39</dd> +<dd><a href="#c15">Logarithms</a> 46</dd> +<dd><a href="#c16">Invention of the Slide Rule; Controversy on Priority of Invention</a> 46</dd> +<dt><a href="#c17"><span class="chn">III.</span> <span class="sc">Minor Works</span></a> 50</dt> +<dt><a href="#c18"><span class="chn">IV.</span> <span class="sc">Oughtred’s Influence upon Mathematical Progress and Teaching</span></a> 57</dt> +<dd><a href="#c19">Oughtred and Harriot</a> 57</dd> +<dd><a href="#c20">Oughtred’s Pupils</a> 58</dd> +<dd><a href="#c21">Oughtred, the “Todhunter of the Seventeenth Century”</a> 60</dd> +<dd><a href="#c22">Was Descartes Indebted to Oughtred?</a> 69</dd> +<dd><a href="#c23">The Spread of Oughtred’s Notations</a> 73</dd> +<dt><a href="#c24"><span class="chn">V.</span> <span class="sc">Oughtred’s Ideas on the Teaching of Mathematics</span></a> 84</dt> +<dd><a href="#c25">General Statement</a> 84</dd> +<dd><a href="#c26">Mathematics, “a Science of the Eye”</a> 85</dd> +<dd><a href="#c27">Rigorous Thinking and the Use of Instruments</a> 87</dd> +<dd><a href="#c28">Newton’s Comments on Oughtred</a> 94</dd> +<dt><a href="#c29"><span class="sc">Index</span></a> 97</dt> +</dl> +<div class="pb" id="Page_1">[1]</div> +<h2 id="c1">INTRODUCTION</h2> +<p>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.</p> +<p>Biographers of Sir Isaac Newton make particular mention +of five mathematical books which he read while a +young student at Cambridge, namely, Euclid’s <i>Elements</i>, +Descartes’s <i>Géométrie</i>, Vieta’s <i>Works</i>, Van Schooten’s +<i>Miscellanies</i>, and Oughtred’s <i>Clavis mathematicae</i>. The +last of these books has been receiving increasing attention +<span class="pb" id="Page_2">[2]</span> +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 <i>Clavis mathematicae</i>, +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.</p> +<p>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.</p> +<p class="jr1">F. C.</p> +<div class="pb" id="Page_3">[3]</div> +<h2 id="c2">CHAPTER I +<br /><span class="small">OUGHTRED’S LIFE</span></h2> +<h3 id="c3">AT SCHOOL AND UNIVERSITY</h3> +<p>William Oughtred, or, as he sometimes wrote his name, +<i>Owtred</i>, 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.”<a class="fn" id="fr_1" href="#fn_1">[1]</a> +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.</p> +<p>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:</p> +<blockquote> +<p>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 +<span class="pb" id="Page_4">[4]</span> +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.<a class="fn" id="fr_2" href="#fn_2">[2]</a></p> +</blockquote> +<p>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.</p> +<blockquote> +<p>“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 +<span class="pb" id="Page_5">[5]</span> +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.”<a class="fn" id="fr_3" href="#fn_3">[3]</a></p> +</blockquote> +<p>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.</p> +<p>At the age of twenty-three Oughtred invented his +<i>Easy Way of Delineating Sun-Dials by Geometry</i>, which, +though not published until about half a century later, +in the first English edition of Oughtred’s <i>Clavis mathematicae</i> +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.</p> +<div class="pb" id="Page_6">[6]</div> +<h3 id="c4">AS RECTOR AND AMATEUR MATHEMATICIAN</h3> +<p>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:</p> +<blockquote> +<p>Lord Napier, in 1614, published at Edinburgh his <i>Mirifici +logarithmorum canonis descriptio</i>. . . . . 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 <i>Treatise of Trigonometry</i> about the same +time, since it is evidently formed upon the plan of Lord Napier’s +<i>Canon</i>.<a class="fn" id="fr_4" href="#fn_4">[4]</a></p> +</blockquote> +<p>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 <i>Descriptio</i>. 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 <i>Constructio</i> +(1619) bound up with a copy of Kepler’s <i>Chilias +logarithmorum</i> (1624), that at the beginning of the <i>Constructio</i> +<span class="pb" id="Page_7">[7]</span> +is a blank leaf, and before this occurs the title-page +only of Napier’s <i>Descriptio</i> (1619), at the top of +which appears Oughtred’s autograph. The history of +this interesting signature is unknown.</p> +<h3 id="c5">HIS WIFE</h3> +<p>In 1606 he married Christ’sgift Caryll, daughter of +Caryll, Esq., of Tangley, in an adjoining parish.<a class="fn" id="fr_5" href="#fn_5">[5]</a> We +know very little about Oughtred’s family life. The +records at King’s College, Cambridge,<a class="fn" id="fr_6" href="#fn_6">[6]</a> 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,<a class="fn" id="fr_7" href="#fn_7">[7]</a> 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.”<a class="fn" id="fr_8" href="#fn_8">[8]</a></p> +<p>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 +<span class="pb" id="Page_8">[8]</span> +accuracy of the following item handed down by Aubrey; +it cannot be a true characterization:</p> +<blockquote> +<p>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.<a class="fn" id="fr_9" href="#fn_9">[9]</a></p> +</blockquote> +<h3 id="c6">IN DANGER OF SEQUESTRATION</h3> +<p>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:</p> +<blockquote> +<p>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.<a class="fn" id="fr_10" href="#fn_10">[10]</a></p> +</blockquote> +<p>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:</p> +<blockquote> +<p>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 <i>Bulstrode Whitlock</i> and others, who, +at the intercession of <i>William Lilye</i> the Astrologer, appeared +in great numbers on his behalf, he had a majority on his side, +and so escaped a sequestration.<a class="fn" id="fr_11" href="#fn_11">[11]</a></p> +</blockquote> +<p>Not without interest is the account of this matter given +by Lilly himself:</p> +<blockquote> +<p>About this Time, the most famous Mathematician of all +Europe, (Mr. William Oughtred, Parson of Aldbury in Surrey) +<span class="pb" id="Page_9">[9]</span> +was in Danger of Sequestration by the Committee of or for +plunder’d Ministers; (<i>Ambo-dexters</i> 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.<a class="fn" id="fr_12" href="#fn_12">[12]</a></p> +</blockquote> +<h3 id="c7">HIS TEACHING</h3> +<p>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 +<i>Apologeticall Epistle</i>, from which we quoted above. This +document contains biographical details, in part as follows:</p> +<blockquote> +<p>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>I have taken so much liberty +to the losse of time, and the neglect of my calling</i> 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 +<span class="pb" id="Page_10">[10]</span> +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.</p> +</blockquote> +<p>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 <i>Clavis mathematicae</i>, upon which his reputation as a +mathematician largely rests:</p> +<blockquote> +<p>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. . . . .</p> +<p>And although I am no <i>mercenary man</i>, 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 +<span class="pb" id="Page_11">[11]</span> +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.</p> +</blockquote> +<p>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.</p> +<p>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:</p> +<blockquote> +<p>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, +<span class="pb" id="Page_12">[12]</span> +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.<a class="fn" id="fr_13" href="#fn_13">[13]</a></p> +</blockquote> +<h3 id="c8">APPEARANCE AND HABITS</h3> +<p>Aubrey gives information about the appearance and +habits of Oughtred:</p> +<blockquote> +<p>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. . . . .</p> +<p>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 <i>quaesitum</i>.</p> +<p>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. . . . .</p> +<p>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. . . . .</p> +<div class="pb" id="Page_13">[13]</div> +<p>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.” . . . .</p> +<p>Nicolaus Mercator, Holsatus . . . . went to see him few +yeares before he dyed. . . . .</p> +<p>The right hon<sup>ble</sup> Thomas Howard, earle of Arundel and +Surrey, Lord High Marshall of England, was his great patron, +and loved him intirely. One time they were like to have been +killed together by the fall at Albury of a grott, which fell +downe but just as they were come out.<a class="fn" id="fr_14" href="#fn_14">[14]</a></p> +</blockquote> +<p>Oughtred’s friends convey the impression that, in the +main, Oughtred enjoyed a comfortable living at Albury. +Only once appear indications of financial embarrassment. +About 1634 one of his pupils, W. Robinson, writes as +follows:</p> +<blockquote> +<p>I protest unto you sincerely, were I as able as some, at whose +hands you have merited exceedingly, or (to speak more absolutely) +as able as willing, I would as freely give you 500 <i>l.</i> +per ann. as 500 pence; and I cannot but be astonished at this +our age, wherein pelf and dross is made their summum bonum, +and the best part of man, with the true ornaments thereof, +science and knowledge, are so slighted. . . . .<a class="fn" id="fr_15" href="#fn_15">[15]</a></p> +</blockquote> +<p>In his letters Oughtred complains several times of the +limitations for work and the infirmities due to his advancing +old age. The impression he made upon others was +quite different. Says one biographer:</p> +<blockquote> +<p>He sometimes amused himself with archery, and sometimes +practised as a surveyor of land. . . . . He was sprightly and +active, when more than eighty years of age.<a class="fn" id="fr_16" href="#fn_16">[16]</a></p> +</blockquote> +<div class="pb" id="Page_14">[14]</div> +<p>Another informant says that Oughtred was</p> +<blockquote> +<p>as facetious in Greek and Latine as solid in Arithmetique, +Astronomy, and the sphere of all Measures, Musick, etc.; exact +in his style as in his judgment; handling his Cube, and other +Instruments at eighty, as steadily, as others did at thirty; +owing this, he said, to temperance and Archery; principling +his people with plain and solid truths, as he did the world with +great and useful Arts; advancing new Inventions in all things +but Religion. Which in its old order and decency he maintained +secure in his privacy, prudence, meekness, simplicity, +resolution, patience, and contentment.<a class="fn" id="fr_17" href="#fn_17">[17]</a></p> +</blockquote> +<h3 id="c9">ALLEGED TRAVEL ABROAD</h3> +<p>According to certain sources of information, Oughtred +traveled on the European Continent and was invited to +change his abode to the Continent. We have seen no +statement from Oughtred himself on this matter. He +seldom referred to himself in his books and letters. The +autobiography contained in his <i>Apologeticall Epistle</i> +was written a quarter of a century before his death. +Aubrey gives the following:</p> +<blockquote> +<p>In the time of the civill warres the duke of Florence invited +him over, and offered him 500 li. per annum; but he would not +accept it, because of his religion.<a class="fn" id="fr_18" href="#fn_18">[18]</a></p> +</blockquote> +<p>A portrait of Oughtred, painted in 1646 by Hollar and +inserted in the English edition of the <i>Clavis</i> of 1647, contains +underneath the following lines:</p> +<div class="verse"> +<p class="t0">“Haec est Oughtredi senio labantis imago</p> +<p class="t0">Itala quam cupiit, Terra Britanna tulit.”</p> +</div> +<p>In the sketch of Oughtred by Owen Manning it is +confessed that “it is not known to what this alludes; but +<span class="pb" id="Page_15">[15]</span> +possibly he might have been in <i>Italy</i> with his patron, the +Earl of Arundel.”<a class="fn" id="fr_19" href="#fn_19">[19]</a> It would seem quite certain either +that Oughtred traveled in Europe or that he received +some sort of an offer to settle in Italy. In view of Aubrey’s +explicit statement and of Oughtred’s well-known habit of +confining himself to his duties and studies in his own +parish, seldom going even as far as London, we strongly +incline to the opinion that he did not travel on the Continent, +but that he received an offer from some patron +of the sciences—possibly some distinguished visitor—to +settle in Italy.</p> +<h3 id="c10">HIS DEATH</h3> +<p>He died at Albury, June 30, 1660, aged about eighty-six +years. Of his last days and death, Aubrey speaks as +follows:</p> +<blockquote> +<p>Before he dyed he burned a world of papers, and sayd that +the world was not worthy of them; he was so superb. He +burned also severall printed bookes, and would not stirre, till +they were consumed. . . . . I myselfe have his Pitiscus, +imbelished with his excellent marginall notes, which I esteeme +as a great rarity. I wish I could also have got his Bilingsley’s +Euclid, which John Collins sayes was full of his annotations. +. . . .</p> +<p>Ralph Greatrex, his great friend, the mathematicall +instrument-maker, sayed he conceived he dyed with joy for +the comeing-in of the king, which was the 29th of May before. +“And are yee sure he is restored?”—“Then give me a glasse +of sack to drinke his sacred majestie’s health.” His spirits +were then quite upon the wing to fly away. . . . .<a class="fn" id="fr_20" href="#fn_20">[20]</a></p> +</blockquote> +<p>In this passage, as in others, due allowance must be +made for Aubrey’s lack of discrimination. He was not +<span class="pb" id="Page_16">[16]</span> +in the habit of sifting facts from mere gossip. That +Oughtred should have declared that the world was not +worthy of his papers or manuscripts is not in consonance +with the sweetness of disposition ordinarily attributed to +him. More probable was the feeling that the papers he +burned—possibly old sermons—were of no particular +value to the world. That he did not destroy a large mass +of mathematical manuscripts is evident from the fact that +a considerable number of them came after his death into +the hands of Sir Charles Scarborough, M.D., under whose +supervision some of them were carefully revised and published +at Oxford in 1677 under the title of <i>Opuscula mathematica +hactenus inedita</i>.</p> +<p>Aubrey’s story of Oughtred’s mode of death has been +as widely circulated in every modern biographical sketch +as has his slander of Mrs. Oughtred by claiming that she +was so penurious that she would deny him the use of +candles to read by. Oughtred died on June 30; the Restoration +occurred on May 29. No doubt Oughtred +rejoiced over the Restoration, but the story of his drinking +“a glass of sack” to his Majesty’s health, and then dying +of joy is surely apocryphal. De Morgan humorously +remarks, “It should be added, by way of excuse, that he +was eighty-six years old.”<a class="fn" id="fr_21" href="#fn_21">[21]</a></p> +<div class="pb" id="Page_17">[17]</div> +<h2 id="c11">CHAPTER II +<br /><span class="small">PRINCIPAL WORKS</span></h2> +<h3 id="c12">“CLAVIS MATHEMATICAE”</h3> +<p>Passing to the consideration of Oughtred’s mathematical +books, we begin with the observation that he +showed a marked disinclination to give his writings to the +press. His first paper on sun-dials was written at the age +of twenty-three, but we are not aware that more than one +brief mathematical manuscript was printed before his +fifty-seventh year. In every instance, publication in +printed form seems to have been due to pressure exerted +by one or more of his patrons, pupils, or friends. Some +of his manuscripts were lent out to his pupils, who prepared +copies for their own use. In some instances they urged +upon him the desirability of publication and assisted in +preparing copy for the printer. The earliest and best-known +book of Oughtred was his <i>Clavis mathematicae</i>, +to which repeated allusion has already been made. As +he himself informs us, he was employed by the Earl of +Arundel about 1628 to instruct the Earl’s son, Lord +William Howard (afterward Viscount Stafford) in the +mathematics. For the use of this young man Oughtred +composed a treatise on algebra which was published in +Latin in the year 1631 at the urgent request of a kinsman +of the young man, Charles Cavendish, a patron of learning.</p> +<p>The <i>Clavis mathematicae</i>,<a class="fn" id="fr_22" href="#fn_22">[22]</a> +in its first edition of 1631, was +a booklet of only 88 small pages. Yet it contained in very +<span class="pb" id="Page_18">[18]</span> +condensed form the essentials of arithmetic and algebra as +known at that time.</p> +<p>Aside from the addition of four tracts, the 1631 edition +underwent some changes in the editions of 1647 and 1648, +which two are much alike. The twenty chapters of 1631 +are reduced to nineteen in 1647 and in all the later editions. +Numerous minute alterations from the 1631 edition occur +in all parts of the books of 1647 and 1648. The material +of the last three chapters of the 1631 edition is rearranged, +with some slight additions here and there. The 1648 +edition has no preface. In the print of 1652 there are only +slight alterations from the 1648 edition; after that the +<span class="pb" id="Page_19">[19]</span> +book underwent hardly any changes, except for the number +of tracts appended, and brief explanatory notes added +at the close of the chapters in the English editions of +1694 and 1702. The 1652 and 1667 editions were seen +through the press by John Wallis; the 1698 impression +contains on the title-page the words: <i>Ex Recognitione D. +Johannis Wallis, S.T.D. Geometriae Professoris Saviliani</i>.</p> +<p>The cost of publishing may be a matter of some interest. +When arranging for the printing of the 1667 edition +of the <i>Clavis</i>, Wallis wrote Collins: “I told you in my last +what price she [Mrs. Lichfield] expects for it, as I have +formerly understood from her, viz., £ 40 for the impression, +which is about 9½<i>d.</i> a book.”<a class="fn" id="fr_23" href="#fn_23">[23]</a></p> +<p>As compared with other contemporary works on algebra, +Oughtred’s distinguishes itself for the amount of +symbolism used, particularly in the treatment of geometric +problems. Extraordinary emphasis was placed +upon what he called in the <i>Clavis</i> the “analytical art.”<a class="fn" id="fr_24" href="#fn_24">[24]</a> +<span class="pb" id="Page_20">[20]</span> +By that term he did not mean our modern analysis or +analytical geometry, but the art “in which by taking the +thing sought as knowne, we finde out that we seeke.”<a class="fn" id="fr_25" href="#fn_25">[25]</a> +He meant to express by it condensed processes of rigid, +logical deduction expressed by appropriate symbols, as +contrasted with mere description or elucidation by passages +fraught with verbosity. In the preface to the first +edition (1631) he says:</p> +<blockquote> +<p>In this little book I make known . . . . the rules relating +to fundamentals, collected together, just like a bundle, and +adapted to the explanation of as many problems as possible.</p> +</blockquote> +<p>As stated in this preface, one of his reasons for publishing +the book, is</p> +<blockquote> +<p>. . . . that like Ariadne I might offer a thread to mathematical +study by which the mysteries of this science might be revealed, +and direction given to the best authors of antiquity, Euclid, +Archimedes, the great geometrician Apollonius of Perga, and +others, so as to be easily and thoroughly understood, their +theorems being added, not only because to many they are the +height and depth of mathematical science (I ignore the would-be +mathematicians who occupy themselves only with the so-called +practice, which is in reality mere juggler’s tricks with instruments, +the surface so to speak, pursued with a disregard of the +great art, a contemptible picture), but also to show with what +keenness they have penetrated, with what mass of equations, +comparisons, reductions, conversions and disquisitions these +heroes have ornamented, increased and invented this most +beautiful science.</p> +</blockquote> +<p>The <i>Clavis</i> opens with an explanation of the Hindu-Arabic +notation and of decimal fractions. Noteworthy is +the absence of the words “million,” “billion,” etc. Though +used on the Continent by certain mathematical writers +long before this, these words did not become current in +<span class="pb" id="Page_21">[21]</span> +English mathematical books until the eighteenth century. +The author was a great admirer of decimal fractions, but +failed to introduce the notation which in later centuries +came to be universally adopted. Oughtred wrote 0.56 +in this manner 0|<span class="u">56</span>; the point he used to designate ratio. +Thus 3:4 was written by him 3·4. The decimal point +(or comma) was first used by the inventor of logarithms, +John Napier, as early as 1616 and 1617. Although +Oughtred had mastered the theory of logarithms soon after +their publication in 1614 and was a great admirer of +Napier, he preferred to use the dot for the designation of +<i>ratio</i>. This notation of ratio is used in all his mathematical +books, except in two instances. The two dots (:) +occur as symbols of ratio in some parts of Oughtred’s +posthumous work, <i>Opuscula mathematica hactenus inedita</i>, +Oxford, 1677, but may have been due to the editors and +not to Oughtred himself. Then again the two dots (:) +are used to designate ratio on the last two pages of the +tables of the Latin edition of Oughtred’s <i>Trigonometria</i> +of 1657. In all other parts of that book the dot (·) is +used. Probably someone who supervised the printing +of the tables introduced the (:) on the last two pages, +following the logarithmic tables, where methods of interpolation +are explained. The probability of this conjecture +is the stronger, because in the English edition of the +<i>Trigonometrie</i>, brought out the same year (1657) but <i>after</i> +the Latin edition, the notation (:) at the end of the book +is replaced by the usual (·), except that in some copies +of the English edition the explanations at the end are +omitted altogether.</p> +<p>Oughtred introduces an interesting, and at the same +time new, feature of an abbreviated multiplication and an +abbreviated division of decimal fractions. On this point +<span class="pb" id="Page_22">[22]</span> +he took a position far in advance of his time. The part +on abbreviated multiplication was rewritten in slightly +enlarged form and with some unimportant alterations +in the later edition of the <i>Clavis</i>. We give it as it occurs +in the revision. Four cases are given. In finding the +product of 246|<span class="u">914</span> and 35|<span class="u">27</span>, “if you would +have the Product without any Parts” +(without any decimal part), “set the place +of Unity of the lesser under the place of +Unity in the greater: as in the Example,” +writing the figures of the lesser number in +<i>inverse order</i>. From the example it will be +seen that he begins by multiplying by 3, the +right-hand digit of the multiplier. In the +first edition of the <i>Clavis</i> he began with 7, +the left digit. Observe also that he “carries” the nearest +tens in the product of each lower digit and the upper digit +one place to its right. For instance, he takes 7×4=28 +and carries 3, then he finds 7×2+3=17 and writes down 17.</p> +<pre> + 2 4 6|<span class="u">9 1 4</span> + <span class="u">7 2</span>|5 3 + ------- + 7 4 0 7 + 1 2 3 5 + 4 9 + 1 7 + ------- + 8 7 0 8 +</pre> +<p>The second case supposes that “you would have the +Product with some places of parts” (decimals), say 4: +“Set the place of Unity of the lesser Number under the +Fourth place of the Parts of the greater.” The multiplication +of 246|<span class="u">914</span> by 35|<span class="u">27</span> is now performed thus:</p> +<pre> + 2 4 6|<span class="u">9 1 4</span> + <span class="u">7 2</span>|5 3 + --------------- + 7 4 0 7 4 2 0 0 + 1 2 3 4 5 7 0 0 + 4 9 3 8 2 8 + 1 7 2 8 4 0 + --------------- + 8 7 0 8|<span class="u">6 5 6</span> 8 +</pre> +<div class="pb" id="Page_23">[23]</div> +<p>In the third and fourth cases are considered factors +which appear as integers, but are in reality decimals; +for instance, the sine of 54° is given in the tables as 80902 +when in reality it is .80902.</p> +<p>Of interest as regards the use of the word “parabola” +is the following: “The Number found by Division is +called the <i>Quotient</i>, or also <i>Parabola</i>, because it arises out +of the Application of a plain Number to a given Longitude, +that a congruous Latitude may be found.”<a class="fn" id="fr_26" href="#fn_26">[26]</a> This is in +harmony with etymological dictionaries which speak of a +parabola as the application of a given area to a given +straight line. The dividend or product is the area; the +divisor or factor is the line.</p> +<p>Oughtred gives two processes of long division. The +first is identical with the modern process, except that the +divisor is written below every remainder, each digit of +the divisor being crossed out as soon as it has been used +in the partial multiplication. The second method of +long division is one of the several types of the old “scratch +method.” This antiquated process held its place by the +side of the modern method in all editions of the <i>Clavis</i>. +The author divides 467023 by 357|<span class="u">0926425</span>, giving the +following instructions: “Take as many of the first Figures +of the Divisor as are necessary, for the first Divisor, and +then in every following particular Division drop one of +the Figures of the Divisor towards the Left Hand, till +you have got a competent Quotient.” He does not explain +abbreviated division as thoroughly as abbreviated multiplication.</p> +<div class="pb" id="Page_24">[24]</div> +<pre> + 17 + 3<span class="xo"≯</span>0<span class="xo"≯</span>3<span class="xo"≯</span> + 2<span class="xo"≯</span>8<span class="xo"≯</span>0<span class="xo"≯</span>3<span class="xo"≯</span> + 1<span class="xo"≯</span>0<span class="xo"≯</span>9<span class="xo"≯</span>9<span class="xo"≯</span>3<span class="xo"≯</span>0<span class="xo"≯</span> + 3̣5̣7̣|<span class="u">0̣9̣2̣64</span>25) 4<span class="xo"≯</span>6<span class="xo"≯</span>7<span class="xo"≯</span>0<span class="xo"≯</span>2<span class="xo"≯</span>3<span class="xo"≯</span> (1307|<span class="u">80</span> + 3<span class="xo"≯</span>5<span class="xo"≯</span>7<span class="xo"≯</span>0<span class="xo"≯</span>9<span class="xo"≯</span>3<span class="xo"≯</span> + 1<span class="xo"≯</span>0<span class="xo"≯</span>7<span class="xo"≯</span>1<span class="xo"≯</span>2<span class="xo"≯</span>7<span class="xo"≯</span> + 2<span class="xo"≯</span>5<span class="xo"≯</span>0<span class="xo"≯</span>0<span class="xo"≯</span> + 2<span class="xo"≯</span>8<span class="xo"≯</span>6<span class="xo"≯</span> +</pre> +<p class="tb">Oughtred does not examine the degree of reliability +or accuracy of his processes of abbreviated multiplication +and division. Here as in other places he gives in condensed +statement the mode of procedure, without further +discussion.</p> +<p>He does not attempt to establish the rules for the addition, +subtraction, multiplication, and division of positive +and negative numbers. “If the Signs are both alike, the +Product will be affirmative, if unlike, negative”; then +he proceeds to applications. This attitude is superior to +that of many writers of the eighteenth and nineteenth +centuries, on pedagogical as well as logical grounds: +pedagogically, because the beginner in the study of algebra +is not in a position to appreciate an abstract train of +thought, as every teacher well knows, and derives better +intellectual exercise from the applications of the rules to +problems; logically, because the rule of signs in multiplication +does not admit of rigorous proof, unless some +other assumption is first made which is no less arbitrary +than the rule itself. It is well known that the proofs +of the rule of signs given by eighteenth-century writers +are invalid. Somewhere they involve some surreptitious +assumption. This criticism applies even to the proof +given by Laplace, which tacitly assumes the distributive +law in multiplication.</p> +<div class="pb" id="Page_25">[25]</div> +<p>A word should be said on Oughtred’s definition of + +and -. He recognizes their double function in algebra by +saying (<i>Clavis</i>, 1631, p. 2): “Signum additionis, sive +affirmationis, est + plus” and “Signum subductionis, +sive negationis est - minus.” They are symbols which +indicate the <i>quality</i> of numbers in some instances and +<i>operations</i> of addition or subtraction in other instances. +In the 1694 edition of the <i>Clavis</i>, thirty-four years after +the death of Oughtred, these symbols are defined as signifying +operations only, but are actually used to signify the +quality of numbers as well. In this respect the 1694 +edition marks a recrudescence.</p> +<p>The characteristic in the <i>Clavis</i> that is most striking +to a modern reader is the total absence of indexes or exponents. +There is much discussion in the leading treatises +of the latter part of the sixteenth and the early part of the +seventeenth century on the theory of indexes, but +the modern exponential notation, <i>aⁿ</i>, is of later date. +The modern notation, for positive integral exponents, first +appears in Descartes’ <i>Géométrie</i>, 1637; fractional and +negative exponents were first used in the modern form +by Sir Isaac Newton, in his announcement of the binomial +formula, in a letter written in 1676. This total absence +of our modern exponential notation in Oughtred’s <i>Clavis</i> +gives it a strange aspect. Like Vieta, Oughtred uses ordinarily +the capital letters, <i>A</i>, <i>B</i>, <i>C</i>, . . . . to designate +given numbers; <i>A</i>² is written <i>Aq</i>, <i>A</i>³ is written <i>Ac</i>; for +<i>A</i>⁴, <i>A</i>⁵, <i>A</i>⁶ he has, respectively, <i>Aqq</i>, <i>Aqc</i>, <i>Acc</i>. Only on +rare occasions, usually when some parallelism in notation +is aimed at, does he use small letters<a class="fn" id="fr_27" href="#fn_27">[27]</a> to represent numbers +or magnitudes. Powers of binomials or polynomials +<span class="pb" id="Page_26">[26]</span> +are marked by prefixing the capital letters <i>Q</i> (for square), +<i>C</i> (for cube), <i>QQ</i> (for the fourth power), <i>QC</i> (for the fifth +power), etc.</p> +<p>Oughtred does not express aggregation by (). Parentheses +had been used by Girard, and by Clavius as +early as 1609,<a class="fn" id="fr_28" href="#fn_28">[28]</a> but did not come into general use in +mathematical language until the time of Leibniz and the +Bernoullis. Oughtred indicates aggregation by writing +a colon (:) at both ends. Thus, <i>Q</i>:<i>A</i>-<i>E</i>: means with +him (<i>A</i>-<i>E</i>)². Similarly, √<i>q</i>:<i>A</i>+<i>E</i>: means √(<i>A</i>+<i>E</i>). +The two dots at the end are frequently omitted when the +part affected includes all the terms of the polynomial to +the end. Thus, <i>C</i>:<i>A</i>+<i>B</i>-<i>E</i>=.. means (<i>A</i>+<i>B</i>-<i>E</i>)³=.. +There are still further departures from this notation, but +they occur so seldom that we incline to the interpretation +that they are simply printer’s errors. For proportion +Oughtred uses the symbol (::). The proportion <i>a</i>:<i>b</i>=<i>c</i>:<i>d</i> +appears in his notation <i>a</i>·<i>b</i>::<i>c</i>·<i>d</i>. Apparently, a +proportion was not fully recognized in this day as being +the expression of an equality of ratios. That probably +explains why he did not use = here as in the notation of +ordinary equations. Yet Oughtred must have been very +close to the interpretation of a proportion as an equality; +for he says in his <i>Elementi decimi Euclidis declaratio</i>, +“proportio, sive ratio aequalis ::” That he introduced +this extra symbol when the one for equality was sufficient +is a misfortune. Simplicity demands that no unnecessary +symbols be introduced. However, Oughtred’s symbolism +is certainly superior to those which preceded. Consider +the notation of Clavius.<a class="fn" id="fr_29" href="#fn_29">[29]</a> He wrote 20:60=4:<i>x</i>, <i>x</i>=12, +<span class="pb" id="Page_27">[27]</span> +thus: “20·60·4? <i>fiunt</i> 12.” The insufficiency of such a +notation in the more involved expressions frequently +arising in algebra is readily seen. Hence Oughtred’s +notation (::) was early adopted by English mathematicians. +It was used by John Wallis at Oxford, by Samuel +Foster at Gresham College, by James Gregory of Edinburgh, +by the translators into English of Rahn’s algebra, +and by many other early writers. Oughtred has been +credited generally with the introduction of St. Andrew’s +cross × as the symbol for multiplication in the <i>Clavis</i> of +1631. We have discovered that this symbol, or rather +the letter <i>x</i> which closely resembles it, occurs as the sign +of multiplication thirteen years earlier in an anonymous +“Appendix to the Logarithmes, shewing the practise of +the Calculation of Triangles etc.” to Edward Wright’s +translation of John Napier’s <i>Descriptio</i>, published in 1618.<a class="fn" id="fr_30" href="#fn_30">[30]</a> +Later we shall give our reasons for believing that Oughtred +is the author of that “Appendix.” The × has survived +as a symbol of multiplication.</p> +<p>Another symbol introduced by Oughtred and found in +modern books is ~, expressing difference; thus <i>C</i>~<i>D</i> +signifies the difference between <i>C</i> and <i>D</i>, even when <i>D</i> is +the larger number.<a class="fn" id="fr_31" href="#fn_31">[31]</a> This symbol was used by John +Wallis in 1657.<a class="fn" id="fr_32" href="#fn_32">[32]</a></p> +<p>Oughtred represented in symbols also certain composite +expressions, as for instance <i>A</i>+<i>E</i>=<i>Z</i>, <i>A</i>-<i>E</i>=<i>X</i>, +where <i>A</i> is greater than <i>E</i>. He represented by a symbol +also each of the following: <i>A</i>²+<i>E</i>², <i>A</i>³+<i>E</i>³, <i>A</i>²-<i>E</i>², +<i>A</i>³-<i>E</i>³.</p> +<div class="pb" id="Page_28">[28]</div> +<p>Oughtred practically translated the tenth book of +Euclid from its ponderous rhetorical form into that of +brief symbolism. An appeal to the eye was a passion with +Oughtred. The present writer has collected the different +mathematical symbols used by Oughtred and has found +more than one hundred and fifty of them.</p> +<p>The differences between the seven different editions of +the <i>Clavis</i> lie mainly in the special parts appended to some +editions and dropped in the latest editions. The part +which originally constituted the <i>Clavis</i> was not materially +altered, except in two or three of the original twenty +chapters. These changes were made in the editions of +1647 and 1648. After the first edition, great stress was +laid upon the theory of indices upon the very first page, +as also in passages farther on. Of course, Oughtred did +not have our modern notation of indices or exponents, +but their theory had been a part of algebra and arithmetic +for some time. Oughtred incorporated this theory in his +brief exposition of the Hindu-Arabic notation and in his +explanation of logarithms. As previously pointed out, +the last three chapters of the 1631 edition were considerably +rearranged in the later editions and combined into +two chapters, so that the <i>Clavis</i> proper had nineteen +chapters instead of twenty in the additions after the first. +These chapters consisted of applications of algebra to +geometry and were so framed as to constitute a severe +test of the student’s grip of the subject. The very last +problem deals with the division of angles into equal parts. +He derives the cubic equation upon which the trisection +depends algebraically, also the equations of the fifth degree +and seventh degree upon which the divisions of the angle +into 5 and 7 equal parts depend, respectively. The +exposition was severely brief, yet accurate. He did not +<span class="pb" id="Page_29">[29]</span> +believe in conducting the reader along level paths or along +slight inclines. He was a guide for mountain-climbers, +and woe unto him who lacked nerve.</p> +<p>Oughtred lays great stress upon expansions of powers of +a binomial. He makes use of these expansions in the +solution of numerical equations. To one who does not +specialize in the history of mathematics such expansions +may create surprise, for did not Newton invent the +binomial theorem after the death of Oughtred? As a +matter of fact, the expansions of positive integral powers +of a binomial were known long before Newton, not only +to seventeenth-century but even to eleventh-century +mathematicians. Oughtred’s <i>Clavis</i> of 1631 gave the +binomial coefficients for all powers up to and including +the tenth. What Newton really accomplished was the +generalization of the binomial expansion which makes it +applicable to negative and fractional exponents and converts +it into an infinite series.</p> +<p>As a specimen of Oughtred’s style of writing we quote +his solution of quadratic equations, accompanied by a +translation into English and into modern mathematical +symbols.</p> +<p>As a preliminary step<a class="fn" id="fr_33" href="#fn_33">[33]</a> he lets</p> +<blockquote> +<p><i>Z</i>=<i>A</i>+<i>E</i> and <i>A</i>><i>E</i>;</p> +</blockquote> +<p>he lets also <i>X</i>=<i>A</i>-<i>E</i>. From these relations he obtains +identities which, in modern notation, are +¼<i>Z</i>²-<i>AE</i>=(½<i>Z</i>-<i>E</i>)²=¼<i>X</i>². +Now, if we know <i>Z</i> and <i>AE</i>, we +can find ½<i>X</i>. Then +½(<i>Z</i>+<i>X</i>)=<i>A</i>, and +½(<i>Z</i>-<i>X</i>)=<i>E</i>, +and</p> +<div class="verse"> +<p class="t0"><i>A</i>=½<i>Z</i>+√<span class="over">¼<i>Z</i>²-<i>AE</i></span>.</p> +</div> +<div class="pb" id="Page_30">[30]</div> +<p>Having established these preliminaries, he proceeds thus:</p> +<blockquote> +<div class="p">Datis igitur linea inaequaliter secta <i>Z</i> (10), & +rectangulo sub segmentis <i>AE</i> (21) qui gnomon est: datur +semidifferentia segmentorum ½<i>X</i>: & per consequens +ipsa segmenta. Nam ponatur alterutrum segmentum <i>A</i>: alterum +erit <i>Z</i>-<i>A</i>: Rectangulum auctem est +<i>ZA</i>-<i>A<sub>q</sub></i>=<i>AE</i>. +Et quia dantur <i>Z</i> & +<i>AE</i>: estque +¼<i>Z<sub>q</sub></i>-<i>AE</i>=¼<i>X<sub>q</sub></i>: +& per 5c. 18, ½<i>Z</i>+½X=<i>A</i>: & +½<i>Z</i>-½<i>X</i>=<i>E</i>: Aequatio sic resoluetur: +½<i>Z</i>±√<i><sub>q</sub></i>:¼<i>Z<sub>q</sub></i>-<i>AE</i>:=<i>A</i> +<table class="inline"><tr><td rowspan="2"><span class="xlarge">{</span></td><td>maius segment<br />minus segment.</td></tr></table></div> +<p>Itaque proposita equatione, in qua sunt tres species aequaliter +in ordine tabellae adscendentes, altissima autem species +ponitur negata: Magnitudo data coefficiens mediam speciem +est linea bisecanda: & magnitudo absoluta data, ad quam sit +aequatio, est rectangulum sub segmentis inaequalibus, sine +gnomon: vt <i>ZA-A<sub>q</sub></i>=<i>AE</i>: in numeris autem 10<i>l</i>-<i>l<sub>q</sub></i>=21: +Estque <i>A</i>, vel 1<i>l</i>, alterutrum segmentum inaequale. Inuenitur +autem sic:</p> +<div class="p">Dimidiata coefficiens median speciem est +<table class="inline"><tr><td><span class="u"><i>Z</i></span><br />2</td></tr></table> +(5); cuius quadratum est +<table class="inline"><tr><td><span class="u"><i>Z<sub>q</sub></i></span><br />4</td></tr></table> +(25): ex hoc tolle <i>AE</i> (21) absolutum: eritque +<table class="inline"><tr><td><span class="u"><i>Z<sub>q</sub></i></span><br />4</td><td rowspan="2">-<i>AE</i></td></tr></table> +(4) quadratum semidifferentiae segmentorum: latus huius quadratum +(2) est semidifferentia: quam si addas ad +<table class="inline"><tr><td><span class="u"><i>Z</i></span><br />2</td></tr></table> +(5) semissem coefficientis, sive lineae bisecandae, erit maius +segment.; sin detrahas, erit minus segment: Dico +<table class="inline"><tr><td><span class="u"><i>Z</i></span><br />2</td><td rowspan="2">±√<i><sub>q</sub></i>:</td><td><span class="u"><i>Z<sub>q</sub></i></span><br />4</td><td rowspan="2">-<i>AE</i>:=<i>A</i></td></tr></table> +<table class="inline"><tr><td rowspan="2"><span class="xlarge">{</span></td><td>maius segmentum<br />minus segmentum.</td></tr></table></div> +</blockquote> +<p>We translate the Latin passage, using the modern +exponential notation and parentheses, as follows:</p> +<blockquote> +<div class="p">Given therefore an unequally divided line <i>Z</i> (10), and a +rectangle beneath the segments <i>AE</i> (21) which is a gnomon. +<div class="pb" id="Page_31">[31]</div> +Half the difference of the segments ½<i>X</i> is given, and consequently +the segment itself. For, if one of the two segments is +placed equal to <i>A</i>, the other will be <i>Z</i>-<i>A</i>. Moreover, the +rectangle is <i>ZA</i>-<i>A</i>²=<i>AE</i>. And because <i>Z</i> and <i>AE</i> are given, +and there is ¼<i>Z</i>²-<i>AE</i>=¼<i>X</i>², and by 5<i>c</i>.18, ½<i>Z</i>+½<i>X</i>=<i>A</i>, and +½<i>Z</i>-½<i>X</i>=<i>E</i>, +the equation will be solved thus: +½<i>Z</i>±√(¼<i>Z</i>²-<i>AE</i>)=<i>A</i> +<table class="inline"><tr><td rowspan="2"><span class="xlarge">{</span></td><td>major segment<br />minor segment.</td></tr></table></div> +<p>And so an equation having been proposed in which three +species (terms) are in equally ascending powers, the highest +species, moreover, being negative, the given magnitude which +constitutes the middle species is the line to be bisected. And +the given absolute magnitude to which it is equal is the rectangle +beneath the unequal segments, without gnomon. As +<i>ZA</i>-<i>A</i>²=<i>AE</i>, or in numbers, 10<i>x</i>-<i>x</i>²=21. And <i>A</i> or <i>x</i> is +one of the two unequal segments. It may be found thus:</p> +<div class="p">The half of the middle species is +<table class="inline"><tr><td><span class="u"><i>Z</i></span><br />2</td></tr></table> +(5), its square is +<table class="inline"><tr><td><span class="u">Z²</span><br />4</td></tr></table> +(25). +From it subtract the absolute term <i>AE</i> (21), and +<table class="inline"><tr><td><span class="u"><i>Z</i>²</span><br />4</td><td rowspan="2">-<i>AE</i></td></tr></table> +(4) will be the square of half the difference of the segments. +The square root of this, +<table class="inline"><tr><td>√</td><td><span class="xxlarge">[(</span></td><td><span class="u"><i>Z</i>²</span><br />2</td><td><span class="xxlarge">)</span></td><td>²<br class="nil" /> </td><td>-<i>AE</i></td><td><span class="xxlarge">]</span></td></tr></table> +(2), is half the difference. If you add it to half the coefficient +<table class="inline"><tr><td><span class="u"><i>Z</i></span><br />2</td></tr></table> +(5), or half the line +to be bisected, the longer segment is obtained; if you subtract +it, the smaller segment is obtained. I say:</div> +<div> +<table class="inline"><tr><td><span class="u"><i>Z</i></span><br />2</td><td>±√</td><td><span class="xxlarge">(</span></td><td><span class="u"><i>Z</i>²</span><br />4</td><td>-<i>AE</i></td><td><span class="xxlarge">)</span></td><td>=<i>A</i></td> +<td><span class="xxlarge">{</span></td><td>major segment<br />minor segment.</td></tr></table></div> +</blockquote> +<p>The quadratic equation <i>Aq</i>+<i>ZA</i>=<i>AE</i> receives similar +treatment. This and the preceding equation, +<i>ZA</i>-<i>Aq</i>=<i>AE</i>, +constitute together a solution of the general quadratic equation, +<i>x</i>²+<i>ax</i>=<i>b</i>, provided that <i>E</i> or +<i>Z</i> are not restricted to positive values, but admit of being either +<span class="pb" id="Page_32">[32]</span> +positive or negative, a case not adequately treated by +Oughtred. Imaginary numbers and imaginary roots receive +no consideration whatever.</p> +<p>A notation suggested by Vieta and favored by Girard +made vowels stand for unknowns and consonants for +knowns. This conventionality was adopted by Oughtred +in parts of his algebra, but not throughout. Near the +beginning he used <i>Q</i> to designate the unknown, though +usually this letter stood with him for the “square” of +the expression after it.<a class="fn" id="fr_34" href="#fn_34">[34]</a></p> +<div class="p">It is of some interest that Oughtred used +<table class="inline"><tr><td><span class="greek" title="{pi/delta}"><span class="u">π</span><br />δ</span></td></tr></table> +to signify the ratio of the circumference to the diameter of a circle. +Very probably this notation is the forerunner of the <span class="greek" title="{pi}">π</span>=3.14159 +. . . . used in 1706 by William Jones. Oughtred first used +<table class="inline"><tr><td><span class="greek" title="{pi/delta}"><span class="u">π</span><br />δ</span></td></tr></table> +in the 1647 edition of the <i>Clavis mathematicae</i>. +In the 1652 edition he says, “Si in circulo sit +7.22::<span class="greek" title="{delta·pi}">δ·π</span>::113.355:erit +<span class="greek" title="{delta·pi}">δ·π</span>::2 <i>R</i>.<i>P</i>: periph.” This +notation was adopted by Isaac Barrow, who used it extensively. +David Gregory<a class="fn" id="fr_35" href="#fn_35">[35]</a> used +<table class="inline"><tr><td><span class="greek" title="{pi/rho}"><span class="u">π</span><br />ρ</span></td></tr></table> +in 1697, and De Moivre<a class="fn" id="fr_36" href="#fn_36">[36]</a> used +<table class="inline"><tr><td><i><span class="u">c</span><br />r</i></td></tr></table> +about 1697, to designate the ratio of the circumference to the radius.</div> +<div class="pb" id="Page_33">[33]</div> +<p>We quote the description of the <i>Clavis</i> that was given +by Oughtred’s greatest pupil, John Wallis. It contains +additional information of interest to us. Wallis devotes +chap. xv of his <i>Treatise of Algebra</i>, London, 1685, pp. 67-69, +to Mr. Oughtred and his <i>Clavis</i>, saying:</p> +<blockquote> +<p>Mr. William Oughtred (our Country-man) in his <i>Clavis +Mathematicae</i>, (or Key of Mathematicks,) first published in the +Year 1631, follows Vieta (as he did Diophantus) in the use of +the Cossick Denominations; omitting (as he had done) the +names of <i>Sursolids</i>, and contenting himself with those of <i>Square</i> +and <i>Cube</i>, and the Compounds of these.</p> +<p>But he doth abridge Vieta’s Characters or Species, using +only the letters q, c, &c. which in Vieta are expressed (at length) +by <i>Quadrate</i>, <i>Cube</i>, &c. For though when Vieta first introduced +this way of Specious Arithmetick, it was more necessary +(the thing being new,) to express it in words at length: Yet +when the thing was once received in practise, Mr. Oughtred +(who affected brevity, and to deliver what he taught as briefly +as might be, and reduce all to a short view,) contented himself +with single Letters instead of those words.</p> +<p>Thus what Vieta would have written</p> +<div class="center"> +<table class="inline"> +<tr><td class="center"><span class="u"><i>A Quadrate</i>, into <i>B Cube</i>,</span><br /><i>CDE Solid</i>,</td></tr></table> +<i>Equal to FG Plane</i>, +</div> +<p>would with him be thus expressed</p> +<div class="center"> +<table class="inline"> +<tr><td><span class="u"><i>A<sub>q</sub> B<sub>c</sub></i></span><br /><i>C D E</i></td></tr></table>=<i>FG</i>. +</div> +<p>And the better to distinguish upon the first view, what +quantities were Known, and what Unknown, he doth (usually) +denote the Known to <i>Consonants</i>, and the Unknown by +<i>Vowels</i>; as Vieta (for the same reason) had done before him.</p> +<p>He doth also (to very great advantage) make use of several +Ligatures, or Compendious Notes, to signify <i>Summs</i>, <i>Differences</i>, +and <i>Rectangles</i> of several Quantities. As for instance, +<span class="pb" id="Page_34">[34]</span> +Of two Quantities A (the Greater), and E (the Lesser), the Sum +he calls Z, the Difference X, the Rectangle AE. . . . .</p> +<p>Which being of (almost) a constant signification with him +throughout, do save a great circumlocution of words, (each +Letter serving instead of a Definition;) and are also made use +of (with very great advantage) to discover the true nature of +divers intricate Operations, arising from the various compositions +of such Parts, Sums, Differences, and Rectangles; (of +which there is great plenty in his <i>Clavis</i>, Cap. 11, 16, 18, 19. +and elsewhere,) which without such Ligatures, or Compendious +Notes, would not be easily discovered or apprehended. . . . .</p> +<p>I know there are who find fault with his <i>Clavis</i>, as too obscure, +because so short, but without cause; for his words be +always full, but not Redundant, and need only a little attention +in the Reader to weigh the force of every word, and the +Syntax of it; . . . . And this, when once apprehended, is +much more easily retained, than if it were expressed with the prolixity +of some other Writers; where a Reader must first be at +the pains to weed out a great deal of superfluous Language, +that he may have a short prospect of what is material; which +is here contracted for him in a short Synopsis. . . . .</p> +<p>Mr. Oughtred in his <i>Clavis</i>, contents himself (for the most +part) with the solution of Quadratick Equations, without proceeding +(or very sparingly) to Cubick Equations, and those of +Higher Powers; having designed that Work for an <i>Introduction</i> +into <i>Algebra</i> so far, leaving the Discussion of Superior Equations +for another work. . . . . He contents himself likewise in +Resolving Equations, to take notice of the <i>Affirmative</i> or <i>Positive +Roots</i>; omitting the <i>Negative</i> or <i>Ablative Roots</i>, and such as +are called <i>Imaginary</i> or <i>Impossible Roots</i>. And of those which, +he calls <i>Ambiguous</i> Equations, (as having more Affirmative +Roots than one,) he doth not (that I remember) any where take +notice of more than <i>Two</i> Affirmative Roots: (Because in +Quadratick Equations, which are those he handleth, there are +indeed no more.) Whereas yet in <i>Cubick</i> Equations, there may +be <i>Three</i>, and in those of Higher Powers, yet more. Which +<span class="pb" id="Page_35">[35]</span> +Vieta was well aware of, and mentioneth in some of his Writings; +and of which Mr. Oughtred could not be ignorant.</p> +</blockquote> +<h3 id="c13">“CIRCLES OF PROPORTION” AND “TRIGONOMETRIE”</h3> +<p>Oughtred wrote and had published three important mathematical books, +the <i>Clavis</i>, the <i>Circles of Proportion</i>,<a class="fn" id="fr_37" href="#fn_37">[37]</a> +and a <i>Trigonometrie</i>.<a class="fn" id="fr_38" href="#fn_38">[38]</a> +This last appeared in the year 1657 at London, in both Latin and +English.</p> +<p>It is claimed that the trigonometry was “neither +finished nor published by himself, but collected out of +his scattered papers; and though he connived at the +printing it, yet imperfectly done, as appears by his MSS.; +and one of the printed Books, corrected by his own +<span class="pb" id="Page_36">[36]</span> +Hand.”<a class="fn" id="fr_39" href="#fn_39">[39]</a> Doubtless more accurate on this point is a letter +of Richard Stokes who saw the book through the press:</p> +<blockquote> +<p>I have procured your Trigonometry to be written over in a +fair hand, which when finished I will send to you, to know if it +be according to your mind; for I intend (since you were pleased +to give your assent) to endeavour to print it with Mr. Briggs +his Tables, and so soon as I can get the Prutenic Tables I will +turn those of the sun and moon, and send them to you.<a class="fn" id="fr_40" href="#fn_40">[40]</a></p> +</blockquote> +<p>In the preface to the Latin edition Stokes writes:</p> +<blockquote> +<p>Since this trigonometry was written for private use without +the intention of having it published, it pleased the Reverend +Author, before allowing it to go to press, to expunge some things, +to change other things and even to make some additions and +insert more lucid methods of exposition.</p> +</blockquote> +<p>This much is certain, the <i>Trigonometry</i> bears the impress +characteristic of Oughtred. Like all his mathematical +writings, the book was very condensed. Aside from +the tables, the text covered only 36 pages. Plane and +spherical triangles were taken up together. The treatise +is known in the history of trigonometry as among the +very earliest works to adopt a condensed symbolism so +that equations involving trigonometric functions could +be easily taken in by the eye. In the work of 1657, contractions +are given as follows: <i>s</i>=sine, <i>t</i>=tangent, <i>se</i>=secant, +<i>s co</i>=cosine (sine complement), <i>t co</i>=cotangent, +<i>se co</i>=cosecant, <i>log</i>=logarithm, <i>Z cru</i>=sum of the sides +of a rectangle or right angle, <i>X cru</i>=difference of these +sides. It has been generally overlooked by historians +that Oughtred used the abbreviations of trigonometric +functions, named above, a quarter of a century earlier, +<span class="pb" id="Page_37">[37]</span> +in his <i>Circles of Proportion</i>, 1632, 1633. Moreover, he +used sometimes also the abbreviations which are current +at the present time, namely sin=sine, tan=tangent, sec=secant. +We know that the <i>Circles of Proportion</i> existed +in manuscript many years before they were published. +The symbol <i>sv</i> for <i>sinus versus</i> occurs in the <i>Clavis</i> of 1631. +The great importance of well-chosen symbols needs no +emphasis to readers of the present day. With reference +to Oughtred’s trigonometric symbols. Augustus De +Morgan said:</p> +<blockquote> +<p>This is so very important a step, simple as it is, that Euler +is justly held to have greatly advanced trigonometry by its +introduction. Nobody that we know of has noticed that +Oughtred was master of the improvement, and willing to have +taught it, if people would have learnt.<a class="fn" id="fr_41" href="#fn_41">[41]</a></p> +</blockquote> +<p>We find, however, that even Oughtred cannot be given +the whole credit in this matter. By or before 1631 +several other writers used abbreviations of the trigonometric +functions. As early as 1624 the contractions +<i>sin</i> for sine and <i>tan</i> for tangent appear on the drawing +representing Gunter’s scale, but Gunter did not use them +in his books, except in the drawing of his scale.<a class="fn" id="fr_42" href="#fn_42">[42]</a> A closer +competitor for the honor of first using these trigonometric +abbreviations is Richard Norwood in his <i>Trigonometrie</i>, +London, 1631, where <i>s</i> stands for sine, <i>t</i> for tangent, <i>sc</i> +for sine complement (cosine), <i>tc</i> for tangent complement +(cotangent), and <i>sec</i> for secant. Norwood was a teacher +of mathematics in London and a well-known writer of +books on navigation. Aside from the abbreviations just +<span class="pb" id="Page_38">[38]</span> +cited Norwood did not use nearly as much symbolism +in his mathematics as did Oughtred.</p> +<p>Mention should be made of trigonometric symbols +used even earlier than any of the preceding, in “An +Appendix to the Logarithmes, shewing the practise of the +Calculation of Triangles, etc.,” printed in Edward Wright’s +edition of Napier’s <i>A Description of the Admirable Table +of Logarithmes</i>, London, 1618. We referred to this “Appendix” +in tracing the origin of the sign ×. It contains, +on p. 4, the following passage: “For the Logarithme of an +arch or an angle I set before (<i>s</i>), for the antilogarithme or +compliment thereof (<i>s</i>*) and for the Differential (<i>t</i>).” In +further explanation of this rather unsatisfactory passage, +the author (Oughtred?) says, “As for example: <i>sB</i>+<i>BC</i>=<i>CA</i>. +that is, the Logarithme of an angle <i>B</i>. at the Base +of a plane right-angled triangle, increased by the addition +of the Logarithm of <i>BC</i>, the hypothenuse thereof, is equall +to the Logarithme of <i>CA</i> the cathetus.”</p> +<p>Here “logarithme of an angle <i>B</i>” evidently means +“log sin <i>B</i>,” just as with Napier, “Logarithms of the +arcs” signifies really “Logarithms of the sines of the +angles.” In Napier’s table, the numbers in the column +marked “Differentiae” signify log sine minus log cosine +of an angle; that is, the logarithms of the tangents. This +explains the contraction (<i>t</i>) in the “Appendix.” The +conclusion of all this is that as early as 1618 the signs <i>s</i>, <i>s</i>*, +<i>t</i> were used for <i>sine</i>, <i>cosine</i>, and <i>tangent</i>, respectively.</p> +<p><i>John Speidell</i>, in his <i>Breefe Treatise of Sphaericall +Triangles</i>, London, 1627, uses <i>Si.</i> for sine, <i>T.</i> and <i>Tan</i> +for tangent, <i>Se.</i> for secant, <i>Si. Co.</i> for cosine, <i>Se. Co.</i> for +cosecant, <i>T. Co.</i> for cotangent.</p> +<p>The innovation of designating the sides and angles of +a triangle by <i>A</i>, <i>B</i>, <i>C</i>, and <i>a</i>, <i>b</i>, <i>c</i>, so that <i>A</i> was opposite +<span class="pb" id="Page_39">[39]</span> +<i>a</i>, <i>B</i> opposite <i>b</i>, and <i>C</i> opposite <i>c</i>, is attributed to Leonard +Euler (1753), but was first used by Richard Rawlinson +of Queen’s College, Oxford, sometimes after 1655 and +before 1668. Oughtred did not use Rawlinson’s notation.<a class="fn" id="fr_43" href="#fn_43">[43]</a></p> +<p>In trigonometry English writers of the first half of the +seventeenth century used contractions more freely than +their continental contemporaries; even more freely, indeed, +than English writers of a later period. Von Braunmühl, +the great historian of trigonometry, gives Oughtred much +praise for his trigonometry, and points out that half a +century later the army of writers on trigonometry had +hardly yet reached the standard set by Oughtred’s +analysis.<a class="fn" id="fr_44" href="#fn_44">[44]</a> Oughtred must be credited also with the first +complete proof that was given to the first two of “Napier’s +analogies.” His trigonometry contains seven-place tables +of sines, tangents, and secants, and six-place tables of +logarithmic sines and tangents; also seven-place logarithmic +tables of numbers. At the time of Oughtred there +was some agitation in favor of a wider introduction of +decimal systems. This movement is reflected in those +tables which contain the centesimal division of the degree, +a practice which is urged for general adoption in our own +day, particularly by the French.</p> +<h3 id="c14">SOLUTION OF NUMERICAL EQUATIONS</h3> +<p>In the solution of numerical equations Oughtred does +not mention the sources from which he drew, but the +method is substantially that of the great French algebraist +Vieta, as explained in a publication which appeared +<span class="pb" id="Page_40">[40]</span> +in 1600 in Paris under the title, <i>De numerosa potestatum +purarum atque adfectarum ad exegesin resolutione tractatus</i>. +In view of the fact that Vieta’s process has been described +inaccurately by leading modern historians including H. +Hankel<a class="fn" id="fr_45" href="#fn_45">[45]</a> and M. Cantor,<a class="fn" id="fr_46" href="#fn_46">[46]</a> it may be worth while to go into +some detail.<a class="fn" id="fr_47" href="#fn_47">[47]</a> By them it is made to appear as identical +with the procedure given later by Newton. The two are +not the same. The difference lies in the divisor used. +What is now called “Newton’s method” is Newton’s +method as modified by Joseph Raphson.<a class="fn" id="fr_48" href="#fn_48">[48]</a> The Newton-Raphson +method of approximation to the roots of an +equation <i>f</i>(<i>x</i>)=0 is usually given the form <i>a</i>-[<i>f</i>(<i>a</i>)/<i>f</i>´(<i>a</i>)], +where <i>a</i> is an approximate value of the required root. +It will be seen that the divisor is <i>f</i>´(<i>a</i>). Vieta’s divisor +is different; it is</p> +<div class="center">|<i>f</i>(<i>a</i>+<i>s</i>₁)-<i>f</i>(<i>a</i>)|-<i>s</i>₁<i>ⁿ</i>,</div> +<p>where <i>f</i>(<i>x</i>) is the left of the equation <i>f</i>(<i>x</i>)=<i>k</i>, <i>n</i> is the degree +of equation, and <i>s</i>₁ is a unit of the denomination of the +digit next to be found. Thus in <i>x</i>³+420000<i>x</i>=247651713, +it can be shown that 417 is approximately a root; suppose +that <i>a</i> has been taken to be 400, then <i>s</i>₁=10; but if, at +the next step of approximation, <i>a</i> is taken to be 410, then +<i>s</i>₁=1. In this example, taking <i>a</i>=400, Vieta’s divisor +<span class="pb" id="Page_41">[41]</span> +would have been 9120000; Newton’s divisor would have +been 900000.</p> +<p>A comparison of Vieta’s method with the Newton-Raphson +method reveals the fact that Vieta’s divisor +is more reliable, but labors under the very great disadvantage +of requiring a much larger amount of computation. +The latter divisor is accurate enough and easier +to compute. Altogether the Newton-Raphson process +marks a decided advance over that of Vieta.</p> +<p>As already stated, it is the method of Vieta that +Oughtred explains. The Englishman’s exposition is an +improvement on that of Vieta, printed forty years earlier. +Nevertheless, Oughtred’s explanation is far from easy +to follow. The theory of equations was at that time still +in its primitive stage of development. Algebraic notation +was not sufficiently developed to enable the argument +to be condensed into a form easily surveyed. So complicated +does Vieta’s process of approximation appear +that M. Cantor failed to recognize that Vieta possessed +a uniform mode of procedure. But when one has in mind +the general expression for Vieta’s divisor which we gave +above, one will recognize that there was marked uniformity +in Vieta’s approximations.</p> +<p>Oughtred allows himself twenty-eight sections in which +to explain the process and at the close cannot forbear +remarking that 28 is a “perfect” number (being equal +to the sum of its divisors, 1, 2, 4, 7, 14).</p> +<p>The early part of his exposition shows how an equation +may be transformed so as to make its roots 10, 100, 1000, or +10<sup>m</sup> times smaller. This simplifies the task of “locating a +root”; that is, of finding between what integers the root lies.</p> +<p>Taking one of Oughtred’s equations, <i>x</i>⁴-72<i>x</i>³+238600<i>x</i>=8725815, +upon dividing 72<i>x</i>³ by 10, 238600<i>x</i> by 1000, +<span class="pb" id="Page_42">[42]</span> +and 8725815 by 10,000, we obtain <i>x</i>⁴-7·2<i>x</i>³+238·6<i>x</i>=872·5. +Dividing both sides by <i>x</i>, we obtain <i>x</i>³+238·6-7·2<i>x</i>²=<i>x</i>)872·5. +Letting <i>x</i>=4, we have 64+238·6-115·2=187·4.</p> +<p>But 4)872·5(218·1; 4 is too small. Next let <i>x</i>=5, +we have 125+238·6-180=183·6.</p> +<p>But 5)872·5(174·5; 5 is too large. We take the lesser +value, <i>x</i>=4, or in the original equation, <i>x</i>=40. This +method may be used to find the second digit in the root. +Oughtred divides both sides of the equation by <i>x</i>², and +obtains <i>x</i>²+<i>x</i>)238600-72<i>x</i>=<i>x</i>²)8725815. He tries <i>x</i>=47 +and <i>x</i>=48, and finds that <i>x</i>=47.</p> +<p>He explains also how the last computation may be +done by logarithms. Thereby he established for himself +the record of being the first to use logarithms in the solution +of affected equations.</p> +<p>As an illustration of Oughtred’s method of approximation +after the root sought has been located, we have +chosen for brevity a cubic in preference to a quartic. We +selected the equation <i>x</i>³+420000<i>x</i>=247651713. By the +process explained above a root is found to lie between +<i>x</i>=400 and <i>x</i>=500. From this point on, the approximation +as given by Oughtred is as shown on <a href="#Page_43">p. 43</a>.</p> +<p>In further explanation of this process, observe that the +given equation is of the form <i>L<sub>c</sub></i>+<i>C<sub>q</sub>L</i>=<i>D<sub>c</sub></i>, where <i>L<sub>c</sub></i> +is our <i>x</i>, <i>C<sub>q</sub></i>=420000, <i>D<sub>c</sub></i>=247651713. In the first step +of approximation, let <i>L</i>=<i>A</i>+<i>E</i>, where <i>A</i>=400 and <i>E</i> is, +as yet, undetermined. We have</p> +<div class="verse"> +<p class="t0"><i>L<sub>c</sub></i>=(<i>A</i>+<i>E</i>)³=<i>A</i>³+3<i>A</i>²<i>E</i>+3<i>AE</i>²+<i>E</i>³</p> +</div> +<p>and</p> +<div class="verse"> +<p class="t0"><i>C<sub>q</sub>L</i>=420000(<i>A</i>+<i>E</i>).</p> +</div> +<p>Subtract from 247651713 the sum of the known terms +<i>A</i>³ (his <i>A<sub>c</sub></i>) and +420000 <i>A</i> (his <i>C<sub>q</sub>A</i>). This sum is 232000000 +the remainder is 15651713.</p> +<div class="pb" id="Page_43">[43]</div> +<p class="center">“<span class="sc">Exemplum II</span></p> +<p class="center">1<i>c</i>+42̣00̣00̣<i>l</i>=247̇651̇7̣1̣3̣̇</p> +<p class="center">Hoc est, <i>L<sub>c</sub>+C<sub>q</sub>L=D<sub>c</sub></i></p> +<pre> + 2 4 7̇ | 6 5 1̇ | 7̣ 1̣ 3̣̇ | ( 4 1 7 + ------+-------+-------+------------ + 4 2 | 0 0 0 | 0 | <i>C<sub>q</sub></i> + ------+-------+-------+------------ + 6 4 | | | <i>A<sub>c</sub></i> + 1 6 8 | 0 0 0 | 0 | <i>C<sub>q</sub> A</i> + ------+-------+-------+------------ + 2 3 2 | 0 0 0 | 0 | Ablatit. + =================================== +<i>R</i> 1 5 | 6 5 1̇ | 7 1 3̣ | + ------+-------+-------+------------ + 4 | 8 | | 3 <i>A<sub>q</sub></i> + | 1 2 | | 3 <i>A</i> + 4 | 2 0 0 | 0 0 | <i>C<sub>q</sub></i> + ------+-------+-------+------------ + 9 | 1 2 0 | 0 0 | Divisor. + ------+-------+-------+------------ + 4 | 8 | | 3 <i>A<sub>q</sub> E</i> + | 1 2 | | 3 <i>A E<sub>q</sub></i> + | 1 | | <i>E<sub>c</sub></i> + 4 | 2 0 0 | 0 0 | <i>C<sub>q</sub> E</i> + ------+-------+-------+------------ + 9 | 1 2 1 | 0 0 | Ablatit. + =================================== +<i>R</i> 6 | 5 3 0 | 7 1 3̣̇ | 4 | 1 | + ------+-------+-------+------------ ----+-----+--- + | 5 0 4 | 3 | 3 <i>A<sub class="ms">q</sub></i> | | + | 1 | 2 3 | 3 <i>A</i> 1 6 | 8 | + | 4 2 0 | 0 0 0 | <i>C<sub class="ms">q</sub></i> | 1 | + ------+-------+-------+------------ ----+-----+--- + | 9 2 5 | 5 3 0 | Divisor. 1 6 8 1 + ------+-------+-------+------------ + 3 | 5 3 0 | 1 | 3 <i>A<sub>q</sub> E</i> + | 6 0 | 2 7 | 3 <i>A E<sub>q</sub></i> + | | 3 4 3 | <i>E<sub>c</sub></i> + 2 | 9 4 0 | 0 0 0 | <i>C<sub>q</sub> E</i> + ------+-------+-------+------------ + 6 | 5 3 0 | 7 1 3 | Ablatit.” +</pre> +<div class="pb" id="Page_44">[44]</div> +<p>Next, he evaluates the coefficients of <i>E</i> in 3<i>A</i>²<i>E</i> and +420000<i>E</i>, also 3<i>A</i>, the coefficient of <i>E</i>². He obtains +3<i>A</i>²=480000, 3<i>A</i>=1200, <i>C<sub>q</sub></i>=420000. He interprets +3<i>A</i>² and <i>C<sub>q</sub></i> as tens, 3<i>A</i> as hundreds. Accordingly, he +obtains as their sum 9120000, which is the divisor for +finding the second digit in the approximation. Observe +that this divisor is the value of +|<i>f</i>(<i>a</i>+<i>s</i>₁)-<i>f</i>(<i>a</i>)|-<i>s</i>₁<i>ⁿ</i> +in our general expression, where <i>a</i>=400, +<i>s</i>₁=10, <i>n</i>=3, +<i>f</i>(<i>x</i>)=<i>x</i>³+420000<i>x</i>.</p> +<p>Dividing the remainder 15651713 by 9120000, he obtains +the integer 1 in ten’s place; thus <i>E</i>=10, approximately. +He now computes the terms 3<i>A</i>²<i>E</i>, 3<i>AE</i>² and +<i>E</i>³ to be, respectively, 4800000, 120000, 1000. Their +sum is 9121000. Subtracting it from the previous +remainder, 15651713, leaves the new remainder, 6530713.</p> +<p>From here on each step is a repetition of the preceding +step. The new <i>A</i> is 410, the new <i>E</i> is to be +determined. We have now in closer approximation, +<i>L</i>=<i>A</i>+<i>E</i>. This time we do not subtract +<i>A</i>³ and <i>C<sub>q</sub>A</i>, +because this subtraction is already affected by the preceding +work.</p> +<p>We find the second trial divisor by computing the sum +of 3<i>A</i>², 3<i>A</i> and <i>C<sub>q</sub></i>; +that is, the sum of 504300, 1230, +420000, which is 925530. Again, this divisor can be computed +by our general expression for divisors, by taking +<i>a</i>=410, <i>s</i>₁=1, <i>n</i>=3.</p> +<p>Dividing 6530713 by 925530 yields the integer 7. Thus +<i>E</i>=7. Computing 3<i>A</i>²<i>E</i>, +3<i>AE</i>², <i>E</i>³ and subtracting their +sum, the remainder is 0. Hence 417 is an exact root of +the given equation.</p> +<p>Since the extraction of a cube root is merely the solution +of a pure cubic equation, <i>x</i>³=<i>n</i>, the process given +above may be utilized in finding cube roots. This is +<span class="pb" id="Page_45">[45]</span> +precisely what Oughtred does in chap. xiv of his <i>Clavis</i>. +If the foregoing computation is modified by putting <i>C<sub>q</sub></i>=0, +the process will yield the approximate cube root of +247651713.</p> +<p>Oughtred solves 16 examples by the process of approximation +here explained. Of these, 9 are cubics, 5 are +quartics, and 2 are quintics. In all cases he finds only +one or two real roots. Of the roots sought, five are irrational, +the remaining are rational and are computed to +their exact values. Three of the computed roots have 2 +figures each, 9 roots have 3 figures each, 4 roots have 4 +figures each. While no attempt is made to secure all the +roots—methods of computing complex roots were invented +much later—he computes roots of equations which involve +large coefficients and some of them are of a degree as high +as the fifth. In view of the fact that many editions of +the <i>Clavis</i> were issued, one impression as late as 1702, it +contributed probably more than any other book to the +popularization of Vieta’s method in England.</p> +<p>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.</p> +<p>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.”</p> +<div class="pb" id="Page_46">[46]</div> +<h3 id="c15">LOGARITHMS</h3> +<p>Oughtred’s treatment of logarithms is quite in accordance +with the more recent practice.<a class="fn" id="fr_49" href="#fn_49">[49]</a> 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 <i>Key of the Mathematicks</i> of +1647; the <i>Clavis</i> of 1631 contains no treatment of +logarithms.</p> +<p>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.”</p> +<h3 id="c16">INVENTION OF THE SLIDE RULE; CONTROVERSY ON PRIORITY OF INVENTION</h3> +<p>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 +<span class="pb" id="Page_47">[47]</span> +own school and college.”<a class="fn" id="fr_50" href="#fn_50">[50]</a> The invention of the slide +rule has, until recently,<a class="fn" id="fr_51" href="#fn_51">[51]</a> 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 <i>Grammelogia; or the Mathematicall Ring</i>. +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, +<span class="pb" id="Page_48">[48]</span> +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 <i>Circles of Proportion</i> 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 <i>Apologeticall Epistle</i> 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.”<a class="fn" id="fr_52" href="#fn_52">[52]</a> As stated previously, +Oughtred’s circular slide rule was described by him in his +<i>Circles of Proportion</i>, London, 1632, which was translated +from Oughtred’s Latin manuscript and then seen through +the press by his pupil, William Forster. In 1633 appeared +<i>An Addition vnto the Vse of the Instrvment called the Circles +<span class="pb" id="Page_49">[49]</span> +of Proportion</i> which contained at the end “The Declaration +of the two Rulers for Calculation,” giving a description +of Oughtred’s rectilinear slide rule. This <i>Addition</i> +was bound with the <i>Circles of Proportion</i> as one volume. +About the same time Oughtred described a modified +form of the rectilinear slide rule, to be used in London for +gauging.<a class="fn" id="fr_53" href="#fn_53">[53]</a></p> +<div class="pb" id="Page_50">[50]</div> +<h2 id="c17">CHAPTER III +<br /><span class="small">MINOR WORKS</span></h2> +<p>Among the minor works of Oughtred must be ranked +his booklet of forty pages to which reference has already +been made, entitled, <i>The New Artificial Gauging Line or +Rod</i>, 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 <i>Horological Dialogues</i> +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 <i>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.</i> London, 1675.</p> +<p>“J. S.” says in his preface:</p> +<blockquote> +<p>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.</p> +</blockquote> +<div class="pb" id="Page_51">[51]</div> +<p>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:</p> +<blockquote> +<p>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. . . . .</p> +<p>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.<a class="fn" id="fr_54" href="#fn_54">[54]</a></p> +</blockquote> +<p>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 <i>Clavis</i> of 1647. The second paper appeared in his +<i>Circles of Proportion</i>.</p> +<p>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 +<span class="pb" id="Page_52">[52]</span> +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).</p> +<p>Portable sun-dials were sometimes carried in pockets, as +we carry watches. Thus Shakespeare, in <i>As You Like It</i>, +Act II, sc. vii:</p> +<div class="verse"> +<p class="t0">“And then he drew a diall from his poke.”</p> +</div> +<p>Watches were first made for carrying in the pocket +about 1658.</p> +<p>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.</p> +<p>Besides the accounts previously noted, there came +from his pen: <i>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.</i>, London, +1636.</p> +<p>The “Horizontall Dyall” and “Horologicall Ring” +appeared again as appendixes to Oughtred’s translation +from the French of a book on mathematical recreations.</p> +<p>The fourth French edition of that work appeared in +1627 at Paris, under the title of <i>Recreations mathematiqve</i>, +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: <i>Mathematical Recreations, or, A Collection of many +Problemes, extracted out of the Ancient and Modern Philosophers, +<span class="pb" id="Page_53">[53]</span> +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.</i> MDCLIII.</p> +<p>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: <i>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.</i></p> +<p>Brookes says in the preface:</p> +<blockquote> +<p>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.</p> +</blockquote> +<p>Interesting as one of our sources from which Oughtred +obtained his knowledge of the conic sections is his study +<span class="pb" id="Page_54">[54]</span> +of Mydorge. A tract which he wrote thereon was published +by Jonas Moore, in his <i>Arithmetick in two books</i> +. . . . [containing also] <i>the two first books of Mydorgius his +conical sections analyzed by that reverend devine Mr. W. +Oughtred, Englished and completed with cuts</i>. London, +1660. Another edition bears the date 1688.</p> +<p>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, <i>Gulielmi +Oughtredi, Etonensis, quondam Collegii Regalis in Cantabrigia +Socii, Opuscula Mathematica hactenus inedita</i>. Its +nine tracts are of little interest to a modern reader.</p> +<p>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 <i>Descriptio</i>, under the title, <i>A Description of the +Admirable Table of Logarithmes</i>, 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 +<i>x</i> in the “Appendix” as the sign of multiplication (to +<span class="pb" id="Page_55">[55]</span> +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 (<i>S</i>*) as the notation for sine complement +(cosine), while Seth Ward, an early pupil of +Oughtred, in his <i>Idea trigonometriae demonstratae</i>, Oxford, +1654, used a similar notation (<i>S</i>’). 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 <i>Trigonometria</i>, p. 2, +Oughtred uses (’) to designate 180°-angle.</p> +<p>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 <i>natural</i> logarithms—the +first of their kind ever published. In this +table we find log 10=2302584; in modern notation, this +is stated, log<sub><i>e</i></sub> 10=2.302584. The first more extended +table of natural logarithms of numbers was published by +John Speidell in the 1622 impression of his <i>New Logarithmes</i>, +which contains, besides trigonometric tables, the +logarithms of the numbers 1-1000.</p> +<p>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 <i>Arithmetica logarithmica</i>, 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±<i>x</i>/10<i>ⁿ</i>. +<span class="pb" id="Page_56">[56]</span> +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.</p> +<p>We conclude with the words of Dr. J. W. L. Glaisher:</p> +<blockquote> +<p>The <i>Appendix</i> 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.<a class="fn" id="fr_55" href="#fn_55">[55]</a></p> +</blockquote> +<div class="pb" id="Page_57">[57]</div> +<h2 id="c18">CHAPTER IV +<br /><span class="small">OUGHTRED’S INFLUENCE UPON MATHEMATICAL PROGRESS AND TEACHING</span></h2> +<h3 id="c19">OUGHTRED AND HARRIOT</h3> +<p>Oughtred’s <i>Clavis mathematicae</i> was the most influential +mathematical publication in Great Britain which appeared +in the interval between John Napier’s <i>Mirifici logarithmorum +canonis descriptio</i>, 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 +<i>Clavis</i>, but also of Thomas Harriot’s <i>Artis analyticae +praxis</i>. 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 <i>magnum opus</i>, or ten years before +the publication of Oughtred’s <i>Clavis</i>. 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 +<i>Artis analyticae praxis</i> was inferior to Oughtred’s <i>Clavis</i>. +The former was a much larger book, not as conveniently +portable, compiled after the author’s death by others, +<span class="pb" id="Page_58">[58]</span> +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.</p> +<h3 id="c20">OUGHTRED’S PUPILS</h3> +<p>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 <i>Clavis</i> and +sought his assistance by correspondence. We permit +Aubrey to enumerate some of these pupils in his own +gossipy style:</p> +<blockquote> +<p>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.</p> +<div class="pb" id="Page_59">[59]</div> +<p>He could not endure to see a scholar write an ill hand; +he taught them all presently to mend their hands.<a class="fn" id="fr_56" href="#fn_56">[56]</a></p> +</blockquote> +<p>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 <i>Circles of Proportion</i>; Arthur +Haughton, who brought out the 1660 Oxford edition of +the <i>Circles of Proportion</i>; Robert Wood, an educator +and politician, who assisted Oughtred in the translation +of the <i>Clavis</i> 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 <i>Clavis mathematicae</i>.</p> +<p>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 <i>Clavis</i>); 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 <i>Clavis</i> +<span class="pb" id="Page_60">[60]</span> +of 1647), and Thomas Wharton, who studied the <i>Clavis</i> +and assisted in the editing of the edition of 1647.</p> +<p>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.</p> +<h3 id="c21">OUGHTRED, THE “TODHUNTER OF THE SEVENTEENTH +<br />CENTURY”</h3> +<p>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 +Wood<a class="fn" id="fr_57" href="#fn_57">[57]</a> 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:</p> +<blockquote> +<p>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 <i>Clavis mathematica</i> +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 +<span class="pb" id="Page_61">[61]</span> +read the <i>Clav. Math.</i> 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.</p> +</blockquote> +<p>Anthony Wood makes a similar statement about +Thomas Henshaw:</p> +<blockquote> +<p>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.<a class="fn" id="fr_58" href="#fn_58">[58]</a></p> +</blockquote> +<p>Extracts from letters of W. Gascoigne to Oughtred, +of the years 1640 and 1641, throw some light upon mathematical +teaching of the time:</p> +<blockquote> +<p>Amongst the mathematical rarities these times have +afforded, there are none of that small number I (a late intruder +into these studies) have yet viewed, which so fully demonstrates +their authors’ great abilities as your Clavis, not richer in +augmentations, than valuable for contraction; . . . .</p> +<p>Your belief that there is in all inventions aliquid divinum, +an infusion beyond human cogitations, I am confident will +appear notably strengthened, if you please to afford this truth +belief, that I entered upon these studies accidentally after I +betook myself to the country, having never had so much aid as +to be taught addition, nor the discourse of an artist (having left +both Oxford and London before I knew what any proposition in +geometry meant) to inform me what were the best authors.<a class="fn" id="fr_59" href="#fn_59">[59]</a></p> +</blockquote> +<div class="pb" id="Page_62">[62]</div> +<p>The following extracts from two letters by W. Robinson, +written before the appearance of the 1647 English +edition of the <i>Clavis</i>, express the feeling of many readers +of the <i>Clavis</i> on its extreme conciseness and brevity of +explanation:</p> +<blockquote> +<p>I shall long exceedingly till I see your <i>Clavis</i> turned into +a pick-lock; and I beseech you enlarge it, and explain it what +you can, for we shall not need to fear either tautology or superfluity; +you are naturally concise, and your clear judgment +makes you both methodical and pithy; and your analytical +way is indeed the only way. . . . .</p> +<p>I will once again earnestly entreat you, that you be rather +diffuse in the setting forth of your English mathematical <i>Clavis</i>, +than concise, considering that the wisest of men noted of old, +and said stultorum infinitus est numerus, these arts cannot be +made too easy, they are so abstruse of themselves, and men +either so lazy or dull, that their fastidious wits take a loathing +at the very entrance of these studies, unless it be sweetened on +with plainness and facility. Brevity may well argue a learned +author, that without any excess or redundance, either of matter +or words, can give the very substance and essence of the thing +treated of; but it seldom makes a learned scholar; and if one +be capable, twenty are not; and if the master sum up in brief +the pith of his own long labours and travails, it is not easy to +imagine that scholars can with less labour than it cost their +masters dive into the depths thereof.<a class="fn" id="fr_60" href="#fn_60">[60]</a></p> +</blockquote> +<p>Here is the judgment of another of Oughtred’s friends:</p> +<blockquote> +<p>. . . . with the character I received from your and my noble +friend Sir Charles Cavendish, then at Paris, of your second +edition of the same piece, made me at my return into England +speedily to get, and diligently peruse the same. Neither +truly did I find my expectation deceived; having with admiration +often considered how it was possible (even in the hardest +<span class="pb" id="Page_63">[63]</span> +things of geometry) to deliver so much matter in so few words, +yet with such demonstrative clearness and perspicuity: and +hath often put me in mind of learned Mersennus his judgment +(since dead) of it, that there was more matter comprehended in +that little book than in Diophantus, and all the ancients. . . . .<a class="fn" id="fr_61" href="#fn_61">[61]</a></p> +</blockquote> +<p>Oughtred’s own feeling was against diffuseness in textbook +writing. In his revisions of his <i>Clavis</i> the original +character of that book was not altered. In his reply to +W. Robinson, Oughtred said:</p> +<blockquote> +<p>. . . . But my art for all such mathematical inventions I +have set down in my Clavis Mathematica, which therefore +in my title I say is tum logisticae cum analyticae adeoque +totius mathematicae quasi clavis, which if any one of a mathematical +genius will carefully study, (and indeed it must be +carefully studied,) he will not admire others, but himself do +wonders. But I (such is my tenuity) have enough fungi vice +cotis, acutum reddere quae ferrum valet, exsors ipsa secandi, +or like the touchstone, which being but a stone, base and little +worth, can shew the excellence and riches of gold.<a class="fn" id="fr_62" href="#fn_62">[62]</a></p> +</blockquote> +<p>John Wallis held Oughtred’s <i>Clavis</i> in high regard. +When in correspondence with John Collins concerning +plans for a new edition, Wallis wrote in 1666-67, six +years after the death of Oughtred:</p> +<blockquote> +<p>. . . . But for the goodness of the book in itself, it is that +(I confess) which I look upon as a very good book, and which +doth in as little room deliver as much of the fundamental and +useful part of geometry (as well as of arithmetic and algebra) +as any book I know; and why it should not be now acceptable +I do not see. It is true, that as in other things so in mathematics, +fashions will daily alter, and that which Mr. Oughtred +designed by great letters may be now by others be designed by +small; but a mathematician will, with the same ease and advantage, +understand <i>A<sub>c</sub></i>, and <i>a</i>³ or <i>aaa</i>. . . . . And the like +<span class="pb" id="Page_64">[64]</span> +I judge of Mr. Oughtred’s Clavis, which I look upon (as those +pieces of Vieta who first went in that way) as lasting books and +classic authors in this kind; to which, notwithstanding, every +day may make new additions. . . . .</p> +<p>But I confess, as to my own judgment, I am not for making +the book bigger, because it is contrary to the design of it, being +intended for a manual or contract; whereas comments, by +enlarging it, do rather destroy it. . . . . But it was by him +intended, in a small epitome, to give the substance of what is +by others delivered in larger volumes. . . . .<a class="fn" id="fr_63" href="#fn_63">[63]</a></p> +</blockquote> +<p>That there continued to be a group of students and +teachers who desired a fuller exposition than is given by +Oughtred is evident from the appearance, over fifty +years after the first publication of the <i>Clavis</i>, of a booklet +by Gilbert Clark, entitled <i>Oughtredus Explicatus</i>, London, +1682. A review of this appeared in the <i>Acta Eruditorum</i> +(Leipzig, 1684), on p. 168, wherein Oughtred is named +“clarissimus Angliae mathematicus.” John Collins wrote +Wallis in 1666-67 that Clark, “who lives with Sir Justinian +Isham, within seven miles of Northampton, . . . . +intimates he wrote a comment on the <i>Clavis</i>, which lay +long in the hands of a printer, by whom he was abused, +meaning Leybourne.”<a class="fn" id="fr_64" href="#fn_64">[64]</a></p> +<p>We shall have occasion below to refer to Oughtred’s +inability to secure a copy of a noted Italian mathematical +work published a few years before. In those days the +condition of the book trade in England must have been +somewhat extraordinary. Dr. J. W. L. Glaisher throws +some light upon this subject.<a class="fn" id="fr_65" href="#fn_65">[65]</a> He found in the <i>Calendar +<span class="pb" id="Page_65">[65]</span> +of State Papers</i>, Domestic Series, 1637, a petition to Archbishop +Laud in which it is set forth that when Hooganhuysen, +a Dutchman, “heretofore complained of in the +High Commission for importing books printed beyond +the seas,” had been bound “not to bring in any more,” +one Vlacq (the computer and publisher of logarithmic +tables) “kept up the same agency and sold books in his +stead. . . . . Vlacq is now preparing to go beyond the +seas to avoid answering his late bringing over nine bales of +books contrary to the decree of the Star Chamber.” Judgment +was passed that, “Considering the ill-consequence and +scandal that would arise by strangers importing and venting +in this kingdom books printed beyond the seas,” certain +importations be prohibited, and seized if brought over.</p> +<p>This want of easy intercommunication of results of +scientific research in Oughtred’s time is revealed in the +following letter, written by Oughtred to Robert Keylway, +in 1645:</p> +<blockquote> +<p>I speak this the rather, and am induced to a better confidence +of your performance, by reason of a geometric-analytical +art or practice found out by one Cavalieri, an Italian, of which +about three years since I received information by a letter from +Paris, wherein was praelibated only a small taste thereof, yet +so that I divine great enlargement of the bounds of the mathematical +empire will ensue. I was then very desirous to see the +author’s own book while my spirits were more free and lightsome, +but I could not get it in France. Since, being more stept +into years, daunted and broken with the sufferings of these +disastrous times, I must content myself to keep home, and not +put out to any foreign discoveries.<a class="fn" id="fr_66" href="#fn_66">[66]</a></p> +</blockquote> +<p>It was in 1655, when Oughtred was about eighty years +old, that John Wallis, the great forerunner of Newton in +<span class="pb" id="Page_66">[66]</span> +Great Britain, began to publish his great researches on +the arithmetic of infinites. Oughtred rejoiced over the +achievements of his former pupil. In 1655, Oughtred +wrote John Wallis as follows:</p> +<blockquote> +<p>I have with unspeakable delight, so far as my necessary +businesses, the infirmness of my health, and the greatness of +my age (approaching now to an end) would permit, perused +your most learned papers, of several choice arguments, which +you sent me: wherein I do first with thankfulness acknowledge +to God, the Father of lights, the great light he hath given you; +and next I congratulate you, even with admiration, the clearness +and perspicacity of your understanding and genius, who +have not only gone, but also opened a way into these profoundest +mysteries of art, unknown and not thought of by the +ancients. With which your mysterious inventions I am the +more affected, because full twenty years ago, the learned patron +of learning, Sir Charles Cavendish, shewed me a paper written, +wherein were some few excellent new theorems, wrought by +the way, as I suppose, of Cavalieri, which I wrought over +again more agreeably to my way. The paper, wherein I +wrought it, I shewed to many, whereof some took copies, but +my own I cannot find. I mention it for this, because I saw +therein a light breaking out for the discovery of wonders to +be revealed to mankind, in this last age of the world: which +light I did salute as afar off, and now at a nearer distance +embrace in your prosperous beginnings. Sir, that you are +pleased to mention my name in your never dying papers, that +is your noble favour to me, who can add nothing to your glory, +but only my applause. . . . .<a class="fn" id="fr_67" href="#fn_67">[67]</a></p> +</blockquote> +<p>The last sentence has reference to Wallis’ appreciative +and eulogistic reference to Oughtred in the preface. It +is of interest to secure the opinion of later English writers +who knew Oughtred only through his books. John +<span class="pb" id="Page_67">[67]</span> +Locke wrote in his journal under the date, June 24, 1681, +“the best algebra yet extant is Outred’s.”<a class="fn" id="fr_68" href="#fn_68">[68]</a> John Collins, +who is known in the history of mathematics chiefly +through his very extensive correspondence with nearly +all mathematicians of his day, was inclined to be more +critical. He wrote Wallis about 1667:</p> +<blockquote> +<p>It was not my intent to disparage the author, though I +know many that did lightly esteem him when living, some +whereof are at rest, as Mr. Foster and Mr. Gibson. . . . . +You grant the author is brief, and therefore obscure, and I +say it is but a collection, which, if himself knew, he had done +well to have quoted his authors, whereto the reader might have +repaired. You do not like those words of Vieta in his theorems, +ex adjunctione plano solidi, plus quadrato quadrati, etc., and +think Mr. Oughtred the first that abridged those expressions +by symbols; but I dissent, and tell you ’twas done before by +Cataldus, Geysius, and Camillus Gloriosus,<a class="fn" id="fr_69" href="#fn_69">[69]</a> who in his first +decade of exercises, (not the first tract,) printed at Naples in +1627, which was four years before the first edition of the Clavis, +proposeth this equation just as I here give it you, viz. +1<i>ccc</i>+16<i>qcc</i>+41<i>qqc</i>-2304<i>cc</i>-18364<i>qc</i>-133000<i>qq</i>-54505<i>c</i>+3728<i>q</i>+8064 +<i>N</i> <i>aequatur</i> 4608, finds <i>N</i> or a root of it to be 24, and composeth +the whole out of it for proof, just in Mr. Oughtred’s +symbols and method. Cataldus on Vieta came out fifteen +years before, and I cannot quote that, as not having it +by me.</p> +<p>. . . . And as for Mr. Oughtred’s method of symbols, +this I say to it; it may be proper for you as a commentator to +follow it, but divers I know, men of inferior rank that have good +skill in algebra, that neither use nor approve it. . . . . Is not +<i>A</i>⁵ sooner wrote than <i>A<sub>qc</sub></i>? Let <i>A</i> be 2, the cube of 2 is 8, +which squared is 64: one of the questions between Maghet +<span class="pb" id="Page_68">[68]</span> +Grisio and Gloriosus is whether 64=<i>A<sub>cc</sub></i> or <i>A<sub>qc</sub></i>. The Cartesian +method tells you it is <i>A</i>⁶, and decides the doubt. . . . .<a class="fn" id="fr_70" href="#fn_70">[70]</a></p> +</blockquote> +<p>There is some ground for the criticisms passed by +Collins. To be sure, the first edition of the <i>Clavis</i> is +dated 1631—six years before Descartes suggested the +exponential notation which came to be adopted as the +symbolism in our modern algebra. But the second edition +of the <i>Clavis</i>, 1647, appeared ten years after Descartes’ +innovation. Had Oughtred seen fit to adopt the new exponential +notation in 1647, the step would have been epoch-making +in the teaching of algebra in England. We have +seen no indication that Oughtred was familiar with Descartes’ +<i>Géométrie</i> of 1637.</p> +<p>The year preceding Oughtred’s death Mr. John Twysden +expressed himself as follows in the preface to his +<i>Miscellanies</i>:</p> +<blockquote> +<p>It remains that I should adde something touching the beginning, +and use of these Sciences. . . . . I shall only, to their +honours, name some of our own Nation yet living, who have +happily laboured upon both stages. That succeeding ages +may understand that in this of ours, there yet remained some +who were neither ignorant of these Arts, as if they had held +them vain, nor condemn them as superfluous. Amongst +them all let Mr. William Oughtred, of Aeton, be named in the +first place, a Person of venerable grey haires, and exemplary +piety, who indeed exceeds all praise we can bestow upon +him. Who by an easie method, and admirable Key, hath +unlocked the hidden things of geometry. Who by an accurate +Trigonometry and furniture of Instruments, hath inriched, +as well geometry, as Astronomy. Let D. John Wallis, +and D. Seth Ward, succeed in the next place, both famous +Persons, and Doctors in Divinity, the one of geometry, the +<span class="pb" id="Page_69">[69]</span> +other of astronomy, Savilian Professors in the University +of Oxford.<a class="fn" id="fr_71" href="#fn_71">[71]</a></p> +</blockquote> +<p>The astronomer Edmund Halley, in his preface to the +1694 English edition of the <i>Clavis</i>, speaks of this book as +one of “so established a reputation, that it were needless +to say anything thereof,” though “the concise Brevity +of the author is such, as in many places to need Explication, +to render it Intelligible to the less knowing Mathematical +matters.”</p> +<p>In closing this part of our monograph, we quote the +testimony of Robert Boyle, the experimental physicist, +as given May 8, 1647, in a letter to Mr. Hartlib:</p> +<blockquote> +<p>The Englishing of, and additions to Oughtred’s <i>Clavis +mathematica</i> does much content me, I having formerly spent +much study on the original of that algebra, which I have long +since esteemed a much more instructive way of logic, than that +of Aristotle.<a class="fn" id="fr_72" href="#fn_72">[72]</a></p> +</blockquote> +<h3 id="c22">WAS DESCARTES INDEBTED TO OUGHTRED?</h3> +<p>This question first arose in the seventeenth century, +when John Wallis, of Oxford, in his <i>Algebra</i> (the English +edition of 1685, and more particularly the Latin edition +of 1693), raised the issue of Descartes’ indebtedness to the +English scientists, Thomas Harriot and William Oughtred. +In discussing matters of priority between Harriot and +Descartes, relating to the theory of equations, Wallis +is generally held to have shown marked partiality to +Harriot. Less attention has been given by historians +<span class="pb" id="Page_70">[70]</span> +of mathematics to Descartes’ indebtedness to Oughtred. +Yet this question is of importance in tracing Oughtred’s +influence upon his time.</p> +<p>On January 8, 1688-89, Samuel Morland addressed a +letter of inquiry to John Wallis, containing a passage +which we translate from the Latin:</p> +<blockquote> +<p>Some time ago I read in the elegant and truly precious book +that you have written on <i>Algebra</i>, about Descartes, this philosopher +so extolled above all for having arrived at a very perfect +system by his own powers, without the aid of others, this +Descartes, I say, who has received in geometry very great light +from our Oughtred and our Harriot, and has followed their +track though he carefully suppressed their names. I stated +this in a conversation with a professor in Utrecht (where I +reside at present). He requested me to indicate to him the +page-numbers in the two authors which justified this accusation. +I admitted that I could not do so. The <i>Géométrie</i> of +Descartes is not sufficiently familiar to me, although with +Oughtred I am fairly familiar. I pray you therefore that you +will assume this burden. Give me at least those references +to passages of the two authors from the comparison of which +the plagiarism by Descartes is the most striking.<a class="fn" id="fr_73" href="#fn_73">[73]</a></p> +</blockquote> +<p>Following Morland’s letter in the <i>De algebra tractatus</i>, +is printed Wallis’ reply, dated March 12, 1688 (“Stilo +Angliae”), which is, in part, as follows:</p> +<blockquote> +<p>I nowhere give him the name of a plagiarist; I would not +appear so impolite. However this I say, the major part of his +algebra (if not all) is found before him in other authors (notably +in our Harriot) whom he does not designate by name. That +algebra may be applied to geometry, and that it is in fact so +applied, is nothing new. Passing the ancients in silence, we +state that this has been done by Vieta, Ghetaldi, Oughtred +<span class="pb" id="Page_71">[71]</span> +and others, before Descartes. They have resolved by algebra +and specious arithmetic [literal arithmetic] many geometrical +problems. . . . . But the question is not as to application of +algebra to geometry (a thing quite old), but of the Cartesian +algebra considered by itself.</p> +</blockquote> +<p>Wallis then indicates in the 1659 edition of Descartes’ +<i>Géométrie</i> where the subjects treated on the first six pages +are found in the writings of earlier algebraists, particularly +of Harriot and Oughtred. For example, what is +found on the first page of Descartes, relating to addition, +subtraction, multiplication, division, and root extraction, +is declared by Wallis to be drawn from Vieta, Ghetaldi, +and Oughtred.</p> +<p>It is true that Descartes makes no mention of modern +writers, except once of Cardan. But it was not the purpose +of Descartes to write a history of algebra. To be +sure, references to such of his immediate predecessors as +he had read would not have been out of place. Nevertheless, +Wallis fails to show that Descartes made illegitimate +use of anything he may have seen in Harriot or +Oughtred.</p> +<p>The first inquiry to be made is, Did Descartes possess +copies of the books of Harriot and Oughtred? It is only +in recent time that this question has been answered as to +Harriot. As to Oughtred, it is still unanswered. It is +now known that Descartes had seen Harriot’s <i>Artis analyticae +praxis</i> (1631). Descartes wrote a letter to Constantin +Huygens in which he states that he is sending +Harriot’s book.<a class="fn" id="fr_74" href="#fn_74">[74]</a></p> +<p>An able discussion of the question, what effect, if +any, Oughtred’s <i>Clavis mathematicae</i> of 1631 had upon +<span class="pb" id="Page_72">[72]</span> +Descartes’<a class="fn" id="fr_75" href="#fn_75">[75]</a> <i>Géométrie</i> of 1637, is given by H. Bosmans in +a recent article. According to Bosmans no evidence has +been found that Descartes possessed a copy of Oughtred’s +book, or that he had examined it. Bosmans believes +nevertheless that Descartes was influenced by the <i>Clavis</i>, +either directly or indirectly. He says:</p> +<blockquote> +<p>If Descartes did not read it carefully, which is not proved, +he was none the less well informed with regard to it. No +one denies his intimate knowledge of the intellectual movement +of his time. The <i>Clavis mathematica</i> enjoyed a rapid +success. It is impossible that, at least indirectly, he did not +know the more original ideas which it contained. Far from +belittling Descartes, as I much desire to repeat, this rather +makes him the greater.<a class="fn" id="fr_76" href="#fn_76">[76]</a></p> +</blockquote> +<p>We ourselves would hardly go as far as does Bosmans. +Unless Descartes actually examined a copy of Oughtred +it is not likely that he was influenced by Oughtred in +appreciable degree. Book reviews were quite unknown +in those days. No evidence has yet been adduced to show +that Descartes obtained a knowledge of Oughtred by +correspondence. A most striking feature about Oughtred’s +<i>Clavis</i> is its notation. No trace of the Englishman’s +symbolism has been pointed out in Descartes’ <i>Géométrie</i> +of 1637. Only six years intervened between the publication +of the <i>Clavis</i> and the <i>Géométrie</i>. It took longer than +this period for the <i>Clavis</i> to show evidence of its influence +upon mathematical books published in <i>England</i>; it is +not probable that <i>abroad</i> the contact was more immediate +<span class="pb" id="Page_73">[73]</span> +than at home. Our study of seventeenth-century algebra +has led us to the conviction that Oughtred deserves a +higher place in the development of this science than is +usually accorded to him; but that it took several decennia +for his influence fully to develop.</p> +<h3 id="c23">THE SPREAD OF OUGHTRED’S NOTATIONS</h3> +<p>An idea of Oughtred’s influence upon mathematical +thought and teaching can be obtained from the spread +of his symbolism. This study indicates that the adoption +was not immediate. The earliest use that we have been +able to find of Oughtred’s notation for proportion, <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i>, +occurs nineteen years after the <i>Clavis mathematicae</i> +of 1631. In 1650 John Kersey brought out in London an +edition of Edmund Wingates’ <i>Arithmetique made easie</i>, +in which this notation is used. After this date publications +employing it became frequent, some of them being +the productions of pupils of Oughtred. We have seen it in +Vincent Wing (1651),<a class="fn" id="fr_77" href="#fn_77">[77]</a> Seth Ward (1653),<a class="fn" id="fr_78" href="#fn_78">[78]</a> John Wallis +(1655),<a class="fn" id="fr_79" href="#fn_79">[79]</a> in “R. B.,” a schoolmaster in Suffolk,<a class="fn" id="fr_80" href="#fn_80">[80]</a> Samuel +Foster (1659),<a class="fn" id="fr_81" href="#fn_81">[81]</a> Jonas Moore (1660),<a class="fn" id="fr_82" href="#fn_82">[82]</a> and Isaac Barrow +(1657).<a class="fn" id="fr_83" href="#fn_83">[83]</a> In the latter part of the seventeenth century +<span class="pb" id="Page_74">[74]</span> +Oughtred’s notation, <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i>, became the prevalent, +though not universal, notation in Great Britain. A tremendous +impetus to their adoption was given by Seth +Ward, Isaac Barrow, and particularly by John Wallis, who +was rising to international eminence as a mathematician.</p> +<p>In France we have noticed Oughtred’s notation for +proportion in Franciscus Dulaurens (1667),<a class="fn" id="fr_84" href="#fn_84">[84]</a> J. Prestet +(1675),<a class="fn" id="fr_85" href="#fn_85">[85]</a> R. P. Bernard Lamy (1684),<a class="fn" id="fr_86" href="#fn_86">[86]</a> Ozanam (1691),<a class="fn" id="fr_87" href="#fn_87">[87]</a> +De l’Hospital (1696),<a class="fn" id="fr_88" href="#fn_88">[88]</a> R. P. Petro Nicolas (1697).<a class="fn" id="fr_89" href="#fn_89">[89]</a></p> +<p>In the Netherlands we have noticed it in R. P. Bernard +Lamy (1680),<a class="fn" id="fr_90" href="#fn_90">[90]</a> and in an anonymous work of 1690.<a class="fn" id="fr_91" href="#fn_91">[91]</a> +In German and Italian works of the seventeenth century +we have not seen Oughtred’s notation for proportion.</p> +<p>In England a modified notation soon sprang up in +which ratio was indicated by two dots instead of a single +dot, thus <i>A</i>:<i>B</i>::<i>C</i>:<i>D</i>. The reason for the change lies +probably in the inclination to use the single dot to designate +decimal fractions. W. W. Beman pointed out that +this modified symbolism (:) for ratio is found as early as +1657 in the end of the trigonometric and logarithmic +<span class="pb" id="Page_75">[75]</span> +tables that were bound with Oughtred’s <i>Trigonometria</i>.<a class="fn" id="fr_92" href="#fn_92">[92]</a> +It is not probable, however, that this notation was used +by Oughtred himself. The <i>Trigonometria</i> proper has +Oughtred’s <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i> throughout. Moreover, in the +English edition of this trigonometry, which appeared the +same year, 1657, but subsequent to the Latin edition, the +passages which contained the colon as the symbol for +ratio, when not omitted, are recast, and the regular +Oughtredian notation is introduced. In Oughtred’s +posthumous work, <i>Opuscula mathematica hactenus inedita</i>, +1677, the colon appears quite often but is most likely due +to the editor of the book.</p> +<p>We have noticed that the notation <i>A</i>:<i>B</i>::<i>C</i>:<i>D</i> antedates +the year 1657. Vincent Wing, the astronomer, +published in 1651 in London the <i>Harmonicon coeleste</i>, in +which is found not only Oughtred’s notation <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i> +but also the modified form of it given above. The two +are used interchangeably. His later works, the <i>Logistica +astronomica</i> (1656), <i>Doctrina spherica</i> (1655), and <i>Doctrina +theorica</i>, published in one volume in London, all use the +symbols <i>A</i>:<i>B</i>::<i>C</i>:<i>D</i> exclusively. The author of a book +entitled, <i>An Idea of Arithmetick at first designed for the +use of the Free Schoole at Thurlow in Suffolk . . . . by +R. B., Schoolmaster there</i>, London, 1655, writes <i>A</i>:<i>a</i>::<i>C</i>:<i>c</i>, +though part of the time he uses Oughtred’s unmodified +notation.</p> +<p>We can best indicate the trend in England by indicating +the authors of the seventeenth century whom we have +found using the notation <i>A</i>:<i>B</i>::<i>C</i>:<i>D</i> and the authors of +the eighteenth century whom we have found using <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i>. +The former notation was the less common during +<span class="pb" id="Page_76">[76]</span> +the seventeenth but the more common during the eighteenth +century. We have observed the symbols <i>A</i>:<i>B</i>::<i>C</i>:<i>D</i> +(besides the authors already named) in John Collins +(1659),<a class="fn" id="fr_93" href="#fn_93">[93]</a> James Gregory (1663),<a class="fn" id="fr_94" href="#fn_94">[94]</a> Christopher Wren (1668-69),<a class="fn" id="fr_95" href="#fn_95">[95]</a> +William Leybourn (1673),<a class="fn" id="fr_96" href="#fn_96">[96]</a> William Sanders (1686),<a class="fn" id="fr_97" href="#fn_97">[97]</a> +John Hawkins (1684),<a class="fn" id="fr_98" href="#fn_98">[98]</a> Joseph Raphson (1697),<a class="fn" id="fr_99" href="#fn_99">[99]</a> E. Wells +(1698),<a class="fn" id="fr_100" href="#fn_100">[100]</a> and John Ward (1698).<a class="fn" id="fr_101" href="#fn_101">[101]</a></p> +<p>Of English eighteenth-century authors the following +still clung to the notation <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i>: John Harris’ +translation of F. Ignatius Gaston Pardies (1701),<a class="fn" id="fr_102" href="#fn_102">[102]</a> George +Shelley (1704),<a class="fn" id="fr_103" href="#fn_103">[103]</a> Sam Cobb (1709),<a class="fn" id="fr_104" href="#fn_104">[104]</a> J. Collins in <i>Commercium +Epistolicum</i> (1712), John Craig (1718),<a class="fn" id="fr_105" href="#fn_105">[105]</a> Jo. +<span class="pb" id="Page_77">[77]</span> +Wilson (1724).<a class="fn" id="fr_106" href="#fn_106">[106]</a> The latest use of <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i> which has +come to our notice is in the translation of the <i>Analytical +Institutions</i> of Maria G. Agnesi, made by John Colson +sometime before 1760, but which was not published until +1801. During the seventeenth century the notation +<i>A</i>:<i>B</i>::<i>C</i>:<i>D</i> acquired almost complete ascendancy in +England.</p> +<p>In France Oughtred’s unmodified notation <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i>, +having been adopted later, was also discarded later than +in England. An approximate idea of the situation appears +from the following data. The notation <i>A</i>.<i>B</i>::<i>C</i>.<i>D</i> was +used by M. Carré (1700),<a class="fn" id="fr_107" href="#fn_107">[107]</a> M. Guisnée (1705),<a class="fn" id="fr_108" href="#fn_108">[108]</a> M. de +Fontenelle (1727),<a class="fn" id="fr_109" href="#fn_109">[109]</a> M. Varignon (1725),<a class="fn" id="fr_110" href="#fn_110">[110]</a> M. Robillard +(1753),<a class="fn" id="fr_111" href="#fn_111">[111]</a> M. Sebastien le Clerc (1764),<a class="fn" id="fr_112" href="#fn_112">[112]</a> Clairaut (1731),<a class="fn" id="fr_113" href="#fn_113">[113]</a> +M. L’Hospital (1781).<a class="fn" id="fr_114" href="#fn_114">[114]</a></p> +<p>In Italy Oughtred’s modified notation <i>a</i>, <i>b</i>::<i>c</i>, <i>d</i> was +used by Maria G. Agnesi in her <i>Instituzioni analitiche</i>, +<span class="pb" id="Page_78">[78]</span> +Milano, 1748. The notation <i>a</i>:<i>b</i>::<i>c</i>:<i>d</i> +found entrance the latter part of the eighteenth century. In Germany +the symbolism <i>a</i>:<i>b</i>=<i>c</i>:<i>d</i>, suggested by +Leibniz, found wider +acceptance.<a class="fn" id="fr_115" href="#fn_115">[115]</a></p> +<p>It is evident from the data presented that Oughtred +proposed his notation for ratio and proportion at a time +when the need of a specific notation began to be generally +felt, that his symbol for ratio <i>a</i>.<i>b</i> was temporarily adopted +in England and France but gave way in the eighteenth +century to the symbol <i>a</i>:<i>b</i>, that Oughtred’s symbol for +proportion :: found almost universal adoption in England +and France and was widely used in Italy, the Netherlands, +the United States, and to some extent in Germany; it has +survived to the present time but is now being gradually +displaced by the sign of equality =.</p> +<p>Oughtred’s notation to express aggregation of terms +has received little attention from historians but is nevertheless +<span class="pb" id="Page_79">[79]</span> +interesting. His books, as well as those of John +Wallis, are full of parentheses but they are not used as +symbols of aggregation in algebra; they are simply marks +of punctuation for parenthetical clauses. We have seen +that Oughtred writes (<i>a</i>+<i>b</i>)² and +√<span class="over"><i>a</i>+<i>b</i></span> thus, +<i>Q</i>:<i>a</i>+<i>b</i>:, +√:<i>a</i>+<i>b</i>:, or <i>Q</i>:<i>a</i>+<i>b</i>, +√:<i>a</i>+<i>b</i>, using on rarer occasions +a single dot in place of the colon. This notation did not +originate with Oughtred, but, in slightly modified form, +occurs in writings from the Netherlands. In 1603 <i>C. +Dibvadii in geometriam Evclidis demonstratio numeralis</i>, +Leyden, contains many expressions of this sort, +√·136+√2048, +signifying √(136+√2048). The dot is used to +indicate that the root of the binomial (not of 136 alone) is +called for. This notation is used extensively in <i>Ludolphi +à Cevlen de circulo</i>, Leyden, 1619, and in <i>Willebrordi +Snellii De circuli dimensione</i>, Leyden, 1621. In place +of the single dot Oughtred used the colon (:), probably +<span class="pb" id="Page_80">[80]</span> +to avoid confusion with his notation for ratio. To avoid +further possibility of uncertainty he usually placed the +colon both before and after the algebraic expression under +aggregation. This notation was adopted by John Wallis +and Isaac Barrow. It is found in the writings of Descartes. +Together with Vieta’s horizontal bar, placed +over two or more terms, it constituted the means used +almost universally for denoting aggregation of terms in +algebra. Before Oughtred the use of parentheses had been suggested by +Clavius<a class="fn" id="fr_116" href="#fn_116">[116]</a> and Girard.<a class="fn" id="fr_117" href="#fn_117">[117]</a> The latter +wrote, for instance, √(2+√3). While parentheses never +became popular in algebra before the time of Leibniz +and the Bernoullis they were by no means lost sight of. +We are able to point to the following authors who made +use of them: I. Errard de Bar-le-Duc (1619),<a class="fn" id="fr_118" href="#fn_118">[118]</a> Jacobo +de Billy (1643),<a class="fn" id="fr_119" href="#fn_119">[119]</a> one of whose books containing this +notation was translated into English, and also the posthumous +works of Samuel Foster.<a class="fn" id="fr_120" href="#fn_120">[120]</a> J. W. L. Glaisher +points out that parentheses were used by Norwood in his +<i>Trigonometrie</i> (1631), p. 30.<a class="fn" id="fr_121" href="#fn_121">[121]</a></p> +<div class="pb" id="Page_81">[81]</div> +<p>The symbol for the arithmetical difference between +two numbers, ~, is usually attributed to John Wallis, +but it occurs in Oughtred’s <i>Clavis mathematicae</i> of 1652, +in the tract on <i>Elementi decimi Euclidis declaratio</i>, at an +earlier date than in any of Wallis’ books. As Wallis +assisted in putting this edition through the press it is +possible, though not probable, that the symbol was inserted +by him. Were the symbol Wallis’, Oughtred would +doubtless have referred to its origin in the preface. During +the eighteenth century the symbol found its way into +foreign texts even in far-off Italy.<a class="fn" id="fr_122" href="#fn_122">[122]</a> It is one of three +symbols presumably invented by Oughtred and which are +still used at the present time. The others are × and ::.</p> +<div class="p">The curious and ill-chosen symbols, +<table class="symbol" summary="|̲̅ ̅"><tr><td class="lb"></td><td class="top"></td></tr></table> +for “greater than,” and +<table class="symbol" summary="̲ ̲̅|"><tr><td class="bot"></td><td class="rb"></td></tr></table> +for “less than,” were certain to succumb in +their struggle for existence against Harriot’s admirably +chosen > and <. Yet such was the reputation of Oughtred +that his symbols were used in England quite extensively +during the seventeenth and the beginning of the eighteenth +century. Considerable confusion has existed among algebraists +and also among historians as to what Oughtred’s +symbols really were. Particularly is this true of the sign for +“less than” which is frequently written +<table class="symbol" summary="̅ ̲̅|"><tr><td class="top"></td><td class="rb"></td></tr></table>. +Oughtred’s symbols, or these symbols turned about in some way, have +been used by Seth Ward,<a class="fn" id="fr_123" href="#fn_123">[123]</a> John Wallis,<a class="fn" id="fr_124" href="#fn_124">[124]</a> Isaac Barrow,<a class="fn" id="fr_125" href="#fn_125">[125]</a> +<div class="pb" id="Page_82">[82]</div> +John Kersey,<a class="fn" id="fr_126" href="#fn_126">[126]</a> E. Wells,<a class="fn" id="fr_127" href="#fn_127">[127]</a> John Hawkins,<a class="fn" id="fr_128" href="#fn_128">[128]</a> Tho. Baker,<a class="fn" id="fr_129" href="#fn_129">[129]</a> +Richard Sault,<a class="fn" id="fr_130" href="#fn_130">[130]</a> Richard Rawlinson,<a class="fn" id="fr_131" href="#fn_131">[131]</a> Franciscus Dulaurens,<a class="fn" id="fr_132" href="#fn_132">[132]</a> +James Milnes,<a class="fn" id="fr_133" href="#fn_133">[133]</a> George Cheyne,<a class="fn" id="fr_134" href="#fn_134">[134]</a> John Craig,<a class="fn" id="fr_135" href="#fn_135">[135]</a> Jo. +Wilson,<a class="fn" id="fr_136" href="#fn_136">[136]</a> and J. Collins.<a class="fn" id="fr_137" href="#fn_137">[137]</a></div> +<p>General acceptance has been accorded to Oughtred’s +symbol ×. The first printed appearance of this symbol +for multiplication in 1618 in the form of the letter <i>x</i> hardly +explains its real origin. The author of the “Appendix” +(be he Oughtred or someone else) may not have used the +letter <i>x</i> at all, but may have written the cross ×, called +the St. Andrew’s cross, while the printer, in the absence +of any type accurately representing that cross, may have +substituted the letter <i>x</i> in its place. The hypothesis +that the symbol × of multiplication owes its origin to +the old habit of using directed bars to indicate that two +<span class="pb" id="Page_83">[83]</span> +numbers are to be combined, as for instance in the multiplication +of 23 and 34, thus,</p> +<table class="center"> +<tr><td>2<br /><span class="xxlarge">|</span><br />3</td> +<td style="line-height:50%; font-size:800%; margin-top:-.5em; margin-bottom:-.3em; margin-right:-.2em; margin-left:-.2em; ">×</td> +<td>3<br /><span class="xxlarge">|</span><br />4</td></tr> +<tr><td colspan="3"><hr style="width:100%" /></td></tr> +<tr><td>7</td><td>8</td><td>2</td></tr> +</table> +<p>has been advanced by two writers, C. Le Paige<a class="fn" id="fr_138" href="#fn_138">[138]</a> and +Gravelaar.<a class="fn" id="fr_139" href="#fn_139">[139]</a> Bosmans is more inclined to the belief that +Oughtred adopted the symbol somewhat arbitrarily, +much as he did the numerous symbols in his <i>Elementi +decimi Euclidis declaratio</i>.<a class="fn" id="fr_140" href="#fn_140">[140]</a></p> +<p>Le Paige’s and Gravelaar’s theory finds some support +in the fact that the cross ×, without the two additional +vertical lines shown above, occurs in a commentary +published by Oswald Schreshensuchs<a class="fn" id="fr_141" href="#fn_141">[141]</a> in 1551, where the +sign is written between two factors placed one above the +other.</p> +<div class="pb" id="Page_84">[84]</div> +<h2 id="c24">CHAPTER V +<br /><span class="small">OUGHTRED’S IDEAS ON THE TEACHING OF MATHEMATICS</span></h2> +<h3 id="c25">GENERAL STATEMENT</h3> +<p>Nowhere has Oughtred given a full and systematic +exposition of his views on mathematical teaching. Nevertheless, +he had very pronounced and clear-cut ideas on the +subject. That a man who was not a teacher by profession +should have mature views on teaching is most interesting. +We gather his ideas from the quality of the books he published, +from his prefaces, and from passages in his controversial +writing against Delamain. As we proceed to +give quotations unfolding Oughtred’s views, we shall +observe that three points receive special emphasis: (1) an +appeal to the eye through suitable symbolism; (2) emphasis +upon rigorous thinking; (3) the postponement of +the use of mathematical instruments until after the +logical foundations of a subject have been thoroughly +mastered.</p> +<p>The importance of these tenets is immensely reinforced +by the conditions of the hour. This voice from the past +speaks wisdom to specialists of today. Recent methods +of determining educational values and the modern cult +of utilitarianism have led some experts to extraordinary +conclusions. Laboratory methods of testing, by the narrowness +of their range, often mislead. Thus far they have +been inferior to the word of a man of experience, insight, +and conviction.</p> +<div class="pb" id="Page_85">[85]</div> +<h3 id="c26">MATHEMATICS, “A SCIENCE OF THE EYE”</h3> +<p>Oughtred was a great admirer of the Greek mathematicians—Euclid, +Archimedes, Apollonius of Perga, +Diophantus. But in reading their works he experienced +keenly what many modern readers have felt, namely, +that the almost total absence of mathematical symbols +renders their writings unnecessarily difficult to read. +Statements that can be compressed into a few well-chosen +symbols which the eye is able to survey as a whole are +expressed in long-drawn-out sentences. A striking illustration +of the importance of symbolism is afforded by the +history of the formula</p> +<div class="verse"> +<p class="t0"><i>ix</i>=log(cos <i>x</i>+<i>i</i> sin <i>x</i>).</p> +</div> +<p>It was given in Roger Cotes’ <i>Harmonia mensurarum</i>, +1722, not in symbols, but expressed in rhetorical form, +destitute of special aids to the eye. The result was that +the theorem remained in the book undetected for 185 +years and was meanwhile rediscovered by others. Owing +to the prominence of Cotes as a mathematician it is very +improbable that such a thing could have happened had the +theorem been thrust into view by the aid of mathematical +symbols.</p> +<p>In studying the ancient authors Oughtred is reported +to have written down on the margin of the printed page +some of the theorems and their proofs, expressed in the +symbolic language of algebra.</p> +<p>In the preface of his <i>Clavis</i> of 1631 and of 1647 he says:</p> +<blockquote> +<p>Wherefore, that I might more clearly behold the things +themselves, I uncasing the Propositions and Demonstrations +out of their covert of words, designed them in notes and species +appearing to the very eye. After that by comparing the divers +<span class="pb" id="Page_86">[86]</span> +affections of Theorems, inequality, proportion, affinity, and +dependence, I tryed to educe new out of them.</p> +</blockquote> +<p>It was this motive which led him to introduce the many +abbreviations in algebra and trigonometry to which +reference has been made in previous pages. The pedagogical +experience of recent centuries has indorsed Oughtred’s +view, provided of course that the pupil is carefully +taught the exact meaning of the symbols. There have +been and there still are those who oppose the intensive use +of symbolism. In our day the new symbolism for all +mathematics, suggested by the school of Peano in Italy, +can hardly be said to be received with enthusiasm. In +Oughtred’s day symbolism was not yet the fashion. To +be convinced of this fact one need only open a book of +Edmund Gunter, with whom Oughtred came in contact +in his youth, or consult the <i>Principia</i> of Sir Isaac Newton, +who flourished after Oughtred. The mathematical works +of Gunter and Newton, particularly the former, are +surprisingly destitute of mathematical symbols. The +philosopher Hobbes, in a controversy with John Wallis, +criticized the latter for that “Scab of Symbols,” whereupon +Wallis replied:</p> +<blockquote> +<p>I wonder how you durst touch M. Oughtred for fear of catching +the Scab. For, doubtlesse, his book is as much covered +over with the Scab of Symbols, as any of mine. . . . . As for +my Treatise of Conick Sections, you say, it is covered over with +the Scab of Symbols, that you had not the patience to examine +whether it is well or ill demonstrated.<a class="fn" id="fr_142" href="#fn_142">[142]</a></p> +</blockquote> +<div class="pb" id="Page_87">[87]</div> +<p>Oughtred maintained his view of the importance of +symbols on many different occasions. Thus, in his <i>Circles +of Proportion</i>, 1632, p. 20:</p> +<blockquote> +<p>This manner of setting downe Theoremes, whether they be +Proportions, or Equations, by Symboles or notes of words, is +most excellent, artificiall, and doctrinall. Wherefore I earnestly +exhort every one, that desireth though but to looke into +these noble Sciences Mathematicall, to accustome themselves +unto it: and indeede it is easie, being most agreeable to reason, +yea even to sence. And out of this working may many singular +consectaries be drawne: which without this would, it may be, +for ever lye hid.</p> +</blockquote> +<h3 id="c27">RIGOROUS THINKING AND THE USE OF INSTRUMENTS</h3> +<p>The author’s elevated concept of mathematical study +as conducive to rigorous thinking shines through the following +extract from his preface to the 1647 <i>Clavis</i>:</p> +<blockquote> +<p>. . . . Which Treatise being not written in the usuall synthetical +manner, nor with verbous expressions, but in the inventive +way of Analitice, and with symboles or notes of things +instead of words, seemed unto many very hard; though indeed +it was but their owne diffidence, being scared by the newnesse +of the delivery; and not any difficulty in the thing it selfe. +For this specious and symbolicall manner, neither racketh the +memory with multiplicity of words, nor chargeth the phantasie +with comparing and laying things together; but plainly presenteth +to the eye the whole course and processe of every operation +and argumentation.</p> +<p>Now my scope and intent in the first Edition of that my +Key was, and in this New Filing, or rather forging of it, is, to +reach out to the ingenious lovers of these Sciences, as it were +Ariadnes thread, to guide them through the intricate Labyrinth +of these studies, and to direct them for the more easie and full +understanding of the best and antientest Authors. . . . . +<span class="pb" id="Page_88">[88]</span> +That they may not only learn their propositions, which is the +highest point of Art that most Students aime at; but also may +perceive with what solertiousnesse, by what engines of aequations, +Interpretations, Comparations, Reductions, and Disquisitions, +those antient Worthies have beautified, enlarged, +and first found out this most excellent Science. . . . . Lastly, +by framing like questions problematically, and in a way of +Analysis, as if they were already done, resolving them into their +principles, I sought out reasons and means whereby they might +be effected. And by this course of practice, not without long +time, and much industry, I found out this way for the helpe +and facilitation of Art.</p> +</blockquote> +<p>Still greater emphasis upon rigorous thinking in mathematics +is laid in the preface to the <i>Circles of Proportion</i> +and in some parts of his <i>Apologeticall Epistle</i> against +Delamain. In that preface William Forster quotes the +reply of Oughtred to the question how he (Oughtred) had +for so many years concealed his invention of the slide +rule from himself (Forster) whom he had taught so many +other things. The reply was:</p> +<blockquote> +<p>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 in-stead of Artists, to make their Scholers only doers +of tricks, and as it were Iuglers: to the despite of Art, losse +of previous time, and betraying of willing and industrious +wits, vnto ignorance, and idlenesse. That the vse 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 +haue me well instructed in the Sciences.”</p> +</blockquote> +<p>Delamain took a different view, arguing that instruments +might very well be placed in the hands of pupils +from the start. At the time of this controversy Delamain +<span class="pb" id="Page_89">[89]</span> +supported himself by teaching mathematics in London +and he advertised his ability to give instruction in mathematics, +including the use of instruments. Delamain +brought the charge against Oughtred of unjustly calling +“many of the [British] Nobility and Gentry doers of trickes +and juglers.” To this Oughtred replies:</p> +<blockquote> +<p>As I did to Delamain and to some others, so I did to +William Forster: I freely gave him my helpe and instruction in +these faculties: only this was the difference, I had the very +first moulding (as I may say) of this latter: But Delamain +was already corrupted with doring upon Instruments, and quite +lost from ever being made an Artist: I suffered not William +Forster for some time so much as speake of any Instrument, +except only the Globe it selfe; and to explicate, and worke +the questions of the Sphaere, by the way of the Analemma: +which also himselfe did describe for the present occasion. And +this my restraint from such pleasing avocations, and holding +him to the strictnesse of percept, brought forth this fruit, that +in short time, even by his owne skill, he could not onely use +any Instrument he should see, but also was able to delineate the +like, and devise others.<a class="fn" id="fr_143" href="#fn_143">[143]</a></p> +</blockquote> +<p>As representing Delamain’s views, we make the following +selection from his <i>Grammelogia</i> (London, about +1633), the part near the end of the book and bearing the +title, “In the behalfe of vulgar Teachers and others,” +where Delamain refers to Oughtred’s charge that the +scholars of “vulgar” teachers are “doers of tricks, as it +were iuglers.” Delamain says:</p> +<blockquote> +<p>. . . . Which words are neither <i>cautelous</i>, nor <i>subterfugious</i>, +but are as downe right in their <i>plainnesse</i>, as they are touching, +and <i>pernitious</i>, by two much derogating from many, and glancing +upon many <i>noble personages</i>, with too <i>grosse</i>, if not too +<i>base</i> an attribute, in tearming them <i>doers of tricks, as it were to +<span class="pb" id="Page_90">[90]</span> +iuggle</i>: because they perhaps make use of a necessitie in the +furnishing of themselves with such knowledge by <i>Practicall +Instrumentall operation</i>, when their more weighty <i>negotiations</i> +will not permit them for <i>Theoreticall figurative demonstration</i>; +those that are guilty of the aspertion, and are touched therewith +may answer for themselves, and studie to be more <i>Theoreticall</i>, +than <i>Practicall</i>: for the <i>Theory</i>, is as the <i>Mother</i> that produceth +the <i>daughter</i>, the very sinewes and life of <i>Practise</i>, the excellencie +and highest degree of true <i>Mathematicall Knowledge</i>: +but for those that would make but a step as it were into that +kind of <i>Learning</i>, whose onely desire is expedition, and facilitie, +both which by the generall consent of all are best effected with +Instrument, rather then with tedious regular demonstrations, it +was ill to checke them so grosly, not onely in what they have +<i>Practised</i>, but abridging them also of their liberties with what +they may <i>Practise</i>, which aspertion may not easily be slighted +off by any <i>glosse</i> or <i>Apologie</i>, without an Ingenuous <i>confession</i>, +or some mentall reservation: To which vilification, howsoever, +in the behalfe of my selfe, and others, I answer; That <i>Instrumentall</i> +operation is not only the Compendiating, and facilitating +of <i>Art</i>, but even the glory of it, whole demonstration both +of the making, and operation is soly in the <i>science</i>, and to an +<i>Artist</i> or disputant proper to be knowne, and so to all, who +would truly know the cause of the <i>Mathematicall operations</i> +in their originall; But, for none to know the use of a <i>Mathematicall +Instrumen</i>[<i>t</i>], except he knowes the cause of its operation, +is somewhat too strict, which would keepe many from +affecting the <i>Art</i>, which of themselves are ready enough every +where, to conceive more harshly of the difficultie, and impossibilitie +of attayning any skill therein, then it deserves, because +they see nothing but obscure propositions, and perplex and +intricate demonstrations before their eyes, whose unsavoury +tartnes, to an unexperienced palate like bitter pills is sweetned +over, and made pleasant with an <i>Instrumentall compendious +facilitie</i>, and made to goe downe the more readily, and yet to +retaine the same vertue, and working; And me thinkes in this +<span class="pb" id="Page_91">[91]</span> +queasy age, all <i>helpes</i> may bee used to procure a <i>stomacke</i>, all +<i>bates</i> and invitations to the declining studie of so noble a <i>Science</i>, +rather then by rigid Method and generall <i>Lawes</i> to scarre men +away. All are not of like disposition, neither all (as was sayd +before) propose the same end, some resolve to <i>wade</i>, others +to put a <i>finger</i> in onely, or wet a <i>hand</i>: now thus to tye them +to an obscure and <i>Theoricall</i> forme of teaching, is to crop their +hope, even in the very bud. . . . . The beginning of a <i>mans +knowledge</i> even in the use of an <i>Instrument</i>, is first founded on +<i>doctrinal precepts</i>, and these precepts may be conceived all +along in its use: and are so farre from being excluded, that they +doe necessarily <i>concomitate</i> and are contained therein: the +<i>practicke</i> being better understood by the <i>doctrinall part</i>, and +this later explained by the <i>Instrumentall</i>, making precepts +obvious unto sense, and the <i>Theory</i> going along with the +<i>Instrument</i>, better informing and inlightning the understanding, +etc. <i>vis vnita fortior</i>, so as if that in <i>Phylosophy</i> bee true, <i>Nihil +est</i> [<i>in</i>] <i>intellectu quod non prius fuit in sensu</i>.</p> +</blockquote> +<p>The difference between Oughtred and Delamain as to +the use of mathematical instruments raises important +questions. Should the slide rule be placed in the hands of +a boy before, or after, he has mastered the theory of logarithms? +Should logarithmic tables be withheld from him +until the theoretical foundation is laid in the mind of the +pupil? Is it a good thing to let a boy use a surveying +instrument unless he first learns trigonometry? Is it +advisable to permit a boy to familiarize himself with the +running of a dynamo before he has mastered the underlying +principles of electricity? Does the use of instruments +ordinarily discourage a boy from mastery of the +theory? Or does such manipulation constitute a natural +and pleasing approach to the abstract? On this particular +point, who showed the profounder psychological insight, +Oughtred or Delamain?</p> +<div class="pb" id="Page_92">[92]</div> +<p>In July, 1914, there was held in Edinburgh a celebration +of the three-hundredth anniversary of the invention +of logarithms. On that occasion there was collected at +Edinburgh university one of the largest exhibits ever seen +of modern instruments of calculation. The opinion was +expressed by an experienced teacher that “weapons as +those exhibited there are for men and not for boys, and +such danger as there may be in them is of the same +character as any form of too early specialization.”</p> +<p>It is somewhat of a paradox that Oughtred, who in his +student days and during his active years felt himself +impelled to invent sun-dials, planispheres, and various +types of slide rules—instruments which represent the +most original contributions which he handed down to +posterity—should discourage the use of such instruments +in teaching mathematics to beginners. That without the +aid of instruments he himself should have succeeded so +well in attracting and inspiring young men constitutes +the strongest evidence of his transcendent teaching ability. +It may be argued that his pedagogic dogma, otherwise +so excellent, here goes contrary to the course he himself +followed instinctively in his self-education along mathematical +lines. We read that Sir Isaac Newton, as a child, +constructed sun-dials, windmills, kites, paper lanterns, +and a wooden clock. Should these activities have been +suppressed? Ordinary children are simply Isaac Newtons +on a smaller intellectual scale. Should their activities +along these lines be encouraged or checked?</p> +<p>On the other hand, it may be argued that the paradox +alluded to above admits of explanation, like all paradoxes, +and that there is no inconsistency between Oughtred’s +pedagogic views and his own course of development. If +he invented sun-dials, he must have had a comprehension +<span class="pb" id="Page_93">[93]</span> +of the cosmic motions involved; if he solved spherical +triangles graphically by the aid of the planisphere, he must +have understood the geometry of the sphere, so far as it +relates to such triangles; if he invented slide rules, he +had beforehand a thorough grasp of logarithms. The +question at issue does not involve so much the invention +of instruments, as the use by the pupil of instruments +already constructed, before he fully understands the +theory which is involved. Nor does Sir Isaac Newton’s +activity as a child establish Delamain’s contention. Of +course, a child should not be discouraged from manual +activity along the line of producing interesting toys in +imitation of structures and machines that he sees, but +to introduce him to the realm of abstract thought by the +aid of instruments is a different proposition, fraught with +danger. A boy may learn to use a slide rule mechanically +and, because of his ability to obtain practical results, +feel justified in foregoing the mastery of underlying theory; +or he may consider the ability of manipulating a surveying +instrument quite sufficient, even though he be ignorant of +geometry and trigonometry; or he may learn how to +operate a dynamo and an electric switchboard and be +altogether satisfied, though having no grasp of electrical +science. Thus instruments draw a youth aside from the +path leading to real intellectual attainments and real +efficiency; they allure him into lanes which are often +blind alleys. Such were the views of Oughtred.</p> +<p>Who was right, Oughtred or Delamain? It may be +claimed that there is a middle ground which more nearly +represents the ideal procedure in teaching. Shall the slide +rule be placed in the student’s hands at the time when +he is engaged in the mastery of principles? Shall there +be an alternate study of the theory of logarithms and of +<span class="pb" id="Page_94">[94]</span> +the slide rule—on the idea of one hand washing the other—until +a mastery of both the theory and the use of the +instrument has been attained? Does this method not +produce the best and most lasting results? Is not this +Delamain’s actual contention? We leave it to the reader +to settle these matters from his own observation, knowledge, +and experience.</p> +<h3 id="c28">NEWTON’S COMMENTS ON OUGHTRED</h3> +<p>Oughtred is an author who has been found to be of +increasing interest to modern historians of mathematics. +But no modern writer has, to our knowledge, pointed out +his importance in the history of the <i>teaching</i> of mathematics. +Yet his importance as a teacher did receive +recognition in the seventeenth century by no less distinguished +a scientist than Sir Isaac Newton. On May 25, +1694, Sir Isaac Newton wrote a long letter in reply to a +request for his recommendation on a proposed new course +of study in mathematics at Christ’s Hospital. Toward +the close of his letter, Newton says:</p> +<blockquote> +<p>And now I have told you my opinion in these things, I will +give you Mr. Oughtred’s, a Man whose judgment (if any man’s) +may be safely relyed upon. For he in his book of the circles +of proposition, in the end of what he writes about Navigation +(page 184) has this exhortation to Seamen. “And if,” saith +he, “the Masters of Ships and Pilots will take the pains in the +Journals of their Voyages diligently and faithfully to set down +in severall columns, not onely the Rumb they goe on and the +measure of the Ships way in degrees, and the observation of +Latitude and variation of their compass; but alsoe their conjectures +and reason of their correction they make of the aberrations +they shall find, and the qualities and condition of their +ship, and the diversities and seasons of the winds, and the +secret motions or agitations of the Seas, when they begin, and +<span class="pb" id="Page_95">[95]</span> +how long they continue, how farr they extend and with what +inequality; and what else they shall observe at Sea worthy +consideration, and will be pleased freely to communicate the +same with Artists, such as are indeed skilfull in the Mathematicks +and lovers and enquirers of the truth: I doubt not +but that there shall be in convenient time, brought to light +many necessary precepts which may tend to y<sup>e</sup> perfecting of +Navigation, and the help and safety of such whose Vocations +doe inforce them to commit their lives and estates in the vast +Ocean to the providence of God.” Thus farr that very good +and judicious man Mr. Oughtred. I will add, that if instead of +sending the Observations of Seamen to able Mathematicians +at Land, the Land would send able Mathematicians to Sea, +it would signify much more to the improvem<sup>t</sup> of Navigation and +safety of Mens lives and estates on that element.<a class="fn" id="fr_144" href="#fn_144">[144]</a></p> +</blockquote> +<p>May Oughtred prove as instructive to the modern +reader as he did to Newton!</p> +<div class="pb" id="Page_96">[96]</div> +<h2>Footnotes</h2> +<div class="fnblock"><div class="fndef"><a class="fn" id="fn_1" href="#fr_1">[1]</a>Aubrey’s <i>Brief Lives</i>, ed. A. Clark, Vol. II, Oxford, 1898, p. 106. +</div><div class="fndef"><a class="fn" id="fn_2" href="#fr_2">[2]</a>“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 <i>Grammelogia</i>, or the Mathematicall Ring, +or <i>Mirifica logarithmorum projectio circularis</i>” [1633?], p. 8. Hereafter +we shall refer to this pamphlet as the <i>Apologeticall Epistle</i>, this +name appearing on the page-headings. +</div><div class="fndef"><a class="fn" id="fn_3" href="#fr_3">[3]</a><i>Companion to the [British] Almanac of 1837</i>, 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.” +</div><div class="fndef"><a class="fn" id="fn_4" href="#fr_4">[4]</a><i>New and General Biographical Dictionary</i> (John Nichols), London, +1784, art. “Oughtred.” +</div><div class="fndef"><a class="fn" id="fn_5" href="#fr_5">[5]</a>Rev. Owen Manning, <i>History of Antiquities in Surrey</i>, Vol. II, +p. 132. +</div><div class="fndef"><a class="fn" id="fn_6" href="#fr_6">[6]</a><i>Skeleton Collegii Regalis Cantab.: Or A Catalogue of All the +Provosts, Fellows and Scholars, of the King’s College . . . . since the +Foundation Thereof</i>, Vol. II, “William Oughtred.” +</div><div class="fndef"><a class="fn" id="fn_7" href="#fr_7">[7]</a>Aubrey, <i>op. cit.</i>, Vol. II, p. 107. +</div><div class="fndef"><a class="fn" id="fn_8" href="#fr_8">[8]</a>Rigaud, <i>Correspondence of Scientific Men of the Seventeenth +Century</i>, Oxford, Vol. I, 1841, p. 5. +</div><div class="fndef"><a class="fn" id="fn_9" href="#fr_9">[9]</a>Aubrey, <i>op. cit.</i>, Vol. II, p. 110. +</div><div class="fndef"><a class="fn" id="fn_10" href="#fr_10">[10]</a><i>Ibid.</i>, p. 111. +</div><div class="fndef"><a class="fn" id="fn_11" href="#fr_11">[11]</a><i>Op. cit.</i>, Vol. II, p. 132. +</div><div class="fndef"><a class="fn" id="fn_12" href="#fr_12">[12]</a><i>Mr. William Lilly’s History of His Life and Times, From the +Year 1602 to 1681</i>, London, 1715, p. 58. +</div><div class="fndef"><a class="fn" id="fn_13" href="#fr_13">[13]</a>Rigaud, <i>op. cit.</i>, Vol. I, p. 60. +</div><div class="fndef"><a class="fn" id="fn_14" href="#fr_14">[14]</a>Aubrey, <i>op. cit.</i>, Vol. II, p. 107. +</div><div class="fndef"><a class="fn" id="fn_15" href="#fr_15">[15]</a>Rigaud, <i>op. cit.</i>, Vol. I, p. 16. +</div><div class="fndef"><a class="fn" id="fn_16" href="#fr_16">[16]</a>Owen Manning, <i>op. cit.</i>, p. 132. +</div><div class="fndef"><a class="fn" id="fn_17" href="#fr_17">[17]</a><i>New and General Biographical Dictionary</i> (John Nichols), +London, 1784, art. “Oughtred.” +</div><div class="fndef"><a class="fn" id="fn_18" href="#fr_18">[18]</a><i>Op. cit.</i>, Vol. II, p. 110. +</div><div class="fndef"><a class="fn" id="fn_19" href="#fr_19">[19]</a>Rev. Owen Manning, <i>The History and Antiquities of Surrey</i>, +Vol. II, London, 1809, p. 132. +</div><div class="fndef"><a class="fn" id="fn_20" href="#fr_20">[20]</a><i>Op. cit.</i>, Vol. II, 1898, p. 111. +</div><div class="fndef"><a class="fn" id="fn_21" href="#fr_21">[21]</a><i>Budget of Paradoxes</i>, London, 1872, p. 451; 2d ed., Chicago and +London, 1915, Vol. II, p. 303. +</div><div class="fndef"><a class="fn" id="fn_22" href="#fr_22">[22]</a>The full title of the <i>Clavis</i> of 1631 is as follows: <i>Arithmeticae +in numeris et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae, +atqve adeo totivs mathematicae, qvasi clavis est.—Ad nobilissimvm +spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur, +Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae & +Svrriae, Comitis Mareschalli Angliae, &c filium.—Londini, Apud +Thomam Harpervm.</i> M.DC.XXXI. +<p class="fncont">In all there appeared five Latin editions, the second in 1648 at London, +the third in 1652 at Oxford, the fourth in 1667 at Oxford, the fifth +in 1693 and 1698 at Oxford. There were two independent English +editions: the first in 1647 at London, translated in greater part by +Robert Wood of Lincoln College, Oxford, as is stated in the preface +to the 1652 Latin edition; the second in 1694 and 1702 is a new translation, +the preface being written and the book recommended by the +astronomer Edmund Halley. The 1694 and 1702 impressions labored +under the defect of many sense-disturbing errors due to careless +reading of the proofs. All the editions of the <i>Clavis</i>, after the first +edition, had one or more of the following tracts added on:</p> +<dl> +<dt><i>Eq.</i>=<i>De Aequationum affectarvm resolvtione in numeris.</i></dt> +<dt><i>Eu.</i>=<i>Elementi decimi Euclidis declaratio.</i></dt> +<dt><i>So.</i>=<i>De Solidis regularibus, tractatus.</i></dt> +<dt><i>An.</i>=<i>De Anatocismo, sive usura composita.</i></dt> +<dt><i>Fa.</i>=<i>Regula falsae positionis.</i></dt> +<dt><i>Ar.</i>=<i>Theorematum in libris Archimedis de Sphaera & cylindro declaratio.</i></dt> +<dt><i>Ho.</i>=<i>Horologia scioterica in plano, geometricè delineandi modus.</i></dt> +</dl> +<p class="fncont">The abbreviated titles given here are, of course, our own. The +lists of tracts added to the <i>Clavis mathematicae</i> of 1631 in its later +editions, given in the order in which the tracts appear in each edition, +are as follows: <i>Clavis</i> of 1647, <i>Eq.</i>, <i>An.</i>, <i>Fa.</i>, <i>Ho.</i>; <i>Clavis</i> of 1648, +<i>Eq.</i>, <i>An.</i>, <i>Fa.</i>, <i>Eu.</i>, <i>So.</i>; <i>Clavis</i> of 1652, <i>Eq.</i>, <i>Eu.</i>, <i>So.</i>, <i>An.</i>, <i>Fa.</i>, +<i>Ar.</i>, <i>Ho.</i>; <i>Clavis</i> of 1667, <i>Eq.</i>, <i>Eu.</i>, <i>So.</i>, <i>An.</i>, <i>Fa.</i>, <i>Ar.</i>, <i>Ho.</i>; <i>Clavis</i> of +1693 and 1698, <i>Eq.</i>, <i>Eu.</i>, <i>So.</i>, <i>An.</i>, <i>Fa.</i>, <i>Ar.</i>, <i>Ho.</i>; <i>Clavis</i> of 1694 and +1702, <i>Eq.</i></p> +<p class="fncont">The title-page of the <i>Clavis</i> was considerably modified after the +first edition. Thus, the 1652 Latin edition has this title-page: +<i>Guilelmi Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia +Socii, Clavis mathematicae denvo limata, sive potius fabricata. Cum +aliis quibusdam ejusdem commentationibus, quae in sequenti pagina +recensentur. Editio tertia auctior & emendatior. Oxoniae, Excudebat +Leon. Lichfield, Veneunt apud Tho. Robinson. 1652.</i></p> +</div><div class="fndef"><a class="fn" id="fn_23" href="#fr_23">[23]</a>Rigaud, <i>op. cit.</i>, Vol. II, p. 476. +</div><div class="fndef"><a class="fn" id="fn_24" href="#fr_24">[24]</a>See, for instance, the <i>Clavis mathematicae</i> of 1652, where he +expresses himself thus (p. 4): “Speciosa haec Arithmetica arti +Analyticae (per quam ex sumptione quaesiti, tanquam noti, +investigatur quaesitum) multo accommodatior est, quam illa +numerosa.” +</div><div class="fndef"><a class="fn" id="fn_25" href="#fr_25">[25]</a>Oughtred, <i>The Key of the Mathematicks</i>, London, 1647, p. 4. +</div><div class="fndef"><a class="fn" id="fn_26" href="#fr_26">[26]</a><i>Clavis</i> 1694, p. 19, and the <i>Clavis</i> of 1631, p. 8. +</div><div class="fndef"><a class="fn" id="fn_27" href="#fr_27">[27]</a>See for instance, Oughtred’s <i>Elementi decimi Euclidis declaratio</i>, +1652, p. 1, where he uses <i>A</i> and <i>E</i>, and also <i>a</i> and <i>e</i>. +</div><div class="fndef"><a class="fn" id="fn_28" href="#fr_28">[28]</a>See <i>Christophori Clavii Bambergensis Operum mathematicorum, +tomus secundus</i>, Moguntiae, M.DC.XI, algebra, p. 39. +</div><div class="fndef"><a class="fn" id="fn_29" href="#fr_29">[29]</a><i>Christophori Clavii operum mathematicorum Tomus Secundus</i>, +Moguntiae, M.DC.XI, <i>Epitome arithmeticae</i>, p. 36. +</div><div class="fndef"><a class="fn" id="fn_30" href="#fr_30">[30]</a>See F. Cajori, “The Cross × as a Symbol of Multiplication,” +in <i>Nature</i>, Vol. XCIV (1914), p. 363. +</div><div class="fndef"><a class="fn" id="fn_31" href="#fr_31">[31]</a>See <i>Elementi decimi Euclidis declaratio</i>, 1652, p. 2. +</div><div class="fndef"><a class="fn" id="fn_32" href="#fr_32">[32]</a>See Johannis Wallisii <i>Operum mathematicorum pars prima</i>, +Oxonii, 1657, p. 247. +</div><div class="fndef"><a class="fn" id="fn_33" href="#fr_33">[33]</a><i>Clavis</i> of 1631, chap. xix, sec. 5, p. 50. +</div><div class="fndef"><a class="fn" id="fn_34" href="#fr_34">[34]</a>We have noticed the representation of known quantities by +consonants and the unknown by vowels in Wingate’s <i>Arithmetick +made easie</i>, edited by John Kersey, London, 1650, algebra, p. 382; +and in the second part, section 19, of Jonas Moore’s <i>Arithmetick in +two parts</i>, London, 1660, Moore suggests as an alternative the use +of <i>z</i>, <i>y</i>, <i>x</i>, etc., for the unknowns. The practice of representing +unknowns by vowels did not spread widely in England. +</div><div class="fndef"><a class="fn" id="fn_35" href="#fr_35">[35]</a><i>Philosophical Transactions</i>, Vol. XIX, No. 231, London, p. 652. +</div><div class="fndef"><a class="fn" id="fn_36" href="#fr_36">[36]</a><i>Ibid.</i>, Vol. XIX, p. 56. +</div><div class="fndef"><a class="fn" id="fn_37" href="#fr_37">[37]</a>There +are two title-pages to the edition of 1632. The first title-page +is as follows: <i>The Circles of Proportion and The Horizontall +Instrument. Both invented, and the vses 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. +T. Cecill Sculp.</i> +<p class="fncont">In 1633 there was added the following, with a separate title-page: +<i>An addition vnto the Vse of the Instrvment called the Circles of +Proportion. . . . . London, 1633</i>, this being followed by +Oughtred’s <i>To the English Gentrie etc.</i> In the British +Museum there is a copy of another impression of the <i>Circles of +Proportion</i>, dated 1639, with the <i>Addition vnto the Vse of +the Instrument etc.</i>, bearing the original date, 1633, and with +the epistle, <i>To the English Gentrie, etc.</i>, inserted +immediately after Forster’s dedication, instead of at the +end of the volume.</p> +</div><div class="fndef"><a class="fn" id="fn_38" href="#fr_38">[38]</a>The complete title of the English edition is as follows: <i>Trigonometrie, +or, The manner of calculating the Sides and Angles of Triangles, +by the Mathematical Canon, demonstrated. By William Oughtred +Etonens. And published by Richard Stokes Fellow of Kings Colledge in +Cambridge, and Arthur Haughton Gentleman. London, Printed by +R. and W. Leybourn, for Thomas Johnson at the Golden Key in St. +Pauls Church-yard.</i> M.DC.LVII. +</div><div class="fndef"><a class="fn" id="fn_39" href="#fr_39">[39]</a>Jer. Collier, <i>The Great Historical, Geographical, Genealogical and +Poetical Dictionary</i>, Vol. II, London, 1701, art. “Oughtred.” +</div><div class="fndef"><a class="fn" id="fn_40" href="#fr_40">[40]</a>Rigaud <i>op. cit.</i>, Vol. I, p. 82. +</div><div class="fndef"><a class="fn" id="fn_41" href="#fr_41">[41]</a>A. De Morgan, <i>Budget of Paradoxes</i>, London, 1872, p. 451; 2d +ed., Chicago, 1915, Vol. II, p. 303. +</div><div class="fndef"><a class="fn" id="fn_42" href="#fr_42">[42]</a>E. Gunter, <i>Description and Use of the Sector, the Crosse-staffe and +other Instruments</i>, London, 1624, second book, p. 31. +</div><div class="fndef"><a class="fn" id="fn_43" href="#fr_43">[43]</a>F. Cajori, “On the History of a Notation in Trigonometry,” +<i>Nature</i>, Vol. XCIV, 1915, pp. 642, 643. +</div><div class="fndef"><a class="fn" id="fn_44" href="#fr_44">[44]</a>A. von Braunmühl, <i>Geschichte der Trigonometrie</i>, 2. Teil, Leipzig, +1903, pp. 42, 91. +</div><div class="fndef"><a class="fn" id="fn_45" href="#fr_45">[45]</a>H. Hankel, <i>Geschichte der Mathematik in Alterthum und Mittelalter</i>, +Leipzig, 1874, pp. 369, 370. +</div><div class="fndef"><a class="fn" id="fn_46" href="#fr_46">[46]</a>M. Cantor, <i>Vorlesungen über Geschichte der Mathematik</i>, II, +1900, pp. 640, 641. +</div><div class="fndef"><a class="fn" id="fn_47" href="#fr_47">[47]</a>This matter has been discussed in a paper by F. Cajori, “A +History of the Arithmetical Methods of Approximation, etc., +<i>Colorado College Publication</i>, General Series No. 51, 1910, pp. 182-84. +Later this subject was again treated by G. Eneström in <i>Bibliotheca +mathematica</i>, 3. Folge, Vol. XI, 1911, pp. 234, 235. +</div><div class="fndef"><a class="fn" id="fn_48" href="#fr_48">[48]</a>See F. Cajori, <i>op. cit.</i>, p. 193. +</div><div class="fndef"><a class="fn" id="fn_49" href="#fr_49">[49]</a>See William Oughtred’s <i>Key of the Mathematicks</i>, London, 1694, +pp. 173-75, tract, “Of the Resolution of the Affected Equations,” +or any edition of the <i>Clavis</i> after the first. +</div><div class="fndef"><a class="fn" id="fn_50" href="#fr_50">[50]</a>A. De Morgan, <i>op. cit.</i>, p. 451; 2d ed., Vol. II, p. 303. +</div><div class="fndef"><a class="fn" id="fn_51" href="#fr_51">[51]</a>See F. Cajori, <i>History of the Logarithmic Slide Rule</i>, New York, +1909, pp. 7-14, Addenda, p. ii. +</div><div class="fndef"><a class="fn" id="fn_52" href="#fr_52">[52]</a>Rigaud, <i>op. cit.</i>, Vol. I, p. 12. +</div><div class="fndef"><a class="fn" id="fn_53" href="#fr_53">[53]</a><i>The New Artificial Gauging Line or Rod: together with rules concerning +the use thereof: Invented and written by William Oughtred</i>, +London, 1633. +</div><div class="fndef"><a class="fn" id="fn_54" href="#fr_54">[54]</a>W. Oughtred, <i>Apologeticall Epistle</i>, p. 13. +</div><div class="fndef"><a class="fn" id="fn_55" href="#fr_55">[55]</a><i>Quarterly Journal of Pure and Applied Mathematics</i>, Vol. XLVI, +(1915), p. 169. In this article Glaisher republishes the “Appendix” +in full. +</div><div class="fndef"><a class="fn" id="fn_56" href="#fr_56">[56]</a>Aubrey, <i>op. cit.</i>, Vol. II, 1898, p. 108. +</div><div class="fndef"><a class="fn" id="fn_57" href="#fr_57">[57]</a>Wood’s <i>Athenae Oxonienses</i> (ed. P. Bliss), Vol. IV, 1820, p. 247. +</div><div class="fndef"><a class="fn" id="fn_58" href="#fr_58">[58]</a>Wood, <i>op. cit.</i>, Vol. II, p. 445. +</div><div class="fndef"><a class="fn" id="fn_59" href="#fr_59">[59]</a>Rigaud, <i>op. cit.</i>, Vol. I, pp. 33, 35. +</div><div class="fndef"><a class="fn" id="fn_60" href="#fr_60">[60]</a>Rigaud, <i>op. cit.</i>, Vol. I, pp. 16, 26. +</div><div class="fndef"><a class="fn" id="fn_61" href="#fr_61">[61]</a>Rigaud, <i>op. cit.</i>, Vol. I, p. 66. +</div><div class="fndef"><a class="fn" id="fn_62" href="#fr_62">[62]</a><i>Ibid.</i>, Vol. I, p. 9. +</div><div class="fndef"><a class="fn" id="fn_63" href="#fr_63">[63]</a>Rigaud, <i>op. cit.</i>, Vol. II, p. 475. +</div><div class="fndef"><a class="fn" id="fn_64" href="#fr_64">[64]</a><i>Ibid.</i>, Vol. II, p. 471. +</div><div class="fndef"><a class="fn" id="fn_65" href="#fr_65">[65]</a>J. W. L. Glaisher, “On Early Logarithmic Tables, and Their +Calculators,” <i>Philosophical Magazine</i>, 4th Ser., Vol. XLV (1873), +pp. 378, 379. +</div><div class="fndef"><a class="fn" id="fn_66" href="#fr_66">[66]</a>Rigaud, <i>op. cit.</i>, Vol. I, p. 65. +</div><div class="fndef"><a class="fn" id="fn_67" href="#fr_67">[67]</a>Rigaud, <i>op. cit.</i>, Vol. I, p. 87. +</div><div class="fndef"><a class="fn" id="fn_68" href="#fr_68">[68]</a>King’s <i>Life of John Locke</i>, Vol. I, London, 1830, p. 227. +</div><div class="fndef"><a class="fn" id="fn_69" href="#fr_69">[69]</a><i>Exercitationum Mathematicarum Decas prima</i>, Naples, 1627, and +probably Cataldus’ <i>Transformatio Geometrica</i>, Bonon., 1612. +</div><div class="fndef"><a class="fn" id="fn_70" href="#fr_70">[70]</a>Rigaud, <i>op. cit.</i>, Vol. II, pp. 477-80. +</div><div class="fndef"><a class="fn" id="fn_71" href="#fr_71">[71]</a><i>Miscellanies: or Mathematical Lucubrations, of Mr. Samuel +Foster, Sometimes publike Professor of Astronomie in Gresham Colledge +in London</i>, by John Twysden, London, 1659. +</div><div class="fndef"><a class="fn" id="fn_72" href="#fr_72">[72]</a><i>The Works of the Honourable Robert Boyle in five volumes, to +which is prefixed the Life of the Author</i>, Vol. I, London, 1744, p. 24. +</div><div class="fndef"><a class="fn" id="fn_73" href="#fr_73">[73]</a>The letter is printed in John Wallis’ <i>De algebra tractatus</i>, 1693, +p. 206. +</div><div class="fndef"><a class="fn" id="fn_74" href="#fr_74">[74]</a>See <i>La Correspondance de Descartes</i>, published by Charles Adam +and Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457. +</div><div class="fndef"><a class="fn" id="fn_75" href="#fr_75">[75]</a>H. Bosmans, S.J., “La première édition de la <i>Clavis Mathematica</i> +d’Oughtred. Son influence sur la Géométrie de Descartes,” <i>Annales +de la société scientifique de Bruxelles</i>, 35th year, 1910-11, Part II, +pp. 24-78. +</div><div class="fndef"><a class="fn" id="fn_76" href="#fr_76">[76]</a><i>Ibid.</i>, p. 78. +</div><div class="fndef"><a class="fn" id="fn_77" href="#fr_77">[77]</a>Vincent Wing, <i>Harmonicon coeleste</i>, London, 1651, p. 5. +</div><div class="fndef"><a class="fn" id="fn_78" href="#fr_78">[78]</a>Seth Ward, <i>In Ismaelis Bullialdi astronomiae philolaicae fundamenta +inquisitio brevis</i>, Oxford, 1653, p. 7. +</div><div class="fndef"><a class="fn" id="fn_79" href="#fr_79">[79]</a>John Wallis, <i>Elenchus geometriae Hobbianae</i>, Oxford, 1655, p. 48. +</div><div class="fndef"><a class="fn" id="fn_80" href="#fr_80">[80]</a><i>An Idea of Arithmetick, at first designed for the use of the Free +Schoole at Thurlow in Suffolk. . . . . By R. B., Schoolmaster there</i>, +London, 1655, p. 6. +</div><div class="fndef"><a class="fn" id="fn_81" href="#fr_81">[81]</a><i>The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel +Foster</i> . . . . by John Twysden, London, 1659, p. 1. +</div><div class="fndef"><a class="fn" id="fn_82" href="#fr_82">[82]</a><i>Moor’s Arithmetick in two Books</i>, London, 1660, p. 89. +</div><div class="fndef"><a class="fn" id="fn_83" href="#fr_83">[83]</a>Isaac Barrow, <i>Euclidis data</i>, Cambridge, 1657, p. 2. +</div><div class="fndef"><a class="fn" id="fn_84" href="#fr_84">[84]</a><i>Francisci Dulaurens Specima mathematica</i>, Paris, 1667, p. 1. +</div><div class="fndef"><a class="fn" id="fn_85" href="#fr_85">[85]</a><i>Elémens des mathématiques</i>, Paris, 1675, Preface signed “J. P.” +</div><div class="fndef"><a class="fn" id="fn_86" href="#fr_86">[86]</a><i>Nouveaux élémens de géométrie</i>, Paris, 1692 (permission to print +1684). +</div><div class="fndef"><a class="fn" id="fn_87" href="#fr_87">[87]</a>Ozanam, <i>Dictionnaire mathématique</i>, Paris, 1691, p. 12. +</div><div class="fndef"><a class="fn" id="fn_88" href="#fr_88">[88]</a><i>Analyse des infiniment petits</i>, Paris, 1696, p. 11. +</div><div class="fndef"><a class="fn" id="fn_89" href="#fr_89">[89]</a>Petro Nicolas, <i>De conchoidibus et cissoidibus exercitationes +geometricae</i>, Toulouse, 1697, p. 17. +</div><div class="fndef"><a class="fn" id="fn_90" href="#fr_90">[90]</a>R. P. Bernard Lamy, <i>Elémens des mathématiques</i>, Amsterdam, +1692 (permission to print 1680). +</div><div class="fndef"><a class="fn" id="fn_91" href="#fr_91">[91]</a><i>Nouveaux élémens de géométrie</i>, 2d ed., The Hague, 1690, +p. 304. +</div><div class="fndef"><a class="fn" id="fn_92" href="#fr_92">[92]</a>W. W. Beman in <i>L’intermédiaire des mathématiciens</i>, Paris, Vol. +IX, 1902, p. 229, question 2424. +</div><div class="fndef"><a class="fn" id="fn_93" href="#fr_93">[93]</a>John Collins, <i>The Mariner’s Plain Scale New Plain’d</i>, London, +1659, p. 25. +</div><div class="fndef"><a class="fn" id="fn_94" href="#fr_94">[94]</a>James Gregory, <i>Optica promota</i>, London, 1663, pp. 19, 48. +</div><div class="fndef"><a class="fn" id="fn_95" href="#fr_95">[95]</a><i>Philosophical Transactions</i>, Vol. III, London, p. 868. +</div><div class="fndef"><a class="fn" id="fn_96" href="#fr_96">[96]</a>William Leybourn, <i>The Line of Proportion</i>, London, 1673, p. 14. +</div><div class="fndef"><a class="fn" id="fn_97" href="#fr_97">[97]</a><i>Elementa geometriae . . . . a Gulielmo Sanders</i>, Glasgow, 1686, +p. 3. +</div><div class="fndef"><a class="fn" id="fn_98" href="#fr_98">[98]</a><i>Cocker’s Decimal Arithmetick</i>, . . . . perused by John Hawkins, +London, 1695 (preface dated 1684), p. 41. +</div><div class="fndef"><a class="fn" id="fn_99" href="#fr_99">[99]</a>Joseph Raphson, <i>Analysis Aequationum universalis</i>, London, +1697, p. 26. +</div><div class="fndef"><a class="fn" id="fn_100" href="#fr_100">[100]</a>E. Wells, <i>Elementa arithmeticae numerosae et speciosae</i>, Oxford, +1698, p. 107. +</div><div class="fndef"><a class="fn" id="fn_101" href="#fr_101">[101]</a>John Ward, <i>A Compendium of Algebra</i>, 2d ed., London, 1698, +p. 62. +</div><div class="fndef"><a class="fn" id="fn_102" href="#fr_102">[102]</a><i>Plain Elements of Geometry and Plain Trigonometry</i>, London, +1701, p. 63. +</div><div class="fndef"><a class="fn" id="fn_103" href="#fr_103">[103]</a>George Shelley, <i>Wingate’s Arithmetick</i>, London, 1704, p. 343. +</div><div class="fndef"><a class="fn" id="fn_104" href="#fr_104">[104]</a><i>A Synopsis of Algebra, Being a posthumous work of John Alexander +of Bern, Swisserland. . . . . Done from the Latin</i> by Sam. +Cobb, London, 1709, p. 16. +</div><div class="fndef"><a class="fn" id="fn_105" href="#fr_105">[105]</a>John Craig, <i>De Calculo fluentium</i>, London, 1718, p. 35. The +notation <i>A</i>:<i>B</i>::<i>C</i>:<i>D</i> is given also. +</div><div class="fndef"><a class="fn" id="fn_106" href="#fr_106">[106]</a><i>Trigonometry</i>, 2d ed., Edinburgh, 1724, p. 11. +</div><div class="fndef"><a class="fn" id="fn_107" href="#fr_107">[107]</a><i>Méthode pour la mésure des surfaces, la dimension des solides +. . . . par M. Carré de l’académie r. des sciences</i>, 1700, p. 59. +</div><div class="fndef"><a class="fn" id="fn_108" href="#fr_108">[108]</a><i>Application de l’algèbre à géométrie</i> . . . . Paris, 1705. +</div><div class="fndef"><a class="fn" id="fn_109" href="#fr_109">[109]</a><i>Elémens de la géométrie de l’infini</i>, by M. de Fontenelle, Paris, +1727, p. 110. +</div><div class="fndef"><a class="fn" id="fn_110" href="#fr_110">[110]</a><i>Eclaircissemens sur l’analyse des infiniment petits</i>, by M. Varignon, +Paris, 1725, p. 87. +</div><div class="fndef"><a class="fn" id="fn_111" href="#fr_111">[111]</a><i>Application de la géométrie ordinaire et des calculs différentiel et +intégral</i>, by M. Robillard, Paris, 1753. +</div><div class="fndef"><a class="fn" id="fn_112" href="#fr_112">[112]</a><i>Traité de géométrie théorique et pratique</i>, new ed., Paris, 1764, +p. 15. +</div><div class="fndef"><a class="fn" id="fn_113" href="#fr_113">[113]</a><i>Recherches sur les courbes à double courbure</i>, Paris, 1731, p. 13. +</div><div class="fndef"><a class="fn" id="fn_114" href="#fr_114">[114]</a><i>Analyse des infiniment petits</i>, by the Marquis de L’Hospital. +New ed. by M. Le Fèvre, Paris, 1781, p. 41. In this volume passages +in fine print, probably supplied by the editor, contain the notation +<i>a</i>:<i>b</i>::<i>c</i>:<i>d</i>; the parts in large type give Oughtred’s original notation. +</div><div class="fndef"><a class="fn" id="fn_115" href="#fr_115">[115]</a>The tendency during the eighteenth century is shown +in part by the following data: +<i>Jacobi Bernoulli Opera, Tomus primus</i>, Geneva, 1744, gives +<i>B</i>.<i>A</i>::<i>D</i>.<i>C</i> on p. 368, the paper having +been first published in 1688; on p. 419 is given +<i>GE</i>:<i>AG</i>=<i>LA</i>:<i>ML</i>, the paper having been first +published in 1689. <i>Bernhardi Nieuwentiit, Considerationes circa +analyseos ad quantitates infinitè parvas applicatae +principia</i>, Amsterdam, 1694, p. 20, and <i>Analysis infinitorum</i>, +Amsterdam, 1695, on p. 276, have <i>x</i>:<i>c</i>::<i>s</i>:<i>r</i>. +Paul Halcken’s <i>Deliciae mathematicae</i>, Hamburg, 1719, +gives <i>a</i>:<i>b</i>::<i>c</i>:<i>d</i>. Johannis Baptistae +Caraccioli, <i>Geometria algebraica universa</i>, Rome, 1759, p. 79, +has <i>a</i>.<i>b</i>::<i>c</i>.<i>d</i>. +<i>Delle corde ouverto fibre elastiche schediasmi fisico-matematici +del conte Giordano Riccati</i>, Bologna, 1767, p. 65, gives +<i>P</i>:<i>b</i>::<i>r</i>:<i>ds</i>. +<i>“Produzioni mathematiche” del Conte Giulio Carlo de +Fagnano</i>, Vol. I, Pesario, 1750, p. 193, has +<i>a</i>.<i>b</i>::<i>c</i>.<i>d</i>. L. Mascheroni, +<i>Géométrie du compas</i>, translated by A. M. Carette, +Paris, 1798, p. 188, gives +√<span class="over">3</span>:2::√<span class="over">2</span>:<i>Lp</i>. +Danielis Melandri and Paulli Frisi, +<i>De theoria lunae commentarii</i>, Parma, 1769, p. 13, has <i>a</i>:<i>b</i>::<i>c</i>:<i>d</i>. +Vicentio Riccato and Hieronymo Saladino, <i>Institutiones analyticae</i>, +Vol. I, Bologna, 1765, p. 47, gives <i>x</i>:<i>a</i>::<i>m</i>:<i>n</i>+<i>m</i>. R. G. Boscovich, +<i>Opera pertinentia ad opticam et astronomiam, Bassani</i>, 1785, p. 409, +uses <i>a</i>:<i>b</i>::<i>c</i>:<i>d</i>. Jacob Bernoulli, <i>Ars Conjectandi</i>, Basel, 1713, has +<i>n</i>-<i>r</i>.<i>n</i>-1::<i>c</i>.<i>d</i>. Pavlini Chelvicii, <i>Institutiones analyticae, editio +post tertiam Romanam prima in Germania</i>, Vienna, 1761, p. 2, <i>a</i>.<i>b</i>::<i>c</i>.<i>d</i>. +Christiani Wolfii, <i>Elementa matheseos universae</i>, Vol. III, +Geneva, 1735, p. 63, has <i>AB</i>:<i>AE</i>=1:<i>q</i>. Johann Bernoulli, <i>Opera +omnia</i>, Vol. I, Lausanne and Geneva, 1742, p. 43, has <i>a</i>:<i>b</i>=<i>c</i>:<i>d</i>. +D. C. Walmesley, <i>Analyse des mesures des rapports et des angles</i>, +Paris, 1749, uses extensively <i>a</i>.<i>b</i>::<i>c</i>.<i>d</i>, later <i>a</i>:<i>b</i>::<i>c</i>:<i>d</i>. G. W. +Krafft, <i>Institutiones geometriae sublimoris</i>, Tübingen, 1753, p. 194, +has <i>a</i>:<i>b</i>=<i>c</i>:<i>d</i>. J. H. Lambert, <i>Photometria</i>, 1760, p. 104, has <i>C</i>:<span class="greek" title="{pi}">π</span>=<i>BC</i>²:<i>MH</i>². +<i>Meccanica sublime del Dott. Domenico Bartaloni</i>, Naples, +1765, has <i>a</i>:<i>b</i>::<i>c</i>:<i>d</i>. Occasionally ratio is not designated by <i>a</i>.<i>b</i>, +nor by <i>a</i>:<i>b</i>, but by <i>a</i>, <i>b</i>, as for instance in A. de Moivre’s <i>Doctrine +of Chance</i>, London, 1756, p. 34, where he writes <i>a</i>, <i>b</i>::1, <i>q</i>. A further +variation in the designation of ratio is found in James Atkinson’s +<i>Epitome of the Art of Navigation</i>, London, 1718, p. 24, namely, +3..2::72..48. Curious notations are given in Rich. Balam’s +<i>Algebra</i>, London, 1653. +</div><div class="fndef"><a class="fn" id="fn_116" href="#fr_116">[116]</a><i>Chr. Clavii Operum mathematicorum tomus secundus</i>, +Mayence, 1611, Algebra, p. 39. +</div><div class="fndef"><a class="fn" id="fn_117" href="#fr_117">[117]</a><i>Invention +nouvelle en l’algèbre</i>, by Albert Girard, Amsterdam, +1629, p. 17. +</div><div class="fndef"><a class="fn" id="fn_118" href="#fr_118">[118]</a><i>La géométrie et pratique générale d’icelle, par I. Errard de Bar-le-Duc, +Ingénieur ordinaire de sa Majesté</i>, 3d ed., revised by D. H. P. +E. M., Paris, 1619, p. 216. +</div><div class="fndef"><a class="fn" id="fn_119" href="#fr_119">[119]</a><i>Novae geometriae clavis algebra, authore P. Jacobo de Billy</i>, +Paris, 1643, p. 157; also an <i>Abridgement of the Precepts of Algebra. +Written in French by James de Billy</i>, London, 1659, p. 346. +</div><div class="fndef"><a class="fn" id="fn_120" href="#fr_120">[120]</a><i>Miscellanies: or Mathematical Lucubrations, of Mr. Samuel +Foster, Sometime publike Professor of Astronomie in Gresham Colledge +in London</i>, London, 1659, p. 7. +</div><div class="fndef"><a class="fn" id="fn_121" href="#fr_121">[121]</a><i>Quarterly Jour. of Pure and Applied Math.</i>, Vol. XLVI (London, +1915), p. 191. +</div><div class="fndef"><a class="fn" id="fn_122" href="#fr_122">[122]</a>Pietro Cossali, <i>Origine, trasporto in Italia primi progressi in +essa dell’ algebra, Vol. I, Parmense</i>, 1797, p. 52. +</div><div class="fndef"><a class="fn" id="fn_123" href="#fr_123">[123]</a><i>In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio +brevis, Auctore Setho Wardo</i>, Oxford, 1653, p. 1. +</div><div class="fndef"><a class="fn" id="fn_124" href="#fr_124">[124]</a>John Wallis, <i>Algebra</i>, London, 1685, p. 321, and in some of his +other works. He makes greater use of Harriot’s symbols. +</div><div class="fndef"><a class="fn" id="fn_125" href="#fr_125">[125]</a><i>Euclidis data</i>, 1657, p. 1; also <i>Euclidis elementorum libris XV</i>, +London, 1659, p. 1. +</div><div class="fndef"><a class="fn" id="fn_126" href="#fr_126">[126]</a>John Kersey, <i>Algebra</i>, London, 1673, p. 321. +</div><div class="fndef"><a class="fn" id="fn_127" href="#fr_127">[127]</a>E. Wells, <i>Elementa arithmeticae numerosae et speciosae</i>, Oxford, +1698, p. 142. +</div><div class="fndef"><a class="fn" id="fn_128" href="#fr_128">[128]</a>Cocker’s <i>Decimal Arithmetick</i>, perused by John Hawkins, +London, 1695 (preface dated 1684), p. 278. +</div><div class="fndef"><a class="fn" id="fn_129" href="#fr_129">[129]</a>Th. Baker, <i>The Geometrical Key</i>, London, 1684, p. 15. +</div><div class="fndef"><a class="fn" id="fn_130" href="#fr_130">[130]</a>Richard Sault, <i>A New Treatise of Algebra</i>, London (no date). +</div><div class="fndef"><a class="fn" id="fn_131" href="#fr_131">[131]</a>Richard Rawlinson in a pamphlet without date, issued sometime +between 1655 and 1668, containing trigonometric formulas. +There is a copy in the British Museum. +</div><div class="fndef"><a class="fn" id="fn_132" href="#fr_132">[132]</a>F. Dulaurens, <i>Specima mathematica</i>, Paris, 1667, p. 1. +</div><div class="fndef"><a class="fn" id="fn_133" href="#fr_133">[133]</a>J. Milnes, <i>Sectionum conicarum elementa</i>, Oxford, 1702, p. 42. +</div><div class="fndef"><a class="fn" id="fn_134" href="#fr_134">[134]</a>Cheyne, <i>Philosophical Principles of Natural Religion</i>, London, +1705, p. 55. +</div><div class="fndef"><a class="fn" id="fn_135" href="#fr_135">[135]</a>J. Craig, <i>De calculo fluentium</i>, London, 1718, p. 86. +</div><div class="fndef"><a class="fn" id="fn_136" href="#fr_136">[136]</a>Jo. Wilson, <i>Trigonometry</i>, 2d ed., Edinburgh, 1724, p. v. +</div><div class="fndef"><a class="fn" id="fn_137" href="#fr_137">[137]</a><i>Commercium Epistolicum</i>, 1712, p. 20. +</div><div class="fndef"><a class="fn" id="fn_138" href="#fr_138">[138]</a>C. Le Paige, “Sur l’origine de certains signes d’opération,” +<i>Annales de la société scientifique de Bruxelles</i>, 16th year, 1891-92, +Part II, pp. 79-82. +</div><div class="fndef"><a class="fn" id="fn_139" href="#fr_139">[139]</a>Gravelaar, “Over den oorsprong van ons maalteeken (×),” +<i>Wiskundig Tijdschrift</i>, 6th year. We have not had access to this +article. +</div><div class="fndef"><a class="fn" id="fn_140" href="#fr_140">[140]</a>H. Bosmans, <i>op. cit.</i>, p. 40. +</div><div class="fndef"><a class="fn" id="fn_141" href="#fr_141">[141]</a><i>Claudii Ptolemaei . . . . annotationes</i>, Bâle, 1551. This reference +is taken from the <i>Encyclopédie des sciences mathématiques</i>, +Tome I, Vol. I, Fasc. 1, p. 40. +</div><div class="fndef"><a class="fn" id="fn_142" href="#fr_142">[142]</a><i>Due Correction for Mr. Hobbes. Or Schoole Discipline, for not +saying his Lessons right. In answer to his Six Lessons, directed to the +Professors of Mathematicks.</i> By the Professor of Geometry. Oxford, +1656, pp. 7, 47, 50. +</div><div class="fndef"><a class="fn" id="fn_143" href="#fr_143">[143]</a>Oughtred, <i>Apologeticall Epistle</i>, p. 27. +</div><div class="fndef"><a class="fn" id="fn_144" href="#fr_144">[144]</a>J. Edleston, <i>Correspondence of Sir Isaac Newton and Professor +Cotes</i>, London, 1850, pp. 279-92. +</div> +</div> +<div class="pb" id="Page_97">[97]</div> +<h2 id="c29">INDEX</h2> +<dl class="index"> +<dt>Adam, Charles, <a href="#Page_71">71</a></dt> +<dt>Agnesi, Maria G., <a href="#Page_77">77</a></dt> +<dt>Alexander, J., <a href="#Page_76">76</a></dt> +<dt>Allen, E., <a href="#Page_35">35</a></dt> +<dt>Analysis, <a href="#Page_19">19</a>, <a href="#Page_20">20</a></dt> +<dt>Apollonius of Perga, <a href="#Page_20">20</a>, <a href="#Page_85">85</a></dt> +<dt>Archimedes, <a href="#Page_18">18</a>, <a href="#Page_20">20</a>, <a href="#Page_85">85</a></dt> +<dt>Aristotle, <a href="#Page_69">69</a></dt> +<dt>Ashmole, E., <a href="#Page_13">13</a></dt> +<dt>Atkinson, J., <a href="#Page_79">79</a></dt> +<dt>Atwood, <a href="#Page_56">56</a></dt> +<dt>Aubrey, <a href="#Page_3">3</a>, <a href="#Page_7">7</a>, <a href="#Page_8">8</a>, <a href="#Page_12">12</a>-16, <a href="#Page_58">58</a>, <a href="#Page_59">59</a></dt> +<dt>Austin, <a href="#Page_58">58</a></dt> +</dl> +<dl class="index"> +<dt>Baker, T., <a href="#Page_82">82</a></dt> +<dt>Balam, R., <a href="#Page_79">79</a></dt> +<dt>Bar-le-Duc, de, <a href="#Page_80">80</a></dt> +<dt>Barrow, S., <a href="#Page_1">1</a>, <a href="#Page_32">32</a>, <a href="#Page_73">73</a>, <a href="#Page_74">74</a>, <a href="#Page_80">80</a>, <a href="#Page_81">81</a></dt> +<dt>Bartaloni, D., <a href="#Page_79">79</a></dt> +<dt>Beman, W. W., <a href="#Page_74">74</a>, <a href="#Page_75">75</a></dt> +<dt>Bernoulli, Jakob, <a href="#Page_78">78</a>-80</dt> +<dt>Bernoulli, John, <a href="#Page_79">79</a>, <a href="#Page_80">80</a></dt> +<dt>Billingsley’s Euclid, <a href="#Page_15">15</a></dt> +<dt>Billion, <a href="#Page_20">20</a></dt> +<dt>Billy, Jacobo de, <a href="#Page_80">80</a></dt> +<dt>Binomial formula, <a href="#Page_25">25</a>, <a href="#Page_29">29</a></dt> +<dt>Bliss, P., <a href="#Page_60">60</a></dt> +<dt>Boscovich, R. G., <a href="#Page_78">78</a></dt> +<dt>Bosmans, H., <a href="#Page_72">72</a>, <a href="#Page_83">83</a></dt> +<dt>Boyle, R., <a href="#Page_1">1</a>, <a href="#Page_69">69</a></dt> +<dt>Braunmühl, von, <a href="#Page_39">39</a></dt> +<dt>Brearly, W., <a href="#Page_59">59</a></dt> +<dt>Briggs, <a href="#Page_6">6</a>, <a href="#Page_36">36</a>, <a href="#Page_55">55</a></dt> +<dt>Brookes, Christopher, <a href="#Page_7">7</a>, <a href="#Page_53">53</a>, <a href="#Page_59">59</a></dt> +</dl> +<dl class="index"> +<dt>Cajori, F., <a href="#Page_27">27</a>, <a href="#Page_39">39</a>, <a href="#Page_40">40</a>, <a href="#Page_47">47</a></dt> +<dt>Cantor, M., <a href="#Page_40">40</a>, <a href="#Page_41">41</a></dt> +<dt>Caraccioli, J. B., <a href="#Page_78">78</a></dt> +<dt>Cardan, <a href="#Page_71">71</a></dt> +<dt>Carré, <a href="#Page_77">77</a></dt> +<dt>Carrete, N. M., <a href="#Page_78">78</a></dt> +<dt>Caryll, C., <a href="#Page_7">7</a></dt> +<dt>Cataldi, <a href="#Page_67">67</a></dt> +<dt>Cavalieri, <a href="#Page_65">65</a>, <a href="#Page_66">66</a></dt> +<dt>Cavendish, Charles, <a href="#Page_17">17</a>, <a href="#Page_62">62</a>, <a href="#Page_66">66</a></dt> +<dt>Charles I, <a href="#Page_9">9</a>, <a href="#Page_60">60</a></dt> +<dt>Chelvicius, P., <a href="#Page_79">79</a></dt> +<dt>Cheyne, G., <a href="#Page_82">82</a></dt> +<dt><i>Circles of Proportion</i>, <a href="#Page_35">35</a>, <a href="#Page_37">37</a>, <a href="#Page_48">48</a>, <a href="#Page_49">49</a>, <a href="#Page_51">51</a>, <a href="#Page_59">59</a>, <a href="#Page_87">87</a>, <a href="#Page_88">88</a></dt> +<dt>Clairaut, <a href="#Page_77">77</a></dt> +<dt>Clark, A., <a href="#Page_3">3</a></dt> +<dt>Clark, G., <a href="#Page_63">63</a></dt> +<dt>Clarke, F. L., <a href="#Page_3">3</a></dt> +<dt><i>Clavis mathematicae</i>, <a href="#Page_1">1</a>, <a href="#Page_5">5</a>, <a href="#Page_10">10</a>, <a href="#Page_14">14</a>, <a href="#Page_17">17</a>-35, <a href="#Page_45">45</a>, <a href="#Page_46">46</a>, <a href="#Page_51">51</a>, <a href="#Page_57">57</a>-63, <a href="#Page_68">68</a>-73, <a href="#Page_81">81</a>, <a href="#Page_85">85</a>, <a href="#Page_87">87</a></dt> +<dt>Clavius, <a href="#Page_26">26</a>, <a href="#Page_80">80</a></dt> +<dt>Clerc, le, <a href="#Page_77">77</a></dt> +<dt>Cobb, S., <a href="#Page_76">76</a></dt> +<dt>Cocker, <a href="#Page_76">76</a>, <a href="#Page_82">82</a></dt> +<dt>Collins, John, <a href="#Page_15">15</a>, <a href="#Page_19">19</a>, <a href="#Page_63">63</a>, <a href="#Page_64">64</a>, <a href="#Page_67">67</a>, <a href="#Page_68">68</a>, <a href="#Page_76">76</a>, <a href="#Page_82">82</a></dt> +<dt>Colson, J., <a href="#Page_77">77</a></dt> +<dt>Conchoid, <a href="#Page_12">12</a></dt> +<dt>Conic sections, <a href="#Page_11">11</a>, <a href="#Page_53">53</a></dt> +<dt>Cossali, P., <a href="#Page_81">81</a></dt> +<dt>Cotes, R., <a href="#Page_1">1</a>, <a href="#Page_85">85</a></dt> +<dt>Craig, J., <a href="#Page_76">76</a>, <a href="#Page_82">82</a></dt> +<dt>Cross, symbol of multiplication, <a href="#Page_27">27</a>, <a href="#Page_38">38</a>, <a href="#Page_55">55</a>, <a href="#Page_56">56</a>, <a href="#Page_82">82</a>, <a href="#Page_83">83</a></dt> +<dt>Cubic equations, <a href="#Page_28">28</a>, <a href="#Page_34">34</a>, <a href="#Page_42">42</a>, <a href="#Page_45">45</a></dt> +</dl> +<dl class="index"> +<dt>Decimal fractions, notation of, <a href="#Page_21">21</a></dt> +<dt>Degree, centesimal division, <a href="#Page_39">39</a></dt> +<dt>Delamain, R., <a href="#Page_4">4</a>, <a href="#Page_9">9</a>, <a href="#Page_10">10</a>, <a href="#Page_11">11</a>, <a href="#Page_47">47</a>, <a href="#Page_48">48</a>, <a href="#Page_51">51</a>, <a href="#Page_60">60</a>, <a href="#Page_84">84</a>, <a href="#Page_88">88</a>, <a href="#Page_89">89</a>, <a href="#Page_91">91</a>, <a href="#Page_93">93</a>, <a href="#Page_94">94</a></dt> +<dt>De Moivre, <a href="#Page_32">32</a>, <a href="#Page_79">79</a></dt> +<dt>De Morgan, A., <a href="#Page_5">5</a>, <a href="#Page_16">16</a>, <a href="#Page_37">37</a>, <a href="#Page_46">46</a>, <a href="#Page_47">47</a>, <a href="#Page_54">54</a></dt> +<dt>Descartes, R., <a href="#Page_1">1</a>, <a href="#Page_25">25</a>, <a href="#Page_47">47</a>, <a href="#Page_57">57</a>, <a href="#Page_68">68</a>-72, <a href="#Page_80">80</a></dt> +<dt>Dibuadius, <a href="#Page_79">79</a></dt> +<dt>Difference, symbol for, <a href="#Page_27">27</a>, <a href="#Page_81">81</a></dt> +<dt>Diophantus, <a href="#Page_63">63</a>, <a href="#Page_85">85</a></dt> +<dt>Division, abbreviated, <a href="#Page_21">21</a>, <a href="#Page_23">23</a>, <a href="#Page_24">24</a></dt> +<dt>Dulaurens, F., <a href="#Page_74">74</a>, <a href="#Page_82">82</a></dt> +</dl> +<dl class="index"> +<dt><span id="arundel">Earl of Arundel</span>, <a href="#Page_10">10</a>, <a href="#Page_13">13</a>, <a href="#Page_15">15</a>, <a href="#Page_17">17</a></dt> +<dt>Edleston, J., <a href="#Page_95">95</a></dt> +<dt>Eneström, G., <a href="#Page_40">40</a></dt> +<dt>Equations, solution of, <a href="#Page_18">18</a>, <a href="#Page_28">28</a>, <a href="#Page_29">29</a>, <a href="#Page_31">31</a>, <a href="#Page_34">34</a>, <a href="#Page_39">39</a>-45, <a href="#Page_87">87</a></dt> +<dt>Errard de Bar-le-Duc, <a href="#Page_80">80</a></dt> +<dt>Eton College, <a href="#Page_3">3</a>, <a href="#Page_4">4</a></dt> +<dt>Euclid, <a href="#Page_1">1</a>, <a href="#Page_15">15</a>, <a href="#Page_18">18</a>, <a href="#Page_20">20</a>, <a href="#Page_25">25</a>, <a href="#Page_27">27</a>, <a href="#Page_28">28</a>, <a href="#Page_79">79</a>, <a href="#Page_81">81</a>, <a href="#Page_83">83</a>, <a href="#Page_85">85</a></dt> +<dt>Euler, L., <a href="#Page_37">37</a>, <a href="#Page_39">39</a></dt> +<dt>Ewart, <a href="#Page_59">59</a></dt> +<dt>Exponents, <a href="#Page_25">25</a>, <a href="#Page_28">28</a>, <a href="#Page_29">29</a></dt> +</dl> +<dl class="index"> +<dt>Fagnano, de, <a href="#Page_78">78</a></dt> +<dt>Flower, <a href="#Page_56">56</a></dt> +<dt>Fontenelle, de, <a href="#Page_77">77</a></dt> +<dt>Forster, W., <a href="#Page_35">35</a>, <a href="#Page_48">48</a>, <a href="#Page_59">59</a>, <a href="#Page_88">88</a></dt> +<dt>Foster, S., <a href="#Page_27">27</a>, <a href="#Page_67">67</a>, <a href="#Page_69">69</a>, <a href="#Page_73">73</a>, <a href="#Page_80">80</a>, <a href="#Page_89">89</a></dt> +<dt>Frisi, P., <a href="#Page_78">78</a></dt> +</dl> +<dl class="index"> +<dt>Gascoigne, <a href="#Page_59">59</a>, <a href="#Page_61">61</a></dt> +<dt>Gauss, C. F., <a href="#Page_48">48</a></dt> +<dt>Geysius, <a href="#Page_67">67</a></dt> +<dt>Ghetaldi, <a href="#Page_70">70</a>, <a href="#Page_71">71</a></dt> +<dt>Gibson, <a href="#Page_67">67</a></dt> +<dt>Girard, A., <a href="#Page_32">32</a>, <a href="#Page_80">80</a></dt> +<dt>Glaisher, J. W. L., <a href="#Page_54">54</a>-56, <a href="#Page_64">64</a>, <a href="#Page_80">80</a></dt> +<dt>Glorioso, <a href="#Page_67">67</a>, <a href="#Page_68">68</a></dt> +<dt><i>Grammelogia</i>, <a href="#Page_4">4</a>, <a href="#Page_47">47</a>, <a href="#Page_89">89</a></dt> +<dt>Gravelaar, <a href="#Page_83">83</a></dt> +<dt>Greater than, symbol for, <a href="#Page_81">81</a></dt> +<dt>Greatrex, R., <a href="#Page_15">15</a></dt> +<dt>Gregory, D., <a href="#Page_32">32</a></dt> +<dt>Gregory, J., <a href="#Page_27">27</a>, <a href="#Page_76">76</a></dt> +<dt>Gresham College, <a href="#Page_1">1</a>, <a href="#Page_6">6</a>, <a href="#Page_27">27</a>, <a href="#Page_59">59</a>, <a href="#Page_61">61</a>, <a href="#Page_80">80</a></dt> +<dt>Guisnée, <a href="#Page_77">77</a></dt> +<dt>Gunter, E., <a href="#Page_37">37</a>, <a href="#Page_47">47</a>, <a href="#Page_86">86</a></dt> +<dt>Gunter’s scale, <a href="#Page_37">37</a></dt> +</dl> +<dl class="index"> +<dt>Halcken, P., <a href="#Page_78">78</a></dt> +<dt>Hales, J., <a href="#Page_7">7</a></dt> +<dt>Halley, E., <a href="#Page_1">1</a>, <a href="#Page_18">18</a>, <a href="#Page_69">69</a></dt> +<dt>Hankel, H., <a href="#Page_40">40</a></dt> +<dt>Harper, T., <a href="#Page_18">18</a></dt> +<dt>Harriot, T., <a href="#Page_45">45</a>, <a href="#Page_47">47</a>, <a href="#Page_57">57</a>, <a href="#Page_58">58</a>, <a href="#Page_69">69</a>-71, <a href="#Page_81">81</a></dt> +<dt>Harris, J., <a href="#Page_76">76</a></dt> +<dt>Hartlib, <a href="#Page_69">69</a></dt> +<dt>Haughton, A., <a href="#Page_35">35</a>, <a href="#Page_59">59</a></dt> +<dt>Hawkins, J., <a href="#Page_76">76</a>, <a href="#Page_82">82</a></dt> +<dt>Hearn, <a href="#Page_56">56</a></dt> +<dt>Helmholtz, <a href="#Page_48">48</a></dt> +<dt>Henry, J., <a href="#Page_48">48</a></dt> +<dt>Henry van Etten, <a href="#Page_52">52</a>, <a href="#Page_53">53</a></dt> +<dt>Henshaw, T., <a href="#Page_8">8</a>, <a href="#Page_58">58</a>, <a href="#Page_61">61</a></dt> +<dt>Hobbes, <a href="#Page_73">73</a>, <a href="#Page_86">86</a></dt> +<dt>Hollar, <a href="#Page_14">14</a></dt> +<dt>Holsatus, <a href="#Page_13">13</a></dt> +<dt>Hooganhuysen, <a href="#Page_64">64</a></dt> +<dt>Hooke, Rb., <a href="#Page_1">1</a></dt> +<dt>Horner’s method, <a href="#Page_45">45</a></dt> +<dt>Horology, <a href="#Page_18">18</a>, <a href="#Page_50">50</a></dt> +<dt>Horrox, J., <a href="#Page_4">4</a></dt> +<dt>Hospital, de l’, <a href="#Page_74">74</a>, <a href="#Page_77">77</a></dt> +<dt>Howard, Th. <i>See</i> <a href="#arundel">Earl of Arundel</a>.</dt> +<dt>Howard, W., <a href="#Page_17">17</a>, <a href="#Page_18">18</a>, <a href="#Page_59">59</a></dt> +<dt>Hutchinson, A., <a href="#Page_6">6</a></dt> +</dl> +<dl class="index"> +<dt>Invisible college, <a href="#Page_1">1</a></dt> +</dl> +<dl class="index"> +<dt>Joule, <a href="#Page_48">48</a></dt> +</dl> +<dl class="index"> +<dt>Kepler, J., <a href="#Page_6">6</a></dt> +<dt>Kersey, J., <a href="#Page_32">32</a>, <a href="#Page_73">73</a>, <a href="#Page_82">82</a></dt> +<dt>Keylway, R., <a href="#Page_65">65</a></dt> +<dt>King, <a href="#Page_67">67</a></dt> +<dt>Kings College, Cambridge, <a href="#Page_3">3</a>, <a href="#Page_35">35</a></dt> +<dt>Krafft, G. W., <a href="#Page_79">79</a></dt> +</dl> +<dl class="index"> +<dt>Lambert, J. H., <a href="#Page_79">79</a></dt> +<dt>Lamy, R. P. B., <a href="#Page_74">74</a></dt> +<dt>Laud, Archbishop, <a href="#Page_65">65</a></dt> +<dt>Leake, W., <a href="#Page_53">53</a></dt> +<dt>Le Clerc, <a href="#Page_77">77</a></dt> +<dt>Leech, W., <a href="#Page_59">59</a></dt> +<dt>Le Fèvre, <a href="#Page_77">77</a></dt> +<dt>Leibniz, <a href="#Page_47">47</a>, <a href="#Page_78">78</a>, <a href="#Page_80">80</a></dt> +<dt>Leonelli, <a href="#Page_56">56</a></dt> +<dt>Le Paige, de, <a href="#Page_83">83</a></dt> +<dt>Less than, symbol for, <a href="#Page_81">81</a></dt> +<dt>Leurechon, <a href="#Page_52">52</a></dt> +<dt>Leybourn, <a href="#Page_35">35</a>, <a href="#Page_64">64</a>, <a href="#Page_76">76</a></dt> +<dt>Lichfield, Mrs., <a href="#Page_19">19</a></dt> +<dt>Lilly, W., <a href="#Page_8">8</a>, <a href="#Page_9">9</a></dt> +<dt>Locke, J., <a href="#Page_67">67</a></dt> +<dt>Logarithms, <a href="#Page_6">6</a>, <a href="#Page_21">21</a>, <a href="#Page_27">27</a>, <a href="#Page_28">28</a>, <a href="#Page_38">38</a>, <a href="#Page_39">39</a>, <a href="#Page_42">42</a>, <a href="#Page_46">46</a>, <a href="#Page_54">54</a>-56, <a href="#Page_65">65</a>, <a href="#Page_92">92</a>, <a href="#Page_93">93</a>;</dt> +<dd>natural, <a href="#Page_55">55</a>;</dd> +<dd>radix method of computing, <a href="#Page_55">55</a>, <a href="#Page_56">56</a></dd> +<dt>Lower, W., <a href="#Page_58">58</a></dt> +<dt>Ludolph à Ceulen, <a href="#Page_79">79</a></dt> +</dl> +<dl class="index"> +<dt>Manning, <a href="#Page_56">56</a></dt> +<dt>Manning, O., <a href="#Page_7">7</a>, <a href="#Page_8">8</a>, <a href="#Page_13">13</a>-15</dt> +<dt>Mascheroni, L., <a href="#Page_78">78</a></dt> +<dt>Mayer, R., <a href="#Page_47">47</a></dt> +<dt>Melandri, D., <a href="#Page_78">78</a></dt> +<dt>Mercator, N., <a href="#Page_13">13</a></dt> +<dt>Mersenne, <a href="#Page_63">63</a></dt> +<dt>Milbourn, W., <a href="#Page_45">45</a></dt> +<dt>Million, <a href="#Page_20">20</a></dt> +<dt>Milnes, J., <a href="#Page_82">82</a></dt> +<dt>Moivre, de, <a href="#Page_32">32</a>, <a href="#Page_79">79</a></dt> +<dt>Moore, Jonas, <a href="#Page_32">32</a>, <a href="#Page_54">54</a>, <a href="#Page_58">58</a>, <a href="#Page_73">73</a></dt> +<dt>Moreland, S., <a href="#Page_70">70</a></dt> +<dt>Morse, R., <a href="#Page_48">48</a></dt> +<dt>Multiplication, abbreviated, <a href="#Page_21">21</a>, <a href="#Page_22">22</a>, <a href="#Page_24">24</a>;</dt> +<dd>symbol for, <a href="#Page_27">27</a>, <a href="#Page_82">82</a>, <a href="#Page_83">83</a></dd> +<dt>Mydorge, <a href="#Page_54">54</a></dt> +</dl> +<dl class="index"> +<dt>Napier, J., <a href="#Page_6">6</a>, <a href="#Page_7">7</a>, <a href="#Page_21">21</a>, <a href="#Page_27">27</a>, <a href="#Page_38">38</a>, <a href="#Page_39">39</a>, <a href="#Page_52">52</a>, <a href="#Page_54">54</a>, <a href="#Page_57">57</a>, <a href="#Page_59">59</a></dt> +<dt>Napier’s analogies, <a href="#Page_39">39</a></dt> +<dt>Newton, Sir Isaac, <a href="#Page_1">1</a>, <a href="#Page_25">25</a>, <a href="#Page_29">29</a>, <a href="#Page_40">40</a>, <a href="#Page_41">41</a>, <a href="#Page_45">45</a>, <a href="#Page_47">47</a>, <a href="#Page_59">59</a>, <a href="#Page_65">65</a>, <a href="#Page_86">86</a>, <a href="#Page_92">92</a>-95</dt> +<dt>Nichols, J., <a href="#Page_6">6</a>, <a href="#Page_14">14</a></dt> +<dt>Nicolas, R. P. P., <a href="#Page_74">74</a></dt> +<dt>Nieuwentiit, B., <a href="#Page_78">78</a></dt> +<dt>Norwood, R., <a href="#Page_37">37</a>, <a href="#Page_38">38</a>, <a href="#Page_80">80</a></dt> +</dl> +<dl class="index"> +<dt><i>Opuscula mathematica hactenus inedita</i>, <a href="#Page_16">16</a>, <a href="#Page_21">21</a>, <a href="#Page_75">75</a></dt> +<dt>Orchard, <a href="#Page_56">56</a></dt> +<dt><i>Oughtredus explicatus</i>, <a href="#Page_64">64</a></dt> +<dt>Ozanam, <a href="#Page_74">74</a></dt> +</dl> +<dl class="index"> +<dt><span class="greek" title="{pi}">π</span>, symbol for, <a href="#Page_32">32</a></dt> +<dt>Paige, C. de, <a href="#Page_83">83</a></dt> +<dt>Pardies, <a href="#Page_76">76</a></dt> +<dt>Parentheses, <a href="#Page_26">26</a>, <a href="#Page_79">79</a>, <a href="#Page_80">80</a></dt> +<dt>Partridge, S., <a href="#Page_47">47</a></dt> +<dt>Peano, <a href="#Page_86">86</a></dt> +<dt>Perfect number, <a href="#Page_41">41</a></dt> +<dt>Pitiscus, <a href="#Page_15">15</a></dt> +<dt>Planisphere, <a href="#Page_53">53</a>, <a href="#Page_92">92</a>, <a href="#Page_93">93</a></dt> +<dt>Prestet, J., <a href="#Page_74">74</a></dt> +<dt>Price, <a href="#Page_11">11</a></dt> +<dt>Proportion, notation for, <a href="#Page_26">26</a>, <a href="#Page_27">27</a>, <a href="#Page_73">73</a>-79</dt> +<dt>Protheroe, <a href="#Page_58">58</a></dt> +<dt>Ptolemy, <a href="#Page_83">83</a></dt> +</dl> +<dl class="index"> +<dt>Quadratic equation, <a href="#Page_29">29</a>, <a href="#Page_31">31</a>, <a href="#Page_34">34</a></dt> +</dl> +<dl class="index"> +<dt>Radix method, <a href="#Page_55">55</a>, <a href="#Page_56">56</a></dt> +<dt>Rahn, <a href="#Page_27">27</a></dt> +<dt>Raphson, J., <a href="#Page_40">40</a>, <a href="#Page_41">41</a>, <a href="#Page_76">76</a></dt> +<dt>Ratio, notation of, <a href="#Page_21">21</a>, <a href="#Page_73">73</a>-80</dt> +<dt>Rawlinson, R., <a href="#Page_39">39</a>, <a href="#Page_82">82</a></dt> +<dt>Regula falsa, <a href="#Page_18">18</a></dt> +<dt>Regular solids, <a href="#Page_18">18</a></dt> +<dt>Riccati, G., <a href="#Page_78">78</a></dt> +<dt>Riccati, V., <a href="#Page_78">78</a></dt> +<dt>Rigaud, <a href="#Page_7">7</a>, <a href="#Page_12">12</a>, <a href="#Page_13">13</a>, <a href="#Page_19">19</a>, <a href="#Page_48">48</a>, <a href="#Page_61">61</a>-66, <a href="#Page_68">68</a></dt> +<dt>Robillard, <a href="#Page_77">77</a></dt> +<dt>Robinson, W., <a href="#Page_13">13</a>, <a href="#Page_48">48</a>, <a href="#Page_59">59</a>, <a href="#Page_62">62</a>, <a href="#Page_63">63</a></dt> +<dt>Rooke, L., <a href="#Page_59">59</a>, <a href="#Page_61">61</a></dt> +</dl> +<dl class="index"> +<dt>Saladini, H., <a href="#Page_78">78</a></dt> +<dt>Sanders, W., <a href="#Page_76">76</a></dt> +<dt>Sault, R., <a href="#Page_82">82</a></dt> +<dt>Scarborough, Charles, <a href="#Page_16">16</a>, <a href="#Page_54">54</a>, <a href="#Page_58">58</a>, <a href="#Page_60">60</a></dt> +<dt>Schooten, Van, <a href="#Page_1">1</a></dt> +<dt>Schreshensuchs, O., <a href="#Page_83">83</a></dt> +<dt>Scratch method, <a href="#Page_23">23</a></dt> +<dt>Shakespeare, <a href="#Page_52">52</a></dt> +<dt>Shelley, G., <a href="#Page_76">76</a></dt> +<dt>Shipley, A. E., <a href="#Page_1">1</a></dt> +<dt>Shuttleworth, <a href="#Page_59">59</a></dt> +<dt>Slide rule, <a href="#Page_9">9</a>, <a href="#Page_46">46</a>-49, <a href="#Page_50">50</a>, <a href="#Page_60">60</a>, <a href="#Page_88">88</a>, <a href="#Page_93">93</a></dt> +<dt>Smethwyck, <a href="#Page_58">58</a></dt> +<dt>Smith, J., <a href="#Page_50">50</a></dt> +<dt>Snellius, W., <a href="#Page_79">79</a></dt> +<dt>Solids, regular, <a href="#Page_18">18</a></dt> +<dt>Speidell, John, <a href="#Page_38">38</a>, <a href="#Page_55">55</a></dt> +<dt>Spherical triangles, <a href="#Page_53">53</a>, <a href="#Page_54">54</a>, <a href="#Page_93">93</a></dt> +<dt>Stokes, R., <a href="#Page_35">35</a>, <a href="#Page_36">36</a>, <a href="#Page_58">58</a></dt> +<dt>Sudell, <a href="#Page_59">59</a></dt> +<dt>Sun dials, <a href="#Page_5">5</a>, <a href="#Page_9">9</a>, <a href="#Page_50">50</a>, <a href="#Page_51">51</a>, <a href="#Page_52">52</a>, <a href="#Page_60">60</a>, <a href="#Page_92">92</a></dt> +</dl> +<dl class="index"> +<dt>Tannery, P., <a href="#Page_71">71</a></dt> +<dt>Todhunter, <a href="#Page_60">60</a></dt> +<dt>Torporley, <a href="#Page_58">58</a></dt> +<dt>Triangles, spherical, <a href="#Page_53">53</a>, <a href="#Page_54">54</a>, <a href="#Page_93">93</a></dt> +<dt><i>Trigonometria</i>, <a href="#Page_21">21</a>, <a href="#Page_36">36</a>, <a href="#Page_55">55</a>, <a href="#Page_75">75</a></dt> +<dt>Trigonometric functions, symbols for, <a href="#Page_36">36</a>, <a href="#Page_37">37</a>, <a href="#Page_55">55</a>, <a href="#Page_56">56</a></dt> +<dt><i>Trigonometrie</i>, <a href="#Page_21">21</a>, <a href="#Page_35">35</a>, <a href="#Page_39">39</a></dt> +<dt>Trisection of angles, <a href="#Page_28">28</a></dt> +<dt>Twysden, <a href="#Page_59">59</a>, <a href="#Page_68">68</a>, <a href="#Page_69">69</a>, <a href="#Page_73">73</a></dt> +</dl> +<dl class="index"> +<dt>Varignon, <a href="#Page_77">77</a></dt> +<dt>Vieta, <a href="#Page_1">1</a>, <a href="#Page_2">2</a>, <a href="#Page_25">25</a>, <a href="#Page_32">32</a>, <a href="#Page_33">33</a>, <a href="#Page_35">35</a>, <a href="#Page_39">39</a>-41, <a href="#Page_45">45</a>, <a href="#Page_63">63</a>, <a href="#Page_67">67</a>, <a href="#Page_70">70</a>, <a href="#Page_71">71</a></dt> +<dt>Vlack, <a href="#Page_65">65</a></dt> +<dt>Von Braunmühl, <a href="#Page_39">39</a></dt> +</dl> +<dl class="index"> +<dt>Wadham College, <a href="#Page_5">5</a>, <a href="#Page_53">53</a></dt> +<dt>Wallis, John, <a href="#Page_1">1</a>, <a href="#Page_19">19</a>, <a href="#Page_27">27</a>, <a href="#Page_33">33</a>, <a href="#Page_45">45</a>, <a href="#Page_57">57</a>-59, <a href="#Page_63">63</a>, <a href="#Page_64">64</a>, <a href="#Page_66">66</a>-74, <a href="#Page_79">79</a>-81, <a href="#Page_86">86</a></dt> +<dt>Walmesley, D. C., <a href="#Page_79">79</a></dt> +<dt>Ward, Bishop, <a href="#Page_13">13</a></dt> +<dt>Ward, John, <a href="#Page_76">76</a></dt> +<dt>Ward, Seth, <a href="#Page_55">55</a>, <a href="#Page_58">58</a>, <a href="#Page_60">60</a>, <a href="#Page_68">68</a>, <a href="#Page_73">73</a>, <a href="#Page_74">74</a>, <a href="#Page_81">81</a></dt> +<dt>Watch-making, <a href="#Page_18">18</a>, <a href="#Page_50">50</a></dt> +<dt>Weber, W. E., <a href="#Page_48">48</a></dt> +<dt>Weddle, <a href="#Page_56">56</a></dt> +<dt>Wells, E., <a href="#Page_76">76</a>, <a href="#Page_82">82</a></dt> +<dt>Wharton, <a href="#Page_60">60</a></dt> +<dt>Whitlock, B., <a href="#Page_8">8</a>, <a href="#Page_9">9</a></dt> +<dt>Wilson, J., <a href="#Page_77">77</a>, <a href="#Page_82">82</a></dt> +<dt>Wing, V., <a href="#Page_73">73</a>, <a href="#Page_75">75</a></dt> +<dt>Wingate, E., <a href="#Page_32">32</a>, <a href="#Page_47">47</a>, <a href="#Page_73">73</a></dt> +<dt>Wolf, Christian, <a href="#Page_79">79</a></dt> +<dt>Wood, A., <a href="#Page_60">60</a>, <a href="#Page_61">61</a></dt> +<dt>Wood, R., <a href="#Page_18">18</a>, <a href="#Page_59">59</a></dt> +<dt>Wren, Christopher, <a href="#Page_5">5</a>, <a href="#Page_58">58</a>, <a href="#Page_59">59</a>, <a href="#Page_76">76</a></dt> +<dt>Wright, E., <a href="#Page_6">6</a>, <a href="#Page_27">27</a>, <a href="#Page_38">38</a>, <a href="#Page_54">54</a></dt> +<dt>Wright, S., <a href="#Page_54">54</a></dt> +</dl> +<h2 id="c30">Transcriber’s Notes</h2> +<p>A handful of typos, mostly misplaced punctuation, were silently +corrected.</p> +<p>HTML and UTF text versions make heavy use of mathematical symbols: +particularly superscripts, subscripts, and combining characters. 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You may copy it, give it away or re-use it under the terms of +the Project Gutenberg License included with this eBook or online at +www.gutenberg.org. If you are not located in the United States, you'll have +to check the laws of the country where you are located before using this ebook. + +Title: William Oughtred + A great Seventeenth-Century Teacher of Mathematics + +Author: Florian Cajori + +Release Date: September 9, 2014 [EBook #46815] + +Language: English + +Character set encoding: ASCII + +*** START OF THIS PROJECT GUTENBERG EBOOK WILLIAM OUGHTRED *** + + + + +Produced by Brenda Lewis, Stephen Hutcheson, and the Online +Distributed Proofreading Canada Team at +http://www.pgdpcanada.net + + + + + + + WILLIAM OUGHTRED + + + + + WILLIAM OUGHTRED + A GREAT SEVENTEENTH-CENTURY + TEACHER OF + MATHEMATICS + + + BY + FLORIAN CAJORI, Ph.D. + Professor of Mathematics + Colorado College + + CHICAGO LONDON + THE OPEN COURT PUBLISHING COMPANY + 1916 + + Copyright 1916 By + The Open Court Publishing Co. + + All Rights Reserved + + Published September 1916 + + + Composed and Printed By + The University of Chicago Press + Chicago, Illinois, U.S.A. + + + + + TABLE OF CONTENTS + + + PAGE + Introduction 1 + CHAPTER + I. Oughtred's Life 3 + At School and University 3 + As Rector and Amateur Mathematician 6 + His Wife 7 + In Danger of Sequestration 8 + His Teaching 9 + Appearance and Habits 12 + Alleged Travel Abroad 14 + His Death 15 + II. Principal Works 17 + Clavis mathematicae 17 + Circles of Proportion and Trigonometrie 35 + Solution of Numerical Equations 39 + Logarithms 46 + Invention of the Slide Rule; Controversy on Priority of Invention 46 + III. Minor Works 50 + IV. Oughtred's Influence upon Mathematical Progress and Teaching 57 + Oughtred and Harriot 57 + Oughtred's Pupils 58 + Oughtred, the "Todhunter of the Seventeenth Century" 60 + Was Descartes Indebted to Oughtred? 69 + The Spread of Oughtred's Notations 73 + V. Oughtred's Ideas on the Teaching of Mathematics 84 + General Statement 84 + Mathematics, "a Science of the Eye" 85 + Rigorous Thinking and the Use of Instruments 87 + Newton's Comments on Oughtred 94 + Index 97 + + + + + 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 Geometrie, 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] 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] + +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] + +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] + +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] We know very little about Oughtred's +family life. The records at King's College, Cambridge,[6] 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] 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] + +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] + + + 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] + +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] + +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] + + + 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] + + + 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. . . . . + + The right hon^ble Thomas Howard, earle of Arundel and Surrey, Lord High + Marshall of England, was his great patron, and loved him intirely. One + time they were like to have been killed together by the fall at Albury + of a grott, which fell downe but just as they were come out.[14] + +Oughtred's friends convey the impression that, in the main, Oughtred +enjoyed a comfortable living at Albury. Only once appear indications of +financial embarrassment. About 1634 one of his pupils, W. Robinson, +writes as follows: + + I protest unto you sincerely, were I as able as some, at whose hands + you have merited exceedingly, or (to speak more absolutely) as able as + willing, I would as freely give you 500 l. per ann. as 500 pence; and I + cannot but be astonished at this our age, wherein pelf and dross is + made their summum bonum, and the best part of man, with the true + ornaments thereof, science and knowledge, are so slighted. . . . .[15] + +In his letters Oughtred complains several times of the limitations for +work and the infirmities due to his advancing old age. The impression he +made upon others was quite different. Says one biographer: + + He sometimes amused himself with archery, and sometimes practised as a + surveyor of land. . . . . He was sprightly and active, when more than + eighty years of age.[16] + +Another informant says that Oughtred was + + as facetious in Greek and Latine as solid in Arithmetique, Astronomy, + and the sphere of all Measures, Musick, etc.; exact in his style as in + his judgment; handling his Cube, and other Instruments at eighty, as + steadily, as others did at thirty; owing this, he said, to temperance + and Archery; principling his people with plain and solid truths, as he + did the world with great and useful Arts; advancing new Inventions in + all things but Religion. Which in its old order and decency he + maintained secure in his privacy, prudence, meekness, simplicity, + resolution, patience, and contentment.[17] + + + ALLEGED TRAVEL ABROAD + +According to certain sources of information, Oughtred traveled on the +European Continent and was invited to change his abode to the Continent. +We have seen no statement from Oughtred himself on this matter. He seldom +referred to himself in his books and letters. The autobiography contained +in his Apologeticall Epistle was written a quarter of a century before +his death. Aubrey gives the following: + + In the time of the civill warres the duke of Florence invited him over, + and offered him 500 li. per annum; but he would not accept it, because + of his religion.[18] + +A portrait of Oughtred, painted in 1646 by Hollar and inserted in the +English edition of the Clavis of 1647, contains underneath the following +lines: + + "Haec est Oughtredi senio labantis imago + Itala quam cupiit, Terra Britanna tulit." + +In the sketch of Oughtred by Owen Manning it is confessed that "it is not +known to what this alludes; but possibly he might have been in Italy with +his patron, the Earl of Arundel."[19] It would seem quite certain either +that Oughtred traveled in Europe or that he received some sort of an +offer to settle in Italy. In view of Aubrey's explicit statement and of +Oughtred's well-known habit of confining himself to his duties and +studies in his own parish, seldom going even as far as London, we +strongly incline to the opinion that he did not travel on the Continent, +but that he received an offer from some patron of the sciences--possibly +some distinguished visitor--to settle in Italy. + + + HIS DEATH + +He died at Albury, June 30, 1660, aged about eighty-six years. Of his +last days and death, Aubrey speaks as follows: + + Before he dyed he burned a world of papers, and sayd that the world was + not worthy of them; he was so superb. He burned also severall printed + bookes, and would not stirre, till they were consumed. . . . . I + myselfe have his Pitiscus, imbelished with his excellent marginall + notes, which I esteeme as a great rarity. I wish I could also have got + his Bilingsley's Euclid, which John Collins sayes was full of his + annotations. . . . . + + Ralph Greatrex, his great friend, the mathematicall instrument-maker, + sayed he conceived he dyed with joy for the comeing-in of the king, + which was the 29th of May before. "And are yee sure he is + restored?"--"Then give me a glasse of sack to drinke his sacred + majestie's health." His spirits were then quite upon the wing to fly + away. . . . .[20] + +In this passage, as in others, due allowance must be made for Aubrey's +lack of discrimination. He was not in the habit of sifting facts from +mere gossip. That Oughtred should have declared that the world was not +worthy of his papers or manuscripts is not in consonance with the +sweetness of disposition ordinarily attributed to him. More probable was +the feeling that the papers he burned--possibly old sermons--were of no +particular value to the world. That he did not destroy a large mass of +mathematical manuscripts is evident from the fact that a considerable +number of them came after his death into the hands of Sir Charles +Scarborough, M.D., under whose supervision some of them were carefully +revised and published at Oxford in 1677 under the title of Opuscula +mathematica hactenus inedita. + +Aubrey's story of Oughtred's mode of death has been as widely circulated +in every modern biographical sketch as has his slander of Mrs. Oughtred +by claiming that she was so penurious that she would deny him the use of +candles to read by. Oughtred died on June 30; the Restoration occurred on +May 29. No doubt Oughtred rejoiced over the Restoration, but the story of +his drinking "a glass of sack" to his Majesty's health, and then dying of +joy is surely apocryphal. De Morgan humorously remarks, "It should be +added, by way of excuse, that he was eighty-six years old."[21] + + + + + CHAPTER II + PRINCIPAL WORKS + + + "CLAVIS MATHEMATICAE" + +Passing to the consideration of Oughtred's mathematical books, we begin +with the observation that he showed a marked disinclination to give his +writings to the press. His first paper on sun-dials was written at the +age of twenty-three, but we are not aware that more than one brief +mathematical manuscript was printed before his fifty-seventh year. In +every instance, publication in printed form seems to have been due to +pressure exerted by one or more of his patrons, pupils, or friends. Some +of his manuscripts were lent out to his pupils, who prepared copies for +their own use. In some instances they urged upon him the desirability of +publication and assisted in preparing copy for the printer. The earliest +and best-known book of Oughtred was his Clavis mathematicae, to which +repeated allusion has already been made. As he himself informs us, he was +employed by the Earl of Arundel about 1628 to instruct the Earl's son, +Lord William Howard (afterward Viscount Stafford) in the mathematics. For +the use of this young man Oughtred composed a treatise on algebra which +was published in Latin in the year 1631 at the urgent request of a +kinsman of the young man, Charles Cavendish, a patron of learning. + +The Clavis mathematicae,[22] in its first edition of 1631, was a booklet +of only 88 small pages. Yet it contained in very condensed form the +essentials of arithmetic and algebra as known at that time. + +Aside from the addition of four tracts, the 1631 edition underwent some +changes in the editions of 1647 and 1648, which two are much alike. The +twenty chapters of 1631 are reduced to nineteen in 1647 and in all the +later editions. Numerous minute alterations from the 1631 edition occur +in all parts of the books of 1647 and 1648. The material of the last +three chapters of the 1631 edition is rearranged, with some slight +additions here and there. The 1648 edition has no preface. In the print +of 1652 there are only slight alterations from the 1648 edition; after +that the book underwent hardly any changes, except for the number of +tracts appended, and brief explanatory notes added at the close of the +chapters in the English editions of 1694 and 1702. The 1652 and 1667 +editions were seen through the press by John Wallis; the 1698 impression +contains on the title-page the words: Ex Recognitione D. Johannis Wallis, +S.T.D. Geometriae Professoris Saviliani. + +The cost of publishing may be a matter of some interest. When arranging +for the printing of the 1667 edition of the Clavis, Wallis wrote Collins: +"I told you in my last what price she [Mrs. Lichfield] expects for it, as +I have formerly understood from her, viz., L 40 for the impression, which +is about 91/2d. a book."[23] + +As compared with other contemporary works on algebra, Oughtred's +distinguishes itself for the amount of symbolism used, particularly in +the treatment of geometric problems. Extraordinary emphasis was placed +upon what he called in the Clavis the "analytical art."[24] By that term +he did not mean our modern analysis or analytical geometry, but the art +"in which by taking the thing sought as knowne, we finde out that we +seeke."[25] He meant to express by it condensed processes of rigid, +logical deduction expressed by appropriate symbols, as contrasted with +mere description or elucidation by passages fraught with verbosity. In +the preface to the first edition (1631) he says: + + In this little book I make known . . . . the rules relating to + fundamentals, collected together, just like a bundle, and adapted to + the explanation of as many problems as possible. + +As stated in this preface, one of his reasons for publishing the book, is + + . . . . that like Ariadne I might offer a thread to mathematical study + by which the mysteries of this science might be revealed, and direction + given to the best authors of antiquity, Euclid, Archimedes, the great + geometrician Apollonius of Perga, and others, so as to be easily and + thoroughly understood, their theorems being added, not only because to + many they are the height and depth of mathematical science (I ignore + the would-be mathematicians who occupy themselves only with the + so-called practice, which is in reality mere juggler's tricks with + instruments, the surface so to speak, pursued with a disregard of the + great art, a contemptible picture), but also to show with what keenness + they have penetrated, with what mass of equations, comparisons, + reductions, conversions and disquisitions these heroes have ornamented, + increased and invented this most beautiful science. + +The Clavis opens with an explanation of the Hindu-Arabic notation and of +decimal fractions. Noteworthy is the absence of the words "million," +"billion," etc. Though used on the Continent by certain mathematical +writers long before this, these words did not become current in English +mathematical books until the eighteenth century. The author was a great +admirer of decimal fractions, but failed to introduce the notation which +in later centuries came to be universally adopted. Oughtred wrote 0.56 in +this manner 0|56; the point he used to designate ratio. Thus 3:4 was +written by him 3.4. The decimal point (or comma) was first used by the +inventor of logarithms, John Napier, as early as 1616 and 1617. Although +Oughtred had mastered the theory of logarithms soon after their +publication in 1614 and was a great admirer of Napier, he preferred to +use the dot for the designation of ratio. This notation of ratio is used +in all his mathematical books, except in two instances. The two dots (:) +occur as symbols of ratio in some parts of Oughtred's posthumous work, +Opuscula mathematica hactenus inedita, Oxford, 1677, but may have been +due to the editors and not to Oughtred himself. Then again the two dots +(:) are used to designate ratio on the last two pages of the tables of +the Latin edition of Oughtred's Trigonometria of 1657. In all other parts +of that book the dot (.) is used. Probably someone who supervised the +printing of the tables introduced the (:) on the last two pages, +following the logarithmic tables, where methods of interpolation are +explained. The probability of this conjecture is the stronger, because in +the English edition of the Trigonometrie, brought out the same year +(1657) but after the Latin edition, the notation (:) at the end of the +book is replaced by the usual (.), except that in some copies of the +English edition the explanations at the end are omitted altogether. + +Oughtred introduces an interesting, and at the same time new, feature of +an abbreviated multiplication and an abbreviated division of decimal +fractions. On this point he took a position far in advance of his time. +The part on abbreviated multiplication was rewritten in slightly enlarged +form and with some unimportant alterations in the later edition of the +Clavis. We give it as it occurs in the revision. Four cases are given. In +finding the product of 246|914 and 35|27, "if you would have the Product +without any Parts" (without any decimal part), "set the place of Unity of +the lesser under the place of Unity in the greater: as in the Example," +writing the figures of the lesser number in inverse order. From the +example it will be seen that he begins by multiplying by 3, the +right-hand digit of the multiplier. In the first edition of the Clavis he +began with 7, the left digit. Observe also that he "carries" the nearest +tens in the product of each lower digit and the upper digit one place to +its right. For instance, he takes 7x4=28 and carries 3, then he finds +7x2+3=17 and writes down 17. + + 2 4 6|9 1 4 + 7 2|5 3 + ------- + 7 4 0 7 + 1 2 3 5 + 4 9 + 1 7 + ------- + 8 7 0 8 + +The second case supposes that "you would have the Product with some +places of parts" (decimals), say 4: "Set the place of Unity of the lesser +Number under the Fourth place of the Parts of the greater." The +multiplication of 246|914 by 35|27 is now performed thus: + + 2 4 6|9 1 4 + 7 2|5 3 + --------------- + 7 4 0 7 4 2 0 0 + 1 2 3 4 5 7 0 0 + 4 9 3 8 2 8 + 1 7 2 8 4 0 + --------------- + 8 7 0 8|6 5 6 8 + +In the third and fourth cases are considered factors which appear as +integers, but are in reality decimals; for instance, the sine of 54^o is +given in the tables as 80902 when in reality it is .80902. + +Of interest as regards the use of the word "parabola" is the following: +"The Number found by Division is called the Quotient, or also Parabola, +because it arises out of the Application of a plain Number to a given +Longitude, that a congruous Latitude may be found."[26] This is in +harmony with etymological dictionaries which speak of a parabola as the +application of a given area to a given straight line. The dividend or +product is the area; the divisor or factor is the line. + +Oughtred gives two processes of long division. The first is identical +with the modern process, except that the divisor is written below every +remainder, each digit of the divisor being crossed out as soon as it has +been used in the partial multiplication. The second method of long +division is one of the several types of the old "scratch method." This +antiquated process held its place by the side of the modern method in all +editions of the Clavis. The author divides 467023 by 357|0926425, giving +the following instructions: "Take as many of the first Figures of the +Divisor as are necessary, for the first Divisor, and then in every +following particular Division drop one of the Figures of the Divisor +towards the Left Hand, till you have got a competent Quotient." He does +not explain abbreviated division as thoroughly as abbreviated +multiplication. + + 17 + 303 + 2803 + 109930 + 357|0926425) 467023 (1307|80 + 357093 + 107127 + 2500 + 286 + + +Oughtred does not examine the degree of reliability or accuracy of his +processes of abbreviated multiplication and division. Here as in other +places he gives in condensed statement the mode of procedure, without +further discussion. + +He does not attempt to establish the rules for the addition, subtraction, +multiplication, and division of positive and negative numbers. "If the +Signs are both alike, the Product will be affirmative, if unlike, +negative"; then he proceeds to applications. This attitude is superior to +that of many writers of the eighteenth and nineteenth centuries, on +pedagogical as well as logical grounds: pedagogically, because the +beginner in the study of algebra is not in a position to appreciate an +abstract train of thought, as every teacher well knows, and derives +better intellectual exercise from the applications of the rules to +problems; logically, because the rule of signs in multiplication does not +admit of rigorous proof, unless some other assumption is first made which +is no less arbitrary than the rule itself. It is well known that the +proofs of the rule of signs given by eighteenth-century writers are +invalid. Somewhere they involve some surreptitious assumption. This +criticism applies even to the proof given by Laplace, which tacitly +assumes the distributive law in multiplication. + +A word should be said on Oughtred's definition of + and -. He recognizes +their double function in algebra by saying (Clavis, 1631, p. 2): "Signum +additionis, sive affirmationis, est + plus" and "Signum subductionis, +sive negationis est - minus." They are symbols which indicate the quality +of numbers in some instances and operations of addition or subtraction in +other instances. In the 1694 edition of the Clavis, thirty-four years +after the death of Oughtred, these symbols are defined as signifying +operations only, but are actually used to signify the quality of numbers +as well. In this respect the 1694 edition marks a recrudescence. + +The characteristic in the Clavis that is most striking to a modern reader +is the total absence of indexes or exponents. There is much discussion in +the leading treatises of the latter part of the sixteenth and the early +part of the seventeenth century on the theory of indexes, but the modern +exponential notation, a^n, is of later date. The modern notation, for +positive integral exponents, first appears in Descartes' Geometrie, 1637; +fractional and negative exponents were first used in the modern form by +Sir Isaac Newton, in his announcement of the binomial formula, in a +letter written in 1676. This total absence of our modern exponential +notation in Oughtred's Clavis gives it a strange aspect. Like Vieta, +Oughtred uses ordinarily the capital letters, A, B, C, . . . . to +designate given numbers; A^2 is written Aq, A^3 is written Ac; for A^4, +A^5, A^6 he has, respectively, Aqq, Aqc, Acc. Only on rare occasions, +usually when some parallelism in notation is aimed at, does he use small +letters[27] to represent numbers or magnitudes. Powers of binomials or +polynomials are marked by prefixing the capital letters Q (for square), C +(for cube), QQ (for the fourth power), QC (for the fifth power), etc. + +Oughtred does not express aggregation by (). Parentheses had been used by +Girard, and by Clavius as early as 1609,[28] but did not come into +general use in mathematical language until the time of Leibniz and the +Bernoullis. Oughtred indicates aggregation by writing a colon (:) at both +ends. Thus, Q:A-E: means with him (A-E)^2. Similarly, {root}q:A+E: means +{root}(A+E). The two dots at the end are frequently omitted when the part +affected includes all the terms of the polynomial to the end. Thus, +C:A+B-E=.. means (A+B-E)^3=.. There are still further departures from +this notation, but they occur so seldom that we incline to the +interpretation that they are simply printer's errors. For proportion +Oughtred uses the symbol (::). The proportion a:b=c:d appears in his +notation a.b::c.d. Apparently, a proportion was not fully recognized in +this day as being the expression of an equality of ratios. That probably +explains why he did not use = here as in the notation of ordinary +equations. Yet Oughtred must have been very close to the interpretation +of a proportion as an equality; for he says in his Elementi decimi +Euclidis declaratio, "proportio, sive ratio aequalis ::" That he +introduced this extra symbol when the one for equality was sufficient is +a misfortune. Simplicity demands that no unnecessary symbols be +introduced. However, Oughtred's symbolism is certainly superior to those +which preceded. Consider the notation of Clavius.[29] He wrote 20:60=4:x, +x=12, thus: "20.60.4? fiunt 12." The insufficiency of such a notation in +the more involved expressions frequently arising in algebra is readily +seen. Hence Oughtred's notation (::) was early adopted by English +mathematicians. It was used by John Wallis at Oxford, by Samuel Foster at +Gresham College, by James Gregory of Edinburgh, by the translators into +English of Rahn's algebra, and by many other early writers. Oughtred has +been credited generally with the introduction of St. Andrew's cross x as +the symbol for multiplication in the Clavis of 1631. We have discovered +that this symbol, or rather the letter x which closely resembles it, +occurs as the sign of multiplication thirteen years earlier in an +anonymous "Appendix to the Logarithmes, shewing the practise of the +Calculation of Triangles etc." to Edward Wright's translation of John +Napier's Descriptio, published in 1618.[30] Later we shall give our +reasons for believing that Oughtred is the author of that "Appendix." The +x has survived as a symbol of multiplication. + +Another symbol introduced by Oughtred and found in modern books is ~, +expressing difference; thus C~D signifies the difference between C and D, +even when D is the larger number.[31] This symbol was used by John Wallis +in 1657.[32] + +Oughtred represented in symbols also certain composite expressions, as +for instance A+E=Z, A-E=X, where A is greater than E. He represented by a +symbol also each of the following: A^2+E^2, A^3+E^3, A^2-E^2, A^3-E^3. + +Oughtred practically translated the tenth book of Euclid from its +ponderous rhetorical form into that of brief symbolism. An appeal to the +eye was a passion with Oughtred. The present writer has collected the +different mathematical symbols used by Oughtred and has found more than +one hundred and fifty of them. + +The differences between the seven different editions of the Clavis lie +mainly in the special parts appended to some editions and dropped in the +latest editions. The part which originally constituted the Clavis was not +materially altered, except in two or three of the original twenty +chapters. These changes were made in the editions of 1647 and 1648. After +the first edition, great stress was laid upon the theory of indices upon +the very first page, as also in passages farther on. Of course, Oughtred +did not have our modern notation of indices or exponents, but their +theory had been a part of algebra and arithmetic for some time. Oughtred +incorporated this theory in his brief exposition of the Hindu-Arabic +notation and in his explanation of logarithms. As previously pointed out, +the last three chapters of the 1631 edition were considerably rearranged +in the later editions and combined into two chapters, so that the Clavis +proper had nineteen chapters instead of twenty in the additions after the +first. These chapters consisted of applications of algebra to geometry +and were so framed as to constitute a severe test of the student's grip +of the subject. The very last problem deals with the division of angles +into equal parts. He derives the cubic equation upon which the trisection +depends algebraically, also the equations of the fifth degree and seventh +degree upon which the divisions of the angle into 5 and 7 equal parts +depend, respectively. The exposition was severely brief, yet accurate. He +did not believe in conducting the reader along level paths or along +slight inclines. He was a guide for mountain-climbers, and woe unto him +who lacked nerve. + +Oughtred lays great stress upon expansions of powers of a binomial. He +makes use of these expansions in the solution of numerical equations. To +one who does not specialize in the history of mathematics such expansions +may create surprise, for did not Newton invent the binomial theorem after +the death of Oughtred? As a matter of fact, the expansions of positive +integral powers of a binomial were known long before Newton, not only to +seventeenth-century but even to eleventh-century mathematicians. +Oughtred's Clavis of 1631 gave the binomial coefficients for all powers +up to and including the tenth. What Newton really accomplished was the +generalization of the binomial expansion which makes it applicable to +negative and fractional exponents and converts it into an infinite +series. + +As a specimen of Oughtred's style of writing we quote his solution of +quadratic equations, accompanied by a translation into English and into +modern mathematical symbols. + +As a preliminary step[33] he lets + + Z=A+E and A>E; + +he lets also X=A-E. From these relations he obtains identities which, in +modern notation, are 1/4Z^2-AE=(1/2Z-E)^2=1/4X^2. Now, if we know Z and +AE, we can find 1/2X. Then 1/2(Z+X)=A, and 1/2(Z-X)=E, and + + A=1/2Z+{root}(1/4Z^2-AE). + +Having established these preliminaries, he proceeds thus: + + Datis igitur linea inaequaliter secta Z (10), & rectangulo sub + segmentis AE (21) qui gnomon est: datur semidifferentia segmentorum + 1/2X: & per consequens ipsa segmenta. Nam ponatur alterutrum segmentum + A: alterum erit Z-A: Rectangulum auctem est ZA-A_q=AE. Et quia dantur Z + & AE: estque 1/4Z_q-AE=1/4X_q: & per 5c. 18, 1/2Z+1/2X=A: & + 1/2Z-1/2X=E: Aequatio sic resoluetur: 1/2Z+/-{root}_q:1/4Z_q-AE:=A + {maius segment/minus segment. + + Itaque proposita equatione, in qua sunt tres species aequaliter in + ordine tabellae adscendentes, altissima autem species ponitur negata: + Magnitudo data coefficiens mediam speciem est linea bisecanda: & + magnitudo absoluta data, ad quam sit aequatio, est rectangulum sub + segmentis inaequalibus, sine gnomon: vt ZA-A_q=AE: in numeris autem + 10l-l_q=21: Estque A, vel 1l, alterutrum segmentum inaequale. Inuenitur + autem sic: + + Dimidiata coefficiens median speciem est Z/2 (5); cuius quadratum est + Z_q/4 (25): ex hoc tolle AE (21) absolutum: eritque Z_q/4-AE (4) + quadratum semidifferentiae segmentorum: latus huius quadratum (2) est + semidifferentia: quam si addas ad Z/2 (5) semissem coefficientis, sive + lineae bisecandae, erit maius segment.; sin detrahas, erit minus + segment: Dico Z/2+/-{root}_q:Z_q/4-AE:=A {maius segmentum/minus + segmentum. + +We translate the Latin passage, using the modern exponential notation and +parentheses, as follows: + + Given therefore an unequally divided line Z (10), and a rectangle + beneath the segments AE (21) which is a gnomon. Half the difference of + the segments 1/2X is given, and consequently the segment itself. For, + if one of the two segments is placed equal to A, the other will be Z-A. + Moreover, the rectangle is ZA-A^2=AE. And because Z and AE are given, + and there is 1/4Z^2-AE=1/4X^2, and by 5c.18, 1/2Z+1/2X=A, and + 1/2Z-1/2X=E, the equation will be solved thus: + 1/2Z+/-{root}(1/4Z^2-AE)=A {major segment/minor segment. + + And so an equation having been proposed in which three species (terms) + are in equally ascending powers, the highest species, moreover, being + negative, the given magnitude which constitutes the middle species is + the line to be bisected. And the given absolute magnitude to which it + is equal is the rectangle beneath the unequal segments, without gnomon. + As ZA-A^2=AE, or in numbers, 10x-x^2=21. And A or x is one of the two + unequal segments. It may be found thus: + + The half of the middle species is Z/2 (5), its square is Z^2/4 (25). + From it subtract the absolute term AE (21), and Z^2/4-AE (4) will be + the square of half the difference of the segments. The square root of + this, {root}[(Z^2/2)^2-AE] (2), is half the difference. If you add it + to half the coefficient Z/2 (5), or half the line to be bisected, the + longer segment is obtained; if you subtract it, the smaller segment is + obtained. I say: Z/2+/-{root}(Z^2/4-AE)=A {major segment/minor segment. + +The quadratic equation Aq+ZA=AE receives similar treatment. This and the +preceding equation, ZA-Aq=AE, constitute together a solution of the +general quadratic equation, x^2+ax=b, provided that E or Z are not +restricted to positive values, but admit of being either positive or +negative, a case not adequately treated by Oughtred. Imaginary numbers +and imaginary roots receive no consideration whatever. + +A notation suggested by Vieta and favored by Girard made vowels stand for +unknowns and consonants for knowns. This conventionality was adopted by +Oughtred in parts of his algebra, but not throughout. Near the beginning +he used Q to designate the unknown, though usually this letter stood with +him for the "square" of the expression after it.[34] + +It is of some interest that Oughtred used {pi/delta} to signify the ratio +of the circumference to the diameter of a circle. Very probably this +notation is the forerunner of the {pi}=3.14159 . . . . used in 1706 by +William Jones. Oughtred first used {pi/delta} in the 1647 edition of the +Clavis mathematicae. In the 1652 edition he says, "Si in circulo sit +7.22::{delta.pi}::113.355:erit {delta.pi}::2 R.P: periph." This notation +was adopted by Isaac Barrow, who used it extensively. David Gregory[35] +used {pi/rho} in 1697, and De Moivre[36] used c/r about 1697, to +designate the ratio of the circumference to the radius. + +We quote the description of the Clavis that was given by Oughtred's +greatest pupil, John Wallis. It contains additional information of +interest to us. Wallis devotes chap. xv of his Treatise of Algebra, +London, 1685, pp. 67-69, to Mr. Oughtred and his Clavis, saying: + + Mr. William Oughtred (our Country-man) in his Clavis Mathematicae, (or + Key of Mathematicks,) first published in the Year 1631, follows Vieta + (as he did Diophantus) in the use of the Cossick Denominations; + omitting (as he had done) the names of Sursolids, and contenting + himself with those of Square and Cube, and the Compounds of these. + + But he doth abridge Vieta's Characters or Species, using only the + letters q, c, &c. which in Vieta are expressed (at length) by Quadrate, + Cube, &c. For though when Vieta first introduced this way of Specious + Arithmetick, it was more necessary (the thing being new,) to express it + in words at length: Yet when the thing was once received in practise, + Mr. Oughtred (who affected brevity, and to deliver what he taught as + briefly as might be, and reduce all to a short view,) contented himself + with single Letters instead of those words. + + Thus what Vieta would have written + + A Quadrate, into B Cube, + ------------------------ Equal to FG Plane, + CDE Solid, + + would with him be thus expressed + + A_q B_c + ------- = FG. + C D E + + And the better to distinguish upon the first view, what quantities were + Known, and what Unknown, he doth (usually) denote the Known to + Consonants, and the Unknown by Vowels; as Vieta (for the same reason) + had done before him. + + He doth also (to very great advantage) make use of several Ligatures, + or Compendious Notes, to signify Summs, Differences, and Rectangles of + several Quantities. As for instance, Of two Quantities A (the Greater), + and E (the Lesser), the Sum he calls Z, the Difference X, the Rectangle + AE. . . . . + + Which being of (almost) a constant signification with him throughout, + do save a great circumlocution of words, (each Letter serving instead + of a Definition;) and are also made use of (with very great advantage) + to discover the true nature of divers intricate Operations, arising + from the various compositions of such Parts, Sums, Differences, and + Rectangles; (of which there is great plenty in his Clavis, Cap. 11, 16, + 18, 19. and elsewhere,) which without such Ligatures, or Compendious + Notes, would not be easily discovered or apprehended. . . . . + + I know there are who find fault with his Clavis, as too obscure, + because so short, but without cause; for his words be always full, but + not Redundant, and need only a little attention in the Reader to weigh + the force of every word, and the Syntax of it; . . . . And this, when + once apprehended, is much more easily retained, than if it were + expressed with the prolixity of some other Writers; where a Reader must + first be at the pains to weed out a great deal of superfluous Language, + that he may have a short prospect of what is material; which is here + contracted for him in a short Synopsis. . . . . + + Mr. Oughtred in his Clavis, contents himself (for the most part) with + the solution of Quadratick Equations, without proceeding (or very + sparingly) to Cubick Equations, and those of Higher Powers; having + designed that Work for an Introduction into Algebra so far, leaving the + Discussion of Superior Equations for another work. . . . . He contents + himself likewise in Resolving Equations, to take notice of the + Affirmative or Positive Roots; omitting the Negative or Ablative Roots, + and such as are called Imaginary or Impossible Roots. And of those + which, he calls Ambiguous Equations, (as having more Affirmative Roots + than one,) he doth not (that I remember) any where take notice of more + than Two Affirmative Roots: (Because in Quadratick Equations, which are + those he handleth, there are indeed no more.) Whereas yet in Cubick + Equations, there may be Three, and in those of Higher Powers, yet more. + Which Vieta was well aware of, and mentioneth in some of his Writings; + and of which Mr. Oughtred could not be ignorant. + + + "CIRCLES OF PROPORTION" AND "TRIGONOMETRIE" + +Oughtred wrote and had published three important mathematical books, the +Clavis, the Circles of Proportion,[37] and a Trigonometrie.[38] This last +appeared in the year 1657 at London, in both Latin and English. + +It is claimed that the trigonometry was "neither finished nor published +by himself, but collected out of his scattered papers; and though he +connived at the printing it, yet imperfectly done, as appears by his +MSS.; and one of the printed Books, corrected by his own Hand."[39] +Doubtless more accurate on this point is a letter of Richard Stokes who +saw the book through the press: + + I have procured your Trigonometry to be written over in a fair hand, + which when finished I will send to you, to know if it be according to + your mind; for I intend (since you were pleased to give your assent) to + endeavour to print it with Mr. Briggs his Tables, and so soon as I can + get the Prutenic Tables I will turn those of the sun and moon, and send + them to you.[40] + +In the preface to the Latin edition Stokes writes: + + Since this trigonometry was written for private use without the + intention of having it published, it pleased the Reverend Author, + before allowing it to go to press, to expunge some things, to change + other things and even to make some additions and insert more lucid + methods of exposition. + +This much is certain, the Trigonometry bears the impress characteristic +of Oughtred. Like all his mathematical writings, the book was very +condensed. Aside from the tables, the text covered only 36 pages. Plane +and spherical triangles were taken up together. The treatise is known in +the history of trigonometry as among the very earliest works to adopt a +condensed symbolism so that equations involving trigonometric functions +could be easily taken in by the eye. In the work of 1657, contractions +are given as follows: s=sine, t=tangent, se=secant, s co=cosine (sine +complement), t co=cotangent, se co=cosecant, log=logarithm, Z cru=sum of +the sides of a rectangle or right angle, X cru=difference of these sides. +It has been generally overlooked by historians that Oughtred used the +abbreviations of trigonometric functions, named above, a quarter of a +century earlier, in his Circles of Proportion, 1632, 1633. Moreover, he +used sometimes also the abbreviations which are current at the present +time, namely sin=sine, tan=tangent, sec=secant. We know that the Circles +of Proportion existed in manuscript many years before they were +published. The symbol sv for sinus versus occurs in the Clavis of 1631. +The great importance of well-chosen symbols needs no emphasis to readers +of the present day. With reference to Oughtred's trigonometric symbols. +Augustus De Morgan said: + + This is so very important a step, simple as it is, that Euler is justly + held to have greatly advanced trigonometry by its introduction. Nobody + that we know of has noticed that Oughtred was master of the + improvement, and willing to have taught it, if people would have + learnt.[41] + +We find, however, that even Oughtred cannot be given the whole credit in +this matter. By or before 1631 several other writers used abbreviations +of the trigonometric functions. As early as 1624 the contractions sin for +sine and tan for tangent appear on the drawing representing Gunter's +scale, but Gunter did not use them in his books, except in the drawing of +his scale.[42] A closer competitor for the honor of first using these +trigonometric abbreviations is Richard Norwood in his Trigonometrie, +London, 1631, where s stands for sine, t for tangent, sc for sine +complement (cosine), tc for tangent complement (cotangent), and sec for +secant. Norwood was a teacher of mathematics in London and a well-known +writer of books on navigation. Aside from the abbreviations just cited +Norwood did not use nearly as much symbolism in his mathematics as did +Oughtred. + +Mention should be made of trigonometric symbols used even earlier than +any of the preceding, in "An Appendix to the Logarithmes, shewing the +practise of the Calculation of Triangles, etc.," printed in Edward +Wright's edition of Napier's A Description of the Admirable Table of +Logarithmes, London, 1618. We referred to this "Appendix" in tracing the +origin of the sign x. It contains, on p. 4, the following passage: "For +the Logarithme of an arch or an angle I set before (s), for the +antilogarithme or compliment thereof (s*) and for the Differential (t)." +In further explanation of this rather unsatisfactory passage, the author +(Oughtred?) says, "As for example: sB+BC=CA. that is, the Logarithme of +an angle B. at the Base of a plane right-angled triangle, increased by +the addition of the Logarithm of BC, the hypothenuse thereof, is equall +to the Logarithme of CA the cathetus." + +Here "logarithme of an angle B" evidently means "log sin B," just as with +Napier, "Logarithms of the arcs" signifies really "Logarithms of the +sines of the angles." In Napier's table, the numbers in the column marked +"Differentiae" signify log sine minus log cosine of an angle; that is, +the logarithms of the tangents. This explains the contraction (t) in the +"Appendix." The conclusion of all this is that as early as 1618 the signs +s, s*, t were used for sine, cosine, and tangent, respectively. + +John Speidell, in his Breefe Treatise of Sphaericall Triangles, London, +1627, uses Si. for sine, T. and Tan for tangent, Se. for secant, Si. Co. +for cosine, Se. Co. for cosecant, T. Co. for cotangent. + +The innovation of designating the sides and angles of a triangle by A, B, +C, and a, b, c, so that A was opposite a, B opposite b, and C opposite c, +is attributed to Leonard Euler (1753), but was first used by Richard +Rawlinson of Queen's College, Oxford, sometimes after 1655 and before +1668. Oughtred did not use Rawlinson's notation.[43] + +In trigonometry English writers of the first half of the seventeenth +century used contractions more freely than their continental +contemporaries; even more freely, indeed, than English writers of a later +period. Von Braunmuehl, the great historian of trigonometry, gives +Oughtred much praise for his trigonometry, and points out that half a +century later the army of writers on trigonometry had hardly yet reached +the standard set by Oughtred's analysis.[44] Oughtred must be credited +also with the first complete proof that was given to the first two of +"Napier's analogies." His trigonometry contains seven-place tables of +sines, tangents, and secants, and six-place tables of logarithmic sines +and tangents; also seven-place logarithmic tables of numbers. At the time +of Oughtred there was some agitation in favor of a wider introduction of +decimal systems. This movement is reflected in those tables which contain +the centesimal division of the degree, a practice which is urged for +general adoption in our own day, particularly by the French. + + + SOLUTION OF NUMERICAL EQUATIONS + +In the solution of numerical equations Oughtred does not mention the +sources from which he drew, but the method is substantially that of the +great French algebraist Vieta, as explained in a publication which +appeared in 1600 in Paris under the title, De numerosa potestatum purarum +atque adfectarum ad exegesin resolutione tractatus. In view of the fact +that Vieta's process has been described inaccurately by leading modern +historians including H. Hankel[45] and M. Cantor,[46] it may be worth +while to go into some detail.[47] By them it is made to appear as +identical with the procedure given later by Newton. The two are not the +same. The difference lies in the divisor used. What is now called +"Newton's method" is Newton's method as modified by Joseph Raphson.[48] +The Newton-Raphson method of approximation to the roots of an equation +f(x)=0 is usually given the form a-[f(a)/f'(a)], where a is an +approximate value of the required root. It will be seen that the divisor +is f'(a). Vieta's divisor is different; it is + + |f(a+s_1)-f(a)|-s_1^n, + +where f(x) is the left of the equation f(x)=k, n is the degree of +equation, and s_1 is a unit of the denomination of the digit next to be +found. Thus in x^3+420000x=247651713, it can be shown that 417 is +approximately a root; suppose that a has been taken to be 400, then +s_1=10; but if, at the next step of approximation, a is taken to be 410, +then s_1=1. In this example, taking a=400, Vieta's divisor would have +been 9120000; Newton's divisor would have been 900000. + +A comparison of Vieta's method with the Newton-Raphson method reveals the +fact that Vieta's divisor is more reliable, but labors under the very +great disadvantage of requiring a much larger amount of computation. The +latter divisor is accurate enough and easier to compute. Altogether the +Newton-Raphson process marks a decided advance over that of Vieta. + +As already stated, it is the method of Vieta that Oughtred explains. The +Englishman's exposition is an improvement on that of Vieta, printed forty +years earlier. Nevertheless, Oughtred's explanation is far from easy to +follow. The theory of equations was at that time still in its primitive +stage of development. Algebraic notation was not sufficiently developed +to enable the argument to be condensed into a form easily surveyed. So +complicated does Vieta's process of approximation appear that M. Cantor +failed to recognize that Vieta possessed a uniform mode of procedure. But +when one has in mind the general expression for Vieta's divisor which we +gave above, one will recognize that there was marked uniformity in +Vieta's approximations. + +Oughtred allows himself twenty-eight sections in which to explain the +process and at the close cannot forbear remarking that 28 is a "perfect" +number (being equal to the sum of its divisors, 1, 2, 4, 7, 14). + +The early part of his exposition shows how an equation may be transformed +so as to make its roots 10, 100, 1000, or 10^m times smaller. This +simplifies the task of "locating a root"; that is, of finding between +what integers the root lies. + +Taking one of Oughtred's equations, x^4-72x^3+238600x=8725815, upon +dividing 72x^3 by 10, 238600x by 1000, and 8725815 by 10,000, we obtain +x^4-7.2x^3+238.6x=872.5. Dividing both sides by x, we obtain +x^3+238.6-7.2x^2=x)872.5. Letting x=4, we have 64+238.6-115.2=187.4. + +But 4)872.5(218.1; 4 is too small. Next let x=5, we have +125+238.6-180=183.6. + +But 5)872.5(174.5; 5 is too large. We take the lesser value, x=4, or in +the original equation, x=40. This method may be used to find the second +digit in the root. Oughtred divides both sides of the equation by x^2, +and obtains x^2+x)238600-72x=x^2)8725815. He tries x=47 and x=48, and +finds that x=47. + +He explains also how the last computation may be done by logarithms. +Thereby he established for himself the record of being the first to use +logarithms in the solution of affected equations. + +As an illustration of Oughtred's method of approximation after the root +sought has been located, we have chosen for brevity a cubic in preference +to a quartic. We selected the equation x^3+420000x=247651713. By the +process explained above a root is found to lie between x=400 and x=500. +From this point on, the approximation as given by Oughtred is as shown on +p. 43. + +In further explanation of this process, observe that the given equation +is of the form L_c+C_qL=D_c, where L_c is our x, C_q=420000, +D_c=247651713. In the first step of approximation, let L=A+E, where A=400 +and E is, as yet, undetermined. We have + + L_c=(A+E)^3=A^3+3A^2E+3AE^2+E^3 + +and + + C_qL=420000(A+E). + +Subtract from 247651713 the sum of the known terms A^3 (his A_c) and +420000 A (his C_qA). This sum is 232000000 the remainder is 15651713. + + "Exemplum II + + 1c+420000l=247651713 + + Hoc est, L_c+C_qL=D_c + + 2 4 7 | 6 5 1 | 7 1 3 | ( 4 1 7 + ------+-------+-------+------------ + 4 2 | 0 0 0 | 0 | C_q + ------+-------+-------+------------ + 6 4 | | | A_c + 1 6 8 | 0 0 0 | 0 | C_q A + ------+-------+-------+------------ + 2 3 2 | 0 0 0 | 0 | Ablatit. + =================================== +R 1 5 | 6 5 1 | 7 1 3 | + ------+-------+-------+------------ + 4 | 8 | | 3 A_q + | 1 2 | | 3 A + 4 | 2 0 0 | 0 0 | C_q + ------+-------+-------+------------ + 9 | 1 2 0 | 0 0 | Divisor. + ------+-------+-------+------------ + 4 | 8 | | 3 A_q E + | 1 2 | | 3 A E_q + | 1 | | E_c + 4 | 2 0 0 | 0 0 | C_q E + ------+-------+-------+------------ + 9 | 1 2 1 | 0 0 | Ablatit. + =================================== +R 6 | 5 3 0 | 7 1 3 | 4 | 1 | + ------+-------+-------+------------ ----+-----+--- + | 5 0 4 | 3 | 3 A_q | | + | 1 | 2 3 | 3 A 1 6 | 8 | + | 4 2 0 | 0 0 0 | C_q | 1 | + ------+-------+-------+------------ ----+-----+--- + | 9 2 5 | 5 3 0 | Divisor. 1 6 8 1 + ------+-------+-------+------------ + 3 | 5 3 0 | 1 | 3 A_q E + | 6 0 | 2 7 | 3 A E_q + | | 3 4 3 | E_c + 2 | 9 4 0 | 0 0 0 | C_q E + ------+-------+-------+------------ + 6 | 5 3 0 | 7 1 3 | Ablatit." + +Next, he evaluates the coefficients of E in 3A^2E and 420000E, also 3A, +the coefficient of E^2. He obtains 3A^2=480000, 3A=1200, C_q=420000. He +interprets 3A^2 and C_q as tens, 3A as hundreds. Accordingly, he obtains +as their sum 9120000, which is the divisor for finding the second digit +in the approximation. Observe that this divisor is the value of +|f(a+s_1)-f(a)|-s_1^n in our general expression, where a=400, s_1=10, +n=3, f(x)=x^3+420000x. + +Dividing the remainder 15651713 by 9120000, he obtains the integer 1 in +ten's place; thus E=10, approximately. He now computes the terms 3A^2E, +3AE^2 and E^3 to be, respectively, 4800000, 120000, 1000. Their sum is +9121000. Subtracting it from the previous remainder, 15651713, leaves the +new remainder, 6530713. + +From here on each step is a repetition of the preceding step. The new A +is 410, the new E is to be determined. We have now in closer +approximation, L=A+E. This time we do not subtract A^3 and C_qA, because +this subtraction is already affected by the preceding work. + +We find the second trial divisor by computing the sum of 3A^2, 3A and +C_q; that is, the sum of 504300, 1230, 420000, which is 925530. Again, +this divisor can be computed by our general expression for divisors, by +taking a=410, s_1=1, n=3. + +Dividing 6530713 by 925530 yields the integer 7. Thus E=7. Computing +3A^2E, 3AE^2, E^3 and subtracting their sum, the remainder is 0. Hence +417 is an exact root of the given equation. + +Since the extraction of a cube root is merely the solution of a pure +cubic equation, x^3=n, the process given above may be utilized in finding +cube roots. This is precisely what Oughtred does in chap. xiv of his +Clavis. If the foregoing computation is modified by putting C_q=0, the +process will yield the approximate cube root of 247651713. + +Oughtred solves 16 examples by the process of approximation here +explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In +all cases he finds only one or two real roots. Of the roots sought, five +are irrational, the remaining are rational and are computed to their +exact values. Three of the computed roots have 2 figures each, 9 roots +have 3 figures each, 4 roots have 4 figures each. While no attempt is +made to secure all the roots--methods of computing complex roots were +invented much later--he computes roots of equations which involve large +coefficients and some of them are of a degree as high as the fifth. In +view of the fact that many editions of the Clavis were issued, one +impression as late as 1702, it contributed probably more than any other +book to the popularization of Vieta's method in England. + +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" [436^2], that "the logarithm of any Potestas +[436^2] 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 Rene 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 x 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^o-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, log_e 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/10^n. 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 x; (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 Rene 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 SEVENTEENTH + CENTURY" + +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 Wood[57] 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: + + Amongst the mathematical rarities these times have afforded, there are + none of that small number I (a late intruder into these studies) have + yet viewed, which so fully demonstrates their authors' great abilities + as your Clavis, not richer in augmentations, than valuable for + contraction; . . . . + + Your belief that there is in all inventions aliquid divinum, an + infusion beyond human cogitations, I am confident will appear notably + strengthened, if you please to afford this truth belief, that I entered + upon these studies accidentally after I betook myself to the country, + having never had so much aid as to be taught addition, nor the + discourse of an artist (having left both Oxford and London before I + knew what any proposition in geometry meant) to inform me what were the + best authors.[59] + +The following extracts from two letters by W. Robinson, written before +the appearance of the 1647 English edition of the Clavis, express the +feeling of many readers of the Clavis on its extreme conciseness and +brevity of explanation: + + I shall long exceedingly till I see your Clavis turned into a + pick-lock; and I beseech you enlarge it, and explain it what you can, + for we shall not need to fear either tautology or superfluity; you are + naturally concise, and your clear judgment makes you both methodical + and pithy; and your analytical way is indeed the only way. . . . . + + I will once again earnestly entreat you, that you be rather diffuse in + the setting forth of your English mathematical Clavis, than concise, + considering that the wisest of men noted of old, and said stultorum + infinitus est numerus, these arts cannot be made too easy, they are so + abstruse of themselves, and men either so lazy or dull, that their + fastidious wits take a loathing at the very entrance of these studies, + unless it be sweetened on with plainness and facility. Brevity may well + argue a learned author, that without any excess or redundance, either + of matter or words, can give the very substance and essence of the + thing treated of; but it seldom makes a learned scholar; and if one be + capable, twenty are not; and if the master sum up in brief the pith of + his own long labours and travails, it is not easy to imagine that + scholars can with less labour than it cost their masters dive into the + depths thereof.[60] + +Here is the judgment of another of Oughtred's friends: + + . . . . with the character I received from your and my noble friend Sir + Charles Cavendish, then at Paris, of your second edition of the same + piece, made me at my return into England speedily to get, and + diligently peruse the same. Neither truly did I find my expectation + deceived; having with admiration often considered how it was possible + (even in the hardest things of geometry) to deliver so much matter in + so few words, yet with such demonstrative clearness and perspicuity: + and hath often put me in mind of learned Mersennus his judgment (since + dead) of it, that there was more matter comprehended in that little + book than in Diophantus, and all the ancients. . . . .[61] + +Oughtred's own feeling was against diffuseness in textbook writing. In +his revisions of his Clavis the original character of that book was not +altered. In his reply to W. Robinson, Oughtred said: + + . . . . But my art for all such mathematical inventions I have set down + in my Clavis Mathematica, which therefore in my title I say is tum + logisticae cum analyticae adeoque totius mathematicae quasi clavis, + which if any one of a mathematical genius will carefully study, (and + indeed it must be carefully studied,) he will not admire others, but + himself do wonders. But I (such is my tenuity) have enough fungi vice + cotis, acutum reddere quae ferrum valet, exsors ipsa secandi, or like + the touchstone, which being but a stone, base and little worth, can + shew the excellence and riches of gold.[62] + +John Wallis held Oughtred's Clavis in high regard. When in correspondence +with John Collins concerning plans for a new edition, Wallis wrote in +1666-67, six years after the death of Oughtred: + + . . . . But for the goodness of the book in itself, it is that (I + confess) which I look upon as a very good book, and which doth in as + little room deliver as much of the fundamental and useful part of + geometry (as well as of arithmetic and algebra) as any book I know; and + why it should not be now acceptable I do not see. It is true, that as + in other things so in mathematics, fashions will daily alter, and that + which Mr. Oughtred designed by great letters may be now by others be + designed by small; but a mathematician will, with the same ease and + advantage, understand A_c, and a^3 or aaa. . . . . And the like I judge + of Mr. Oughtred's Clavis, which I look upon (as those pieces of Vieta + who first went in that way) as lasting books and classic authors in + this kind; to which, notwithstanding, every day may make new additions. + . . . . + + But I confess, as to my own judgment, I am not for making the book + bigger, because it is contrary to the design of it, being intended for + a manual or contract; whereas comments, by enlarging it, do rather + destroy it. . . . . But it was by him intended, in a small epitome, to + give the substance of what is by others delivered in larger volumes. . + . . .[63] + +That there continued to be a group of students and teachers who desired a +fuller exposition than is given by Oughtred is evident from the +appearance, over fifty years after the first publication of the Clavis, +of a booklet by Gilbert Clark, entitled Oughtredus Explicatus, London, +1682. A review of this appeared in the Acta Eruditorum (Leipzig, 1684), +on p. 168, wherein Oughtred is named "clarissimus Angliae mathematicus." +John Collins wrote Wallis in 1666-67 that Clark, "who lives with Sir +Justinian Isham, within seven miles of Northampton, . . . . intimates he +wrote a comment on the Clavis, which lay long in the hands of a printer, +by whom he was abused, meaning Leybourne."[64] + +We shall have occasion below to refer to Oughtred's inability to secure a +copy of a noted Italian mathematical work published a few years before. +In those days the condition of the book trade in England must have been +somewhat extraordinary. Dr. J. W. L. Glaisher throws some light upon this +subject.[65] He found in the Calendar of State Papers, Domestic Series, +1637, a petition to Archbishop Laud in which it is set forth that when +Hooganhuysen, a Dutchman, "heretofore complained of in the High +Commission for importing books printed beyond the seas," had been bound +"not to bring in any more," one Vlacq (the computer and publisher of +logarithmic tables) "kept up the same agency and sold books in his stead. +. . . . Vlacq is now preparing to go beyond the seas to avoid answering +his late bringing over nine bales of books contrary to the decree of the +Star Chamber." Judgment was passed that, "Considering the ill-consequence +and scandal that would arise by strangers importing and venting in this +kingdom books printed beyond the seas," certain importations be +prohibited, and seized if brought over. + +This want of easy intercommunication of results of scientific research in +Oughtred's time is revealed in the following letter, written by Oughtred +to Robert Keylway, in 1645: + + I speak this the rather, and am induced to a better confidence of your + performance, by reason of a geometric-analytical art or practice found + out by one Cavalieri, an Italian, of which about three years since I + received information by a letter from Paris, wherein was praelibated + only a small taste thereof, yet so that I divine great enlargement of + the bounds of the mathematical empire will ensue. I was then very + desirous to see the author's own book while my spirits were more free + and lightsome, but I could not get it in France. Since, being more + stept into years, daunted and broken with the sufferings of these + disastrous times, I must content myself to keep home, and not put out + to any foreign discoveries.[66] + +It was in 1655, when Oughtred was about eighty years old, that John +Wallis, the great forerunner of Newton in Great Britain, began to publish +his great researches on the arithmetic of infinites. Oughtred rejoiced +over the achievements of his former pupil. In 1655, Oughtred wrote John +Wallis as follows: + + I have with unspeakable delight, so far as my necessary businesses, the + infirmness of my health, and the greatness of my age (approaching now + to an end) would permit, perused your most learned papers, of several + choice arguments, which you sent me: wherein I do first with + thankfulness acknowledge to God, the Father of lights, the great light + he hath given you; and next I congratulate you, even with admiration, + the clearness and perspicacity of your understanding and genius, who + have not only gone, but also opened a way into these profoundest + mysteries of art, unknown and not thought of by the ancients. With + which your mysterious inventions I am the more affected, because full + twenty years ago, the learned patron of learning, Sir Charles + Cavendish, shewed me a paper written, wherein were some few excellent + new theorems, wrought by the way, as I suppose, of Cavalieri, which I + wrought over again more agreeably to my way. The paper, wherein I + wrought it, I shewed to many, whereof some took copies, but my own I + cannot find. I mention it for this, because I saw therein a light + breaking out for the discovery of wonders to be revealed to mankind, in + this last age of the world: which light I did salute as afar off, and + now at a nearer distance embrace in your prosperous beginnings. Sir, + that you are pleased to mention my name in your never dying papers, + that is your noble favour to me, who can add nothing to your glory, but + only my applause. . . . .[67] + +The last sentence has reference to Wallis' appreciative and eulogistic +reference to Oughtred in the preface. It is of interest to secure the +opinion of later English writers who knew Oughtred only through his +books. John Locke wrote in his journal under the date, June 24, 1681, +"the best algebra yet extant is Outred's."[68] John Collins, who is known +in the history of mathematics chiefly through his very extensive +correspondence with nearly all mathematicians of his day, was inclined to +be more critical. He wrote Wallis about 1667: + + It was not my intent to disparage the author, though I know many that + did lightly esteem him when living, some whereof are at rest, as Mr. + Foster and Mr. Gibson. . . . . You grant the author is brief, and + therefore obscure, and I say it is but a collection, which, if himself + knew, he had done well to have quoted his authors, whereto the reader + might have repaired. You do not like those words of Vieta in his + theorems, ex adjunctione plano solidi, plus quadrato quadrati, etc., + and think Mr. Oughtred the first that abridged those expressions by + symbols; but I dissent, and tell you 'twas done before by Cataldus, + Geysius, and Camillus Gloriosus,[69] who in his first decade of + exercises, (not the first tract,) printed at Naples in 1627, which was + four years before the first edition of the Clavis, proposeth this + equation just as I here give it you, viz. + 1ccc+16qcc+41qqc-2304cc-18364qc-133000qq-54505c+3728q+8064 N aequatur + 4608, finds N or a root of it to be 24, and composeth the whole out of + it for proof, just in Mr. Oughtred's symbols and method. Cataldus on + Vieta came out fifteen years before, and I cannot quote that, as not + having it by me. + + . . . . And as for Mr. Oughtred's method of symbols, this I say to it; + it may be proper for you as a commentator to follow it, but divers I + know, men of inferior rank that have good skill in algebra, that + neither use nor approve it. . . . . Is not A^5 sooner wrote than A_qc? + Let A be 2, the cube of 2 is 8, which squared is 64: one of the + questions between Maghet Grisio and Gloriosus is whether 64=A_cc or + A_qc. The Cartesian method tells you it is A^6, and decides the doubt. + . . . .[70] + +There is some ground for the criticisms passed by Collins. To be sure, +the first edition of the Clavis is dated 1631--six years before Descartes +suggested the exponential notation which came to be adopted as the +symbolism in our modern algebra. But the second edition of the Clavis, +1647, appeared ten years after Descartes' innovation. Had Oughtred seen +fit to adopt the new exponential notation in 1647, the step would have +been epoch-making in the teaching of algebra in England. We have seen no +indication that Oughtred was familiar with Descartes' Geometrie of 1637. + +The year preceding Oughtred's death Mr. John Twysden expressed himself as +follows in the preface to his Miscellanies: + + It remains that I should adde something touching the beginning, and use + of these Sciences. . . . . I shall only, to their honours, name some of + our own Nation yet living, who have happily laboured upon both stages. + That succeeding ages may understand that in this of ours, there yet + remained some who were neither ignorant of these Arts, as if they had + held them vain, nor condemn them as superfluous. Amongst them all let + Mr. William Oughtred, of Aeton, be named in the first place, a Person + of venerable grey haires, and exemplary piety, who indeed exceeds all + praise we can bestow upon him. Who by an easie method, and admirable + Key, hath unlocked the hidden things of geometry. Who by an accurate + Trigonometry and furniture of Instruments, hath inriched, as well + geometry, as Astronomy. Let D. John Wallis, and D. Seth Ward, succeed + in the next place, both famous Persons, and Doctors in Divinity, the + one of geometry, the other of astronomy, Savilian Professors in the + University of Oxford.[71] + +The astronomer Edmund Halley, in his preface to the 1694 English edition +of the Clavis, speaks of this book as one of "so established a +reputation, that it were needless to say anything thereof," though "the +concise Brevity of the author is such, as in many places to need +Explication, to render it Intelligible to the less knowing Mathematical +matters." + +In closing this part of our monograph, we quote the testimony of Robert +Boyle, the experimental physicist, as given May 8, 1647, in a letter to +Mr. Hartlib: + + The Englishing of, and additions to Oughtred's Clavis mathematica does + much content me, I having formerly spent much study on the original of + that algebra, which I have long since esteemed a much more instructive + way of logic, than that of Aristotle.[72] + + + WAS DESCARTES INDEBTED TO OUGHTRED? + +This question first arose in the seventeenth century, when John Wallis, +of Oxford, in his Algebra (the English edition of 1685, and more +particularly the Latin edition of 1693), raised the issue of Descartes' +indebtedness to the English scientists, Thomas Harriot and William +Oughtred. In discussing matters of priority between Harriot and +Descartes, relating to the theory of equations, Wallis is generally held +to have shown marked partiality to Harriot. Less attention has been given +by historians of mathematics to Descartes' indebtedness to Oughtred. Yet +this question is of importance in tracing Oughtred's influence upon his +time. + +On January 8, 1688-89, Samuel Morland addressed a letter of inquiry to +John Wallis, containing a passage which we translate from the Latin: + + Some time ago I read in the elegant and truly precious book that you + have written on Algebra, about Descartes, this philosopher so extolled + above all for having arrived at a very perfect system by his own + powers, without the aid of others, this Descartes, I say, who has + received in geometry very great light from our Oughtred and our + Harriot, and has followed their track though he carefully suppressed + their names. I stated this in a conversation with a professor in + Utrecht (where I reside at present). He requested me to indicate to him + the page-numbers in the two authors which justified this accusation. I + admitted that I could not do so. The Geometrie of Descartes is not + sufficiently familiar to me, although with Oughtred I am fairly + familiar. I pray you therefore that you will assume this burden. Give + me at least those references to passages of the two authors from the + comparison of which the plagiarism by Descartes is the most + striking.[73] + +Following Morland's letter in the De algebra tractatus, is printed +Wallis' reply, dated March 12, 1688 ("Stilo Angliae"), which is, in part, +as follows: + + I nowhere give him the name of a plagiarist; I would not appear so + impolite. However this I say, the major part of his algebra (if not + all) is found before him in other authors (notably in our Harriot) whom + he does not designate by name. That algebra may be applied to geometry, + and that it is in fact so applied, is nothing new. Passing the ancients + in silence, we state that this has been done by Vieta, Ghetaldi, + Oughtred and others, before Descartes. They have resolved by algebra + and specious arithmetic [literal arithmetic] many geometrical problems. + . . . . But the question is not as to application of algebra to + geometry (a thing quite old), but of the Cartesian algebra considered + by itself. + +Wallis then indicates in the 1659 edition of Descartes' Geometrie where +the subjects treated on the first six pages are found in the writings of +earlier algebraists, particularly of Harriot and Oughtred. For example, +what is found on the first page of Descartes, relating to addition, +subtraction, multiplication, division, and root extraction, is declared +by Wallis to be drawn from Vieta, Ghetaldi, and Oughtred. + +It is true that Descartes makes no mention of modern writers, except once +of Cardan. But it was not the purpose of Descartes to write a history of +algebra. To be sure, references to such of his immediate predecessors as +he had read would not have been out of place. Nevertheless, Wallis fails +to show that Descartes made illegitimate use of anything he may have seen +in Harriot or Oughtred. + +The first inquiry to be made is, Did Descartes possess copies of the +books of Harriot and Oughtred? It is only in recent time that this +question has been answered as to Harriot. As to Oughtred, it is still +unanswered. It is now known that Descartes had seen Harriot's Artis +analyticae praxis (1631). Descartes wrote a letter to Constantin Huygens +in which he states that he is sending Harriot's book.[74] + +An able discussion of the question, what effect, if any, Oughtred's +Clavis mathematicae of 1631 had upon Descartes'[75] Geometrie of 1637, is +given by H. Bosmans in a recent article. According to Bosmans no evidence +has been found that Descartes possessed a copy of Oughtred's book, or +that he had examined it. Bosmans believes nevertheless that Descartes was +influenced by the Clavis, either directly or indirectly. He says: + + If Descartes did not read it carefully, which is not proved, he was + none the less well informed with regard to it. No one denies his + intimate knowledge of the intellectual movement of his time. The Clavis + mathematica enjoyed a rapid success. It is impossible that, at least + indirectly, he did not know the more original ideas which it contained. + Far from belittling Descartes, as I much desire to repeat, this rather + makes him the greater.[76] + +We ourselves would hardly go as far as does Bosmans. Unless Descartes +actually examined a copy of Oughtred it is not likely that he was +influenced by Oughtred in appreciable degree. Book reviews were quite +unknown in those days. No evidence has yet been adduced to show that +Descartes obtained a knowledge of Oughtred by correspondence. A most +striking feature about Oughtred's Clavis is its notation. No trace of the +Englishman's symbolism has been pointed out in Descartes' Geometrie of +1637. Only six years intervened between the publication of the Clavis and +the Geometrie. It took longer than this period for the Clavis to show +evidence of its influence upon mathematical books published in England; +it is not probable that abroad the contact was more immediate than at +home. Our study of seventeenth-century algebra has led us to the +conviction that Oughtred deserves a higher place in the development of +this science than is usually accorded to him; but that it took several +decennia for his influence fully to develop. + + + THE SPREAD OF OUGHTRED'S NOTATIONS + +An idea of Oughtred's influence upon mathematical thought and teaching +can be obtained from the spread of his symbolism. This study indicates +that the adoption was not immediate. The earliest use that we have been +able to find of Oughtred's notation for proportion, A.B::C.D, occurs +nineteen years after the Clavis mathematicae of 1631. In 1650 John Kersey +brought out in London an edition of Edmund Wingates' Arithmetique made +easie, in which this notation is used. After this date publications +employing it became frequent, some of them being the productions of +pupils of Oughtred. We have seen it in Vincent Wing (1651),[77] Seth Ward +(1653),[78] John Wallis (1655),[79] in "R. B.," a schoolmaster in +Suffolk,[80] Samuel Foster (1659),[81] Jonas Moore (1660),[82] and Isaac +Barrow (1657).[83] In the latter part of the seventeenth century +Oughtred's notation, A.B::C.D, became the prevalent, though not +universal, notation in Great Britain. A tremendous impetus to their +adoption was given by Seth Ward, Isaac Barrow, and particularly by John +Wallis, who was rising to international eminence as a mathematician. + +In France we have noticed Oughtred's notation for proportion in +Franciscus Dulaurens (1667),[84] J. Prestet (1675),[85] R. P. Bernard +Lamy (1684),[86] Ozanam (1691),[87] De l'Hospital (1696),[88] R. P. Petro +Nicolas (1697).[89] + +In the Netherlands we have noticed it in R. P. Bernard Lamy (1680),[90] +and in an anonymous work of 1690.[91] In German and Italian works of the +seventeenth century we have not seen Oughtred's notation for proportion. + +In England a modified notation soon sprang up in which ratio was +indicated by two dots instead of a single dot, thus A:B::C:D. The reason +for the change lies probably in the inclination to use the single dot to +designate decimal fractions. W. W. Beman pointed out that this modified +symbolism (:) for ratio is found as early as 1657 in the end of the +trigonometric and logarithmic tables that were bound with Oughtred's +Trigonometria.[92] It is not probable, however, that this notation was +used by Oughtred himself. The Trigonometria proper has Oughtred's +A.B::C.D throughout. Moreover, in the English edition of this +trigonometry, which appeared the same year, 1657, but subsequent to the +Latin edition, the passages which contained the colon as the symbol for +ratio, when not omitted, are recast, and the regular Oughtredian notation +is introduced. In Oughtred's posthumous work, Opuscula mathematica +hactenus inedita, 1677, the colon appears quite often but is most likely +due to the editor of the book. + +We have noticed that the notation A:B::C:D antedates the year 1657. +Vincent Wing, the astronomer, published in 1651 in London the Harmonicon +coeleste, in which is found not only Oughtred's notation A.B::C.D but +also the modified form of it given above. The two are used +interchangeably. His later works, the Logistica astronomica (1656), +Doctrina spherica (1655), and Doctrina theorica, published in one volume +in London, all use the symbols A:B::C:D exclusively. The author of a book +entitled, An Idea of Arithmetick at first designed for the use of the +Free Schoole at Thurlow in Suffolk . . . . by R. B., Schoolmaster there, +London, 1655, writes A:a::C:c, though part of the time he uses Oughtred's +unmodified notation. + +We can best indicate the trend in England by indicating the authors of +the seventeenth century whom we have found using the notation A:B::C:D +and the authors of the eighteenth century whom we have found using +A.B::C.D. The former notation was the less common during the seventeenth +but the more common during the eighteenth century. We have observed the +symbols A:B::C:D (besides the authors already named) in John Collins +(1659),[93] James Gregory (1663),[94] Christopher Wren (1668-69),[95] +William Leybourn (1673),[96] William Sanders (1686),[97] John Hawkins +(1684),[98] Joseph Raphson (1697),[99] E. Wells (1698),[100] and John +Ward (1698).[101] + +Of English eighteenth-century authors the following still clung to the +notation A.B::C.D: John Harris' translation of F. Ignatius Gaston Pardies +(1701),[102] George Shelley (1704),[103] Sam Cobb (1709),[104] J. Collins +in Commercium Epistolicum (1712), John Craig (1718),[105] Jo. Wilson +(1724).[106] The latest use of A.B::C.D which has come to our notice is +in the translation of the Analytical Institutions of Maria G. Agnesi, +made by John Colson sometime before 1760, but which was not published +until 1801. During the seventeenth century the notation A:B::C:D acquired +almost complete ascendancy in England. + +In France Oughtred's unmodified notation A.B::C.D, having been adopted +later, was also discarded later than in England. An approximate idea of +the situation appears from the following data. The notation A.B::C.D was +used by M. Carre (1700),[107] M. Guisnee (1705),[108] M. de Fontenelle +(1727),[109] M. Varignon (1725),[110] M. Robillard (1753),[111] M. +Sebastien le Clerc (1764),[112] Clairaut (1731),[113] M. L'Hospital +(1781).[114] + +In Italy Oughtred's modified notation a, b::c, d was used by Maria G. +Agnesi in her Instituzioni analitiche, Milano, 1748. The notation +a:b::c:d found entrance the latter part of the eighteenth century. In +Germany the symbolism a:b=c:d, suggested by Leibniz, found wider +acceptance.[115] + +It is evident from the data presented that Oughtred proposed his notation +for ratio and proportion at a time when the need of a specific notation +began to be generally felt, that his symbol for ratio a.b was temporarily +adopted in England and France but gave way in the eighteenth century to +the symbol a:b, that Oughtred's symbol for proportion :: found almost +universal adoption in England and France and was widely used in Italy, +the Netherlands, the United States, and to some extent in Germany; it has +survived to the present time but is now being gradually displaced by the +sign of equality =. + +Oughtred's notation to express aggregation of terms has received little +attention from historians but is nevertheless interesting. His books, as +well as those of John Wallis, are full of parentheses but they are not +used as symbols of aggregation in algebra; they are simply marks of +punctuation for parenthetical clauses. We have seen that Oughtred writes +(a+b)^2 and {root}(a+b) thus, Q:a+b:, {root}:a+b:, or Q:a+b, {root}:a+b, +using on rarer occasions a single dot in place of the colon. This +notation did not originate with Oughtred, but, in slightly modified form, +occurs in writings from the Netherlands. In 1603 C. Dibvadii in +geometriam Evclidis demonstratio numeralis, Leyden, contains many +expressions of this sort, {root}.136+{root}2048, signifying +{root}(136+{root}2048). The dot is used to indicate that the root of the +binomial (not of 136 alone) is called for. This notation is used +extensively in Ludolphi a Cevlen de circulo, Leyden, 1619, and in +Willebrordi Snellii De circuli dimensione, Leyden, 1621. In place of the +single dot Oughtred used the colon (:), probably to avoid confusion with +his notation for ratio. To avoid further possibility of uncertainty he +usually placed the colon both before and after the algebraic expression +under aggregation. This notation was adopted by John Wallis and Isaac +Barrow. It is found in the writings of Descartes. Together with Vieta's +horizontal bar, placed over two or more terms, it constituted the means +used almost universally for denoting aggregation of terms in algebra. +Before Oughtred the use of parentheses had been suggested by Clavius[116] +and Girard.[117] The latter wrote, for instance, {root}(2+{root}3). While +parentheses never became popular in algebra before the time of Leibniz +and the Bernoullis they were by no means lost sight of. We are able to +point to the following authors who made use of them: I. Errard de +Bar-le-Duc (1619),[118] Jacobo de Billy (1643),[119] one of whose books +containing this notation was translated into English, and also the +posthumous works of Samuel Foster.[120] J. W. L. Glaisher points out that +parentheses were used by Norwood in his Trigonometrie (1631), p. 30.[121] + +The symbol for the arithmetical difference between two numbers, ~, is +usually attributed to John Wallis, but it occurs in Oughtred's Clavis +mathematicae of 1652, in the tract on Elementi decimi Euclidis +declaratio, at an earlier date than in any of Wallis' books. As Wallis +assisted in putting this edition through the press it is possible, though +not probable, that the symbol was inserted by him. Were the symbol +Wallis', Oughtred would doubtless have referred to its origin in the +preface. During the eighteenth century the symbol found its way into +foreign texts even in far-off Italy.[122] It is one of three symbols +presumably invented by Oughtred and which are still used at the present +time. The others are x and ::. + +The curious and ill-chosen symbols, {symbol} for "greater than," and +{symbol} for "less than," were certain to succumb in their struggle for +existence against Harriot's admirably chosen > and <. Yet such was the +reputation of Oughtred that his symbols were used in England quite +extensively during the seventeenth and the beginning of the eighteenth +century. Considerable confusion has existed among algebraists and also +among historians as to what Oughtred's symbols really were. Particularly +is this true of the sign for "less than" which is frequently written +{symbol}. Oughtred's symbols, or these symbols turned about in some way, +have been used by Seth Ward,[123] John Wallis,[124] Isaac Barrow,[125] +John Kersey,[126] E. Wells,[127] John Hawkins,[128] Tho. Baker,[129] +Richard Sault,[130] Richard Rawlinson,[131] Franciscus Dulaurens,[132] +James Milnes,[133] George Cheyne,[134] John Craig,[135] Jo. Wilson,[136] +and J. Collins.[137] + +General acceptance has been accorded to Oughtred's symbol x. The first +printed appearance of this symbol for multiplication in 1618 in the form +of the letter x hardly explains its real origin. The author of the +"Appendix" (be he Oughtred or someone else) may not have used the letter +x at all, but may have written the cross x, called the St. Andrew's +cross, while the printer, in the absence of any type accurately +representing that cross, may have substituted the letter x in its place. +The hypothesis that the symbol x of multiplication owes its origin to the +old habit of using directed bars to indicate that two numbers are to be +combined, as for instance in the multiplication of 23 and 34, thus, + + 2 3 + |\ /| + | x | + |/ \| + 3 4 + ------- + 7 8 2 + +has been advanced by two writers, C. Le Paige[138] and Gravelaar.[139] +Bosmans is more inclined to the belief that Oughtred adopted the symbol +somewhat arbitrarily, much as he did the numerous symbols in his Elementi +decimi Euclidis declaratio.[140] + +Le Paige's and Gravelaar's theory finds some support in the fact that the +cross x, without the two additional vertical lines shown above, occurs in +a commentary published by Oswald Schreshensuchs[141] in 1551, where the +sign is written between two factors placed one above the other. + + + + + CHAPTER V + OUGHTRED'S IDEAS ON THE TEACHING OF MATHEMATICS + + + GENERAL STATEMENT + +Nowhere has Oughtred given a full and systematic exposition of his views +on mathematical teaching. Nevertheless, he had very pronounced and +clear-cut ideas on the subject. That a man who was not a teacher by +profession should have mature views on teaching is most interesting. We +gather his ideas from the quality of the books he published, from his +prefaces, and from passages in his controversial writing against +Delamain. As we proceed to give quotations unfolding Oughtred's views, we +shall observe that three points receive special emphasis: (1) an appeal +to the eye through suitable symbolism; (2) emphasis upon rigorous +thinking; (3) the postponement of the use of mathematical instruments +until after the logical foundations of a subject have been thoroughly +mastered. + +The importance of these tenets is immensely reinforced by the conditions +of the hour. This voice from the past speaks wisdom to specialists of +today. Recent methods of determining educational values and the modern +cult of utilitarianism have led some experts to extraordinary +conclusions. Laboratory methods of testing, by the narrowness of their +range, often mislead. Thus far they have been inferior to the word of a +man of experience, insight, and conviction. + + + MATHEMATICS, "A SCIENCE OF THE EYE" + +Oughtred was a great admirer of the Greek mathematicians--Euclid, +Archimedes, Apollonius of Perga, Diophantus. But in reading their works +he experienced keenly what many modern readers have felt, namely, that +the almost total absence of mathematical symbols renders their writings +unnecessarily difficult to read. Statements that can be compressed into a +few well-chosen symbols which the eye is able to survey as a whole are +expressed in long-drawn-out sentences. A striking illustration of the +importance of symbolism is afforded by the history of the formula + + ix=log(cos x+i sin x). + +It was given in Roger Cotes' Harmonia mensurarum, 1722, not in symbols, +but expressed in rhetorical form, destitute of special aids to the eye. +The result was that the theorem remained in the book undetected for 185 +years and was meanwhile rediscovered by others. Owing to the prominence +of Cotes as a mathematician it is very improbable that such a thing could +have happened had the theorem been thrust into view by the aid of +mathematical symbols. + +In studying the ancient authors Oughtred is reported to have written down +on the margin of the printed page some of the theorems and their proofs, +expressed in the symbolic language of algebra. + +In the preface of his Clavis of 1631 and of 1647 he says: + + Wherefore, that I might more clearly behold the things themselves, I + uncasing the Propositions and Demonstrations out of their covert of + words, designed them in notes and species appearing to the very eye. + After that by comparing the divers affections of Theorems, inequality, + proportion, affinity, and dependence, I tryed to educe new out of them. + +It was this motive which led him to introduce the many abbreviations in +algebra and trigonometry to which reference has been made in previous +pages. The pedagogical experience of recent centuries has indorsed +Oughtred's view, provided of course that the pupil is carefully taught +the exact meaning of the symbols. There have been and there still are +those who oppose the intensive use of symbolism. In our day the new +symbolism for all mathematics, suggested by the school of Peano in Italy, +can hardly be said to be received with enthusiasm. In Oughtred's day +symbolism was not yet the fashion. To be convinced of this fact one need +only open a book of Edmund Gunter, with whom Oughtred came in contact in +his youth, or consult the Principia of Sir Isaac Newton, who flourished +after Oughtred. The mathematical works of Gunter and Newton, particularly +the former, are surprisingly destitute of mathematical symbols. The +philosopher Hobbes, in a controversy with John Wallis, criticized the +latter for that "Scab of Symbols," whereupon Wallis replied: + + I wonder how you durst touch M. Oughtred for fear of catching the Scab. + For, doubtlesse, his book is as much covered over with the Scab of + Symbols, as any of mine. . . . . As for my Treatise of Conick Sections, + you say, it is covered over with the Scab of Symbols, that you had not + the patience to examine whether it is well or ill demonstrated.[142] + +Oughtred maintained his view of the importance of symbols on many +different occasions. Thus, in his Circles of Proportion, 1632, p. 20: + + This manner of setting downe Theoremes, whether they be Proportions, or + Equations, by Symboles or notes of words, is most excellent, + artificiall, and doctrinall. Wherefore I earnestly exhort every one, + that desireth though but to looke into these noble Sciences + Mathematicall, to accustome themselves unto it: and indeede it is + easie, being most agreeable to reason, yea even to sence. And out of + this working may many singular consectaries be drawne: which without + this would, it may be, for ever lye hid. + + + RIGOROUS THINKING AND THE USE OF INSTRUMENTS + +The author's elevated concept of mathematical study as conducive to +rigorous thinking shines through the following extract from his preface +to the 1647 Clavis: + + . . . . Which Treatise being not written in the usuall synthetical + manner, nor with verbous expressions, but in the inventive way of + Analitice, and with symboles or notes of things instead of words, + seemed unto many very hard; though indeed it was but their owne + diffidence, being scared by the newnesse of the delivery; and not any + difficulty in the thing it selfe. For this specious and symbolicall + manner, neither racketh the memory with multiplicity of words, nor + chargeth the phantasie with comparing and laying things together; but + plainly presenteth to the eye the whole course and processe of every + operation and argumentation. + + Now my scope and intent in the first Edition of that my Key was, and in + this New Filing, or rather forging of it, is, to reach out to the + ingenious lovers of these Sciences, as it were Ariadnes thread, to + guide them through the intricate Labyrinth of these studies, and to + direct them for the more easie and full understanding of the best and + antientest Authors. . . . . That they may not only learn their + propositions, which is the highest point of Art that most Students aime + at; but also may perceive with what solertiousnesse, by what engines of + aequations, Interpretations, Comparations, Reductions, and + Disquisitions, those antient Worthies have beautified, enlarged, and + first found out this most excellent Science. . . . . Lastly, by framing + like questions problematically, and in a way of Analysis, as if they + were already done, resolving them into their principles, I sought out + reasons and means whereby they might be effected. And by this course of + practice, not without long time, and much industry, I found out this + way for the helpe and facilitation of Art. + +Still greater emphasis upon rigorous thinking in mathematics is laid in +the preface to the Circles of Proportion and in some parts of his +Apologeticall Epistle against Delamain. In that preface William Forster +quotes the reply of Oughtred to the question how he (Oughtred) had for so +many years concealed his invention of the slide rule from himself +(Forster) whom he had taught so many other things. The reply was: + + 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 in-stead of Artists, to + make their Scholers only doers of tricks, and as it were Iuglers: to + the despite of Art, losse of previous time, and betraying of willing + and industrious wits, vnto ignorance, and idlenesse. That the vse 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 haue me + well instructed in the Sciences." + +Delamain took a different view, arguing that instruments might very well +be placed in the hands of pupils from the start. At the time of this +controversy Delamain supported himself by teaching mathematics in London +and he advertised his ability to give instruction in mathematics, +including the use of instruments. Delamain brought the charge against +Oughtred of unjustly calling "many of the [British] Nobility and Gentry +doers of trickes and juglers." To this Oughtred replies: + + As I did to Delamain and to some others, so I did to William Forster: I + freely gave him my helpe and instruction in these faculties: only this + was the difference, I had the very first moulding (as I may say) of + this latter: But Delamain was already corrupted with doring upon + Instruments, and quite lost from ever being made an Artist: I suffered + not William Forster for some time so much as speake of any Instrument, + except only the Globe it selfe; and to explicate, and worke the + questions of the Sphaere, by the way of the Analemma: which also + himselfe did describe for the present occasion. And this my restraint + from such pleasing avocations, and holding him to the strictnesse of + percept, brought forth this fruit, that in short time, even by his owne + skill, he could not onely use any Instrument he should see, but also + was able to delineate the like, and devise others.[143] + +As representing Delamain's views, we make the following selection from +his Grammelogia (London, about 1633), the part near the end of the book +and bearing the title, "In the behalfe of vulgar Teachers and others," +where Delamain refers to Oughtred's charge that the scholars of "vulgar" +teachers are "doers of tricks, as it were iuglers." Delamain says: + + . . . . Which words are neither cautelous, nor subterfugious, but are + as downe right in their plainnesse, as they are touching, and + pernitious, by two much derogating from many, and glancing upon many + noble personages, with too grosse, if not too base an attribute, in + tearming them doers of tricks, as it were to iuggle: because they + perhaps make use of a necessitie in the furnishing of themselves with + such knowledge by Practicall Instrumentall operation, when their more + weighty negotiations will not permit them for Theoreticall figurative + demonstration; those that are guilty of the aspertion, and are touched + therewith may answer for themselves, and studie to be more + Theoreticall, than Practicall: for the Theory, is as the Mother that + produceth the daughter, the very sinewes and life of Practise, the + excellencie and highest degree of true Mathematicall Knowledge: but for + those that would make but a step as it were into that kind of Learning, + whose onely desire is expedition, and facilitie, both which by the + generall consent of all are best effected with Instrument, rather then + with tedious regular demonstrations, it was ill to checke them so + grosly, not onely in what they have Practised, but abridging them also + of their liberties with what they may Practise, which aspertion may not + easily be slighted off by any glosse or Apologie, without an Ingenuous + confession, or some mentall reservation: To which vilification, + howsoever, in the behalfe of my selfe, and others, I answer; That + Instrumentall operation is not only the Compendiating, and facilitating + of Art, but even the glory of it, whole demonstration both of the + making, and operation is soly in the science, and to an Artist or + disputant proper to be knowne, and so to all, who would truly know the + cause of the Mathematicall operations in their originall; But, for none + to know the use of a Mathematicall Instrumen[t], except he knowes the + cause of its operation, is somewhat too strict, which would keepe many + from affecting the Art, which of themselves are ready enough every + where, to conceive more harshly of the difficultie, and impossibilitie + of attayning any skill therein, then it deserves, because they see + nothing but obscure propositions, and perplex and intricate + demonstrations before their eyes, whose unsavoury tartnes, to an + unexperienced palate like bitter pills is sweetned over, and made + pleasant with an Instrumentall compendious facilitie, and made to goe + downe the more readily, and yet to retaine the same vertue, and + working; And me thinkes in this queasy age, all helpes may bee used to + procure a stomacke, all bates and invitations to the declining studie + of so noble a Science, rather then by rigid Method and generall Lawes + to scarre men away. All are not of like disposition, neither all (as + was sayd before) propose the same end, some resolve to wade, others to + put a finger in onely, or wet a hand: now thus to tye them to an + obscure and Theoricall forme of teaching, is to crop their hope, even + in the very bud. . . . . The beginning of a mans knowledge even in the + use of an Instrument, is first founded on doctrinal precepts, and these + precepts may be conceived all along in its use: and are so farre from + being excluded, that they doe necessarily concomitate and are contained + therein: the practicke being better understood by the doctrinall part, + and this later explained by the Instrumentall, making precepts obvious + unto sense, and the Theory going along with the Instrument, better + informing and inlightning the understanding, etc. vis vnita fortior, so + as if that in Phylosophy bee true, Nihil est [in] intellectu quod non + prius fuit in sensu. + +The difference between Oughtred and Delamain as to the use of +mathematical instruments raises important questions. Should the slide +rule be placed in the hands of a boy before, or after, he has mastered +the theory of logarithms? Should logarithmic tables be withheld from him +until the theoretical foundation is laid in the mind of the pupil? Is it +a good thing to let a boy use a surveying instrument unless he first +learns trigonometry? Is it advisable to permit a boy to familiarize +himself with the running of a dynamo before he has mastered the +underlying principles of electricity? Does the use of instruments +ordinarily discourage a boy from mastery of the theory? Or does such +manipulation constitute a natural and pleasing approach to the abstract? +On this particular point, who showed the profounder psychological +insight, Oughtred or Delamain? + +In July, 1914, there was held in Edinburgh a celebration of the +three-hundredth anniversary of the invention of logarithms. On that +occasion there was collected at Edinburgh university one of the largest +exhibits ever seen of modern instruments of calculation. The opinion was +expressed by an experienced teacher that "weapons as those exhibited +there are for men and not for boys, and such danger as there may be in +them is of the same character as any form of too early specialization." + +It is somewhat of a paradox that Oughtred, who in his student days and +during his active years felt himself impelled to invent sun-dials, +planispheres, and various types of slide rules--instruments which +represent the most original contributions which he handed down to +posterity--should discourage the use of such instruments in teaching +mathematics to beginners. That without the aid of instruments he himself +should have succeeded so well in attracting and inspiring young men +constitutes the strongest evidence of his transcendent teaching ability. +It may be argued that his pedagogic dogma, otherwise so excellent, here +goes contrary to the course he himself followed instinctively in his +self-education along mathematical lines. We read that Sir Isaac Newton, +as a child, constructed sun-dials, windmills, kites, paper lanterns, and +a wooden clock. Should these activities have been suppressed? Ordinary +children are simply Isaac Newtons on a smaller intellectual scale. Should +their activities along these lines be encouraged or checked? + +On the other hand, it may be argued that the paradox alluded to above +admits of explanation, like all paradoxes, and that there is no +inconsistency between Oughtred's pedagogic views and his own course of +development. If he invented sun-dials, he must have had a comprehension +of the cosmic motions involved; if he solved spherical triangles +graphically by the aid of the planisphere, he must have understood the +geometry of the sphere, so far as it relates to such triangles; if he +invented slide rules, he had beforehand a thorough grasp of logarithms. +The question at issue does not involve so much the invention of +instruments, as the use by the pupil of instruments already constructed, +before he fully understands the theory which is involved. Nor does Sir +Isaac Newton's activity as a child establish Delamain's contention. Of +course, a child should not be discouraged from manual activity along the +line of producing interesting toys in imitation of structures and +machines that he sees, but to introduce him to the realm of abstract +thought by the aid of instruments is a different proposition, fraught +with danger. A boy may learn to use a slide rule mechanically and, +because of his ability to obtain practical results, feel justified in +foregoing the mastery of underlying theory; or he may consider the +ability of manipulating a surveying instrument quite sufficient, even +though he be ignorant of geometry and trigonometry; or he may learn how +to operate a dynamo and an electric switchboard and be altogether +satisfied, though having no grasp of electrical science. Thus instruments +draw a youth aside from the path leading to real intellectual attainments +and real efficiency; they allure him into lanes which are often blind +alleys. Such were the views of Oughtred. + +Who was right, Oughtred or Delamain? It may be claimed that there is a +middle ground which more nearly represents the ideal procedure in +teaching. Shall the slide rule be placed in the student's hands at the +time when he is engaged in the mastery of principles? Shall there be an +alternate study of the theory of logarithms and of the slide rule--on the +idea of one hand washing the other--until a mastery of both the theory +and the use of the instrument has been attained? Does this method not +produce the best and most lasting results? Is not this Delamain's actual +contention? We leave it to the reader to settle these matters from his +own observation, knowledge, and experience. + + + NEWTON'S COMMENTS ON OUGHTRED + +Oughtred is an author who has been found to be of increasing interest to +modern historians of mathematics. But no modern writer has, to our +knowledge, pointed out his importance in the history of the teaching of +mathematics. Yet his importance as a teacher did receive recognition in +the seventeenth century by no less distinguished a scientist than Sir +Isaac Newton. On May 25, 1694, Sir Isaac Newton wrote a long letter in +reply to a request for his recommendation on a proposed new course of +study in mathematics at Christ's Hospital. Toward the close of his +letter, Newton says: + + And now I have told you my opinion in these things, I will give you Mr. + Oughtred's, a Man whose judgment (if any man's) may be safely relyed + upon. For he in his book of the circles of proposition, in the end of + what he writes about Navigation (page 184) has this exhortation to + Seamen. "And if," saith he, "the Masters of Ships and Pilots will take + the pains in the Journals of their Voyages diligently and faithfully to + set down in severall columns, not onely the Rumb they goe on and the + measure of the Ships way in degrees, and the observation of Latitude + and variation of their compass; but alsoe their conjectures and reason + of their correction they make of the aberrations they shall find, and + the qualities and condition of their ship, and the diversities and + seasons of the winds, and the secret motions or agitations of the Seas, + when they begin, and how long they continue, how farr they extend and + with what inequality; and what else they shall observe at Sea worthy + consideration, and will be pleased freely to communicate the same with + Artists, such as are indeed skilfull in the Mathematicks and lovers and + enquirers of the truth: I doubt not but that there shall be in + convenient time, brought to light many necessary precepts which may + tend to y^e perfecting of Navigation, and the help and safety of such + whose Vocations doe inforce them to commit their lives and estates in + the vast Ocean to the providence of God." Thus farr that very good and + judicious man Mr. Oughtred. I will add, that if instead of sending the + Observations of Seamen to able Mathematicians at Land, the Land would + send able Mathematicians to Sea, it would signify much more to the + improvem^t of Navigation and safety of Mens lives and estates on that + element.[144] + +May Oughtred prove as instructive to the modern reader as he did to +Newton! + + + + + Footnotes + + +[1]Aubrey's Brief Lives, ed. A. Clark, Vol. II, Oxford, 1898, p. 106. + +[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. + +[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." + +[4]New and General Biographical Dictionary (John Nichols), London, 1784, + art. "Oughtred." + +[5]Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, p. 132. + +[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." + +[7]Aubrey, op. cit., Vol. II, p. 107. + +[8]Rigaud, Correspondence of Scientific Men of the Seventeenth Century, + Oxford, Vol. I, 1841, p. 5. + +[9]Aubrey, op. cit., Vol. II, p. 110. + +[10]Ibid., p. 111. + +[11]Op. cit., Vol. II, p. 132. + +[12]Mr. William Lilly's History of His Life and Times, From the Year 1602 + to 1681, London, 1715, p. 58. + +[13]Rigaud, op. cit., Vol. I, p. 60. + +[14]Aubrey, op. cit., Vol. II, p. 107. + +[15]Rigaud, op. cit., Vol. I, p. 16. + +[16]Owen Manning, op. cit., p. 132. + +[17]New and General Biographical Dictionary (John Nichols), London, 1784, + art. "Oughtred." + +[18]Op. cit., Vol. II, p. 110. + +[19]Rev. Owen Manning, The History and Antiquities of Surrey, Vol. II, + London, 1809, p. 132. + +[20]Op. cit., Vol. II, 1898, p. 111. + +[21]Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago and + London, 1915, Vol. II, p. 303. + +[22]The full title of the Clavis of 1631 is as follows: Arithmeticae in + numeris et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae, + atqve adeo totivs mathematicae, qvasi clavis est.--Ad nobilissimvm + spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur, + Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae & + Svrriae, Comitis Mareschalli Angliae, &c filium.--Londini, Apud Thomam + Harpervm. M.DC.XXXI. + + In all there appeared five Latin editions, the second in 1648 at + London, the third in 1652 at Oxford, the fourth in 1667 at Oxford, the + fifth in 1693 and 1698 at Oxford. There were two independent English + editions: the first in 1647 at London, translated in greater part by + Robert Wood of Lincoln College, Oxford, as is stated in the preface to + the 1652 Latin edition; the second in 1694 and 1702 is a new + translation, the preface being written and the book recommended by the + astronomer Edmund Halley. The 1694 and 1702 impressions labored under + the defect of many sense-disturbing errors due to careless reading of + the proofs. All the editions of the Clavis, after the first edition, + had one or more of the following tracts added on: + + Eq.=De Aequationum affectarvm resolvtione in numeris. + Eu.=Elementi decimi Euclidis declaratio. + So.=De Solidis regularibus, tractatus. + An.=De Anatocismo, sive usura composita. + Fa.=Regula falsae positionis. + Ar.=Theorematum in libris Archimedis de Sphaera & cylindro declaratio. + Ho.=Horologia scioterica in plano, geometrice delineandi modus. + + The abbreviated titles given here are, of course, our own. The lists + of tracts added to the Clavis mathematicae of 1631 in its later + editions, given in the order in which the tracts appear in each + edition, are as follows: Clavis of 1647, Eq., An., Fa., Ho.; Clavis of + 1648, Eq., An., Fa., Eu., So.; Clavis of 1652, Eq., Eu., So., An., + Fa., Ar., Ho.; Clavis of 1667, Eq., Eu., So., An., Fa., Ar., Ho.; + Clavis of 1693 and 1698, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of + 1694 and 1702, Eq. + + The title-page of the Clavis was considerably modified after the first + edition. Thus, the 1652 Latin edition has this title-page: Guilelmi + Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia Socii, + Clavis mathematicae denvo limata, sive potius fabricata. Cum aliis + quibusdam ejusdem commentationibus, quae in sequenti pagina + recensentur. Editio tertia auctior & emendatior. Oxoniae, Excudebat + Leon. Lichfield, Veneunt apud Tho. Robinson. 1652. + +[23]Rigaud, op. cit., Vol. II, p. 476. + +[24]See, for instance, the Clavis mathematicae of 1652, where he + expresses himself thus (p. 4): "Speciosa haec Arithmetica arti + Analyticae (per quam ex sumptione quaesiti, tanquam noti, investigatur + quaesitum) multo accommodatior est, quam illa numerosa." + +[25]Oughtred, The Key of the Mathematicks, London, 1647, p. 4. + +[26]Clavis 1694, p. 19, and the Clavis of 1631, p. 8. + +[27]See for instance, Oughtred's Elementi decimi Euclidis declaratio, + 1652, p. 1, where he uses A and E, and also a and e. + +[28]See Christophori Clavii Bambergensis Operum mathematicorum, tomus + secundus, Moguntiae, M.DC.XI, algebra, p. 39. + +[29]Christophori Clavii operum mathematicorum Tomus Secundus, Moguntiae, + M.DC.XI, Epitome arithmeticae, p. 36. + +[30]See F. Cajori, "The Cross x as a Symbol of Multiplication," in + Nature, Vol. XCIV (1914), p. 363. + +[31]See Elementi decimi Euclidis declaratio, 1652, p. 2. + +[32]See Johannis Wallisii Operum mathematicorum pars prima, Oxonii, 1657, + p. 247. + +[33]Clavis of 1631, chap. xix, sec. 5, p. 50. + +[34]We have noticed the representation of known quantities by consonants + and the unknown by vowels in Wingate's Arithmetick made easie, edited + by John Kersey, London, 1650, algebra, p. 382; and in the second part, + section 19, of Jonas Moore's Arithmetick in two parts, London, 1660, + Moore suggests as an alternative the use of z, y, x, etc., for the + unknowns. The practice of representing unknowns by vowels did not + spread widely in England. + +[35]Philosophical Transactions, Vol. XIX, No. 231, London, p. 652. + +[36]Ibid., Vol. XIX, p. 56. + +[37]There are two title-pages to the edition of 1632. The first + title-page is as follows: The Circles of Proportion and The + Horizontall Instrument. Both invented, and the vses 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. T. Cecill Sculp. + + In 1633 there was added the following, with a separate title-page: An + addition vnto the Vse of the Instrvment called the Circles of + Proportion. . . . . London, 1633, this being followed by Oughtred's To + the English Gentrie etc. In the British Museum there is a copy of + another impression of the Circles of Proportion, dated 1639, with the + Addition vnto the Vse of the Instrument etc., bearing the original + date, 1633, and with the epistle, To the English Gentrie, etc., + inserted immediately after Forster's dedication, instead of at the end + of the volume. + +[38]The complete title of the English edition is as follows: + Trigonometrie, or, The manner of calculating the Sides and Angles of + Triangles, by the Mathematical Canon, demonstrated. By William + Oughtred Etonens. And published by Richard Stokes Fellow of Kings + Colledge in Cambridge, and Arthur Haughton Gentleman. London, Printed + by R. and W. Leybourn, for Thomas Johnson at the Golden Key in St. + Pauls Church-yard. M.DC.LVII. + +[39]Jer. Collier, The Great Historical, Geographical, Genealogical and + Poetical Dictionary, Vol. II, London, 1701, art. "Oughtred." + +[40]Rigaud op. cit., Vol. I, p. 82. + +[41]A. De Morgan, Budget of Paradoxes, London, 1872, p. 451; 2d ed., + Chicago, 1915, Vol. II, p. 303. + +[42]E. Gunter, Description and Use of the Sector, the Crosse-staffe and + other Instruments, London, 1624, second book, p. 31. + +[43]F. Cajori, "On the History of a Notation in Trigonometry," Nature, + Vol. XCIV, 1915, pp. 642, 643. + +[44]A. von Braunmuehl, Geschichte der Trigonometrie, 2. Teil, Leipzig, + 1903, pp. 42, 91. + +[45]H. Hankel, Geschichte der Mathematik in Alterthum und Mittelalter, + Leipzig, 1874, pp. 369, 370. + +[46]M. Cantor, Vorlesungen ueber Geschichte der Mathematik, II, 1900, pp. + 640, 641. + +[47]This matter has been discussed in a paper by F. Cajori, "A History of + the Arithmetical Methods of Approximation, etc., Colorado College + Publication, General Series No. 51, 1910, pp. 182-84. Later this + subject was again treated by G. Enestroem in Bibliotheca mathematica, + 3. Folge, Vol. XI, 1911, pp. 234, 235. + +[48]See F. Cajori, op. cit., p. 193. + +[49]See William Oughtred's Key of the Mathematicks, London, 1694, pp. + 173-75, tract, "Of the Resolution of the Affected Equations," or any + edition of the Clavis after the first. + +[50]A. De Morgan, op. cit., p. 451; 2d ed., Vol. II, p. 303. + +[51]See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909, + pp. 7-14, Addenda, p. ii. + +[52]Rigaud, op. cit., Vol. I, p. 12. + +[53]The New Artificial Gauging Line or Rod: together with rules + concerning the use thereof: Invented and written by William Oughtred, + London, 1633. + +[54]W. Oughtred, Apologeticall Epistle, p. 13. + +[55]Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, (1915), + p. 169. In this article Glaisher republishes the "Appendix" in full. + +[56]Aubrey, op. cit., Vol. II, 1898, p. 108. + +[57]Wood's Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247. + +[58]Wood, op. cit., Vol. II, p. 445. + +[59]Rigaud, op. cit., Vol. I, pp. 33, 35. + +[60]Rigaud, op. cit., Vol. I, pp. 16, 26. + +[61]Rigaud, op. cit., Vol. I, p. 66. + +[62]Ibid., Vol. I, p. 9. + +[63]Rigaud, op. cit., Vol. II, p. 475. + +[64]Ibid., Vol. II, p. 471. + +[65]J. W. L. Glaisher, "On Early Logarithmic Tables, and Their + Calculators," Philosophical Magazine, 4th Ser., Vol. XLV (1873), pp. + 378, 379. + +[66]Rigaud, op. cit., Vol. I, p. 65. + +[67]Rigaud, op. cit., Vol. I, p. 87. + +[68]King's Life of John Locke, Vol. I, London, 1830, p. 227. + +[69]Exercitationum Mathematicarum Decas prima, Naples, 1627, and probably + Cataldus' Transformatio Geometrica, Bonon., 1612. + +[70]Rigaud, op. cit., Vol. II, pp. 477-80. + +[71]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, + Sometimes publike Professor of Astronomie in Gresham Colledge in + London, by John Twysden, London, 1659. + +[72]The Works of the Honourable Robert Boyle in five volumes, to which is + prefixed the Life of the Author, Vol. I, London, 1744, p. 24. + +[73]The letter is printed in John Wallis' De algebra tractatus, 1693, p. + 206. + +[74]See La Correspondance de Descartes, published by Charles Adam and + Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457. + +[75]H. Bosmans, S.J., "La premiere edition de la Clavis Mathematica + d'Oughtred. Son influence sur la Geometrie de Descartes," Annales de + la societe scientifique de Bruxelles, 35th year, 1910-11, Part II, pp. + 24-78. + +[76]Ibid., p. 78. + +[77]Vincent Wing, Harmonicon coeleste, London, 1651, p. 5. + +[78]Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenta + inquisitio brevis, Oxford, 1653, p. 7. + +[79]John Wallis, Elenchus geometriae Hobbianae, Oxford, 1655, p. 48. + +[80]An Idea of Arithmetick, at first designed for the use of the Free + Schoole at Thurlow in Suffolk. . . . . By R. B., Schoolmaster there, + London, 1655, p. 6. + +[81]The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster + . . . . by John Twysden, London, 1659, p. 1. + +[82]Moor's Arithmetick in two Books, London, 1660, p. 89. + +[83]Isaac Barrow, Euclidis data, Cambridge, 1657, p. 2. + +[84]Francisci Dulaurens Specima mathematica, Paris, 1667, p. 1. + +[85]Elemens des mathematiques, Paris, 1675, Preface signed "J. P." + +[86]Nouveaux elemens de geometrie, Paris, 1692 (permission to print + 1684). + +[87]Ozanam, Dictionnaire mathematique, Paris, 1691, p. 12. + +[88]Analyse des infiniment petits, Paris, 1696, p. 11. + +[89]Petro Nicolas, De conchoidibus et cissoidibus exercitationes + geometricae, Toulouse, 1697, p. 17. + +[90]R. P. Bernard Lamy, Elemens des mathematiques, Amsterdam, 1692 + (permission to print 1680). + +[91]Nouveaux elemens de geometrie, 2d ed., The Hague, 1690, p. 304. + +[92]W. W. Beman in L'intermediaire des mathematiciens, Paris, Vol. IX, + 1902, p. 229, question 2424. + +[93]John Collins, The Mariner's Plain Scale New Plain'd, London, 1659, p. + 25. + +[94]James Gregory, Optica promota, London, 1663, pp. 19, 48. + +[95]Philosophical Transactions, Vol. III, London, p. 868. + +[96]William Leybourn, The Line of Proportion, London, 1673, p. 14. + +[97]Elementa geometriae . . . . a Gulielmo Sanders, Glasgow, 1686, p. 3. + +[98]Cocker's Decimal Arithmetick, . . . . perused by John Hawkins, + London, 1695 (preface dated 1684), p. 41. + +[99]Joseph Raphson, Analysis Aequationum universalis, London, 1697, p. + 26. + +[100]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, + 1698, p. 107. + +[101]John Ward, A Compendium of Algebra, 2d ed., London, 1698, p. 62. + +[102]Plain Elements of Geometry and Plain Trigonometry, London, 1701, p. + 63. + +[103]George Shelley, Wingate's Arithmetick, London, 1704, p. 343. + +[104]A Synopsis of Algebra, Being a posthumous work of John Alexander of + Bern, Swisserland. . . . . Done from the Latin by Sam. Cobb, London, + 1709, p. 16. + +[105]John Craig, De Calculo fluentium, London, 1718, p. 35. The notation + A:B::C:D is given also. + +[106]Trigonometry, 2d ed., Edinburgh, 1724, p. 11. + +[107]Methode pour la mesure des surfaces, la dimension des solides . . . + . par M. Carre de l'academie r. des sciences, 1700, p. 59. + +[108]Application de l'algebre a geometrie . . . . Paris, 1705. + +[109]Elemens de la geometrie de l'infini, by M. de Fontenelle, Paris, + 1727, p. 110. + +[110]Eclaircissemens sur l'analyse des infiniment petits, by M. Varignon, + Paris, 1725, p. 87. + +[111]Application de la geometrie ordinaire et des calculs differentiel et + integral, by M. Robillard, Paris, 1753. + +[112]Traite de geometrie theorique et pratique, new ed., Paris, 1764, p. + 15. + +[113]Recherches sur les courbes a double courbure, Paris, 1731, p. 13. + +[114]Analyse des infiniment petits, by the Marquis de L'Hospital. New ed. + by M. Le Fevre, Paris, 1781, p. 41. In this volume passages in fine + print, probably supplied by the editor, contain the notation a:b::c:d; + the parts in large type give Oughtred's original notation. + +[115]The tendency during the eighteenth century is shown in part by the + following data: Jacobi Bernoulli Opera, Tomus primus, Geneva, 1744, + gives B.A::D.C on p. 368, the paper having been first published in + 1688; on p. 419 is given GE:AG=LA:ML, the paper having been first + published in 1689. Bernhardi Nieuwentiit, Considerationes circa + analyseos ad quantitates infinite parvas applicatae principia, + Amsterdam, 1694, p. 20, and Analysis infinitorum, Amsterdam, 1695, on + p. 276, have x:c::s:r. Paul Halcken's Deliciae mathematicae, Hamburg, + 1719, gives a:b::c:d. Johannis Baptistae Caraccioli, Geometria + algebraica universa, Rome, 1759, p. 79, has a.b::c.d. Delle corde + ouverto fibre elastiche schediasmi fisico-matematici del conte + Giordano Riccati, Bologna, 1767, p. 65, gives P:b::r:ds. "Produzioni + mathematiche" del Conte Giulio Carlo de Fagnano, Vol. I, Pesario, + 1750, p. 193, has a.b::c.d. L. Mascheroni, Geometrie du compas, + translated by A. M. Carette, Paris, 1798, p. 188, gives + {root}(3):2::{root}(2):Lp. Danielis Melandri and Paulli Frisi, De + theoria lunae commentarii, Parma, 1769, p. 13, has a:b::c:d. Vicentio + Riccato and Hieronymo Saladino, Institutiones analyticae, Vol. I, + Bologna, 1765, p. 47, gives x:a::m:n+m. R. G. Boscovich, Opera + pertinentia ad opticam et astronomiam, Bassani, 1785, p. 409, uses + a:b::c:d. Jacob Bernoulli, Ars Conjectandi, Basel, 1713, has + n-r.n-1::c.d. Pavlini Chelvicii, Institutiones analyticae, editio post + tertiam Romanam prima in Germania, Vienna, 1761, p. 2, a.b::c.d. + Christiani Wolfii, Elementa matheseos universae, Vol. III, Geneva, + 1735, p. 63, has AB:AE=1:q. Johann Bernoulli, Opera omnia, Vol. I, + Lausanne and Geneva, 1742, p. 43, has a:b=c:d. D. C. Walmesley, + Analyse des mesures des rapports et des angles, Paris, 1749, uses + extensively a.b::c.d, later a:b::c:d. G. W. Krafft, Institutiones + geometriae sublimoris, Tuebingen, 1753, p. 194, has a:b=c:d. J. H. + Lambert, Photometria, 1760, p. 104, has C:{pi}=BC^2:MH^2. Meccanica + sublime del Dott. Domenico Bartaloni, Naples, 1765, has a:b::c:d. + Occasionally ratio is not designated by a.b, nor by a:b, but by a, b, + as for instance in A. de Moivre's Doctrine of Chance, London, 1756, p. + 34, where he writes a, b::1, q. A further variation in the designation + of ratio is found in James Atkinson's Epitome of the Art of + Navigation, London, 1718, p. 24, namely, 3..2::72..48. Curious + notations are given in Rich. Balam's Algebra, London, 1653. + +[116]Chr. Clavii Operum mathematicorum tomus secundus, Mayence, 1611, + Algebra, p. 39. + +[117]Invention nouvelle en l'algebre, by Albert Girard, Amsterdam, 1629, + p. 17. + +[118]La geometrie et pratique generale d'icelle, par I. Errard de + Bar-le-Duc, Ingenieur ordinaire de sa Majeste, 3d ed., revised by D. + H. P. E. M., Paris, 1619, p. 216. + +[119]Novae geometriae clavis algebra, authore P. Jacobo de Billy, Paris, + 1643, p. 157; also an Abridgement of the Precepts of Algebra. Written + in French by James de Billy, London, 1659, p. 346. + +[120]Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster, + Sometime publike Professor of Astronomie in Gresham Colledge in + London, London, 1659, p. 7. + +[121]Quarterly Jour. of Pure and Applied Math., Vol. XLVI (London, 1915), + p. 191. + +[122]Pietro Cossali, Origine, trasporto in Italia primi progressi in essa + dell' algebra, Vol. I, Parmense, 1797, p. 52. + +[123]In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio + brevis, Auctore Setho Wardo, Oxford, 1653, p. 1. + +[124]John Wallis, Algebra, London, 1685, p. 321, and in some of his other + works. He makes greater use of Harriot's symbols. + +[125]Euclidis data, 1657, p. 1; also Euclidis elementorum libris XV, + London, 1659, p. 1. + +[126]John Kersey, Algebra, London, 1673, p. 321. + +[127]E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, + 1698, p. 142. + +[128]Cocker's Decimal Arithmetick, perused by John Hawkins, London, 1695 + (preface dated 1684), p. 278. + +[129]Th. Baker, The Geometrical Key, London, 1684, p. 15. + +[130]Richard Sault, A New Treatise of Algebra, London (no date). + +[131]Richard Rawlinson in a pamphlet without date, issued sometime + between 1655 and 1668, containing trigonometric formulas. There is a + copy in the British Museum. + +[132]F. Dulaurens, Specima mathematica, Paris, 1667, p. 1. + +[133]J. Milnes, Sectionum conicarum elementa, Oxford, 1702, p. 42. + +[134]Cheyne, Philosophical Principles of Natural Religion, London, 1705, + p. 55. + +[135]J. Craig, De calculo fluentium, London, 1718, p. 86. + +[136]Jo. Wilson, Trigonometry, 2d ed., Edinburgh, 1724, p. v. + +[137]Commercium Epistolicum, 1712, p. 20. + +[138]C. Le Paige, "Sur l'origine de certains signes d'operation," Annales + de la societe scientifique de Bruxelles, 16th year, 1891-92, Part II, + pp. 79-82. + +[139]Gravelaar, "Over den oorsprong van ons maalteeken (x)," Wiskundig + Tijdschrift, 6th year. We have not had access to this article. + +[140]H. Bosmans, op. cit., p. 40. + +[141]Claudii Ptolemaei . . . . annotationes, Bale, 1551. This reference + is taken from the Encyclopedie des sciences mathematiques, Tome I, + Vol. I, Fasc. 1, p. 40. + +[142]Due Correction for Mr. Hobbes. Or Schoole Discipline, for not saying + his Lessons right. In answer to his Six Lessons, directed to the + Professors of Mathematicks. By the Professor of Geometry. Oxford, + 1656, pp. 7, 47, 50. + +[143]Oughtred, Apologeticall Epistle, p. 27. + +[144]J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes, + London, 1850, pp. 279-92. + + + + + INDEX + + + Adam, Charles, 71 + Agnesi, Maria G., 77 + Alexander, J., 76 + Allen, E., 35 + Analysis, 19, 20 + Apollonius of Perga, 20, 85 + Archimedes, 18, 20, 85 + Aristotle, 69 + Ashmole, E., 13 + Atkinson, J., 79 + Atwood, 56 + Aubrey, 3, 7, 8, 12-16, 58, 59 + Austin, 58 + + Baker, T., 82 + Balam, R., 79 + Bar-le-Duc, de, 80 + Barrow, S., 1, 32, 73, 74, 80, 81 + Bartaloni, D., 79 + Beman, W. W., 74, 75 + Bernoulli, Jakob, 78-80 + Bernoulli, John, 79, 80 + Billingsley's Euclid, 15 + Billion, 20 + Billy, Jacobo de, 80 + Binomial formula, 25, 29 + Bliss, P., 60 + Boscovich, R. G., 78 + Bosmans, H., 72, 83 + Boyle, R., 1, 69 + Braunmuehl, von, 39 + Brearly, W., 59 + Briggs, 6, 36, 55 + Brookes, Christopher, 7, 53, 59 + + Cajori, F., 27, 39, 40, 47 + Cantor, M., 40, 41 + Caraccioli, J. B., 78 + Cardan, 71 + Carre, 77 + Carrete, N. M., 78 + Caryll, C., 7 + Cataldi, 67 + Cavalieri, 65, 66 + Cavendish, Charles, 17, 62, 66 + Charles I, 9, 60 + Chelvicius, P., 79 + Cheyne, G., 82 + _Circles of Proportion_, 35, 37, 48, 49, 51, 59, 87, 88 + Clairaut, 77 + Clark, A., 3 + Clark, G., 63 + Clarke, F. L., 3 + _Clavis mathematicae_, 1, 5, 10, 14, 17-35, 45, 46, 51, 57-63, + 68-73, 81, 85, 87 + Clavius, 26, 80 + Clerc, le, 77 + Cobb, S., 76 + Cocker, 76, 82 + Collins, John, 15, 19, 63, 64, 67, 68, 76, 82 + Colson, J., 77 + Conchoid, 12 + Conic sections, 11, 53 + Cossali, P., 81 + Cotes, R., 1, 85 + Craig, J., 76, 82 + Cross, symbol of multiplication, 27, 38, 55, 56, 82, 83 + Cubic equations, 28, 34, 42, 45 + + Decimal fractions, notation of, 21 + Degree, centesimal division, 39 + Delamain, R., 4, 9, 10, 11, 47, 48, 51, 60, 84, 88, 89, 91, 93, 94 + De Moivre, 32, 79 + De Morgan, A., 5, 16, 37, 46, 47, 54 + Descartes, R., 1, 25, 47, 57, 68-72, 80 + Dibuadius, 79 + Difference, symbol for, 27, 81 + Diophantus, 63, 85 + Division, abbreviated, 21, 23, 24 + Dulaurens, F., 74, 82 + + Earl of Arundel, 10, 13, 15, 17 + Edleston, J., 95 + Enestroem, G., 40 + Equations, solution of, 18, 28, 29, 31, 34, 39-45, 87 + Errard de Bar-le-Duc, 80 + Eton College, 3, 4 + Euclid, 1, 15, 18, 20, 25, 27, 28, 79, 81, 83, 85 + Euler, L., 37, 39 + Ewart, 59 + Exponents, 25, 28, 29 + + Fagnano, de, 78 + Flower, 56 + Fontenelle, de, 77 + Forster, W., 35, 48, 59, 88 + Foster, S., 27, 67, 69, 73, 80, 89 + Frisi, P., 78 + + Gascoigne, 59, 61 + Gauss, C. F., 48 + Geysius, 67 + Ghetaldi, 70, 71 + Gibson, 67 + Girard, A., 32, 80 + Glaisher, J. W. L., 54-56, 64, 80 + Glorioso, 67, 68 + _Grammelogia_, 4, 47, 89 + Gravelaar, 83 + Greater than, symbol for, 81 + Greatrex, R., 15 + Gregory, D., 32 + Gregory, J., 27, 76 + Gresham College, 1, 6, 27, 59, 61, 80 + Guisnee, 77 + Gunter, E., 37, 47, 86 + Gunter's scale, 37 + + Halcken, P., 78 + Hales, J., 7 + Halley, E., 1, 18, 69 + Hankel, H., 40 + Harper, T., 18 + Harriot, T., 45, 47, 57, 58, 69-71, 81 + Harris, J., 76 + Hartlib, 69 + Haughton, A., 35, 59 + Hawkins, J., 76, 82 + Hearn, 56 + Helmholtz, 48 + Henry, J., 48 + Henry van Etten, 52, 53 + Henshaw, T., 8, 58, 61 + Hobbes, 73, 86 + Hollar, 14 + Holsatus, 13 + Hooganhuysen, 64 + Hooke, Rb., 1 + Horner's method, 45 + Horology, 18, 50 + Horrox, J., 4 + Hospital, de l', 74, 77 + Howard, Th. _See_ Earl of Arundel. + Howard, W., 17, 18, 59 + Hutchinson, A., 6 + + Invisible college, 1 + + Joule, 48 + + Kepler, J., 6 + Kersey, J., 32, 73, 82 + Keylway, R., 65 + King, 67 + Kings College, Cambridge, 3, 35 + Krafft, G. W., 79 + + Lambert, J. H., 79 + Lamy, R. P. B., 74 + Laud, Archbishop, 65 + Leake, W., 53 + Le Clerc, 77 + Leech, W., 59 + Le Fevre, 77 + Leibniz, 47, 78, 80 + Leonelli, 56 + Le Paige, de, 83 + Less than, symbol for, 81 + Leurechon, 52 + Leybourn, 35, 64, 76 + Lichfield, Mrs., 19 + Lilly, W., 8, 9 + Locke, J., 67 + Logarithms, 6, 21, 27, 28, 38, 39, 42, 46, 54-56, 65, 92, 93; + natural, 55; + radix method of computing, 55, 56 + Lower, W., 58 + Ludolph a Ceulen, 79 + + Manning, 56 + Manning, O., 7, 8, 13-15 + Mascheroni, L., 78 + Mayer, R., 47 + Melandri, D., 78 + Mercator, N., 13 + Mersenne, 63 + Milbourn, W., 45 + Million, 20 + Milnes, J., 82 + Moivre, de, 32, 79 + Moore, Jonas, 32, 54, 58, 73 + Moreland, S., 70 + Morse, R., 48 + Multiplication, abbreviated, 21, 22, 24; + symbol for, 27, 82, 83 + Mydorge, 54 + + Napier, J., 6, 7, 21, 27, 38, 39, 52, 54, 57, 59 + Napier's analogies, 39 + Newton, Sir Isaac, 1, 25, 29, 40, 41, 45, 47, 59, 65, 86, 92-95 + Nichols, J., 6, 14 + Nicolas, R. P. P., 74 + Nieuwentiit, B., 78 + Norwood, R., 37, 38, 80 + + _Opuscula mathematica hactenus inedita_, 16, 21, 75 + Orchard, 56 + _Oughtredus explicatus_, 64 + Ozanam, 74 + + {pi}, symbol for, 32 + Paige, C. de, 83 + Pardies, 76 + Parentheses, 26, 79, 80 + Partridge, S., 47 + Peano, 86 + Perfect number, 41 + Pitiscus, 15 + Planisphere, 53, 92, 93 + Prestet, J., 74 + Price, 11 + Proportion, notation for, 26, 27, 73-79 + Protheroe, 58 + Ptolemy, 83 + + Quadratic equation, 29, 31, 34 + + Radix method, 55, 56 + Rahn, 27 + Raphson, J., 40, 41, 76 + Ratio, notation of, 21, 73-80 + Rawlinson, R., 39, 82 + Regula falsa, 18 + Regular solids, 18 + Riccati, G., 78 + Riccati, V., 78 + Rigaud, 7, 12, 13, 19, 48, 61-66, 68 + Robillard, 77 + Robinson, W., 13, 48, 59, 62, 63 + Rooke, L., 59, 61 + + Saladini, H., 78 + Sanders, W., 76 + Sault, R., 82 + Scarborough, Charles, 16, 54, 58, 60 + Schooten, Van, 1 + Schreshensuchs, O., 83 + Scratch method, 23 + Shakespeare, 52 + Shelley, G., 76 + Shipley, A. E., 1 + Shuttleworth, 59 + Slide rule, 9, 46-49, 50, 60, 88, 93 + Smethwyck, 58 + Smith, J., 50 + Snellius, W., 79 + Solids, regular, 18 + Speidell, John, 38, 55 + Spherical triangles, 53, 54, 93 + Stokes, R., 35, 36, 58 + Sudell, 59 + Sun dials, 5, 9, 50, 51, 52, 60, 92 + + Tannery, P., 71 + Todhunter, 60 + Torporley, 58 + Triangles, spherical, 53, 54, 93 + _Trigonometria_, 21, 36, 55, 75 + Trigonometric functions, symbols for, 36, 37, 55, 56 + _Trigonometrie_, 21, 35, 39 + Trisection of angles, 28 + Twysden, 59, 68, 69, 73 + + Varignon, 77 + Vieta, 1, 2, 25, 32, 33, 35, 39-41, 45, 63, 67, 70, 71 + Vlack, 65 + Von Braunmuehl, 39 + + Wadham College, 5, 53 + Wallis, John, 1, 19, 27, 33, 45, 57-59, 63, 64, 66-74, 79-81, 86 + Walmesley, D. C., 79 + Ward, Bishop, 13 + Ward, John, 76 + Ward, Seth, 55, 58, 60, 68, 73, 74, 81 + Watch-making, 18, 50 + Weber, W. E., 48 + Weddle, 56 + Wells, E., 76, 82 + Wharton, 60 + Whitlock, B., 8, 9 + Wilson, J., 77, 82 + Wing, V., 73, 75 + Wingate, E., 32, 47, 73 + Wolf, Christian, 79 + Wood, A., 60, 61 + Wood, R., 18, 59 + Wren, Christopher, 5, 58, 59, 76 + Wright, E., 6, 27, 38, 54 + Wright, S., 54 + + + + + Transcriber's Notes + + +A handful of typos, mostly misplaced punctuation, were silently +corrected. + +HTML and UTF text versions make heavy use of mathematical symbols: +particularly superscripts, subscripts, and combining characters. Some +viewers may require user assistance to find fonts containing these +characters. + +The text versions miss much of the formatting, especially in mathematical +formulas: + + +--Several arithmetic examples must be viewed in a monospaced font (which + recognizes combining characters) to be legible. + +--Formulas under the horizontal line of a square root symbol are + parenthesized. + +--Subscripts are preceded by "_". + +--Superscripts are preceded by "^". + +--Italics, used primarily in formulas and bibliographical entries, are + not indicated in the text. + +--Italics in the index are delimited by "_". + +--Underlines are not indicated (in particular, in the fractional part of + a decimal number in Oughtred's notation). + +--The idiosyncratic "greater than" and "less than" symbols are indicated + as {symbol} in the ASCII version. + +--Overdots, underdots, and slashmarks around digits (in the long division + example) are not indicated in the ASCII version. + +--Greek letters are spelled out within {curly brackets} in the ASCII + version. + + + + + + + +End of the Project Gutenberg EBook of William Oughtred, by Florian Cajori + +*** END OF THIS PROJECT GUTENBERG EBOOK WILLIAM OUGHTRED *** + +***** This file should be named 46815.txt or 46815.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/4/6/8/1/46815/ + +Produced by Brenda Lewis, Stephen Hutcheson, and the Online +Distributed Proofreading Canada Team at +http://www.pgdpcanada.net + +Updated editions will replace the previous one--the old editions will +be renamed. + +Creating the works from print editions not protected by U.S. copyright +law means that no one owns a United States copyright in these works, +so the Foundation (and you!) can copy and distribute it in the United +States without permission and without paying copyright +royalties. 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