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+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: 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 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 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' 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; 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 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+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 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 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 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 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' 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 {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 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, {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, 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 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 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
+ {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, Tübingen, 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'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 (x)," 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
+
+ {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 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
+
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