,,william ,,oughtred ,,william ,,oughtred ,a ,,great ,,seventeenth-,,century ,,teacher ,,of ,,mathematics ,,by ,,florian ,,cajori1 ,ph4,d4 ,professor of ,mathematics ,colorado ,college ,,chicago ,,london ,,the ,,open ,,court ,,publishing ,,company #aiaf ,copyright #aiaf ,by ,the ,open ,court ,publishing ,co4 ,all ,rights ,reserved ,published ,september #aiaf ,composed and ,printed ,by ,the ,university of ,chicago ,press ,chicago1 ,illinois1 ,u4,s4,a4 ,,table ,,of ,,contents ,,page ,introduction #a ,,chapter ,i4 ,oughtred's ,life #c ,at ,school and ,university #c ,as ,rector and ,amateur ,mathematician #f ,his ,wife #g ,in ,danger of ,sequestration #h ,his ,teaching #i ,appearance and ,habits #ab ,alleged ,travel ,abroad #ad ,his ,death #ae ,,ii4 ,principal ,works #ag ,clavis mathematicae #ag ,circles of ,proportion and ,trigonometrie #ce ,solution of ,numerical ,equations #ci ,logarithms #df ,invention of the ,slide ,rule2 ,controversy on ,priority of ,invention #df ,,iii4 ,minor ,works #ej ,,iv4 ,oughtred's ,influence upon ,mathematical ,progress and ,teaching #eg ,oughtred and ,harriot #eg ,oughtred's ,pupils #eh ,oughtred1 the 8,todhunter of the ,seventeenth ,century0 #fj ,was ,descartes ,indebted to ,oughtred8 #fi ,the ,spread of ,oughtred's ,notations #gc ,v4 ,oughtred's ,ideas on the ,teaching of ,mathematics #hd ,general ,statement #hd ,mathematics1 8a ,science of the ,eye0 #he ,rigorous ,thinking and the ,use of ,instruments #hg ,newton's ,comments on ,oughtred #id ,index #ig ,,introduction ,in the year #affj the ,royal ,society was founded by royal favor in ,london1 although in reality its inception took place in #afde when the ,philosophical ,society 7or1 as ,boyle called it1 the 8,invisible ,college07 came into being1 which held meetings at ,gresham ,college in ,london and later in ,oxford4 ,it was during the second half of the seventeenth century that ,sir ,isaac ,newton1 surrounded by a group of great men--,wallis1 ,hooke1 ,barrow1 ,halley1 ,cotes--carried on his epoch-making researches in mathematics1 astronomy1 and physics4 ,but it is not this half-century of science in ,england1 nor any of its great men1 that especially engage our attention in this monograph4 ,it is rather the half-century preceding1 an epoch of preparation1 when in the early times of the ,house of ,stuart the sciences began to flourish in ,england4 ,says ,dr4 ,a4 ,e4 ,shipley3 8,whatever were the political and moral deficiencies of the ,stuart kings1 no one of them lacked intelligence in things artistic and scientific40 ,it was at this time that mathematics1 and particularly algebra1 began to be cultivated with greater zeal1 when elementary algebra with its symbolism as we know it now began to take its shape4 ,biographers of ,sir ,isaac ,newton make particular mention of five mathematical books which he read while a young student at ,cambridge1 namely1 ,euclid's ,elements1 ,descartes's ,gomtrie1 ,vieta's ,works1 ,van ,schooten's ,miscellanies1 and ,oughtred's ,clavis mathematicae4 ,the last of these books has been receiving increasing attention from the historians of algebra in recent years4 ,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 treatment4 ,historians have discussed his share in the development of symbolic algebra1 but some have fallen into errors1 due to inability to examine the original editions of ,oughtred's ,clavis mathematicae1 which are quite rare and inaccessible to most readers4 ,moreover1 historians have failed utterly to recognize his inventions of mathematical instruments1 particularly the slide rule2 they have completely overlooked his educational views and his ideas on mathematical teaching4 ,the modern reader may pause with profit to consider briefly the career of this interesting man4 ,oughtred was not a professional mathematician4 ,he did not make his livelihood as a teacher of mathematics or as a writer1 nor as an engineer who applies mathematics to the control and use of nature's forces4 ,oughtred was by profession a minister of the gospel4 ,with him the study of mathematics was a side issue1 a pleasure1 a recreation4 ,like the great ,french algebraist1 ,vieta1 from whom he drew much of his inspiration1 he was an amateur mathematician4 ,the word 8amateur0 must not be taken here in the sense of superficial or unthorough4 ,great ,britain has had many men distinguished in science who pursued science as amateurs4 ,of such men ,oughtred is one of the very earliest4 ,f4 ,c4 ,,chapter ,i ,,oughtred',s ,,life ,,at ,,school ,,and ,,university ,william ,oughtred1 or1 as he sometimes wrote his name1 ,owtred1 was born at ,eton1 the seat of ,eton ,college1 the year of his birth being variously given as #aegc1 #aegd1 and #aege4 8,his father10 says ,aubrey1 8taught to write at ,eaton1 and was a scrivener2 and understood common arithmetique1 and 'twas no small helpe and furtherance to his son to be instructed in it when a schoole-boy40#a ,he was a boy at ,eton in the year of the ,spanish ,armada4 ,at this famous school1 which prepared boys for the universities1 young ,oughtred received thorough training in classical learning4 ,according to information received from ,f4 ,l4 ,clarke1 ,bursar and ,clerk of ,king's ,college1 ,cambridge1 ,oughtred was admitted at ,king's a scholar from ,eton on ,september #a1 #aeib1 at the age of seventeen4 ,he was made ,fellow at ,king's on ,september #a1 #aeie1 while ,elizabeth was still on the throne4 ,he received in #aeif the degree of ,bachelor of ,arts and in #afjj that of ,master of ,arts4 ,he vacated his fellowship about the beginning of ,august1 #afjc4 ,his career at the ,university of ,cambridge we present in his own words4 ,he says3 ,next after ,eaton schoole1 ,i was bred up in ,cambridge in ,kings ,colledge3 of which society ,i was a member about eleven or twelve yeares3 wherein how ,i behaved my selfe1 going hand in hand with the rest of my ranke in the ordinary ,academicall studies and exercises1 and with what approbation1 is well knowne and remembered by many3 the time which over and above those usuall studies ,i employed upon the ,mathematicall sciences1 ,i redeemed night by night from my naturall sleep1 defrauding my body1 and inuring it to watching1 cold1 and labour1 while most others tooke their rest4 ,neither did ,i therein seek only my private content1 but the benefit of many3 and by inciting1 assisting1 and instructing others1 brought many into the love and study of those ,arts1 not only in our own1 but in some other ,colledges also3 which some at this time 7men far better than my selfe in learning1 degree1 and preferment7 will most lovingly acknowledge4#b ,these words describe the struggles which every youth not endowed with the highest genius must make to achieve success4 ,they show1 moreover1 the kindly feeling toward others and the delight he took throughout life in assisting anyone interested in mathematics4 ,oughtred's passion for this study is the more remarkable as neither at ,eton nor at ,cambridge did it receive emphasis4 ,even after his time at ,cambridge mathematical studies and their applications were neglected there4 ,jeremiah ,horrox was at ,cambridge in #afcc-ce1 desiring to make himself an astronomer4 8,but many impediments10 says ,horrox1 8presented themselves3 the tedious difficulty of the study itself deterred a mind not yet formed2 the want of means oppressed1 and still oppresses1 the aspirations of my mind3 but that which gave me most concern was that there was no one who could instruct me in the art1 who could even help my endeavours by joining me in the study2 such was the sloth and languor which had seized all4 4 4 4 4 ,i found that books must be used instead of teachers40#c ,some attention was given to ,greek mathematicians1 but the works of ,italian1 ,german1 and ,french algebraists of the latter part of the sixteenth and beginning of the seventeenth century were quite unknown at ,cambridge in ,oughtred's day4 ,it was part of his life-work as a mathematician to make algebra1 as it was being developed in his time1 accessible to ,english youths4 ,at the age of twenty-three ,oughtred invented his ,easy ,way of ,delineating ,sun-,dials by ,geometry1 which1 though not published until about half a century later1 in the first ,english edition of ,oughtred's ,clavis mathematicae in #afdg1 was in the meantime translated into ,latin by ,christopher ,wren1 then a ,gentleman ,commoner of ,wadham ,college1 ,oxford1 now best known through his architectural creations4 ,in #afjj ,oughtred wrote a monograph on the construction of sun-dials upon a plane of any inclination1 but that paper was withheld by him from publication until #afcb4 ,sun-dials were interesting objects of study1 since watches and pendulum clocks were then still unknown4 ,all sorts of sun-dials1 portable and non-portable1 were used at that time and long afterward4 ,several of the college buildings at ,oxford and ,cambridge have sun-dials even at the present time4 ,,as ,,rector ,,and ,,amateur ,,mathematician ,it was in #afjd that ,oughtred entered upon his professional life-work as a preacher1 being instituted to the vicarage of ,shalford in ,surrey4 ,in #afaj he was made rector of ,albury1 where he spent the remainder of his long life4 ,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 ,horsley4 ,oughtred continued to devote his spare time to mathematics1 as he had done in college4 ,a great mathematical invention made by a ,scotchman soon commanded his attention--the invention of logarithms4 ,an informant writes as follows3 ,lord ,napier1 in #afad1 published at ,edinburgh his ,mirifici logarithmorum canonis descriptio4 4 4 4 4 ,it presently fell into the hands of ,mr4 ,briggs1 then geometry-reader at ,gresham ,college in ,london3 and that gentleman1 forming a design to perfect ,lord ,napier's plan1 consulted ,oughtred upon it2 who probably wrote his ,treatise of ,trigonometry about the same time1 since it is evidently formed upon the plan of ,lord ,napier's ,canon4#d ,it will be shown later that ,oughtred is very probably the author of an 8,appendix0 which appeared in the #afah edition of ,edward ,wright's translation into ,english of ,john ,napier's ,descriptio4 ,this 8,appendix0 relates to logarithms and is an able document1 containing several points of historical interest4 ,mr4 ,arthur ,hutchinson of ,pembroke ,college informs me that in the university library at ,cambridge there is a copy of ,napier's ,constructio 7#afai7 bound up with a copy of ,kepler's ,chilias logarithmorum 7#afbd71 that at the beginning of the ,constructio is a blank leaf1 and before this occurs the title-page only of ,napier's ,descriptio 7#afai71 at the top of which appears ,oughtred's autograph4 ,the history of this interesting signature is unknown4 ,,his ,,wife ,in #afjf he married ,christ'sgift ,caryll1 daughter of ,caryll1 ,esq41 of ,tangley1 in an adjoining parish4#e ,we know very little about ,oughtred's family life4 ,the records at ,king's ,college1 ,cambridge1#f mention a son1 but it is certain that there were more children4 ,a daughter was married to ,christopher ,brookes4 ,but there is no confirmation of ,aubrey's statements1#g according to which ,oughtred had nine sons and four daughters4 ,reference to the wife and children is sometimes made in the correspondence with ,oughtred4 ,in #afaf ,j4 ,hales writes1 8,i pray let me be remembered1 though unknown1 to ,mistress ,oughtred40#h ,as we shall see later1 ,oughtred had a great many young men who came to his house and remained there free of charge to receive instruction in mathematics1 which was likewise gratuitous4 ,this being the case1 certainly great appreciation was due to ,mrs4 ,oughtred1 upon whom the burden of hospitality must have fallen4 ,yet chroniclers are singularly silent in regard to her4 ,hers was evidently a life of obscurity and service4 ,we greatly doubt the accuracy of the following item handed down by ,aubrey2 it cannot be a true characterization3 ,his wife was a penurious woman1 and would not allow him to burne candle after supper1 by which meanes many a good notion is lost1 and many a probleme unsolved2 so that ,mr4 ,thomas ,henshawe1 when he was there1 bought candle1 which was a great comfort to the old man4#i ,,in ,,danger ,,of ,,sequestration ,oughtred spent his years in 8unremitted attention to his favourite study10 sometimes1 it has been whispered1 to the neglect of his rectorial duties4 ,says ,aubrey3 ,i have heard his neighbour ministers say that he was a pittiful preacher2 the reason was because he never studyed it1 but bent all his thoughts on the mathematiques2 but when he was in danger of being sequestred for a royalist1 he fell to the study of divinity1 and preacht 7they sayd7 admirably well1 even in his old age4#aj ,this remark on sequestration brings to mind one of the political and religious struggles of the time1 the episcopacy against the independent movements4 ,says ,manning3 ,in #afdf he was cited before the ,committee for ,ecclesiastical ,affairs1 where many articles had been deposed against him2 but1 by the favour of ,sir ,bulstrode ,whitlock and others1 who1 at the intercession of ,william ,lilye the ,astrologer1 appeared in great numbers on his behalf1 he had a majority on his side1 and so escaped a sequestration4#aa ,not without interest is the account of this matter given by ,lilly himself3 ,about this ,time1 the most famous ,mathematician of all ,europe1 7,mr4 ,william ,oughtred1 ,parson of ,aldbury in ,surrey7 was in ,danger of ,sequestration by the ,committee of or for plunder'd ,ministers2 7,ambo-dexters they were27 several inconsiderable ,articles were deposed and sworn against him1 material enough to have sequestred him1 but that1 upon his ,day of hearing1 ,i applied my self to ,sir ,bolstrode ,whitlock1 and all my own old ,friends1 who in such ,numbers appeared in his ,behalf1 that though the ,chairman and many other ,presbyterian ,members were stiff against him1 yet he was cleared by the major ,number4 ,the truth is1 he had a considerable ,parsonage1 and that only was enough to sequester any moderate ,judgment3 ,he was also well known to affect his ,majesty ,charles ,i4 ,in these ,times many worthy ,ministers lost their ,livings or ,benefices1 for not complying with the ,three-penny ,directory4#ab ,,his ,,teaching ,oughtred had few personal enemies4 ,his pupils held him in highest esteem and showed deep gratitude2 only one pupil must be excepted1 ,richard ,delamain4 ,against him arose a bitter controversy which saddened the life of ,oughtred1 then an old man4 ,it involved1 as we shall see later1 the priority of invention of the circular slide rule and of a horizontal instrument or portable sun-dial4 ,in defense of himself1 ,oughtred wrote in #afcc or #afcd the ,apologeticall ,epistle1 from which we quoted above4 ,this document contains biographical details1 in part as follows3 ,ever since my departure from the ,vniversity1 which is about thirty yeares1 ,i have lived neere to the ,towne of ,guildford in ,surrey3 where1 whether ,i have taken so much liberty to the losse of time1 and the neglect of my calling the whole ,countrey thereabout1 both ,gentry and others1 to whom ,i am full well knowne1 will quickely informe him2 my house being not past three and twenty miles from ,london3 and yet ,i so hid my selve at home1 that ,i seldomly travelled so farre as ,london once in a yeare4 ,indeed the life and mind of man cannot endure without some interchangeablenesse of recreation1 and pawses from the intensive actions of our severall callings2 and every man is drawne with his owne delight4 ,my recreations have been diversity of studies3 and as oft as ,i was toyled with the labour of my owne profession1 ,i have allayed that tediousnesse by walking in the pleasant and more then ,elysian fields of the diverse and various parts of humane learning1 and not the ,mathematics onely4 ,even the opponents of ,delamain must be grateful to him for having been the means of drawing from ,oughtred such interesting biographical details4 ,oughtred proceeds to tell how1 about #afbh1 he was induced to write his ,clavis mathematicae1 upon which his reputation as a mathematician largely rests3 ,about five yeares since1 the ,earle of ,arundell my most honourable ,lord in a time of his private retiring to his house in the countrey then at ,west ,horsley1 foure small miles from me 7though since he hath a house in ,aldebury the parish where ,i live7 hearing of me 7by what meanes ,i know not7 was pleased to send for me3 and afterward at ,london to appoint mee a ,chamber of his owne house3 where1 at such times1 and in such manner as it seemed him good to imploy me1 and when ,i might not inconveniently be spared from my charge1 ,i have been most ready to present my selfe in all humble and affectionate service3 ,i hope also without the offence of ,god1 the transgression of the good ,lawes of this ,land1 neglect of my calling1 or the deserved scandall of any good man4 4 4 4 4 ,and although ,i am no mercenary man1 nor make profession to teach any one in these arts for gaine and recompence1 but as ,i serve at the ,altar1 so ,i live onely of the ,altar3 yet in those interims that ,i am at ,london in my ,lords service1 ,i have been still much frequented both by ,natives and ,strangers1 for my resolution and instruction in many difficult poynts of ,art2 and have most freely and lovingly imparted my selfe and my skill1 such as ,i had1 to their contentments1 and much honourable acknowledgement of their obligation to my ,lord for bringing mee to ,london1 hath beene testifyed by many4 ,of which my liberallity and unwearyed readinesse to doe good to all1 scarce any one can give more ample testimony then ,r4 ,d4 himselfe can3 would he be but pleased to allay the shame of this his hot and eager contention1 blowne up onely with the full bellowes of intended glory and gaine2 4 4 4 4 they the subjects in which ,delamain received assistance from ,oughtred were the first elements of ,astronomie concerning the second motions of the fixed starres1 and of the ,sunne and ,moone2 they were the first elements of ,conics1 to delineate those sections3 they were the first elements of ,optics1 ,catoptrics1 and ,dioptrics3 of all which you knew nothing at all4 ,these last passages are instructive as showing what topics were taken up for study with some of his pupils4 ,the chief subject of interest with most of them was algebra1 which at that time was just beginning to draw the attention of ,english lovers of mathematics4 ,oughtred carried on an extensive correspondence on mathematical subjects4 ,he was frequently called upon to assist in the solution of knotty problems--sometimes to his annoyance1 perhaps1 as is shown by the following letter which he wrote in #afdb to a stranger1 named ,price3 ,it is true that ,i have bestowed such vacant time1 as ,i could gain from the study of divinity1 7which is my calling17 upon human knowledges1 and1 amongst other1 upon the mathematics1 wherein the little skill ,i have attained1 being compared with others of my profession1 who for the most part contenting themselves only with their own way1 refuse to tread these salebrous and uneasy paths1 may peradventure seem the more4 ,but now being in years and mindful of mine end1 and having paid dearly for my former delights both in my health and state1 besides the prejudice of such1 who not considering what incessant labour may produce1 reckon so much wanting unto me in my proper calling1 as they think ,i have acquired in other sciences2 by which opinion 7not of the vulgar only7 ,i have suffered both disrespect1 and also hinderance in some small perferments ,i have aimed at4 ,i have therefore now learned to spare myself1 and am not willing to descend again in arenam1 and to serve such ungrateful muses4 ,yet1 sir1 at your request ,i have perused your problem4 4 4 4 4 ,your problem is easily wrought per ,nicomedis conchoidem lineam4#ac ,,appearance ,,and ,,habits ,aubrey gives information about the appearance and habits of ,oughtred3 ,he was a little man1 had black haire1 and blacke eies 7with a great deal of spirit74 ,his head was always working4 ,he would drawe lines and diagrams on the dust4 4 4 4 4 ,he his oldest son ,benjamin told me that his father did use to lye a bed till eleaven or twelve a clock1 with his doublet on1 ever since he can remember4 ,studyed late at night2 went not to bed till #aa a clock2 had his tinder box by him2 and on the top of his bed-staffe1 he had his inke-horne fix't4 ,he slept but little4 ,sometimes he went not to bed in two or three nights1 and would not come downe to meales till he had found out the quaesitum4 ,he was more famous abroad for his learning1 and more esteemed1 then at home4 ,severall great mathematicians came over into ,england on purpose to converse with him4 ,his countrey neighbours 7though they understood not his worth7 knew that there must be extraordinary worth in him1 that he was so visited by foreigners4 4 4 4 4 ,when learned foreigners came and sawe how privately he lived1 they did admire and blesse themselves1 that a person of so much worth and learning should not be better provided for4 4 4 4 4 ,he has told bishop ,ward1 and ,mr4 ,elias ,ashmole 7who was his neighbour71 that 8on this spott of ground0 7or 8leaning against this oake0 or 8that ashe071 8the solution of such or such a probleme came into my head1 as if infused by a divine genius1 after ,i had thought on it without successe for a yeare1 two1 or three40 4 4 4 4 ,nicolaus ,mercator1 ,holsatus 4 4 4 4 went to see him few yeares before he dyed4 4 4 4 4 ,the right honble ,thomas ,howard1 earle of ,arundel and ,surrey1 ,lord ,high ,marshall of ,england1 was his great patron1 and loved him intirely4 ,one time they were like to have been killed together by the fall at ,albury of a grott1 which fell downe but just as they were come out4#ad ,oughtred's friends convey the impression that1 in the main1 ,oughtred enjoyed a comfortable living at ,albury4 ,only once appear indications of financial embarrassment4 ,about #afcd one of his pupils1 ,w4 ,robinson1 writes as follows3 ,i protest unto you sincerely1 were ,i as able as some1 at whose hands you have merited exceedingly1 or 7to speak more absolutely7 as able as willing1 ,i would as freely give you #ejj l4 per ann4 as #ejj pence2 and ,i cannot but be astonished at this our age1 wherein pelf and dross is made their summum bonum1 and the best part of man1 with the true ornaments thereof1 science and knowledge1 are so slighted4 4 4 4 4#ae ,in his letters ,oughtred complains several times of the limitations for work and the infirmities due to his advancing old age4 ,the impression he made upon others was quite different4 ,says one biographer3 ,he sometimes amused himself with archery1 and sometimes practised as a surveyor of land4 4 4 4 4 ,he was sprightly and active1 when more than eighty years of age4#af ,another informant says that ,oughtred was as facetious in ,greek and ,latine as solid in ,arithmetique1 ,astronomy1 and the sphere of all ,measures1 ,musick1 etc42 exact in his style as in his judgment2 handling his ,cube1 and other ,instruments at eighty1 as steadily1 as others did at thirty2 owing this1 he said1 to temperance and ,archery2 principling his people with plain and solid truths1 as he did the world with great and useful ,arts2 advancing new ,inventions in all things but ,religion4 ,which in its old order and decency he maintained secure in his privacy1 prudence1 meekness1 simplicity1 resolution1 patience1 and contentment4#ag ,,alleged ,,travel ,,abroad ,according to certain sources of information1 ,oughtred traveled on the ,european ,continent and was invited to change his abode to the ,continent4 ,we have seen no statement from ,oughtred himself on this matter4 ,he seldom referred to himself in his books and letters4 ,the autobiography contained in his ,apologeticall ,epistle was written a quarter of a century before his death4 ,aubrey gives the following3 ,in the time of the civill warres the duke of ,florence invited him over1 and offered him #ejj li4 per annum2 but he would not accept it1 because of his religion4#ah ,a portrait of ,oughtred1 painted in #afdf by ,hollar and inserted in the ,english edition of the ,clavis of #afdg1 contains underneath the following lines3 8,haec est ,oughtredi senio labantis imago ,itala quam cupiit1 ,terra ,britanna tulit40 ,in the sketch of ,oughtred by ,owen ,manning it is confessed that 8it is not known to what this alludes2 but possibly he might have been in ,italy with his patron1 the ,earl of ,arundel40#ai ,it would seem quite certain either that ,oughtred traveled in ,europe or that he received some sort of an offer to settle in ,italy4 ,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 parish1 seldom going even as far as ,london1 we strongly incline to the opinion that he did not travel on the ,continent1 but that he received an offer from some patron of the sciences--possibly some distinguished visitor--to settle in ,italy4 ,,his ,,death ,he died at ,albury1 ,june #cj1 #affj1 aged about eighty-six years4 ,of his last days and death1 ,aubrey speaks as follows3 ,before he dyed he burned a world of papers1 and sayd that the world was not worthy of them2 he was so superb4 ,he burned also severall printed bookes1 and would not stirre1 till they were consumed4 4 4 4 4 ,i myselfe have his ,pitiscus1 imbelished with his excellent marginall notes1 which ,i esteeme as a great rarity4 ,i wish ,i could also have got his ,bilingsley's ,euclid1 which ,john ,collins sayes was full of his annotations4 4 4 4 4 ,ralph ,greatrex1 his great friend1 the mathematicall instrument-maker1 sayed he conceived he dyed with joy for the comeing-in of the king1 which was the #bith of ,may before4 8,and are yee sure he is restored80--8,then give me a glasse of sack to drinke his sacred majestie's health40 ,his spirits were then quite upon the wing to fly away4 4 4 4 4#bj ,in this passage1 as in others1 due allowance must be made for ,aubrey's lack of discrimination4 ,he was not in the habit of sifting facts from mere gossip4 ,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 him4 ,more probable was the feeling that the papers he burned--possibly old sermons--were of no particular value to the world4 ,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 ,scarborough1 ,m4,d41 under whose supervision some of them were carefully revised and published at ,oxford in #afgg under the title of ,opuscula mathematica hactenus inedita4 ,aubrey's story of ,oughtred's mode of death has been as widely circulated in every modern biographical sketch as has his slander of ,mrs4 ,oughtred by claiming that she was so penurious that she would deny him the use of candles to read by4 ,oughtred died on ,june #cj2 the ,restoration occurred on ,may #bi4 ,no doubt ,oughtred rejoiced over the ,restoration1 but the story of his drinking 8a glass of sack0 to his ,majesty's health1 and then dying of joy is surely apocryphal4 ,de ,morgan humorously remarks1 8,it should be added1 by way of excuse1 that he was eighty-six years old40#ba ,,chapter ,,ii ,,principal ,,works 8,,clavis ,,mathematicae0 ,passing to the consideration of ,oughtred's mathematical books1 we begin with the observation that he showed a marked disinclination to give his writings to the press4 ,his first paper on sun-dials was written at the age of twenty-three1 but we are not aware that more than one brief mathematical manuscript was printed before his fifty-seventh year4 ,in every instance1 publication in printed form seems to have been due to pressure exerted by one or more of his patrons1 pupils1 or friends4 ,some of his manuscripts were lent out to his pupils1 who prepared copies for their own use4 ,in some instances they urged upon him the desirability of publication and assisted in preparing copy for the printer4 ,the earliest and best-known book of ,oughtred was his ,clavis mathematicae1 to which repeated allusion has already been made4 ,as he himself informs us1 he was employed by the ,earl of ,arundel about #afbh to instruct the ,earl's son1 ,lord ,william ,howard 7afterward ,viscount ,stafford7 in the mathematics4 ,for the use of this young man ,oughtred composed a treatise on algebra which was published in ,latin in the year #afca at the urgent request of a kinsman of the young man1 ,charles ,cavendish1 a patron of learning4 ,the ,clavis mathematicae1#bb in its first edition of #afca1 was a booklet of only #hh small pages4 ,yet it contained in very condensed form the essentials of arithmetic and algebra as known at that time4 ,aside from the addition of four tracts1 the #afca edition underwent some changes in the editions of #afdg and #afdh1 which two are much alike4 ,the twenty chapters of #afca are reduced to nineteen in #afdg and in all the later editions4 ,numerous minute alterations from the #afca edition occur in all parts of the books of #afdg and #afdh4 ,the material of the last three chapters of the #afca edition is rearranged1 with some slight additions here and there4 ,the #afdh edition has no preface4 ,in the print of #afeb there are only slight alterations from the #afdh edition2 after that the book underwent hardly any changes1 except for the number of tracts appended1 and brief explanatory notes added at the close of the chapters in the ,english editions of #afid and #agjb4 ,the #afeb and #affg editions were seen through the press by ,john ,wallis2 the #afih impression contains on the title-page the words3 ,ex ,recognitione ,d4 ,johannis ,wallis1 ,s4,t4,d4 ,geometriae ,professoris ,saviliani4 ,the cost of publishing may be a matter of some interest4 ,when arranging for the printing of the #affg edition of the ,clavis1 ,wallis wrote ,collins3 8,i told you in my last what price she ,mrs4 ,lichfield expects for it1 as ,i have formerly understood from her1 viz41 #dj for the impression1 which is about #i;d4 a book40#bc ,as compared with other contemporary works on algebra1 ,oughtred's distinguishes itself for the amount of symbolism used1 particularly in the treatment of geometric problems4 ,extraordinary emphasis was placed upon what he called in the ,clavis the 8analytical art40#bd ,by that term he did not mean our modern analysis or analytical geometry1 but the art 8in which by taking the thing sought as knowne1 we finde out that we seeke40#be ,he meant to express by it condensed processes of rigid1 logical deduction expressed by appropriate symbols1 as contrasted with mere description or elucidation by passages fraught with verbosity4 ,in the preface to the first edition 7#afca7 he says3 ,in this little book ,i make known 4 4 4 4 the rules relating to fundamentals1 collected together1 just like a bundle1 and adapted to the explanation of as many problems as possible4 ,as stated in this preface1 one of his reasons for publishing the book1 is 4 4 4 4 that like ,ariadne ,i might offer a thread to mathematical study by which the mysteries of this science might be revealed1 and direction given to the best authors of antiquity1 ,euclid1 ,archimedes1 the great geometrician ,apollonius of ,perga1 and others1 so as to be easily and thoroughly understood1 their theorems being added1 not only because to many they are the height and depth of mathematical science 7,i ignore the would-be mathematicians who occupy themselves only with the so-called practice1 which is in reality mere juggler's tricks with instruments1 the surface so to speak1 pursued with a disregard of the great art1 a contemptible picture71 but also to show with what keenness they have penetrated1 with what mass of equations1 comparisons1 reductions1 conversions and disquisitions these heroes have ornamented1 increased and invented this most beautiful science4 ,the ,clavis opens with an explanation of the ,hindu-,arabic notation and of decimal fractions4 ,noteworthy is the absence of the words 8million10 8billion10 etc4 ,though used on the ,continent by certain mathematical writers long before this1 these words did not become current in ,english mathematical books until the eighteenth century4 ,the author was a great admirer of decimal fractions1 but failed to introduce the notation which in later centuries came to be universally adopted4 ,oughtred wrote #j4ef in this manner #jef2 the point he used to designate ratio4 ,thus #c3d was written by him #cd4 ,the decimal point 7or comma7 was first used by the inventor of logarithms1 ,john ,napier1 as early as #afaf and #afag4 ,although ,oughtred had mastered the theory of logarithms soon after their publication in #afad and was a great admirer of ,napier1 he preferred to use the dot for the designation of ratio4 ,this notation of ratio is used in all his mathematical books1 except in two instances4 ,the two dots 737 occur as symbols of ratio in some parts of ,oughtred's posthumous work1 ,opuscula mathematica hactenus inedita1 ,oxford1 #afgg1 but may have been due to the editors and not to ,oughtred himself4 ,then again the two dots 737 are used to designate ratio on the last two pages of the tables of the ,latin edition of ,oughtred's ,trigonometria of #afeg4 ,in all other parts of that book the dot 77 is used4 ,probably someone who supervised the printing of the tables introduced the 737 on the last two pages1 following the logarithmic tables1 where methods of interpolation are explained4 ,the probability of this conjecture is the stronger1 because in the ,english edition of the ,trigonometrie1 brought out the same year 7#afeg7 but after the ,latin edition1 the notation 737 at the end of the book is replaced by the usual 771 except that in some copies of the ,english edition the explanations at the end are omitted altogether4 ,oughtred introduces an interesting1 and at the same time new1 feature of an abbreviated multiplication and an abbreviated division of decimal fractions4 ,on this point he took a position far in advance of his time4 ,the part on abbreviated multiplication was rewritten in slightly enlarged form and with some unimportant alterations in the later edition of the ,clavis4 ,we give it as it occurs in the revision4 ,four cases are given4 ,in finding the product of #bdfiad and #cebg1 8if you would have the ,product without any ,parts0 7without any decimal part71 8set the place of ,unity of the lesser under the place of ,unity in the greater3 as in the ,example10 writing the figures of the lesser number in inverse order4 ,from the example it will be seen that he begins by multiplying by #c1 the right-hand digit of the multiplier4 ,in the first edition of the ,clavis he began with #g1 the left digit4 ,observe also that he 8carries0 the nearest tens in the product of each lower digit and the upper digit one place to its right4 ,for instance1 he takes #gdbh and carries #c1 then he finds #gbcag and writes down #ag4 #b #d #fi #a #d #g #be #c ------- #g #d #j #g #a #b #c #e #d #i #a #g ------- #h #g #j #h ,the second case supposes that 8you would have the ,product with some places of parts0 7decimals71 say #d3 8,set the place of ,unity of the lesser ,number under the ,fourth place of the ,parts of the greater40 ,the multiplication of #bdfiad by #cebg is now performed thus3 #b #d #fi #a #d #g #be #c --------------- #g #d #j #g #d #b #j #j #a #b #c #d #e #g #j #j #d #i #c #h #b #h #a #g #b #h #d #j --------------- #h #g #j #hf #e #f #h ,in the third and fourth cases are considered factors which appear as integers1 but are in reality decimals2 for instance1 the sine of #ed is given in the tables as #hjijb when in reality it is 4#hjijb4 ,of interest as regards the use of the word 8parabola0 is the following3 8,the ,number found by ,division is called the ,quotient1 or also ,parabola1 because it arises out of the ,application of a plain ,number to a given ,longitude1 that a congruous ,latitude may be found40#bf ,this is in harmony with etymological dictionaries which speak of a parabola as the application of a given area to a given straight line4 ,the dividend or product is the area2 the divisor or factor is the line4 ,oughtred gives two processes of long division4 ,the first is identical with the modern process1 except that the divisor is written below every remainder1 each digit of the divisor being crossed out as soon as it has been used in the partial multiplication4 ,the second method of long division is one of the several types of the old 8scratch method40 ,this antiquated process held its place by the side of the modern method in all editions of the ,clavis4 ,the author divides #dfgjbc by #cegjibfdbe1 giving the following instructions3 8,take as many of the first ,figures of the ,divisor as are necessary1 for the first ,divisor1 and then in every following particular ,division drop one of the ,figures of the ,divisor towards the ,left ,hand1 till you have got a competent ,quotient40 ,he does not explain abbreviated division as thoroughly as abbreviated multiplication4 #ag #cjc #bhjc #ajiicj #cegjibfdbe7 #dfgjbc 7#acjghj #cegjic #ajgabg #bejj #bhf ,oughtred does not examine the degree of reliability or accuracy of his processes of abbreviated multiplication and division4 ,here as in other places he gives in condensed statement the mode of procedure1 without further discussion4 ,he does not attempt to establish the rules for the addition1 subtraction1 multiplication1 and division of positive and negative numbers4 8,if the ,signs are both alike1 the ,product will be affirmative1 if unlike1 negative02 then he proceeds to applications4 ,this attitude is superior to that of many writers of the eighteenth and nineteenth centuries1 on pedagogical as well as logical grounds3 pedagogically1 because the beginner in the study of algebra is not in a position to appreciate an abstract train of thought1 as every teacher well knows1 and derives better intellectual exercise from the applications of the rules to problems2 logically1 because the rule of signs in multiplication does not admit of rigorous proof1 unless some other assumption is first made which is no less arbitrary than the rule itself4 ,it is well known that the proofs of the rule of signs given by eighteenth-century writers are invalid4 ,somewhere they involve some surreptitious assumption4 ,this criticism applies even to the proof given by ,laplace1 which tacitly assumes the distributive law in multiplication4 ,a word should be said on ,oughtred's definition of and -4 ,he recognizes their double function in algebra by saying 7,clavis1 #afca1 p4 #b73 8,signum additionis1 sive affirmationis1 est plus0 and 8,signum subductionis1 sive negationis est - minus40 ,they are symbols which indicate the quality of numbers in some instances and operations of addition or subtraction in other instances4 ,in the #afid edition of the ,clavis1 thirty-four years after the death of ,oughtred1 these symbols are defined as signifying operations only1 but are actually used to signify the quality of numbers as well4 ,in this respect the #afid edition marks a recrudescence4 ,the characteristic in the ,clavis that is most striking to a modern reader is the total absence of indexes or exponents4 ,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 indexes1 but the modern exponential notation1 a1 is of later date4 ,the modern notation1 for positive integral exponents1 first appears in ,descartes' ,gomtrie1 #afcg2 fractional and negative exponents were first used in the modern form by ,sir ,isaac ,newton1 in his announcement of the binomial formula1 in a letter written in #afgf4 ,this total absence of our modern exponential notation in ,oughtred's ,clavis gives it a strange aspect4 ,like ,vieta1 ,oughtred uses ordinarily the capital letters1 ,a1 ,b1 ,c1 4 4 4 4 to designate given numbers2 ,a is written ,aq1 ,a is written ,ac2 for ,a1 ,a1 ,a he has1 respectively1 ,aqq1 ,aqc1 ,acc4 ,only on rare occasions1 usually when some parallelism in notation is aimed at1 does he use small letters#bg to represent numbers or magnitudes4 ,powers of binomials or polynomials are marked by prefixing the capital letters ,q 7for square71 ,c 7for cube71 ,,qq 7for the fourth power71 ,,qc 7for the fifth power71 etc4 ,oughtred does not express aggregation by 774 ,parentheses had been used by ,girard1 and by ,clavius as early as #afji1bh but did not come into general use in mathematical language until the time of ,leibniz and the ,bernoullis4 ,oughtred indicates aggregation by writing a colon 737 at both ends4 ,thus1 ,q3,a-,e3 means with him 7,a-,e74 ,similarly1 q3,a,e3 means 7,a,e74 ,the two dots at the end are frequently omitted when the part affected includes all the terms of the polynomial to the end4 ,thus1 ,c3,a,b-,e44 means 7,a,b-,e744 ,there are still further departures from this notation1 but they occur so seldom that we incline to the interpretation that they are simply printer's errors4 ,for proportion ,oughtred uses the symbol 73374 ,the proportion a3bc3d appears in his notation ab33cd4 ,apparently1 a proportion was not fully recognized in this day as being the expression of an equality of ratios4 ,that probably explains why he did not use here as in the notation of ordinary equations4 ,yet ,oughtred must have been very close to the interpretation of a proportion as an equality2 for he says in his ,elementi decimi ,euclidis declaratio1 8proportio1 sive ratio aequalis 330 ,that he introduced this extra symbol when the one for equality was sufficient is a misfortune4 ,simplicity demands that no unnecessary symbols be introduced4 ,however1 ,oughtred's symbolism is certainly superior to those which preceded4 ,consider the notation of ,clavius4#bi ,he wrote #bj3fjd3x1 x#ab1 thus3 8#bjfjd8 fiunt #ab40 ,the insufficiency of such a notation in the more involved expressions frequently arising in algebra is readily seen4 ,hence ,oughtred's notation 7337 was early adopted by ,english mathematicians4 ,it was used by ,john ,wallis at ,oxford1 by ,samuel ,foster at ,gresham ,college1 by ,james ,gregory of ,edinburgh1 by the translators into ,english of ,rahn's algebra1 and by many other early writers4 ,oughtred has been credited generally with the introduction of ,st4 ,andrew's cross as the symbol for multiplication in the ,clavis of #afca4 ,we have discovered that this symbol1 or rather the letter x which closely resembles it1 occurs as the sign of multiplication thirteen years earlier in an anonymous 8,appendix to the ,logarithmes1 shewing the practise of the ,calculation of ,triangles etc40 to ,edward ,wright's translation of ,john ,napier's ,descriptio1 published in #afah4cj ,later we shall give our reasons for believing that ,oughtred is the author of that 8,appendix40 ,the has survived as a symbol of multiplication4 ,another symbol introduced by ,oughtred and found in modern books is 1 expressing difference2 thus ,c,d signifies the difference between ,c and ,d1 even when ,d is the larger number4#ca ,this symbol was used by ,john ,wallis in #afeg4cb ,oughtred represented in symbols also certain composite expressions1 as for instance ,a,e,z1 ,a-,e,x1 where ,a is greater than ,e4 ,he represented by a symbol also each of the following3 ,a,e1 ,a,e1 ,a-,e1 ,a-,e4 ,oughtred practically translated the tenth book of ,euclid from its ponderous rhetorical form into that of brief symbolism4 ,an appeal to the eye was a passion with ,oughtred4 ,the present writer has collected the different mathematical symbols used by ,oughtred and has found more than one hundred and fifty of them4 ,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 editions4 ,the part which originally constituted the ,clavis was not materially altered1 except in two or three of the original twenty chapters4 ,these changes were made in the editions of #afdg and #afdh4 ,after the first edition1 great stress was laid upon the theory of indices upon the very first page1 as also in passages farther on4 ,of course1 ,oughtred did not have our modern notation of indices or exponents1 but their theory had been a part of algebra and arithmetic for some time4 ,oughtred incorporated this theory in his brief exposition of the ,hindu-,arabic notation and in his explanation of logarithms4 ,as previously pointed out1 the last three chapters of the #afca edition were considerably rearranged in the later editions and combined into two chapters1 so that the ,clavis proper had nineteen chapters instead of twenty in the additions after the first4 ,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 subject4 ,the very last problem deals with the division of angles into equal parts4 ,he derives the cubic equation upon which the trisection depends algebraically1 also the equations of the fifth degree and seventh degree upon which the divisions of the angle into #e and #g equal parts depend1 respectively4 ,the exposition was severely brief1 yet accurate4 ,he did not believe in conducting the reader along level paths or along slight inclines4 ,he was a guide for mountain-climbers1 and woe unto him who lacked nerve4 ,oughtred lays great stress upon expansions of powers of a binomial4 ,he makes use of these expansions in the solution of numerical equations4 ,to one who does not specialize in the history of mathematics such expansions may create surprise1 for did not ,newton invent the binomial theorem after the death of ,oughtred8 ,as a matter of fact1 the expansions of positive integral powers of a binomial were known long before ,newton1 not only to seventeenth-century but even to eleventh-century mathematicians4 ,oughtred's ,clavis of #afca gave the binomial coefficients for all powers up to and including the tenth4 ,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 series4 ,as a specimen of ,oughtred's style of writing we quote his solution of quadratic equations1 accompanied by a translation into ,english and into modern mathematical symbols4 ,as a preliminary step#cc he lets ,z,a,e and ,a,e2 he lets also ,x,a-,e4 ,from these relations he obtains identities which1 in modern notation1 are ,z-,,ae7,z-,e7,x4 ,now1 if we know ,z and ,,ae1 we can find ,x4 ,then 7,z,x7,a1 and 7,z-,x7,e1 and ,a,z7,z-,,ae74 ,having established these preliminaries1 he proceeds thus3 ,datis igitur linea inaequaliter secta ,z 7#aj71 rectangulo sub segmentis ,,ae 7#ba7 qui gnomon est3 datur semidifferentia segmentorum ,x3 per consequens ipsa segmenta4 ,nam ponatur alterutrum segmentum ,a3 alterum erit ,z-,a3 ,rectangulum auctem est ,,za-,aq,,ae4 ,et quia dantur ,z ,,ae3 estque ,zq-,,ae,xq3 per #e;c4 #ah1 ,z,x,a3 ,z-,x,e3 ,aequatio sic resoluetur3 ,zq3,zq-,,ae3,a maius segment_/minus segment4 ,itaque proposita equatione1 in qua sunt tres species aequaliter in ordine tabellae adscendentes1 altissima autem species ponitur negata3 ,magnitudo data coefficiens mediam speciem est linea bisecanda3 magnitudo absoluta data1 ad quam sit aequatio1 est rectangulum sub segmentis inaequalibus1 sine gnomon3 vt ,,za-,aq,,ae3 in numeris autem #ajl-lq#ba3 ,estque ,a1 vel #al1 alterutrum segmentum inaequale4 ,inuenitur autem sic3 ,dimidiata coefficiens median speciem est ,z_/#b 7#e72 cuius quadratum est ,zq_/#d 7#be73 ex hoc tolle ,,ae 7#ba7 absolutum3 eritque ,zq_/#d-;,,ae 7#d7 quadratum semidifferentiae segmentorum3 latus huius quadratum 7#b7 est semidifferentia3 quam si addas ad ,z_/#b 7#e7 semissem coefficientis1 sive lineae bisecandae1 erit maius segment42 sin detrahas1 erit minus segment3 ,dico ,z_/#bq3,zq_/#d-;,,ae3,a maius segmentum_/minus segmentum4 ,we translate the ,latin passage1 using the modern exponential notation and parentheses1 as follows3 ,given therefore an unequally divided line ,z 7#aj71 and a rectangle beneath the segments ,,ae 7#ba7 which is a gnomon4 ,half the difference of the segments ,x is given1 and consequently the segment itself4 ,for1 if one of the two segments is placed equal to ,a1 the other will be ,z-,a4 ,moreover1 the rectangle is ,,za-,a,,ae4 ,and because ,z and ,,ae are given1 and there is ,z-,,ae,x1 and by #e;c4#ah1 ,z,x,a1 and ,z-,x,e1 the equation will be solved thus3 ,z7,z-,,ae7,a major segment_/minor segment4 ,and so an equation having been proposed in which three species 7terms7 are in equally ascending powers1 the highest species1 moreover1 being negative1 the given magnitude which constitutes the middle species is the line to be bisected4 ,and the given absolute magnitude to which it is equal is the rectangle beneath the unequal segments1 without gnomon4 ,as ,,za-,a,,ae1 or in numbers1 #ajx-x#ba4 ,and ,a or x is one of the two unequal segments4 ,it may be found thus3 ,the half of the middle species is ,z_/#b 7#e71 its square is ,z_/#d 7#be74 ,from it subtract the absolute term ,,ae 7#ba71 and ,z_/#d-;,,ae 7#d7 will be the square of half the difference of the segments4 ,the square root of this1 7,z_/#b7-;,,ae 7#b71 is half the difference4 ,if you add it to half the coefficient ,z_/#b 7#e71 or half the line to be bisected1 the longer segment is obtained2 if you subtract it1 the smaller segment is obtained4 ,i say3 ,z_/#b7,z_/#d-;,,ae7,a major segment_/minor segment4 ,the quadratic equation ,aq,,za,,ae receives similar treatment4 ,this and the preceding equation1 ,,za-,aq,,ae1 constitute together a solution of the general quadratic equation1 xaxb1 provided that ,e or ,z are not restricted to positive values1 but admit of being either positive or negative1 a case not adequately treated by ,oughtred4 ,imaginary numbers and imaginary roots receive no consideration whatever4 ,a notation suggested by ,vieta and favored by ,girard made vowels stand for unknowns and consonants for knowns4 ,this conventionality was adopted by ,oughtred in parts of his algebra1 but not throughout4 ,near the beginning he used ,q to designate the unknown1 though usually this letter stood with him for the 8square0 of the expression after it4#cd ,it is of some interest that ,oughtred used _/ to signify the ratio of the circumference to the diameter of a circle4 ,very probably this notation is the forerunner of the #c4adaei 4 4 4 4 used in #agjf by ,william ,jones4 ,oughtred first used _/ in the #afdg edition of the ,clavis mathematicae4 ,in the #afeb edition he says1 8,si in circulo sit #g4bb3333aac4cee3;erit 33#b ,r4,p3 periph40 ,this notation was adopted by ,isaac ,barrow1 who used it extensively4 ,david ,gregory#ce used _/ in #afig1 and ,de ,moivre#cf used c_/r about #afig1 to designate the ratio of the circumference to the radius4 ,we quote the description of the ,clavis that was given by ,oughtred's greatest pupil1 ,john ,wallis4 ,it contains additional information of interest to us4 ,wallis devotes chap4 xv of his ,treatise of ,algebra1 ,london1 #afhe1 pp4 #fg-fi1 to ,mr4 ,oughtred and his ,clavis1 saying3 ,mr4 ,william ,oughtred 7our ,country-man7 in his ,clavis ,mathematicae1 7or ,key of ,mathematicks17 first published in the ,year #afca1 follows ,vieta 7as he did ,diophantus7 in the use of the ,cossick ,denominations2 omitting 7as he had done7 the names of ,sursolids1 and contenting himself with those of ,square and ,cube1 and the ,compounds of these4 ,but he doth abridge ,vieta's ,characters or ,species1 using only the letters q1 c1 c4 which in ,vieta are expressed 7at length7 by ,quadrate1 ,cube1 c4 ,for though when ,vieta first introduced this way of ,specious ,arithmetick1 it was more necessary 7the thing being new17 to express it in words at length3 ,yet when the thing was once received in practise1 ,mr4 ,oughtred 7who affected brevity1 and to deliver what he taught as briefly as might be1 and reduce all to a short view17 contented himself with single ,letters instead of those words4 ,thus what ,vieta would have written ,a ,quadrate1 into ,b ,cube1 ------------------------ ,equal to ,,fg ,plane1 ,,cde ,solid1 would with him be thus expressed ,aq ,bc ------- ,,fg4 ,c ,d ,e ,and the better to distinguish upon the first view1 what quantities were ,known1 and what ,unknown1 he doth 7usually7 denote the ,known to ,consonants1 and the ,unknown by ,vowels2 as ,vieta 7for the same reason7 had done before him4 ,he doth also 7to very great advantage7 make use of several ,ligatures1 or ,compendious ,notes1 to signify ,summs1 ,differences1 and ,rectangles of several ,quantities4 ,as for instance1 ,of two ,quantities ,a 7the ,greater71 and ,e 7the ,lesser71 the ,sum he calls ,z1 the ,difference ,x1 the ,rectangle ,,ae4 4 4 4 4 ,which being of 7almost7 a constant signification with him throughout1 do save a great circumlocution of words1 7each ,letter serving instead of a ,definition27 and are also made use of 7with very great advantage7 to discover the true nature of divers intricate ,operations1 arising from the various compositions of such ,parts1 ,sums1 ,differences1 and ,rectangles2 7of which there is great plenty in his ,clavis1 ,cap4 #aa1 #af1 #ah1 #ai4 and elsewhere17 which without such ,ligatures1 or ,compendious ,notes1 would not be easily discovered or apprehended4 4 4 4 4 ,i know there are who find fault with his ,clavis1 as too obscure1 because so short1 but without cause2 for his words be always full1 but not ,redundant1 and need only a little attention in the ,reader to weigh the force of every word1 and the ,syntax of it2 4 4 4 4 ,and this1 when once apprehended1 is much more easily retained1 than if it were expressed with the prolixity of some other ,writers2 where a ,reader must first be at the pains to weed out a great deal of superfluous ,language1 that he may have a short prospect of what is material2 which is here contracted for him in a short ,synopsis4 4 4 4 4 ,mr4 ,oughtred in his ,clavis1 contents himself 7for the most part7 with the solution of ,quadratick ,equations1 without proceeding 7or very sparingly7 to ,cubick ,equations1 and those of ,higher ,powers2 having designed that ,work for an ,introduction into ,algebra so far1 leaving the ,discussion of ,superior ,equations for another work4 4 4 4 4 ,he contents himself likewise in ,resolving ,equations1 to take notice of the ,affirmative or ,positive ,roots2 omitting the ,negative or ,ablative ,roots1 and such as are called ,imaginary or ,impossible ,roots4 ,and of those which1 he calls ,ambiguous ,equations1 7as having more ,affirmative ,roots than one17 he doth not 7that ,i remember7 any where take notice of more than ,two ,affirmative ,roots3 7,because in ,quadratick ,equations1 which are those he handleth1 there are indeed no more47 ,whereas yet in ,cubick ,equations1 there may be ,three1 and in those of ,higher ,powers1 yet more4 ,which ,vieta was well aware of1 and mentioneth in some of his ,writings2 and of which ,mr4 ,oughtred could not be ignorant4 8,,circles ,,of ,,proportion0 ,,and 8,,trigonometrie0 ,oughtred wrote and had published three important mathematical books1 the ,clavis1 the ,circles of ,proportion1#cg and a ,trigonometrie4#ch ,this last appeared in the year #afeg at ,london1 in both ,latin and ,english4 ,it is claimed that the trigonometry was 8neither finished nor published by himself1 but collected out of his scattered papers2 and though he connived at the printing it1 yet imperfectly done1 as appears by his ,,mss42 and one of the printed ,books1 corrected by his own ,hand40#ci ,doubtless more accurate on this point is a letter of ,richard ,stokes who saw the book through the press3 ,i have procured your ,trigonometry to be written over in a fair hand1 which when finished ,i will send to you1 to know if it be according to your mind2 for ,i intend 7since you were pleased to give your assent7 to endeavour to print it with ,mr4 ,briggs his ,tables1 and so soon as ,i can get the ,prutenic ,tables ,i will turn those of the sun and moon1 and send them to you4#dj ,in the preface to the ,latin edition ,stokes writes3 ,since this trigonometry was written for private use without the intention of having it published1 it pleased the ,reverend ,author1 before allowing it to go to press1 to expunge some things1 to change other things and even to make some additions and insert more lucid methods of exposition4 ,this much is certain1 the ,trigonometry bears the impress characteristic of ,oughtred4 ,like all his mathematical writings1 the book was very condensed4 ,aside from the tables1 the text covered only #cf pages4 ,plane and spherical triangles were taken up together4 ,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 eye4 ,in the work of #afeg1 contractions are given as follows3 ssine1 ttangent1 sesecant1 s cocosine 7sine complement71 t cocotangent1 se cocosecant1 loglogarithm1 ,z crusum of the sides of a rectangle or right angle1 ,x crudifference of these sides4 ,it has been generally overlooked by historians that ,oughtred used the abbreviations of trigonometric functions1 named above1 a quarter of a century earlier1 in his ,circles of ,proportion1 #afcb1 #afcc4 ,moreover1 he used sometimes also the abbreviations which are current at the present time1 namely sinsine1 tantangent1 secsecant4 ,we know that the ,circles of ,proportion existed in manuscript many years before they were published4 ,the symbol sv for sinus versus occurs in the ,clavis of #afca4 ,the great importance of well-chosen symbols needs no emphasis to readers of the present day4 ,with reference to ,oughtred's trigonometric symbols4 ,augustus ,de ,morgan said3 ,this is so very important a step1 simple as it is1 that ,euler is justly held to have greatly advanced trigonometry by its introduction4 ,nobody that we know of has noticed that ,oughtred was master of the improvement1 and willing to have taught it1 if people would have learnt4#da ,we find1 however1 that even ,oughtred cannot be given the whole credit in this matter4 ,by or before #afca several other writers used abbreviations of the trigonometric functions4 ,as early as #afbd the contractions sin for sine and tan for tangent appear on the drawing representing ,gunter's scale1 but ,gunter did not use them in his books1 except in the drawing of his scale4#db ,a closer competitor for the honor of first using these trigonometric abbreviations is ,richard ,norwood in his ,trigonometrie1 ,london1 #afca1 where s stands for sine1 t for tangent1 sc for sine complement 7cosine71 tc for tangent complement 7cotangent71 and sec for secant4 ,norwood was a teacher of mathematics in ,london and a well-known writer of books on navigation4 ,aside from the abbreviations just cited ,norwood did not use nearly as much symbolism in his mathematics as did ,oughtred4 ,mention should be made of trigonometric symbols used even earlier than any of the preceding1 in 8,an ,appendix to the ,logarithmes1 shewing the practise of the ,calculation of ,triangles1 etc410 printed in ,edward ,wright's edition of ,napier's ,a ,description of the ,admirable ,table of ,logarithmes1 ,london1 #afah4 ,we referred to this 8,appendix0 in tracing the origin of the sign 4 ,it contains1 on p4 #d1 the following passage3 8,for the ,logarithme of an arch or an angle ,i set before 7s71 for the antilogarithme or compliment thereof 7s997 and for the ,differential 7t740 ,in further explanation of this rather unsatisfactory passage1 the author 7,oughtred87 says1 8,as for example3 s,b,,bc,,ca4 that is1 the ,logarithme of an angle ,b4 at the ,base of a plane right-angled triangle1 increased by the addition of the ,logarithm of ,,bc1 the hypothenuse thereof1 is equall to the ,logarithme of ,,ca the cathetus40 ,here 8logarithme of an angle ,b0 evidently means 8log sin ,b10 just as with ,napier1 8,logarithms of the arcs0 signifies really 8,logarithms of the sines of the angles40 ,in ,napier's table1 the numbers in the column marked 8,differentiae0 signify log sine minus log cosine of an angle2 that is1 the logarithms of the tangents4 ,this explains the contraction 7t7 in the 8,appendix40 ,the conclusion of all this is that as early as #afah the signs s1 s991 t were used for sine1 cosine1 and tangent1 respectively4 ,john ,speidell1 in his ,breefe ,treatise of ,sphaericall ,triangles1 ,london1 #afbg1 uses ,si4 for sine1 ,t4 and ,tan for tangent1 ,se4 for secant1 ,si4 ,co4 for cosine1 ,se4 ,co4 for cosecant1 ,t4 ,co4 for cotangent4 ,the innovation of designating the sides and angles of a triangle by ,a1 ,b1 ,c1 and a1 b1 c1 so that ,a was opposite a1 ,b opposite b1 and ,c opposite c1 is attributed to ,leonard ,euler 7#agec71 but was first used by ,richard ,rawlinson of ,queen's ,college1 ,oxford1 sometimes after #afee and before #affh4 ,oughtred did not use ,rawlinson's notation4#dc ,in trigonometry ,english writers of the first half of the seventeenth century used contractions more freely than their continental contemporaries2 even more freely1 indeed1 than ,english writers of a later period4 ,von ,braunmhl1 the great historian of trigonometry1 gives ,oughtred much praise for his trigonometry1 and points out that half a century later the army of writers on trigonometry had hardly yet reached the standard set by ,oughtred's analysis4#dd ,oughtred must be credited also with the first complete proof that was given to the first two of 8,napier's analogies40 ,his trigonometry contains seven-place tables of sines1 tangents1 and secants1 and six-place tables of logarithmic sines and tangents2 also seven-place logarithmic tables of numbers4 ,at the time of ,oughtred there was some agitation in favor of a wider introduction of decimal systems4 ,this movement is reflected in those tables which contain the centesimal division of the degree1 a practice which is urged for general adoption in our own day1 particularly by the ,french4 ,,solution ,,of ,,numerical ,,equations ,in the solution of numerical equations ,oughtred does not mention the sources from which he drew1 but the method is substantially that of the great ,french algebraist ,vieta1 as explained in a publication which appeared in #afjj in ,paris under the title1 ,de numerosa potestatum purarum atque adfectarum ad exegesin resolutione tractatus4 ,in view of the fact that ,vieta's process has been described inaccurately by leading modern historians including ,h4 ,hankel#de and ,m4 ,cantor1#df it may be worth while to go into some detail4#dg ,by them it is made to appear as identical with the procedure given later by ,newton4 ,the two are not the same4 ,the difference lies in the divisor used4 ,what is now called 8,newton's method0 is ,newton's method as modified by ,joseph ,raphson4#dh ,the ,newton-,raphson method of approximation to the roots of an equation f7x7#j is usually given the form a-f7a7_/f7a71 where a is an approximate value of the required root4 ,it will be seen that the divisor is f7a74 ,vieta's divisor is different2 it is f7as7-f7a7-s1 where f7x7 is the left of the equation f7x7k1 n is the degree of equation1 and s is a unit of the denomination of the digit next to be found4 ,thus in x#dbjjjjx#bdgfeagac1 it can be shown that #dag is approximately a root2 suppose that a has been taken to be #djj1 then s#aj2 but if1 at the next step of approximation1 a is taken to be #daj1 then s#a4 ,in this example1 taking a#djj1 ,vieta's divisor would have been #iabjjjj2 ,newton's divisor would have been #ijjjjj4 ,a comparison of ,vieta's method with the ,newton-,raphson method reveals the fact that ,vieta's divisor is more reliable1 but labors under the very great disadvantage of requiring a much larger amount of computation4 ,the latter divisor is accurate enough and easier to compute4 ,altogether the ,newton-,raphson process marks a decided advance over that of ,vieta4 ,as already stated1 it is the method of ,vieta that ,oughtred explains4 ,the ,englishman's exposition is an improvement on that of ,vieta1 printed forty years earlier4 ,nevertheless1 ,oughtred's explanation is far from easy to follow4 ,the theory of equations was at that time still in its primitive stage of development4 ,algebraic notation was not sufficiently developed to enable the argument to be condensed into a form easily surveyed4 ,so complicated does ,vieta's process of approximation appear that ,m4 ,cantor failed to recognize that ,vieta possessed a uniform mode of procedure4 ,but when one has in mind the general expression for ,vieta's divisor which we gave above1 one will recognize that there was marked uniformity in ,vieta's approximations4 ,oughtred allows himself twenty-eight sections in which to explain the process and at the close cannot forbear remarking that #bh is a 8perfect0 number 7being equal to the sum of its divisors1 #a1 #b1 #d1 #g1 #ad74 ,the early part of his exposition shows how an equation may be transformed so as to make its roots #aj1 #ajj1 #ajjj1 or #ajm times smaller4 ,this simplifies the task of 8locating a root02 that is1 of finding between what integers the root lies4 ,taking one of ,oughtred's equations1 x-#gbx#bchfjjx#hgbehae1 upon dividing #gbx by #aj1 #bchfjjx by #ajjj1 and #hgbehae by #aj1jjj1 we obtain x-#gbx#bchfx#hgbe4 ,dividing both sides by x1 we obtain x#bchf-gbxx7#hgbe4 ,letting x#d1 we have #fdbchf-aaebahgd4 ,but #d7hgbe7baha2 #d is too small4 ,next let x#e1 we have #abebchf-ahjahcf4 ,but #e7hgbe7agde2 #e is too large4 ,we take the lesser value1 x#d1 or in the original equation1 x#dj4 ,this method may be used to find the second digit in the root4 ,oughtred divides both sides of the equation by x1 and obtains xx7#bchfjj-gbxx7#hgbehae4 ,he tries x#dg and x#dh1 and finds that x#dg4 ,he explains also how the last computation may be done by logarithms4 ,thereby he established for himself the record of being the first to use logarithms in the solution of affected equations4 ,as an illustration of ,oughtred's method of approximation after the root sought has been located1 we have chosen for brevity a cubic in preference to a quartic4 ,we selected the equation x#dbjjjjx#bdgfeagac4 ,by the process explained above a root is found to lie between x#djj and x#ejj4 ,from this point on1 the approximation as given by ,oughtred is as shown on p4 #dc4 ,in further explanation of this process1 observe that the given equation is of the form ,lc,cq,l,dc1 where ,lc is our x1 ,cq#dbjjjj1 ,dc#bdgfeagac4 ,in the first step of approximation1 let ,l,a,e1 where ,a#djj and ,e is1 as yet1 undetermined4 ,we have ,lc7,a,e7,a#c;,a,e#c;,,ae,e and ,cq,l#dbjjjj7;,a,e74 ,subtract from #bdgfeagac the sum of the known terms ,a 7his ,ac7 and #dbjjjj ,a 7his ,cq,a74 ,this sum is #bcbjjjjjj the remainder is #aefeagac4 8,exemplum ,,ii #a;c#dbjjjjl#bdgfeagac ,hoc est1 ,lc,cq,l,dc #b #d #g #f #e #a #g #a #c 7 #d #a #g -------------------------------- #d #b #j #j #j #j ,cq -------------------------------- #f #d ,ac #a #f #h #j #j #j #j ,cq ,a -------------------------------- #b #c #b #j #j #j #j ,ablatit4 ,r #a #e #f #e #a #g #a #c -------------------------------- #d #h #c ,aq #a #b #c ,a #d #b #j #j #j #j ,cq -------------------------------- #i #a #b #j #j #j ,divisor4 -------------------------------- #d #h #c ,aq ,e #a #b #c ,a ,eq #a ,ec #d #b #j #j #j #j ,cq ,e -------------------------------- #i #a #b #a #j #j ,ablatit4 ,r #f #e #c #j #g #a #c #d #a -------------------------------- ------------ #e #j #d #c #c ,aq #a #b #c #c ,a #a #f #h #d #b #j #j #j #j ,cq #a -------------------------------- ------------ #i #b #e #e #c #j ,divisor4 #a #f #h #a -------------------------------- #c #e #c #j #a #c ,aq ,e #f #j #b #g #c ,a ,eq #c #d #c ,ec #b #i #d #j #j #j #j ,cq ,e -------------------------------- #f #e #c #j #g #a #c ,ablatit40 ,next1 he evaluates the coefficients of ,e in #c;,a,e and #dbjjjj;,e1 also #c;,a1 the coefficient of ,e4 ,he obtains #c;,a#dhjjjj1 #c;,a#abjj1 ,cq#dbjjjj4 ,he interprets #c;,a and ,cq as tens1 #c;,a as hundreds4 ,accordingly1 he obtains as their sum #iabjjjj1 which is the divisor for finding the second digit in the approximation4 ,observe that this divisor is the value of f7as7-f7a7-s in our general expression1 where a#djj1 s#aj1 n#c1 f7x7x#dbjjjjx4 ,dividing the remainder #aefeagac by #iabjjjj1 he obtains the integer #a in ten's place2 thus ,e#aj1 approximately4 ,he now computes the terms #c;,a,e1 #c;,,ae and ,e to be1 respectively1 #dhjjjjj1 #abjjjj1 #ajjj4 ,their sum is #iabajjj4 ,subtracting it from the previous remainder1 #aefeagac1 leaves the new remainder1 #fecjgac4 ,from here on each step is a repetition of the preceding step4 ,the new ,a is #daj1 the new ,e is to be determined4 ,we have now in closer approximation1 ,l,a,e4 ,this time we do not subtract ,a and ,cq,a1 because this subtraction is already affected by the preceding work4 ,we find the second trial divisor by computing the sum of #c;,a1 #c;,a and ,cq2 that is1 the sum of #ejdcjj1 #abcj1 #dbjjjj1 which is #ibeecj4 ,again1 this divisor can be computed by our general expression for divisors1 by taking a#daj1 s#a1 n#c4 ,dividing #fecjgac by #ibeecj yields the integer #g4 ,thus ,e#g4 ,computing #c;,a,e1 #c;,,ae1 ,e and subtracting their sum1 the remainder is #j4 ,hence #dag is an exact root of the given equation4 ,since the extraction of a cube root is merely the solution of a pure cubic equation1 xn1 the process given above may be utilized in finding cube roots4 ,this is precisely what ,oughtred does in chap4 xiv of his ,clavis4 ,if the foregoing computation is modified by putting ,cq#j1 the process will yield the approximate cube root of #bdgfeagac4 ,oughtred solves #af examples by the process of approximation here explained4 ,of these1 #i are cubics1 #e are quartics1 and #b are quintics4 ,in all cases he finds only one or two real roots4 ,of the roots sought1 five are irrational1 the remaining are rational and are computed to their exact values4 ,three of the computed roots have #b figures each1 #i roots have #c figures each1 #d roots have #d figures each4 ,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 fifth4 ,in view of the fact that many editions of the ,clavis were issued1 one impression as late as #agjb1 it contributed probably more than any other book to the popularization of ,vieta's method in ,england4 ,before ,oughtred1 ,thomas ,harriot and ,william ,milbourn are the only ,englishmen known to have solved numerical equations of higher degrees4 ,milbourn published nothing4 ,harriot slightly modified ,vieta's process by simplifying somewhat the formation of the trial divisor4 ,this method of approximation was the best in existence in ,europe until the publication by ,wallis in #afhe of ,newton's method of approximation4 ,it should be stated that1 before the time of ,newton1 the best method of approximation to the roots of numerical equations existed1 not in ,europe1 but in ,china4 ,as early as the thirteenth century the ,chinese possessed a method which is almost identical with what is known today as 8,horner's method40 ,,logarithms ,oughtred's treatment of logarithms is quite in accordance with the more recent practice4#di ,he explains the finding of the 8index0 7our 8characteristic072 he states that 8the sum of two ,logarithms is the ,logarithm of the ,product of their ,valors2 and their difference is the ,logarithm of the ,quotient10 that 8the ,logarithm of the side #dcf drawn upon the ,index number #b of dimensions of any ,potestas is the logarithm of the same ,potestas0 #dcf1 that 8the logarithm of any ,potestas #dcf divided by the number of its dimensions #b affordeth the ,logarithm of its ,root #dcf40 ,these statements of ,oughtred occur for the first time in the ,key of the ,mathematicks of #afdg2 the ,clavis of #afca contains no treatment of logarithms4 ,if the characteristic of a logarithm is negative1 ,oughtred indicates this fact by placing the - above the characteristic4 ,he separates the characteristic and mantissa by a comma1 but still uses the sign to indicate decimal fractions4 ,he uses the contraction 8log40 ,,invention ,,of ,,the ,,slide ,,rule2 ,,controversy ,,on ,,priority ,,of ,,invention ,oughtred's most original line of scientific activity is the one least known to the present generation4 ,augustus ,de ,morgan1 in speaking of ,oughtred1 who was sometimes called 8,oughtred ,aetonensis10 remarks3 8,he is an animal of extinct race1 an ,eton mathematician4 ,few ,eton men1 even of the minority which knows what a sliding rule is1 are aware that the inventor was of their own school and college40#ej ,the invention of the slide rule has1 until recently1#ea been a matter of dispute2 it has been erroneously ascribed to ,edmund ,gunter1 ,edmund ,wingate1 ,seth ,partridge1 and others4 ,we have been able to establish that ,william ,oughtred was the first inventor of slide rules1 though not the first to publish thereon4 ,we shall see that ,oughtred invented slide rules about #afbb1 but the descriptions of his instruments were not put into print before #afcb and #afcc4 ,meanwhile one of his own pupils1 ,richard ,delamain1 who probably invented the circular slide rule independently1 published a description in #afcj1 at ,london1 in a pamphlet of #cb pages entitled ,grammelogia2 or the ,mathematicall ,ring4 ,in editions of this pamphlet which appeared during the following three or four years1 various parts were added on1 and some parts of the first and second editions eliminated4 ,thus ,delamain antedates ,oughtred two years in the publication of a description of a circular slide rule4 ,but ,oughtred had invented also a rectilinear slide rule1 a description of which appeared in #afcc4 ,to the invention of this ,oughtred has a clear title4 ,a bitter controversy sprang up between ,delamain on one hand1 and ,oughtred and some of his pupils on the other1 on the priority and independence of invention of the circular slide rule4 ,few inventors and scientific men are so fortunate as to escape contests4 ,the reader needs only to recall the disputes which have arisen1 involving the researches of ,sir ,isaac ,newton and ,leibniz on the differential and integral calculus1 of ,thomas ,harriot and ,ren ,descartes relating to the theory of equations1 of ,robert ,mayer1 ,hermann von ,helmholtz1 and ,joule on the principle of the conservation of energy1 or of ,robert ,morse1 ,joseph ,henry1 ,gauss and ,weber1 and others on the telegraph1 to see that questions of priority and independence are not uncommon4 ,the controversy between ,oughtred and ,delamain embittered ,oughtred's life for many years4 ,he refers to it in print on more than one occasion4 ,we shall confine ourselves at present to the statement that it is by no means clear that ,delamain stole the invention from ,oughtred2 ,delamain was probably an independent inventor4 ,moreover1 it is highly probable that the controversy would never have arisen1 had not some of ,oughtred's pupils urged and forced him into it4 ,william ,forster stated in the preface to the ,circles of ,proportion of #afcb that while he had been carefully preparing the manuscript for the press1 8another to whom the ,author ,oughtred in a louing confidence discouered this intent1 using more hast then good speed1 went about to preocupate40 ,it was this passage which started the conflagration4 ,another pupil1 ,w4 ,robinson1 wrote to ,oughtred1 when the latter was preparing his ,apologeticall ,epistle as a reply to ,delamain's countercharges3 8,good sir1 let me be beholden to you for your ,apology whensoever it comes forth1 and 7if ,i speak not too late7 let me entreat you1 whip ignorance well on the blind side1 and we may turn him round1 and see what part of him is free40#eb ,as stated previously1 ,oughtred's circular slide rule was described by him in his ,circles of ,proportion1 ,london1 #afcb1 which was translated from ,oughtred's ,latin manuscript and then seen through the press by his pupil1 ,william ,forster4 ,in #afcc appeared ,an ,addition vnto the ,vse of the ,instrvment called the ,circles of ,proportion which contained at the end 8,the ,declaration of the two ,rulers for ,calculation10 giving a description of ,oughtred's rectilinear slide rule4 ,this ,addition was bound with the ,circles of ,proportion as one volume4 ,about the same time ,oughtred described a modified form of the rectilinear slide rule1 to be used in ,london for gauging4#ec ,,chapter ,,iii ,,minor ,,works ,among the minor works of ,oughtred must be ranked his booklet of forty pages to which reference has already been made1 entitled1 ,the ,new ,artificial ,gauging ,line or ,rod1 ,london1 #afcc4 ,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 instruments4 ,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 8,j4 ,s40 7,john ,smith874 ,oughtred's tract appeared with its own title-page1 but with pagination continued from the preceding part1 as ,an ,appendix wherein is contained a ,method of ,calculating all ,numbers for ,watches4 ,written originally by that famous ,mathematician ,mr4 ,william ,oughtred1 and now made ,publick4 ,by ,j4 ,s4 of ,london1 ,clock-maker4 ,london1 #afge4 8,j4 ,s40 says in his preface3 ,the method following was many years since ,compiled by ,mr4 ,oughtred for the use of some ,ingenious ,gentlemen his friends1 who for recreation at the ,university1 studied to find out the reason and ,knowledge of ,watch-work1 which seemed also to be a thing with which ,mr4 ,oughtred himself was much affected1 as may in part appear by his putting out of his own ,son to the same ,trade1 for whose use 7as ,i am informed7 he did compile a larger tract1 but what became of it cannot be known4 ,notwithstanding ,oughtred's marked activity in the design of mathematical instruments1 and his use of surveying instruments1 he always spoke in deprecating terms of their importance and their educational value4 ,in his epistle against ,delamain he says3 ,the ,instruments ,i doe not value or weigh one single penny4 ,if ,i had been ambitious of praise1 or had thought them 7or better then they7 worthy1 at which to have taken my rise1 out of my secure and quiet obscuritie1 to mount up into glory1 and the knowledge of men3 ,i could have done it many yeares before4 4 4 4 4 ,long agoe1 when ,i was a young student of the ,mathematicall ,sciences1 ,i tryed many wayes and devices to fit my selve with some good ,diall or ,instrument portable for my pocket1 to finde the houre1 and try other conclusions by1 and accordingly framed for that my purpose both ,quadrants1 and ,rings1 and ,cylinders1 and many other composures4 ,yet not to my full content and satisfaction2 for either they performed but little1 or els were patched up with a diversity of lines by an unnaturall and forced contexture4 ,at last ,i 4 4 4 4 found what ,i had before with much studie and paines in vaine sought for4#ed ,mention has been made in the previous pages of two of his papers on sun-dials1 prepared 7as he says7 when he was in his twenty-third year4 ,the first was published in the ,clavis of #afdg4 ,the second paper appeared in his ,circles of ,proportion4 ,both before and after the time of ,oughtred much was written on sun-dials4 ,such instruments were set up against the walls of prominent buildings1 much as the faces of clocks in our time4 ,the inscriptions that were put upon sun-dials are often very clever3 8,i count only the hours of sunshine10 8,alas1 how fleeting40 ,a sun-dial on the grounds of ,merchiston ,castle1 in ,edinburgh1 where the inventor of logarithms1 ,john ,napier1 lived for many years1 bears the inscription1 8,ere time be tint1 tak tent of time0 7,ere time be lost1 take heed of time74 ,portable sun-dials were sometimes carried in pockets1 as we carry watches4 ,thus ,shakespeare1 in ,as ,you ,like ,it1 ,act ,,ii1 sc4 vii3 8,and then he drew a diall from his poke40 ,watches were first made for carrying in the pocket about #afeh4 ,because of this literary1 scientific1 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-measurement4 ,besides the accounts previously noted1 there came from his pen3 ,the ,description and ,use of the double ,horizontall ,dyall3 ,whereby not onely the hower of the day is shewne2 but also the ,meridian ,line is found3 ,and most ,astronomical ,questions1 which may be done by the ,globe1 are resolved4 ,invented and written by ,w4 ,o41 ,london1 #afcf4 ,the 8,horizontall ,dyall0 and 8,horologicall ,ring0 appeared again as appendixes to ,oughtred's translation from the ,french of a book on mathematical recreations4 ,the fourth ,french edition of that work appeared in #afbg at ,paris1 under the title of ,recreations mathematiqve1 written by 8,henry van ,etten10 a pseudonym for the ,french ,jesuit ,jean ,leurechon 7#aeia-afij74 ,english editions appeared in #afcc1 #afec1 and #afgd4 ,the full title of the #afec edition conveys an idea of the contents of the text3 ,mathematical ,recreations1 or1 ,a ,collection of many ,problemes1 extracted out of the ,ancient and ,modern ,philosophers1 as ,secrets and ,experiments in ,arithmetick1 ,geometry1 ,cosmographie1 ,horologiographie1 ,astronomie1 ,navigation1 ,musick1 ,opticks1 ,architecture1 ,statick1 ,mechanicks1 ,chemistry1 ,water-works1 ,fire-works1 c4 ,not vulgarly manifest till now4 ,written first in ,greek and ,latin1 lately compil'd in ,french1 by ,henry ,van ,etten1 and now in ,english1 with the ,examinations and ,augmentations of divers ,modern ,mathematicians4 ,whereunto is added the ,description and ,use of the ,generall ,horologicall ,ring4 ,and ,the ,double ,horizontall ,diall4 ,invented and written by ,william ,oughtred4 ,london1 ,printed for ,william ,leake1 at the ,signe of the ,crown in ,fleet-street1 between the two ,temple-,gates4 ,,mdcliii4 ,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-law1 ,christopher ,brookes1 under the following title3 ,the ,solution of all ,sphaerical ,triangles both right and oblique ,by the ,planisphaere3 ,whereby two of the ,sphaerical partes sought1 are at one position most easily found out4 ,published with consent of the ,author1 ,by ,christopher ,brookes1 ,mathematique ,instrument-maker1 and ,manciple of ,wadham ,colledge1 in ,oxford4 ,brookes says in the preface3 ,i have oftentimes seen my ,reverend friend ,mr4 ,w4 ,o4 in his resolution of all sphaericall triangles both right and oblique1 to use a planisphaere1 without the tedious labour of ,trigonometry by the ordinary ,canons3 which planisphaere he had delineated with his own hands1 and used in his calculations more than ,forty years before4 ,interesting as one of our sources from which ,oughtred obtained his knowledge of the conic sections is his study of ,mydorge4 ,a tract which he wrote thereon was published by ,jonas ,moore1 in his ,arithmetick in two books 4 4 4 4 containing also the two first books of ,mydorgius his conical sections analyzed by that reverend devine ,mr4 ,w4 ,oughtred1 ,englished and completed with cuts4 ,london1 #affj4 ,another edition bears the date #afhh4 ,to be noted among the minor works of ,oughtred are his posthumous papers4 ,he left a considerable number of mathematical papers which his friend ,sir ,charles ,scarborough had revised under his direction and published at ,oxford in #afgf in one volume under the title1 ,gulielmi ,oughtredi1 ,etonensis1 quondam ,collegii ,regalis in ,cantabrigia ,socii1 ,opuscula ,mathematica hactenus inedita4 ,its nine tracts are of little interest to a modern reader4 ,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 which1 as previously noted1 appeared as an 8,appendix0 to ,edward ,wright's translation into ,english of ,john ,napier's ,descriptio1 under the title1 ,a ,description of the ,admirable ,table of ,logarithmes1 ,london1 #afah4 ,the 8,appendix0 bears the title1 8,an ,appendix to the ,logarithmes1 showing the practise of the ,calculation of ,triangles1 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 ,canons40 ,it is an able tract4 ,a natural guess is that the editor of the book1 ,samuel ,wright1 a son of ,edward ,wright1 composed this 8,appendix40 ,more probable is the conjecture which 7,dr4 ,j4 ,w4 ,l4 ,glaisher informs me7 was made by ,augustus ,de ,morgan1 attributing the authorship to ,oughtred4 ,two reasons in support of this are advanced by ,dr4 ,glaisher1 the use of x in the 8,appendix0 as the sign of multiplication 7to ,oughtred is generally attributed the introduction of the cross for multiplication in #afca71 and the then unusual designation 8cathetus0 for the vertical leg of a right triangle1 a term appearing in ,oughtred's books4 ,we are able to advance a third argument1 namely1 the occurrence in the 8,appendix0 of 7,s997 as the notation for sine complement 7cosine71 while ,seth ,ward1 an early pupil of ,oughtred1 in his ,idea trigonometriae demonstratae1 ,oxford1 #afed1 used a similar notation 7,s'74 ,it has been stated elsewhere that ,oughtred claimed ,seth ,ward's exposition of trigonometry as virtually his own4 ,attention should be called also to the fact that1 in his ,trigonometria1 p4 #b1 ,oughtred uses 7'7 to designate #ahj-;angle4 ,dr4 ,j4 ,w4 ,l4 ,glaisher is the first to call attention to other points of interest in this 8,appendix40 ,the interpolations are effected with the aid of a small table containing the logarithms of #gb sines4 ,except for the omission of the decimal point1 these logarithms are natural logarithms--the first of their kind ever published4 ,in this table we find log #ajbcjbehd2 in modern notation1 this is stated1 loge #ajb4cjbehd4 ,the first more extended table of natural logarithms of numbers was published by ,john ,speidell in the #afbb impression of his ,new ,logarithmes1 which contains1 besides trigonometric tables1 the logarithms of the numbers #a-ajjj4 ,the 8,appendix0 contains also the first account of a method of computing logarithms1 called the 8radix method10 which is usually attributed to ,briggs who applied it in his ,arithmetica logarithmica1 #afbd4 ,in general1 this method consists in multiplying or dividing a number1 whose logarithm is sought1 by a suitable factor and resolving the result into factors of the form #ax_/#aj4 ,the logarithm of the number is then obtained by adding the previously calculated logarithms of the factors4 ,the method has been repeatedly rediscovered1 by ,flower in #agga1 ,atwood in #aghf1 ,leonelli in #ahjb1 ,manning in #ahjf1 ,weddle in #ahde1 ,hearn in #ahdg1 and ,orchard in #ahdh4 ,we conclude with the words of ,dr4 ,j4 ,w4 ,l4 ,glaisher3 ,the ,appendix was an interesting and remarkable contribution to mathematics1 for in its sixteen small pages it contains 7#a7 the first use of the sign 2 7#b7 the first abbreviations1 or symbols1 for the sine1 tangent1 cosine1 and cotangent2 7#c7 the invention of the radix method of calculating logarithms2 7#d7 the first table of hyperbolic logarithms4#ee ,,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 descriptio1 ,edinburgh1 #afad1 and the time1 forty years later1 when ,john ,wallis began to publish his important researches at ,oxford4 ,the year #afca is of interest as the date of publication1 not only of ,oughtred's ,clavis1 but also of ,thomas ,harriot's ,artis analyticae praxis4 ,we have no evidence that these two mathematicians ever met4 ,through their writings they did not influence each other4 ,harriot died ten years before the appearance of his magnum opus1 or ten years before the publication of ,oughtred's ,clavis4 ,strangely1 ,oughtred1 who survived ,harriot thirty-nine years1 never mentions him4 ,there is no doubt that1 of the two1 ,harriot was the more original mind1 more capable of penetrating into new fields of research4 ,but he had the misfortune of having a strong competitor in ,ren ,descartes in the development of algebra1 so that no single algebraic achievement stands out strongly and conspicuously as ,harriot's own contribution to algebraic science4 ,as a text to serve as an introduction to algebra1 ,harriot's ,artis analyticae praxis was inferior to ,oughtred's ,clavis4 ,the former was a much larger book1 not as conveniently portable1 compiled after the author's death by others1 and not prepared with the care in the development of the details1 nor with the coherence and unity and the profound pedagogic insight which distinguish the work of ,oughtred4 ,nor was ,harriot's position in life such as to be surrounded by so wide a circle of pupils as was ,oughtred4 ,to be sure1 ,harriot had such followers as ,torporley1 ,william ,lower1 and ,protheroe in ,wales1 but this group is small as compared with ,oughtred's4 ,,oughtred',s ,,pupils ,there was a large number of distinguished men who1 in their youth1 either visited ,oughtred's home and studied under his roof or else read his ,clavis and sought his assistance by correspondence4 ,we permit ,aubrey to enumerate some of these pupils in his own gossipy style3 ,seth ,ward1 ,m4,a41 a fellow of ,sydney ,colledge in ,cambridge 7now bishop of ,sarum71 came to him1 and lived with him halfe a yeare 7and he would not take a farthing for his diet71 and learned all his mathematiques of him4 ,sir ,jonas ,more was with him a good while1 and learn't2 he was but an ordinary logist before4 ,sir ,charles ,scarborough was his scholar2 so ,dr4 ,john ,wallis was his scholar2 so was ,christopher ,wren his scholar1 so was ,mr4 4 4 4 4 ,smethwyck1 ,regiae ,societatis ,socius4 ,one ,mr4 ,austin 7a most ingeniose man7 was his scholar1 and studyed so much that he became mad1 fell a laughing1 and so dyed1 to the great griefe of the old gentleman4 ,mr4 4 4 4 4 ,stokes1 another scholar1 fell mad1 and dream't that the good old gentleman came to him1 and gave him good advice1 and so he recovered1 and is still well4 ,mr4 ,thomas ,henshawe1 ,regiae ,societatis ,socius1 was his scholar 7then a young gentleman74 ,but he did not so much like any as those that tugged and tooke paines to worke out questions4 ,he taught all free4 ,he could not endure to see a scholar write an ill hand2 he taught them all presently to mend their hands4#ef ,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 ,newton1 the other one of the greatest architects of ,england--he would have earned profound gratitude4 ,but the foregoing list embraces nine men1 most of them distinguished in their day4 ,and yet ,aubrey's list is very incomplete4 ,it is easy to more than double it by adding the names of ,william ,forster1 who translated from ,latin into ,english ,oughtred's ,circles of ,proportion2 ,arthur ,haughton1 who brought out the #affj ,oxford edition of the ,circles of ,proportion2 ,robert ,wood1 an educator and politician1 who assisted ,oughtred in the translation of the ,clavis from ,latin into ,english for the edition of #afdg2 ,w4 ,gascoigne1 a man of promise1 who fell in #afdd at ,marston ,moor2 ,john ,twysden1 who was active as a publisher2 ,william ,sudell1 ,n4 ,ewart1 ,richard ,shuttleworth1 ,william ,robinson1 and ,william ,howard1 the son of the ,earl of ,arundel1 for whose instruction ,oughtred originally prepared the manuscript treatise that was published in #afca as the ,clavis mathematicae4 ,nor must we overlook the names of ,lawrence ,rooke 7who 8did admirably well read in ,gresham ,coll4 on the sixth chapt4 of the said book10 the ,clavis72 ,christopher ,brookes 7a maker of mathematical instruments who married a daughter of the famous mathematician72 ,william ,leech and ,william ,brearly 7who with ,robert ,wood 8have been ready and helpfull incouragers of me ,oughtred in this labour0 of preparing the ,english ,clavis of #afdg71 and ,thomas ,wharton1 who studied the ,clavis and assisted in the editing of the edition of #afdg4 ,the devotion of these pupils offers eloquent testimony1 not only of ,oughtred's ability as a mathematician1 but also of his power of drawing young men to him--of his personal magnetism4 ,nor should we omit from the list ,richard ,delamain1 a teacher of mathematics in ,london1 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-dial4 ,delamain became later a tutor in mathematics to ,king ,charles ,i1 and perished in the civil war1 before #afde4 ,,oughtred1 ,,the 8,,todhunter ,,of ,,the ,,seventeenth ,,century0 ,to afford a clearer view of ,oughtred as a teacher and mathematical expositor we quote some passages from various writers and from his correspondence4 ,anthony ,wood#eg 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 algebra3 ,mr4 ,cha4 ,scarborough1 then an ingenious young student and fellow of ,caius ,coll4 in the same university1 was his ,seth ,ward's great acquaintance1 and both being equally students in that faculty and desirous to perfect themselves1 they took a journey to ,mr4 ,will4 ,oughtred living then at ,albury in ,surrey1 to be informed in many things in his ,clavis mathematica which seemed at that time very obscure to them4 ,mr4 ,oughtred treated them with great humanity1 being very much pleased to see such ingenious young men apply themselves to these studies1 and in short time he sent them away well satisfied in their desires4 ,when they returned to ,cambridge1 they afterwards read the ,clav4 ,math4 to their pupils1 which was the first time that book was read in the said university4 ,mr4 ,laur4 ,rook1 a disciple of ,oughtred1 ,i think1 and ,mr4 ,ward's friend1 did admirably well read in ,gresham ,coll4 on the sixth chap4 of the said book1 which obtained him great repute from some and greater from ,mr4 ,ward1 who ever after had an especial favour for him4 ,anthony ,wood makes a similar statement about ,thomas ,henshaw3 ,while he remained in that coll4 ,university ,college1 ,oxford which was five years 4 4 4 4 he made an excursion for about #i months to the famous mathematician ,will4 ,oughtred parson of ,aldbury in ,surrey1 by whom he was initiated in the study of mathematics1 and afterwards retiring to his coll4 for a time1 he at length went to ,london1 was entered a student in the ,middle ,temple4#eh ,extracts from letters of ,w4 ,gascoigne to ,oughtred1 of the years #afdj and #afda1 throw some light upon mathematical teaching of the time3 ,amongst the mathematical rarities these times have afforded1 there are none of that small number ,i 7a late intruder into these studies7 have yet viewed1 which so fully demonstrates their authors' great abilities as your ,clavis1 not richer in augmentations1 than valuable for contraction2 4 4 4 4 ,your belief that there is in all inventions aliquid divinum1 an infusion beyond human cogitations1 ,i am confident will appear notably strengthened1 if you please to afford this truth belief1 that ,i entered upon these studies accidentally after ,i betook myself to the country1 having never had so much aid as to be taught addition1 nor the discourse of an artist 7having left both ,oxford and ,london before ,i knew what any proposition in geometry meant7 to inform me what were the best authors4#ei ,the following extracts from two letters by ,w4 ,robinson1 written before the appearance of the #afdg ,english edition of the ,clavis1 express the feeling of many readers of the ,clavis on its extreme conciseness and brevity of explanation3 ,i shall long exceedingly till ,i see your ,clavis turned into a pick-lock2 and ,i beseech you enlarge it1 and explain it what you can1 for we shall not need to fear either tautology or superfluity2 you are naturally concise1 and your clear judgment makes you both methodical and pithy2 and your analytical way is indeed the only way4 4 4 4 4 ,i will once again earnestly entreat you1 that you be rather diffuse in the setting forth of your ,english mathematical ,clavis1 than concise1 considering that the wisest of men noted of old1 and said stultorum infinitus est numerus1 these arts cannot be made too easy1 they are so abstruse of themselves1 and men either so lazy or dull1 that their fastidious wits take a loathing at the very entrance of these studies1 unless it be sweetened on with plainness and facility4 ,brevity may well argue a learned author1 that without any excess or redundance1 either of matter or words1 can give the very substance and essence of the thing treated of2 but it seldom makes a learned scholar2 and if one be capable1 twenty are not2 and if the master sum up in brief the pith of his own long labours and travails1 it is not easy to imagine that scholars can with less labour than it cost their masters dive into the depths thereof4#fj ,here is the judgment of another of ,oughtred's friends3 4 4 4 4 with the character ,i received from your and my noble friend ,sir ,charles ,cavendish1 then at ,paris1 of your second edition of the same piece1 made me at my return into ,england speedily to get1 and diligently peruse the same4 ,neither truly did ,i find my expectation deceived2 having with admiration often considered how it was possible 7even in the hardest things of geometry7 to deliver so much matter in so few words1 yet with such demonstrative clearness and perspicuity3 and hath often put me in mind of learned ,mersennus his judgment 7since dead7 of it1 that there was more matter comprehended in that little book than in ,diophantus1 and all the ancients4 4 4 4 4#fa ,oughtred's own feeling was against diffuseness in textbook writing4 ,in his revisions of his ,clavis the original character of that book was not altered4 ,in his reply to ,w4 ,robinson1 ,oughtred said3 4 4 4 4 ,but my art for all such mathematical inventions ,i have set down in my ,clavis ,mathematica1 which therefore in my title ,i say is tum logisticae cum analyticae adeoque totius mathematicae quasi clavis1 which if any one of a mathematical genius will carefully study1 7and indeed it must be carefully studied17 he will not admire others1 but himself do wonders4 ,but ,i 7such is my tenuity7 have enough fungi vice cotis1 acutum reddere quae ferrum valet1 exsors ipsa secandi1 or like the touchstone1 which being but a stone1 base and little worth1 can shew the excellence and riches of gold4#fb ,john ,wallis held ,oughtred's ,clavis in high regard4 ,when in correspondence with ,john ,collins concerning plans for a new edition1 ,wallis wrote in #afff-fg1 six years after the death of ,oughtred3 4 4 4 4 ,but for the goodness of the book in itself1 it is that 7,i confess7 which ,i look upon as a very good book1 and which doth in as little room deliver as much of the fundamental and useful part of geometry 7as well as of arithmetic and algebra7 as any book ,i know2 and why it should not be now acceptable ,i do not see4 ,it is true1 that as in other things so in mathematics1 fashions will daily alter1 and that which ,mr4 ,oughtred designed by great letters may be now by others be designed by small2 but a mathematician will1 with the same ease and advantage1 understand ,ac1 and a or aaa4 4 4 4 4 ,and the like ,i judge of ,mr4 ,oughtred's ,clavis1 which ,i look upon 7as those pieces of ,vieta who first went in that way7 as lasting books and classic authors in this kind2 to which1 notwithstanding1 every day may make new additions4 4 4 4 4 ,but ,i confess1 as to my own judgment1 ,i am not for making the book bigger1 because it is contrary to the design of it1 being intended for a manual or contract2 whereas comments1 by enlarging it1 do rather destroy it4 4 4 4 4 ,but it was by him intended1 in a small epitome1 to give the substance of what is by others delivered in larger volumes4 4 4 4 4#fc ,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 appearance1 over fifty years after the first publication of the ,clavis1 of a booklet by ,gilbert ,clark1 entitled ,oughtredus ,explicatus1 ,london1 #afhb4 ,a review of this appeared in the ,acta ,eruditorum 7,leipzig1 #afhd71 on p4 #afh1 wherein ,oughtred is named 8clarissimus ,angliae mathematicus40 ,john ,collins wrote ,wallis in #afff-fg that ,clark1 8who lives with ,sir ,justinian ,isham1 within seven miles of ,northampton1 4 4 4 4 intimates he wrote a comment on the ,clavis1 which lay long in the hands of a printer1 by whom he was abused1 meaning ,leybourne40#fd ,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 before4 ,in those days the condition of the book trade in ,england must have been somewhat extraordinary4 ,dr4 ,j4 ,w4 ,l4 ,glaisher throws some light upon this subject4#fe ,he found in the ,calendar of ,state ,papers1 ,domestic ,series1 #afcg1 a petition to ,archbishop ,laud in which it is set forth that when ,hooganhuysen1 a ,dutchman1 8heretofore complained of in the ,high ,commission for importing books printed beyond the seas10 had been bound 8not to bring in any more10 one ,vlacq 7the computer and publisher of logarithmic tables7 8kept up the same agency and sold books in his stead4 4 4 4 4 ,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 ,chamber40 ,judgment was passed that1 8,considering the ill-consequence and scandal that would arise by strangers importing and venting in this kingdom books printed beyond the seas10 certain importations be prohibited1 and seized if brought over4 ,this want of easy intercommunication of results of scientific research in ,oughtred's time is revealed in the following letter1 written by ,oughtred to ,robert ,keylway1 in #afde3 ,i speak this the rather1 and am induced to a better confidence of your performance1 by reason of a geometric-analytical art or practice found out by one ,cavalieri1 an ,italian1 of which about three years since ,i received information by a letter from ,paris1 wherein was praelibated only a small taste thereof1 yet so that ,i divine great enlargement of the bounds of the mathematical empire will ensue4 ,i was then very desirous to see the author's own book while my spirits were more free and lightsome1 but ,i could not get it in ,france4 ,since1 being more stept into years1 daunted and broken with the sufferings of these disastrous times1 ,i must content myself to keep home1 and not put out to any foreign discoveries4#ff ,it was in #afee1 when ,oughtred was about eighty years old1 that ,john ,wallis1 the great forerunner of ,newton in ,great ,britain1 began to publish his great researches on the arithmetic of infinites4 ,oughtred rejoiced over the achievements of his former pupil4 ,in #afee1 ,oughtred wrote ,john ,wallis as follows3 ,i have with unspeakable delight1 so far as my necessary businesses1 the infirmness of my health1 and the greatness of my age 7approaching now to an end7 would permit1 perused your most learned papers1 of several choice arguments1 which you sent me3 wherein ,i do first with thankfulness acknowledge to ,god1 the ,father of lights1 the great light he hath given you2 and next ,i congratulate you1 even with admiration1 the clearness and perspicacity of your understanding and genius1 who have not only gone1 but also opened a way into these profoundest mysteries of art1 unknown and not thought of by the ancients4 ,with which your mysterious inventions ,i am the more affected1 because full twenty years ago1 the learned patron of learning1 ,sir ,charles ,cavendish1 shewed me a paper written1 wherein were some few excellent new theorems1 wrought by the way1 as ,i suppose1 of ,cavalieri1 which ,i wrought over again more agreeably to my way4 ,the paper1 wherein ,i wrought it1 ,i shewed to many1 whereof some took copies1 but my own ,i cannot find4 ,i mention it for this1 because ,i saw therein a light breaking out for the discovery of wonders to be revealed to mankind1 in this last age of the world3 which light ,i did salute as afar off1 and now at a nearer distance embrace in your prosperous beginnings4 ,sir1 that you are pleased to mention my name in your never dying papers1 that is your noble favour to me1 who can add nothing to your glory1 but only my applause4 4 4 4 4#fg ,the last sentence has reference to ,wallis' appreciative and eulogistic reference to ,oughtred in the preface4 ,it is of interest to secure the opinion of later ,english writers who knew ,oughtred only through his books4 ,john ,locke wrote in his journal under the date1 ,june #bd1 #afha1 8the best algebra yet extant is ,outred's40#fh ,john ,collins1 who is known in the history of mathematics chiefly through his very extensive correspondence with nearly all mathematicians of his day1 was inclined to be more critical4 ,he wrote ,wallis about #affg3 ,it was not my intent to disparage the author1 though ,i know many that did lightly esteem him when living1 some whereof are at rest1 as ,mr4 ,foster and ,mr4 ,gibson4 4 4 4 4 ,you grant the author is brief1 and therefore obscure1 and ,i say it is but a collection1 which1 if himself knew1 he had done well to have quoted his authors1 whereto the reader might have repaired4 ,you do not like those words of ,vieta in his theorems1 ex adjunctione plano solidi1 plus quadrato quadrati1 etc41 and think ,mr4 ,oughtred the first that abridged those expressions by symbols2 but ,i dissent1 and tell you 'twas done before by ,cataldus1 ,geysius1 and ,camillus ,gloriosus1#fi who in his first decade of exercises1 7not the first tract17 printed at ,naples in #afbg1 which was four years before the first edition of the ,clavis1 proposeth this equation just as ,i here give it you1 viz4 #a;ccc#afqcc#daqqc-#bcjd;cc-#ahcfdqc-#ac cjjjqq-#edeje;c#cgbhq#hjfd ,n aequatur #dfjh1 finds ,n or a root of it to be #bd1 and composeth the whole out of it for proof1 just in ,mr4 ,oughtred's symbols and method4 ,cataldus on ,vieta came out fifteen years before1 and ,i cannot quote that1 as not having it by me4 4 4 4 4 ,and as for ,mr4 ,oughtred's method of symbols1 this ,i say to it2 it may be proper for you as a commentator to follow it1 but divers ,i know1 men of inferior rank that have good skill in algebra1 that neither use nor approve it4 4 4 4 4 ,is not ,a sooner wrote than ,aqc8 ,let ,a be #b1 the cube of #b is #h1 which squared is #fd3 one of the questions between ,maghet ,grisio and ,gloriosus is whether #fd;,acc or ,aqc4 ,the ,cartesian method tells you it is ,a1 and decides the doubt4 4 4 4 4#gj ,there is some ground for the criticisms passed by ,collins4 ,to be sure1 the first edition of the ,clavis is dated #afca--six years before ,descartes suggested the exponential notation which came to be adopted as the symbolism in our modern algebra4 ,but the second edition of the ,clavis1 #afdg1 appeared ten years after ,descartes' innovation4 ,had ,oughtred seen fit to adopt the new exponential notation in #afdg1 the step would have been epoch-making in the teaching of algebra in ,england4 ,we have seen no indication that ,oughtred was familiar with ,descartes' ,gomtrie of #afcg4 ,the year preceding ,oughtred's death ,mr4 ,john ,twysden expressed himself as follows in the preface to his ,miscellanies3 ,it remains that ,i should adde something touching the beginning1 and use of these ,sciences4 4 4 4 4 ,i shall only1 to their honours1 name some of our own ,nation yet living1 who have happily laboured upon both stages4 ,that succeeding ages may understand that in this of ours1 there yet remained some who were neither ignorant of these ,arts1 as if they had held them vain1 nor condemn them as superfluous4 ,amongst them all let ,mr4 ,william ,oughtred1 of ,aeton1 be named in the first place1 a ,person of venerable grey haires1 and exemplary piety1 who indeed exceeds all praise we can bestow upon him4 ,who by an easie method1 and admirable ,key1 hath unlocked the hidden things of geometry4 ,who by an accurate ,trigonometry and furniture of ,instruments1 hath inriched1 as well geometry1 as ,astronomy4 ,let ,d4 ,john ,wallis1 and ,d4 ,seth ,ward1 succeed in the next place1 both famous ,persons1 and ,doctors in ,divinity1 the one of geometry1 the other of astronomy1 ,savilian ,professors in the ,university of ,oxford4#ga ,the astronomer ,edmund ,halley1 in his preface to the #afid ,english edition of the ,clavis1 speaks of this book as one of 8so established a reputation1 that it were needless to say anything thereof10 though 8the concise ,brevity of the author is such1 as in many places to need ,explication1 to render it ,intelligible to the less knowing ,mathematical matters40 ,in closing this part of our monograph1 we quote the testimony of ,robert ,boyle1 the experimental physicist1 as given ,may #h1 #afdg1 in a letter to ,mr4 ,hartlib3 ,the ,englishing of1 and additions to ,oughtred's ,clavis mathematica does much content me1 ,i having formerly spent much study on the original of that algebra1 which ,i have long since esteemed a much more instructive way of logic1 than that of ,aristotle4#gb ,,was ,,descartes ,,indebted ,,to ,,oughtred8 ,this question first arose in the seventeenth century1 when ,john ,wallis1 of ,oxford1 in his ,algebra 7the ,english edition of #afhe1 and more particularly the ,latin edition of #afic71 raised the issue of ,descartes' indebtedness to the ,english scientists1 ,thomas ,harriot and ,william ,oughtred4 ,in discussing matters of priority between ,harriot and ,descartes1 relating to the theory of equations1 ,wallis is generally held to have shown marked partiality to ,harriot4 ,less attention has been given by historians of mathematics to ,descartes' indebtedness to ,oughtred4 ,yet this question is of importance in tracing ,oughtred's influence upon his time4 ,on ,january #h1 #afhh-hi1 ,samuel ,morland addressed a letter of inquiry to ,john ,wallis1 containing a passage which we translate from the ,latin3 ,some time ago ,i read in the elegant and truly precious book that you have written on ,algebra1 about ,descartes1 this philosopher so extolled above all for having arrived at a very perfect system by his own powers1 without the aid of others1 this ,descartes1 ,i say1 who has received in geometry very great light from our ,oughtred and our ,harriot1 and has followed their track though he carefully suppressed their names4 ,i stated this in a conversation with a professor in ,utrecht 7where ,i reside at present74 ,he requested me to indicate to him the page-numbers in the two authors which justified this accusation4 ,i admitted that ,i could not do so4 ,the ,gomtrie of ,descartes is not sufficiently familiar to me1 although with ,oughtred ,i am fairly familiar4 ,i pray you therefore that you will assume this burden4 ,give me at least those references to passages of the two authors from the comparison of which the plagiarism by ,descartes is the most striking4#gc ,following ,morland's letter in the ,de algebra tractatus1 is printed ,wallis' reply1 dated ,march #ab1 #afhh 78,stilo ,angliae071 which is1 in part1 as follows3 ,i nowhere give him the name of a plagiarist2 ,i would not appear so impolite4 ,however this ,i say1 the major part of his algebra 7if not all7 is found before him in other authors 7notably in our ,harriot7 whom he does not designate by name4 ,that algebra may be applied to geometry1 and that it is in fact so applied1 is nothing new4 ,passing the ancients in silence1 we state that this has been done by ,vieta1 ,ghetaldi1 ,oughtred and others1 before ,descartes4 ,they have resolved by algebra and specious arithmetic literal arithmetic many geometrical problems4 4 4 4 4 ,but the question is not as to application of algebra to geometry 7a thing quite old71 but of the ,cartesian algebra considered by itself4 ,wallis then indicates in the #afei edition of ,descartes' ,gomtrie where the subjects treated on the first six pages are found in the writings of earlier algebraists1 particularly of ,harriot and ,oughtred4 ,for example1 what is found on the first page of ,descartes1 relating to addition1 subtraction1 multiplication1 division1 and root extraction1 is declared by ,wallis to be drawn from ,vieta1 ,ghetaldi1 and ,oughtred4 ,it is true that ,descartes makes no mention of modern writers1 except once of ,cardan4 ,but it was not the purpose of ,descartes to write a history of algebra4 ,to be sure1 references to such of his immediate predecessors as he had read would not have been out of place4 ,nevertheless1 ,wallis fails to show that ,descartes made illegitimate use of anything he may have seen in ,harriot or ,oughtred4 ,the first inquiry to be made is1 ,did ,descartes possess copies of the books of ,harriot and ,oughtred8 ,it is only in recent time that this question has been answered as to ,harriot4 ,as to ,oughtred1 it is still unanswered4 ,it is now known that ,descartes had seen ,harriot's ,artis analyticae praxis 7#afca74 ,descartes wrote a letter to ,constantin ,huygens in which he states that he is sending ,harriot's book4#gd ,an able discussion of the question1 what effect1 if any1 ,oughtred's ,clavis mathematicae of #afca had upon ,descartes'#ge ,gomtrie of #afcg1 is given by ,h4 ,bosmans in a recent article4 ,according to ,bosmans no evidence has been found that ,descartes possessed a copy of ,oughtred's book1 or that he had examined it4 ,bosmans believes nevertheless that ,descartes was influenced by the ,clavis1 either directly or indirectly4 ,he says3 ,if ,descartes did not read it carefully1 which is not proved1 he was none the less well informed with regard to it4 ,no one denies his intimate knowledge of the intellectual movement of his time4 ,the ,clavis mathematica enjoyed a rapid success4 ,it is impossible that1 at least indirectly1 he did not know the more original ideas which it contained4 ,far from belittling ,descartes1 as ,i much desire to repeat1 this rather makes him the greater4#gf ,we ourselves would hardly go as far as does ,bosmans4 ,unless ,descartes actually examined a copy of ,oughtred it is not likely that he was influenced by ,oughtred in appreciable degree4 ,book reviews were quite unknown in those days4 ,no evidence has yet been adduced to show that ,descartes obtained a knowledge of ,oughtred by correspondence4 ,a most striking feature about ,oughtred's ,clavis is its notation4 ,no trace of the ,englishman's symbolism has been pointed out in ,descartes' ,gomtrie of #afcg4 ,only six years intervened between the publication of the ,clavis and the ,gomtrie4 ,it took longer than this period for the ,clavis to show evidence of its influence upon mathematical books published in ,england2 it is not probable that abroad the contact was more immediate than at home4 ,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 him2 but that it took several decennia for his influence fully to develop4 ,,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 symbolism4 ,this study indicates that the adoption was not immediate4 ,the earliest use that we have been able to find of ,oughtred's notation for proportion1 ,a4,b33,c4,d1 occurs nineteen years after the ,clavis mathematicae of #afca4 ,in #afej ,john ,kersey brought out in ,london an edition of ,edmund ,wingates' ,arithmetique made easie1 in which this notation is used4 ,after this date publications employing it became frequent1 some of them being the productions of pupils of ,oughtred4 ,we have seen it in ,vincent ,wing 7#afea71gg ,seth ,ward 7#afec71gh ,john ,wallis 7#afee71gi in 8,r4 ,b410 a schoolmaster in ,suffolk1#hj ,samuel ,foster 7#afei71ha ,jonas ,moore 7#affj71hb and ,isaac ,barrow 7#afeg74hc ,in the latter part of the seventeenth century ,oughtred's notation1 ,a4,b33,c4,d1 became the prevalent1 though not universal1 notation in ,great ,britain4 ,a tremendous impetus to their adoption was given by ,seth ,ward1 ,isaac ,barrow1 and particularly by ,john ,wallis1 who was rising to international eminence as a mathematician4 ,in ,france we have noticed ,oughtred's notation for proportion in ,franciscus ,dulaurens 7#affg71hd ,j4 ,prestet 7#afge71he ,r4 ,p4 ,bernard ,lamy 7#afhd71hf ,ozanam 7#afia71hg ,de l',hospital 7#afif71hh ,r4 ,p4 ,petro ,nicolas 7#afig74hi ,in the ,netherlands we have noticed it in ,r4 ,p4 ,bernard ,lamy 7#afhj71ij and in an anonymous work of #afij4ia ,in ,german and ,italian works of the seventeenth century we have not seen ,oughtred's notation for proportion4 ,in ,england a modified notation soon sprang up in which ratio was indicated by two dots instead of a single dot1 thus ,a3,b33,c3,d4 ,the reason for the change lies probably in the inclination to use the single dot to designate decimal fractions4 ,w4 ,w4 ,beman pointed out that this modified symbolism 737 for ratio is found as early as #afeg in the end of the trigonometric and logarithmic tables that were bound with ,oughtred's ,trigonometria4#ib ,it is not probable1 however1 that this notation was used by ,oughtred himself4 ,the ,trigonometria proper has ,oughtred's ,a4,b33,c4,d throughout4 ,moreover1 in the ,english edition of this trigonometry1 which appeared the same year1 #afeg1 but subsequent to the ,latin edition1 the passages which contained the colon as the symbol for ratio1 when not omitted1 are recast1 and the regular ,oughtredian notation is introduced4 ,in ,oughtred's posthumous work1 ,opuscula mathematica hactenus inedita1 #afgg1 the colon appears quite often but is most likely due to the editor of the book4 ,we have noticed that the notation ,a3,b33,c3,d antedates the year #afeg4 ,vincent ,wing1 the astronomer1 published in #afea in ,london the ,harmonicon coeleste1 in which is found not only ,oughtred's notation ,a4,b33,c4,d but also the modified form of it given above4 ,the two are used interchangeably4 ,his later works1 the ,logistica astronomica 7#afef71 ,doctrina spherica 7#afee71 and ,doctrina theorica1 published in one volume in ,london1 all use the symbols ,a3,b33,c3,d exclusively4 ,the author of a book entitled1 ,an ,idea of ,arithmetick at first designed for the use of the ,free ,schoole at ,thurlow in ,suffolk 4 4 4 4 by ,r4 ,b41 ,schoolmaster there1 ,london1 #afee1 writes ,a3a33,c3c1 though part of the time he uses ,oughtred's unmodified notation4 ,we can best indicate the trend in ,england by indicating the authors of the seventeenth century whom we have found using the notation ,a3,b33,c3,d and the authors of the eighteenth century whom we have found using ,a4,b33,c4,d4 ,the former notation was the less common during the seventeenth but the more common during the eighteenth century4 ,we have observed the symbols ,a3,b33,c3,d 7besides the authors already named7 in ,john ,collins 7#afei71ic ,james ,gregory 7#affc71id ,christopher ,wren 7#affh-fi71ie ,william ,leybourn 7#afgc71if ,william ,sanders 7#afhf71ig ,john ,hawkins 7#afhd71ih ,joseph ,raphson 7#afig71ii ,e4 ,wells 7#afih71ajj and ,john ,ward 7#afih74aja ,of ,english eighteenth-century authors the following still clung to the notation ,a4,b33,c4,d3 ,john ,harris' translation of ,f4 ,ignatius ,gaston ,pardies 7#agja71ajb ,george ,shelley 7#agjd71ajc ,sam ,cobb 7#agji71ajd ,j4 ,collins in ,commercium ,epistolicum 7#agab71 ,john ,craig 7#agah71aje ,jo4 ,wilson 7#agbd74ajf ,the latest use of ,a4,b33,c4,d which has come to our notice is in the translation of the ,analytical ,institutions of ,maria ,g4 ,agnesi1 made by ,john ,colson sometime before #agfj1 but which was not published until #ahja4 ,during the seventeenth century the notation ,a3,b33,c3,d acquired almost complete ascendancy in ,england4 ,in ,france ,oughtred's unmodified notation ,a4,b33,c4,d1 having been adopted later1 was also discarded later than in ,england4 ,an approximate idea of the situation appears from the following data4 ,the notation ,a4,b33,c4,d was used by ,m4 ,carr 7#agjj71ajg ,m4 ,guisne 7#agje71ajh ,m4 de ,fontenelle 7#agbg71aji ,m4 ,varignon 7#agbe71aaj ,m4 ,robillard 7#agec71aaa ,m4 ,sebastien le ,clerc 7#agfd71aab ,clairaut 7#agca71aac ,m4 ,l',hospital 7#agha74aad ,in ,italy ,oughtred's modified notation a1 b33c1 d was used by ,maria ,g4 ,agnesi in her ,instituzioni analitiche1 ,milano1 #agdh4 ,the notation a3b33c3d found entrance the latter part of the eighteenth century4 ,in ,germany the symbolism a3bc3d1 suggested by ,leibniz1 found wider acceptance4#aae ,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 felt1 that his symbol for ratio a4b was temporarily adopted in ,england and ,france but gave way in the eighteenth century to the symbol a3b1 that ,oughtred's symbol for proportion 33 found almost universal adoption in ,england and ,france and was widely used in ,italy1 the ,netherlands1 the ,united ,states1 and to some extent in ,germany2 it has survived to the present time but is now being gradually displaced by the sign of equality 4 ,oughtred's notation to express aggregation of terms has received little attention from historians but is nevertheless interesting4 ,his books1 as well as those of ,john ,wallis1 are full of parentheses but they are not used as symbols of aggregation in algebra2 they are simply marks of punctuation for parenthetical clauses4 ,we have seen that ,oughtred writes 7ab7 and 7ab7 thus1 ,q3ab31 3ab31 or ,q3ab1 3ab1 using on rarer occasions a single dot in place of the colon4 ,this notation did not originate with ,oughtred1 but1 in slightly modified form1 occurs in writings from the ,netherlands4 ,in #afjc ,c4 ,dibvadii in geometriam ,evclidis demonstratio numeralis1 ,leyden1 contains many expressions of this sort1 #acfbjdh1 signifying 7#acfbjdh74 ,the dot is used to indicate that the root of the binomial 7not of #acf alone7 is called for4 ,this notation is used extensively in ,ludolphi ,cevlen de circulo1 ,leyden1 #afai1 and in ,willebrordi ,snellii ,de circuli dimensione1 ,leyden1 #afba4 ,in place of the single dot ,oughtred used the colon 7371 probably to avoid confusion with his notation for ratio4 ,to avoid further possibility of uncertainty he usually placed the colon both before and after the algebraic expression under aggregation4 ,this notation was adopted by ,john ,wallis and ,isaac ,barrow4 ,it is found in the writings of ,descartes4 ,together with ,vieta's horizontal bar1 placed over two or more terms1 it constituted the means used almost universally for denoting aggregation of terms in algebra4 ,before ,oughtred the use of parentheses had been suggested by ,clavius#aaf and ,girard4#aag ,the latter wrote1 for instance1 7#bc74 ,while parentheses never became popular in algebra before the time of ,leibniz and the ,bernoullis they were by no means lost sight of4 ,we are able to point to the following authors who made use of them3 ,i4 ,errard de ,bar-le-,duc 7#afai71aah ,jacobo de ,billy 7#afdc71aai one of whose books containing this notation was translated into ,english1 and also the posthumous works of ,samuel ,foster4#abj ,j4 ,w4 ,l4 ,glaisher points out that parentheses were used by ,norwood in his ,trigonometrie 7#afca71 p4 #cj4aba ,the symbol for the arithmetical difference between two numbers1 1 is usually attributed to ,john ,wallis1 but it occurs in ,oughtred's ,clavis mathematicae of #afeb1 in the tract on ,elementi decimi ,euclidis declaratio1 at an earlier date than in any of ,wallis' books4 ,as ,wallis assisted in putting this edition through the press it is possible1 though not probable1 that the symbol was inserted by him4 ,were the symbol ,wallis'1 ,oughtred would doubtless have referred to its origin in the preface4 ,during the eighteenth century the symbol found its way into foreign texts even in far-off ,italy4#abb ,it is one of three symbols presumably invented by ,oughtred and which are still used at the present time4 ,the others are and 334 ,the curious and ill-chosen symbols1 for 8greater than10 and for 8less than10 were certain to succumb in their struggle for existence against ,harriot's admirably chosen and 4 ,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 century4 ,considerable confusion has existed among algebraists and also among historians as to what ,oughtred's symbols really were4 ,particularly is this true of the sign for 8less than0 which is frequently written 4 ,oughtred's symbols1 or these symbols turned about in some way1 have been used by ,seth ,ward1#abc ,john ,wallis1#abd ,isaac ,barrow1#abe ,john ,kersey1#abf ,e4 ,wells1#abg ,john ,hawkins1#abh ,tho4 ,baker1#abi ,richard ,sault1#acj ,richard ,rawlinson1#aca ,franciscus ,dulaurens1#acb ,james ,milnes1#acc ,george ,cheyne1#acd ,john ,craig1#ace ,jo4 ,wilson1#acf and ,j4 ,collins4#acg ,general acceptance has been accorded to ,oughtred's symbol 4 ,the first printed appearance of this symbol for multiplication in #afah in the form of the letter x hardly explains its real origin4 ,the author of the 8,appendix0 7be he ,oughtred or someone else7 may not have used the letter x at all1 but may have written the cross 1 called the ,st4 ,andrew's cross1 while the printer1 in the absence of any type accurately representing that cross1 may have substituted the letter x in its place4 ,the hypothesis that the symbol of multiplication owes its origin to the old habit of using directed bars to indicate that two numbers are to be combined1 as for instance in the multiplication of #bc and #cd1 thus1 #b #c _/ x _/ #c #d ------- #g #h #b has been advanced by two writers1 ,c4 ,le ,paige#ach and ,gravelaar4#aci ,bosmans is more inclined to the belief that ,oughtred adopted the symbol somewhat arbitrarily1 much as he did the numerous symbols in his ,elementi decimi ,euclidis declaratio4#adj ,le ,paige's and ,gravelaar's theory finds some support in the fact that the cross 1 without the two additional vertical lines shown above1 occurs in a commentary published by ,oswald ,schreshensuchs#ada in #aeea1 where the sign is written between two factors placed one above the other4 ,,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 teaching4 ,nevertheless1 he had very pronounced and clear-cut ideas on the subject4 ,that a man who was not a teacher by profession should have mature views on teaching is most interesting4 ,we gather his ideas from the quality of the books he published1 from his prefaces1 and from passages in his controversial writing against ,delamain4 ,as we proceed to give quotations unfolding ,oughtred's views1 we shall observe that three points receive special emphasis3 7#a7 an appeal to the eye through suitable symbolism2 7#b7 emphasis upon rigorous thinking2 7#c7 the postponement of the use of mathematical instruments until after the logical foundations of a subject have been thoroughly mastered4 ,the importance of these tenets is immensely reinforced by the conditions of the hour4 ,this voice from the past speaks wisdom to specialists of today4 ,recent methods of determining educational values and the modern cult of utilitarianism have led some experts to extraordinary conclusions4 ,laboratory methods of testing1 by the narrowness of their range1 often mislead4 ,thus far they have been inferior to the word of a man of experience1 insight1 and conviction4 ,,mathematics1 8,a ,,science ,,of ,,the ,,eye0 ,oughtred was a great admirer of the ,greek mathematicians--,euclid1 ,archimedes1 ,apollonius of ,perga1 ,diophantus4 ,but in reading their works he experienced keenly what many modern readers have felt1 namely1 that the almost total absence of mathematical symbols renders their writings unnecessarily difficult to read4 ,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 sentences4 ,a striking illustration of the importance of symbolism is afforded by the history of the formula ixlog7cos xi sin x74 ,it was given in ,roger ,cotes' ,harmonia mensurarum1 #agbb1 not in symbols1 but expressed in rhetorical form1 destitute of special aids to the eye4 ,the result was that the theorem remained in the book undetected for #ahe years and was meanwhile rediscovered by others4 ,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 symbols4 ,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 proofs1 expressed in the symbolic language of algebra4 ,in the preface of his ,clavis of #afca and of #afdg he says3 ,wherefore1 that ,i might more clearly behold the things themselves1 ,i uncasing the ,propositions and ,demonstrations out of their covert of words1 designed them in notes and species appearing to the very eye4 ,after that by comparing the divers affections of ,theorems1 inequality1 proportion1 affinity1 and dependence1 ,i tryed to educe new out of them4 ,it was this motive which led him to introduce the many abbreviations in algebra and trigonometry to which reference has been made in previous pages4 ,the pedagogical experience of recent centuries has indorsed ,oughtred's view1 provided of course that the pupil is carefully taught the exact meaning of the symbols4 ,there have been and there still are those who oppose the intensive use of symbolism4 ,in our day the new symbolism for all mathematics1 suggested by the school of ,peano in ,italy1 can hardly be said to be received with enthusiasm4 ,in ,oughtred's day symbolism was not yet the fashion4 ,to be convinced of this fact one need only open a book of ,edmund ,gunter1 with whom ,oughtred came in contact in his youth1 or consult the ,principia of ,sir ,isaac ,newton1 who flourished after ,oughtred4 ,the mathematical works of ,gunter and ,newton1 particularly the former1 are surprisingly destitute of mathematical symbols4 ,the philosopher ,hobbes1 in a controversy with ,john ,wallis1 criticized the latter for that 8,scab of ,symbols10 whereupon ,wallis replied3 ,i wonder how you durst touch ,m4 ,oughtred for fear of catching the ,scab4 ,for1 doubtlesse1 his book is as much covered over with the ,scab of ,symbols1 as any of mine4 4 4 4 4 ,as for my ,treatise of ,conick ,sections1 you say1 it is covered over with the ,scab of ,symbols1 that you had not the patience to examine whether it is well or ill demonstrated4#adb ,oughtred maintained his view of the importance of symbols on many different occasions4 ,thus1 in his ,circles of ,proportion1 #afcb1 p4 #bj3 ,this manner of setting downe ,theoremes1 whether they be ,proportions1 or ,equations1 by ,symboles or notes of words1 is most excellent1 artificiall1 and doctrinall4 ,wherefore ,i earnestly exhort every one1 that desireth though but to looke into these noble ,sciences ,mathematicall1 to accustome themselves unto it3 and indeede it is easie1 being most agreeable to reason1 yea even to sence4 ,and out of this working may many singular consectaries be drawne3 which without this would1 it may be1 for ever lye hid4 ,,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 #afdg ,clavis3 4 4 4 4 ,which ,treatise being not written in the usuall synthetical manner1 nor with verbous expressions1 but in the inventive way of ,analitice1 and with symboles or notes of things instead of words1 seemed unto many very hard2 though indeed it was but their owne diffidence1 being scared by the newnesse of the delivery2 and not any difficulty in the thing it selfe4 ,for this specious and symbolicall manner1 neither racketh the memory with multiplicity of words1 nor chargeth the phantasie with comparing and laying things together2 but plainly presenteth to the eye the whole course and processe of every operation and argumentation4 ,now my scope and intent in the first ,edition of that my ,key was1 and in this ,new ,filing1 or rather forging of it1 is1 to reach out to the ingenious lovers of these ,sciences1 as it were ,ariadnes thread1 to guide them through the intricate ,labyrinth of these studies1 and to direct them for the more easie and full understanding of the best and antientest ,authors4 4 4 4 4 ,that they may not only learn their propositions1 which is the highest point of ,art that most ,students aime at2 but also may perceive with what solertiousnesse1 by what engines of aequations1 ,interpretations1 ,comparations1 ,reductions1 and ,disquisitions1 those antient ,worthies have beautified1 enlarged1 and first found out this most excellent ,science4 4 4 4 4 ,lastly1 by framing like questions problematically1 and in a way of ,analysis1 as if they were already done1 resolving them into their principles1 ,i sought out reasons and means whereby they might be effected4 ,and by this course of practice1 not without long time1 and much industry1 ,i found out this way for the helpe and facilitation of ,art4 ,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 ,delamain4 ,in that preface ,william ,forster quotes the reply of ,oughtred to the question how he 7,oughtred7 had for so many years concealed his invention of the slide rule from himself 7,forster7 whom he had taught so many other things4 ,the reply was3 ,that the true way of ,art is not by ,instruments1 but by ,demonstration3 and that it is a preposterous course of vulgar ,teachers1 to begin with ,instruments1 and not with the ,sciences1 and so in-stead of ,artists1 to make their ,scholers only doers of tricks1 and as it were ,iuglers3 to the despite of ,art1 losse of previous time1 and betraying of willing and industrious wits1 vnto ignorance1 and idlenesse4 ,that the vse of ,instruments is indeed excellent1 if a man be an ,artist3 but contemptible1 being set and opposed to ,art4 ,and lastly1 that he meant to commend to me1 the skill of ,instruments1 but first he would haue me well instructed in the ,sciences40 ,delamain took a different view1 arguing that instruments might very well be placed in the hands of pupils from the start4 ,at the time of this controversy ,delamain supported himself by teaching mathematics in ,london and he advertised his ability to give instruction in mathematics1 including the use of instruments4 ,delamain brought the charge against ,oughtred of unjustly calling 8many of the ,british ,nobility and ,gentry doers of trickes and juglers40 ,to this ,oughtred replies3 ,as ,i did to ,delamain and to some others1 so ,i did to ,william ,forster3 ,i freely gave him my helpe and instruction in these faculties3 only this was the difference1 ,i had the very first moulding 7as ,i may say7 of this latter3 ,but ,delamain was already corrupted with doring upon ,instruments1 and quite lost from ever being made an ,artist3 ,i suffered not ,william ,forster for some time so much as speake of any ,instrument1 except only the ,globe it selfe2 and to explicate1 and worke the questions of the ,sphaere1 by the way of the ,analemma3 which also himselfe did describe for the present occasion4 ,and this my restraint from such pleasing avocations1 and holding him to the strictnesse of percept1 brought forth this fruit1 that in short time1 even by his owne skill1 he could not onely use any ,instrument he should see1 but also was able to delineate the like1 and devise others4#adc ,as representing ,delamain's views1 we make the following selection from his ,grammelogia 7,london1 about #afcc71 the part near the end of the book and bearing the title1 8,in the behalfe of vulgar ,teachers and others10 where ,delamain refers to ,oughtred's charge that the scholars of 8vulgar0 teachers are 8doers of tricks1 as it were iuglers40 ,delamain says3 4 4 4 4 ,which words are neither cautelous1 nor subterfugious1 but are as downe right in their plainnesse1 as they are touching1 and pernitious1 by two much derogating from many1 and glancing upon many noble personages1 with too grosse1 if not too base an attribute1 in tearming them doers of tricks1 as it were to iuggle3 because they perhaps make use of a necessitie in the furnishing of themselves with such knowledge by ,practicall ,instrumentall operation1 when their more weighty negotiations will not permit them for ,theoreticall figurative demonstration2 those that are guilty of the aspertion1 and are touched therewith may answer for themselves1 and studie to be more ,theoreticall1 than ,practicall3 for the ,theory1 is as the ,mother that produceth the daughter1 the very sinewes and life of ,practise1 the excellencie and highest degree of true ,mathematicall ,knowledge3 but for those that would make but a step as it were into that kind of ,learning1 whose onely desire is expedition1 and facilitie1 both which by the generall consent of all are best effected with ,instrument1 rather then with tedious regular demonstrations1 it was ill to checke them so grosly1 not onely in what they have ,practised1 but abridging them also of their liberties with what they may ,practise1 which aspertion may not easily be slighted off by any glosse or ,apologie1 without an ,ingenuous confession1 or some mentall reservation3 ,to which vilification1 howsoever1 in the behalfe of my selfe1 and others1 ,i answer2 ,that ,instrumentall operation is not only the ,compendiating1 and facilitating of ,art1 but even the glory of it1 whole demonstration both of the making1 and operation is soly in the science1 and to an ,artist or disputant proper to be knowne1 and so to all1 who would truly know the cause of the ,mathematicall operations in their originall2 ,but1 for none to know the use of a ,mathematicall ,instrument1 except he knowes the cause of its operation1 is somewhat too strict1 which would keepe many from affecting the ,art1 which of themselves are ready enough every where1 to conceive more harshly of the difficultie1 and impossibilitie of attayning any skill therein1 then it deserves1 because they see nothing but obscure propositions1 and perplex and intricate demonstrations before their eyes1 whose unsavoury tartnes1 to an unexperienced palate like bitter pills is sweetned over1 and made pleasant with an ,instrumentall compendious facilitie1 and made to goe downe the more readily1 and yet to retaine the same vertue1 and working2 ,and me thinkes in this queasy age1 all helpes may bee used to procure a stomacke1 all bates and invitations to the declining studie of so noble a ,science1 rather then by rigid ,method and generall ,lawes to scarre men away4 ,all are not of like disposition1 neither all 7as was sayd before7 propose the same end1 some resolve to wade1 others to put a finger in onely1 or wet a hand3 now thus to tye them to an obscure and ,theoricall forme of teaching1 is to crop their hope1 even in the very bud4 4 4 4 4 ,the beginning of a mans knowledge even in the use of an ,instrument1 is first founded on doctrinal precepts1 and these precepts may be conceived all along in its use3 and are so farre from being excluded1 that they doe necessarily concomitate and are contained therein3 the practicke being better understood by the doctrinall part1 and this later explained by the ,instrumentall1 making precepts obvious unto sense1 and the ,theory going along with the ,instrument1 better informing and inlightning the understanding1 etc4 vis vnita fortior1 so as if that in ,phylosophy bee true1 ,nihil est in intellectu quod non prius fuit in sensu4 ,the difference between ,oughtred and ,delamain as to the use of mathematical instruments raises important questions4 ,should the slide rule be placed in the hands of a boy before1 or after1 he has mastered the theory of logarithms8 ,should logarithmic tables be withheld from him until the theoretical foundation is laid in the mind of the pupil8 ,is it a good thing to let a boy use a surveying instrument unless he first learns trigonometry8 ,is it advisable to permit a boy to familiarize himself with the running of a dynamo before he has mastered the underlying principles of electricity8 ,does the use of instruments ordinarily discourage a boy from mastery of the theory8 ,or does such manipulation constitute a natural and pleasing approach to the abstract8 ,on this particular point1 who showed the profounder psychological insight1 ,oughtred or ,delamain8 ,in ,july1 #aiad1 there was held in ,edinburgh a celebration of the three-hundredth anniversary of the invention of logarithms4 ,on that occasion there was collected at ,edinburgh university one of the largest exhibits ever seen of modern instruments of calculation4 ,the opinion was expressed by an experienced teacher that 8weapons as those exhibited there are for men and not for boys1 and such danger as there may be in them is of the same character as any form of too early specialization40 ,it is somewhat of a paradox that ,oughtred1 who in his student days and during his active years felt himself impelled to invent sun-dials1 planispheres1 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 beginners4 ,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 ability4 ,it may be argued that his pedagogic dogma1 otherwise so excellent1 here goes contrary to the course he himself followed instinctively in his self-education along mathematical lines4 ,we read that ,sir ,isaac ,newton1 as a child1 constructed sun-dials1 windmills1 kites1 paper lanterns1 and a wooden clock4 ,should these activities have been suppressed8 ,ordinary children are simply ,isaac ,newtons on a smaller intellectual scale4 ,should their activities along these lines be encouraged or checked8 ,on the other hand1 it may be argued that the paradox alluded to above admits of explanation1 like all paradoxes1 and that there is no inconsistency between ,oughtred's pedagogic views and his own course of development4 ,if he invented sun-dials1 he must have had a comprehension of the cosmic motions involved2 if he solved spherical triangles graphically by the aid of the planisphere1 he must have understood the geometry of the sphere1 so far as it relates to such triangles2 if he invented slide rules1 he had beforehand a thorough grasp of logarithms4 ,the question at issue does not involve so much the invention of instruments1 as the use by the pupil of instruments already constructed1 before he fully understands the theory which is involved4 ,nor does ,sir ,isaac ,newton's activity as a child establish ,delamain's contention4 ,of course1 a child should not be discouraged from manual activity along the line of producing interesting toys in imitation of structures and machines that he sees1 but to introduce him to the realm of abstract thought by the aid of instruments is a different proposition1 fraught with danger4 ,a boy may learn to use a slide rule mechanically and1 because of his ability to obtain practical results1 feel justified in foregoing the mastery of underlying theory2 or he may consider the ability of manipulating a surveying instrument quite sufficient1 even though he be ignorant of geometry and trigonometry2 or he may learn how to operate a dynamo and an electric switchboard and be altogether satisfied1 though having no grasp of electrical science4 ,thus instruments draw a youth aside from the path leading to real intellectual attainments and real efficiency2 they allure him into lanes which are often blind alleys4 ,such were the views of ,oughtred4 ,who was right1 ,oughtred or ,delamain8 ,it may be claimed that there is a middle ground which more nearly represents the ideal procedure in teaching4 ,shall the slide rule be placed in the student's hands at the time when he is engaged in the mastery of principles8 ,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 attained8 ,does this method not produce the best and most lasting results8 ,is not this ,delamain's actual contention8 ,we leave it to the reader to settle these matters from his own observation1 knowledge1 and experience4 ,,newton',s ,,comments ,,on ,,oughtred ,oughtred is an author who has been found to be of increasing interest to modern historians of mathematics4 ,but no modern writer has1 to our knowledge1 pointed out his importance in the history of the teaching of mathematics4 ,yet his importance as a teacher did receive recognition in the seventeenth century by no less distinguished a scientist than ,sir ,isaac ,newton4 ,on ,may #be1 #afid1 ,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 ,hospital4 ,toward the close of his letter1 ,newton says3 ,and now ,i have told you my opinion in these things1 ,i will give you ,mr4 ,oughtred's1 a ,man whose judgment 7if any man's7 may be safely relyed upon4 ,for he in his book of the circles of proposition1 in the end of what he writes about ,navigation 7page #ahd7 has this exhortation to ,seamen4 8,and if10 saith he1 8the ,masters of ,ships and ,pilots will take the pains in the ,journals of their ,voyages diligently and faithfully to set down in severall columns1 not onely the ,rumb they goe on and the measure of the ,ships way in degrees1 and the observation of ,latitude and variation of their compass2 but alsoe their conjectures and reason of their correction they make of the aberrations they shall find1 and the qualities and condition of their ship1 and the diversities and seasons of the winds1 and the secret motions or agitations of the ,seas1 when they begin1 and how long they continue1 how farr they extend and with what inequality2 and what else they shall observe at ,sea worthy consideration1 and will be pleased freely to communicate the same with ,artists1 such as are indeed skilfull in the ,mathematicks and lovers and enquirers of the truth3 ,i doubt not but that there shall be in convenient time1 brought to light many necessary precepts which may tend to ye perfecting of ,navigation1 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 ,god40 ,thus farr that very good and judicious man ,mr4 ,oughtred4 ,i will add1 that if instead of sending the ,observations of ,seamen to able ,mathematicians at ,land1 the ,land would send able ,mathematicians to ,sea1 it would signify much more to the improvemt of ,navigation and safety of ,mens lives and estates on that element4#add ,may ,oughtred prove as instructive to the modern reader as he did to ,newton6 ,footnotes #a;,aubrey's ,brief ,lives1 ed4 ,a4 ,clark1 ,vol4 ,,ii1 ,oxford1 #ahih1 p4 #ajf4 #b8,to the ,english ,gentrie1 and all others studious of the ,mathematicks1 which shall bee ,readers hereof4 ,the just ,apologie of ,wil3 ,ovghtred1 against the slaunderous insimulations of ,richard ,delamain1 in a ,pamphlet called ,grammelogia1 or the ,mathematicall ,ring1 or ,mirifica logarithmorum projectio circularis0 #afcc81 p4 #h4 ,hereafter we shall refer to this pamphlet as the ,apologeticall ,epistle1 this name appearing on the page-headings4 #c;,companion to the ,british ,almanac of #ahcg1 p4 #bh1 in an article by ,augustus ,de ,morgan on 8,notices of ,english ,mathematical and ,astronomical ,writers between the ,norman ,conquest and the ,year #afjj40 #d,new and ,general ,biographical ,dictionary 7,john ,nichols71 ,london1 #aghd1 art4 8,oughtred40 #e,rev4 ,owen ,manning1 ,history of ,antiquities in ,surrey1 ,vol4 ,,ii1 p4 #acb4 #f,skeleton ,collegii ,regalis ,cantab43 ,or ,a ,catalogue of ,all the ,provosts1 ,fellows and ,scholars1 of the ,king's ,college 4 4 4 4 since the ,foundation ,thereof1 ,vol4 ,,ii1 8,william ,oughtred40 #g;,aubrey1 op4 cit41 ,vol4 ,,ii1 p4 #ajg4 #h,rigaud1 ,correspondence of ,scientific ,men of the ,seventeenth ,century1 ,oxford1 ,vol4 ,i1 #ahda1 p4 #e4 #i;,aubrey1 op4 cit41 ,vol4 ,,ii1 p4 #aaj4 #aj;,ibid41 p4 #aaa4 #aa,op4 cit41 ,vol4 ,,ii1 p4 #acb4 #ab,mr4 ,william ,lilly's ,history of ,his ,life and ,times1 ,from the ,year #afjb to #afha1 ,london1 #agae1 p4 #eh4 #ac,rigaud1 op4 cit41 ,vol4 ,i1 p4 #fj4 #ad;,aubrey1 op4 cit41 ,vol4 ,,ii1 p4 #ajg4 #ae,rigaud1 op4 cit41 ,vol4 ,i1 p4 #af4 #af,owen ,manning1 op4 cit41 p4 #acb4 #ag,new and ,general ,biographical ,dictionary 7,john ,nichols71 ,london1 #aghd1 art4 8,oughtred40 #ah,op4 cit41 ,vol4 ,,ii1 p4 #aaj4 #ai,rev4 ,owen ,manning1 ,the ,history and ,antiquities of ,surrey1 ,vol4 ,,ii1 ,london1 #ahji1 p4 #acb4 #bj,op4 cit41 ,vol4 ,,ii1 #ahih1 p4 #aaa4 #ba;,budget of ,paradoxes1 ,london1 #ahgb1 p4 #dea2 #b;d ed41 ,chicago and ,london1 #aiae1 ,vol4 ,,ii1 p4 #cjc4 #bb,the full title of the ,clavis of #afca is as follows3 ,arithmeticae in numeris et speciebvs institvtio3 ,qvae tvm logisticae1 tvm analyticae1 atqve adeo totivs mathematicae1 qvasi clavis est4--,ad nobilissimvm spectatissimumque invenem ,dn4 ,gvilelmvm ,howard1 ,ordinis qui dicitur1 ,balnei ,equitem1 honoratissimi ,dn4 ,thomae1 ,comitis ,arvndeliae ,svrriae1 ,comitis ,mareschalli ,angliae1 c filium4--,londini1 ,apud ,thomam ,harpervm4 ,m4,,dc4,,xxxi4 ,in all there appeared five ,latin editions1 the second in #afdh at ,london1 the third in #afeb at ,oxford1 the fourth in #affg at ,oxford1 the fifth in #afic and #afih at ,oxford4 ,there were two independent ,english editions3 the first in #afdg at ,london1 translated in greater part by ,robert ,wood of ,lincoln ,college1 ,oxford1 as is stated in the preface to the #afeb ,latin edition2 the second in #afid and #agjb is a new translation1 the preface being written and the book recommended by the astronomer ,edmund ,halley4 ,the #afid and #agjb impressions labored under the defect of many sense-disturbing errors due to careless reading of the proofs4 ,all the editions of the ,clavis1 after the first edition1 had one or more of the following tracts added on3 ,eq4,de ,aequationum affectarvm resolvtione in numeris4 ,eu4,elementi decimi ,euclidis declaratio4 ,so4,de ,solidis regularibus1 tractatus4 ,an4,de ,anatocismo1 sive usura composita4 ,fa4,regula falsae positionis4 ,ar4,theorematum in libris ,archimedis de ,sphaera cylindro declaratio4 ,ho4,horologia scioterica in plano1 geometric delineandi modus4 ,the abbreviated titles given here are1 of course1 our own4 ,the lists of tracts added to the ,clavis mathematicae of #afca in its later editions1 given in the order in which the tracts appear in each edition1 are as follows3 ,clavis of #afdg1 ,eq41 ,an41 ,fa41 ,ho42 ,clavis of #afdh1 ,eq41 ,an41 ,fa41 ,eu41 ,so42 ,clavis of #afeb1 ,eq41 ,eu41 ,so41 ,an41 ,fa41 ,ar41 ,ho42 ,clavis of #affg1 ,eq41 ,eu41 ,so41 ,an41 ,fa41 ,ar41 ,ho42 ,clavis of #afic and #afih1 ,eq41 ,eu41 ,so41 ,an41 ,fa41 ,ar41 ,ho42 ,clavis of #afid and #agjb1 ,eq4 ,the title-page of the ,clavis was considerably modified after the first edition4 ,thus1 the #afeb ,latin edition has this title-page3 ,guilelmi ,oughtred ,aetonensis1 quondam ,collegii ,regalis in ,cantabrigia ,socii1 ,clavis mathematicae denvo limata1 sive potius fabricata4 ,cum aliis quibusdam ejusdem commentationibus1 quae in sequenti pagina recensentur4 ,editio tertia auctior emendatior4 ,oxoniae1 ,excudebat ,leon4 ,lichfield1 ,veneunt apud ,tho4 ,robinson4 #afeb4 #bc,rigaud1 op4 cit41 ,vol4 ,,ii1 p4 #dgf4 #bd,see1 for instance1 the ,clavis mathematicae of #afeb1 where he expresses himself thus 7p4 #d73 8,speciosa haec ,arithmetica arti ,analyticae 7per quam ex sumptione quaesiti1 tanquam noti1 investigatur quaesitum7 multo accommodatior est1 quam illa numerosa40 #be,oughtred1 ,the ,key of the ,mathematicks1 ,london1 #afdg1 p4 #d4 #bf;,clavis #afid1 p4 #ai1 and the ,clavis of #afca1 p4 #h4 #bg,see for instance1 ,oughtred's ,elementi decimi ,euclidis declaratio1 #afeb1 p4 #a1 where he uses ,a and ,e1 and also a and e4 #bh,see ,christophori ,clavii ,bambergensis ,operum mathematicorum1 tomus secundus1 ,moguntiae1 ,m4,,dc4,,xi1 algebra1 p4 #ci4 #bi;,christophori ,clavii operum mathematicorum ,tomus ,secundus1 ,moguntiae1 ,m4,,dc4,,xi1 ,epitome arithmeticae1 p4 #cf4 #cj,see ,f4 ,cajori1 8,the ,cross as a ,symbol of ,multiplication10 in ,nature1 ,vol4 ,,xciv 7#aiad71 p4 #cfc4 #ca,see ,elementi decimi ,euclidis declaratio1 #afeb1 p4 #b4 #cb,see ,johannis ,wallisii ,operum mathematicorum pars prima1 ,oxonii1 #afeg1 p4 #bdg4 #cc;,clavis of #afca1 chap4 xix1 sec4 #e1 p4 #ej4 #cd,we have noticed the representation of known quantities by consonants and the unknown by vowels in ,wingate's ,arithmetick made easie1 edited by ,john ,kersey1 ,london1 #afej1 algebra1 p4 #chb2 and in the second part1 section #ai1 of ,jonas ,moore's ,arithmetick in two parts1 ,london1 #affj1 ,moore suggests as an alternative the use of z1 y1 x1 etc41 for the unknowns4 ,the practice of representing unknowns by vowels did not spread widely in ,england4 #ce,philosophical ,transactions1 ,vol4 ,,xix1 ,no4 #bca1 ,london1 p4 #feb4 #cf;,ibid41 ,vol4 ,,xix1 p4 #ef4 #cg,there are two title-pages to the edition of #afcb4 ,the first title-page is as follows3 ,the ,circles of ,proportion and ,the ,horizontall ,instrument4 ,both invented1 and the vses of both ,written in ,latine by ,mr4 ,w4 ,o4 ,translated into ,english3 and set forth for the publique benefit by ,william ,forster4 ,london4 ,printed for ,elias ,allen maker of these and all other mathematical ,instruments1 and are to be sold at his shop over against ,st4 ,clements church with out ,temple-barr4 #afcb4 ,t4 ,cecill ,sculp4 ,in #afcc there was added the following1 with a separate title-page3 ,an addition vnto the ,vse of the ,instrvment called the ,circles of ,proportion4 4 4 4 4 ,london1 #afcc1 this being followed by ,oughtred's ,to the ,english ,gentrie etc4 ,in the ,british ,museum there is a copy of another impression of the ,circles of ,proportion1 dated #afci1 with the ,addition vnto the ,vse of the ,instrument etc41 bearing the original date1 #afcc1 and with the epistle1 ,to the ,english ,gentrie1 etc41 inserted immediately after ,forster's dedication1 instead of at the end of the volume4 #ch,the complete title of the ,english edition is as follows3 ,trigonometrie1 or1 ,the manner of calculating the ,sides and ,angles of ,triangles1 by the ,mathematical ,canon1 demonstrated4 ,by ,william ,oughtred ,etonens4 ,and published by ,richard ,stokes ,fellow of ,kings ,colledge in ,cambridge1 and ,arthur ,haughton ,gentleman4 ,london1 ,printed by ,r4 and ,w4 ,leybourn1 for ,thomas ,johnson at the ,golden ,key in ,st4 ,pauls ,church-yard4 ,m4,,dc4,,lvii4 #ci;,jer4 ,collier1 ,the ,great ,historical1 ,geographical1 ,genealogical and ,poetical ,dictionary1 ,vol4 ,,ii1 ,london1 #agja1 art4 8,oughtred40 #dj,rigaud op4 cit41 ,vol4 ,i1 p4 #hb4 #da;,a4 ,de ,morgan1 ,budget of ,paradoxes1 ,london1 #ahgb1 p4 #dea2 #b;d ed41 ,chicago1 #aiae1 ,vol4 ,,ii1 p4 #cjc4 #db;,e4 ,gunter1 ,description and ,use of the ,sector1 the ,crosse-staffe and other ,instruments1 ,london1 #afbd1 second book1 p4 #ca4 #dc;,f4 ,cajori1 8,on the ,history of a ,notation in ,trigonometry10 ,nature1 ,vol4 ,,xciv1 #aiae1 pp4 #fdb1 #fdc4 #dd;,a4 von ,braunmhl1 ,geschichte der ,trigonometrie1 #b4 ,teil1 ,leipzig1 #aijc1 pp4 #db1 #ia4 #de;,h4 ,hankel1 ,geschichte der ,mathematik in ,alterthum und ,mittelalter1 ,leipzig1 #ahgd1 pp4 #cfi1 #cgj4 #df,m4 ,cantor1 ,vorlesungen ber ,geschichte der ,mathematik1 ,,ii1 #aijj1 pp4 #fdj1 #fda4 #dg,this matter has been discussed in a paper by ,f4 ,cajori1 8,a ,history of the ,arithmetical ,methods of ,approximation1 etc41 ,colorado ,college ,publication1 ,general ,series ,no4 #ea1 #aiaj1 pp4 #ahb-hd4 ,later this subject was again treated by ,g4 ,enestrm in ,bibliotheca mathematica1 #c4 ,folge1 ,vol4 ,,xi1 #aiaa1 pp4 #bcd1 #bce4 #dh,see ,f4 ,cajori1 op4 cit41 p4 #aic4 #di,see ,william ,oughtred's ,key of the ,mathematicks1 ,london1 #afid1 pp4 #agc-ge1 tract1 8,of the ,resolution of the ,affected ,equations10 or any edition of the ,clavis after the first4 #ej;,a4 ,de ,morgan1 op4 cit41 p4 #dea2 #b;d ed41 ,vol4 ,,ii1 p4 #cjc4 #ea,see ,f4 ,cajori1 ,history of the ,logarithmic ,slide ,rule1 ,new ,york1 #aiji1 pp4 #g-ad1 ,addenda1 p4 ii4 #eb,rigaud1 op4 cit41 ,vol4 ,i1 p4 #ab4 #ec,the ,new ,artificial ,gauging ,line or ,rod3 together with rules concerning the use thereof3 ,invented and written by ,william ,oughtred1 ,london1 #afcc4 #ed,w4 ,oughtred1 ,apologeticall ,epistle1 p4 #ac4 #ee,quarterly ,journal of ,pure and ,applied ,mathematics1 ,vol4 ,,xlvi1 7#aiae71 p4 #afi4 ,in this article ,glaisher republishes the 8,appendix0 in full4 #ef;,aubrey1 op4 cit41 ,vol4 ,,ii1 #ahih1 p4 #ajh4 #eg,wood's ,athenae ,oxonienses 7ed4 ,p4 ,bliss71 ,vol4 ,,iv1 #ahbj1 p4 #bdg4 #eh,wood1 op4 cit41 ,vol4 ,,ii1 p4 #dde4 #ei,rigaud1 op4 cit41 ,vol4 ,i1 pp4 #cc1 #ce4 #fj,rigaud1 op4 cit41 ,vol4 ,i1 pp4 #af1 #bf4 #fa,rigaud1 op4 cit41 ,vol4 ,i1 p4 #ff4 #fb;,ibid41 ,vol4 ,i1 p4 #i4 #fc,rigaud1 op4 cit41 ,vol4 ,,ii1 p4 #dge4 #fd;,ibid41 ,vol4 ,,ii1 p4 #dga4 #fe;,j4 ,w4 ,l4 ,glaisher1 8,on ,early ,logarithmic ,tables1 and ,their ,calculators10 ,philosophical ,magazine1 #dth ,ser41 ,vol4 ,,xlv 7#ahgc71 pp4 #cgh1 #cgi4 #ff,rigaud1 op4 cit41 ,vol4 ,i1 p4 #fe4 #fg,rigaud1 op4 cit41 ,vol4 ,i1 p4 #hg4 #fh,king's ,life of ,john ,locke1 ,vol4 ,i1 ,london1 #ahcj1 p4 #bbg4 #fi;,exercitationum ,mathematicarum ,decas prima1 ,naples1 #afbg1 and probably ,cataldus' ,transformatio ,geometrica1 ,bonon41 #afab4 #gj,rigaud1 op4 cit41 ,vol4 ,,ii1 pp4 #dgg-hj4 #ga,miscellanies3 or ,mathematical ,lucubrations1 of ,mr4 ,samuel ,foster1 ,sometimes publike ,professor of ,astronomie in ,gresham ,colledge in ,london1 by ,john ,twysden1 ,london1 #afei4 #gb,the ,works of the ,honourable ,robert ,boyle in five volumes1 to which is prefixed the ,life of the ,author1 ,vol4 ,i1 ,london1 #agdd1 p4 #bd4 #gc,the letter is printed in ,john ,wallis' ,de algebra tractatus1 #afic1 p4 #bjf4 #gd,see ,la ,correspondance de ,descartes1 published by ,charles ,adam and ,paul ,tannery1 ,vol4 ,,ii1 ,paris1 #ahih1 pp4 #def and #deg4 #ge;,h4 ,bosmans1 ,s4,j41 8,la premire dition de la ,clavis ,mathematica d',oughtred4 ,son influence sur la ,gomtrie de ,descartes10 ,annales de la socit scientifique de ,bruxelles1 #ceth year1 #aiaj-aa1 ,part ,,ii1 pp4 #bd-gh4 #gf;,ibid41 p4 #gh4 #gg,vincent ,wing1 ,harmonicon coeleste1 ,london1 #afea1 p4 #e4 #gh,seth ,ward1 ,in ,ismaelis ,bullialdi astronomiae philolaicae fundamenta inquisitio brevis1 ,oxford1 #afec1 p4 #g4 #gi;,john ,wallis1 ,elenchus geometriae ,hobbianae1 ,oxford1 #afee1 p4 #dh4 #hj;,an ,idea of ,arithmetick1 at first designed for the use of the ,free ,schoole at ,thurlow in ,suffolk4 4 4 4 4 ,by ,r4 ,b41 ,schoolmaster there1 ,london1 #afee1 p4 #f4 #ha,the ,miscellanies3 or ,mathematical ,lucubrations1 of ,mr4 ,samuel ,foster 4 4 4 4 by ,john ,twysden1 ,london1 #afei1 p4 #a4 #hb,moor's ,arithmetick in two ,books1 ,london1 #affj1 p4 #hi4 #hc;,isaac ,barrow1 ,euclidis data1 ,cambridge1 #afeg1 p4 #b4 #hd;,francisci ,dulaurens ,specima mathematica1 ,paris1 #affg1 p4 #a4 #he;,elmens des mathmatiques1 ,paris1 #afge1 ,preface signed 8,j4 ,p40 #hf,nouveaux lmens de gomtrie1 ,paris1 #afib 7permission to print #afhd74 #hg,ozanam1 ,dictionnaire mathmatique1 ,paris1 #afia1 p4 #ab4 #hh;,analyse des infiniment petits1 ,paris1 #afif1 p4 #aa4 #hi,petro ,nicolas1 ,de conchoidibus et cissoidibus exercitationes geometricae1 ,toulouse1 #afig1 p4 #ag4 #ij,r4 ,p4 ,bernard ,lamy1 ,elmens des mathmatiques1 ,amsterdam1 #afib 7permission to print #afhj74 #ia,nouveaux lmens de gomtrie1 #b;d ed41 ,the ,hague1 #afij1 p4 #cjd4 #ib,w4 ,w4 ,beman in ,l'intermdiaire des mathmaticiens1 ,paris1 ,vol4 ,,ix1 #aijb1 p4 #bbi1 question #bdbd4 #ic;,john ,collins1 ,the ,mariner's ,plain ,scale ,new ,plain'd1 ,london1 #afei1 p4 #be4 #id;,james ,gregory1 ,optica promota1 ,london1 #affc1 pp4 #ai1 #dh4 #ie,philosophical ,transactions1 ,vol4 ,,iii1 ,london1 p4 #hfh4 #if,william ,leybourn1 ,the ,line of ,proportion1 ,london1 #afgc1 p4 #ad4 #ig;,elementa geometriae 4 4 4 4 a ,gulielmo ,sanders1 ,glasgow1 #afhf1 p4 #c4 #ih;,cocker's ,decimal ,arithmetick1 4 4 4 4 perused by ,john ,hawkins1 ,london1 #afie 7preface dated #afhd71 p4 #da4 #ii;,joseph ,raphson1 ,analysis ,aequationum universalis1 ,london1 #afig1 p4 #bf4 #ajj;,e4 ,wells1 ,elementa arithmeticae numerosae et speciosae1 ,oxford1 #afih1 p4 #ajg4 #aja;,john ,ward1 ,a ,compendium of ,algebra1 #b;d ed41 ,london1 #afih1 p4 #fb4 #ajb,plain ,elements of ,geometry and ,plain ,trigonometry1 ,london1 #agja1 p4 #fc4 #ajc;,george ,shelley1 ,wingate's ,arithmetick1 ,london1 #agjd1 p4 #cdc4 #ajd;,a ,synopsis of ,algebra1 ,being a posthumous work of ,john ,alexander of ,bern1 ,swisserland4 4 4 4 4 ,done from the ,latin by ,sam4 ,cobb1 ,london1 #agji1 p4 #af4 #aje;,john ,craig1 ,de ,calculo fluentium1 ,london1 #agah1 p4 #ce4 ,the notation ,a3,b33,c3,d is given also4 #ajf,trigonometry1 #b;d ed41 ,edinburgh1 #agbd1 p4 #aa4 #ajg,mthode pour la msure des surfaces1 la dimension des solides 4 4 4 4 par ,m4 ,carr de l'acadmie r4 des sciences1 #agjj1 p4 #ei4 #ajh;,application de l'algbre gomtrie 4 4 4 4 ,paris1 #agje4 #aji;,elmens de la gomtrie de l'infini1 by ,m4 de ,fontenelle1 ,paris1 #agbg1 p4 #aaj4 #aaj;,eclaircissemens sur l'analyse des infiniment petits1 by ,m4 ,varignon1 ,paris1 #agbe1 p4 #hg4 #aaa;,application de la gomtrie ordinaire et des calculs diffrentiel et intgral1 by ,m4 ,robillard1 ,paris1 #agec4 #aab,trait de gomtrie thorique et pratique1 new ed41 ,paris1 #agfd1 p4 #ae4 #aac,recherches sur les courbes double courbure1 ,paris1 #agca1 p4 #ac4 #aad;,analyse des infiniment petits1 by the ,marquis de ,l',hospital4 ,new ed4 by ,m4 ,le ,fvre1 ,paris1 #agha1 p4 #da4 ,in this volume passages in fine print1 probably supplied by the editor1 contain the notation a3b33c3d2 the parts in large type give ,oughtred's original notation4 #aae,the tendency during the eighteenth century is shown in part by the following data3 ,jacobi ,bernoulli ,opera1 ,tomus primus1 ,geneva1 #agdd1 gives ,b4,a33,d4,c on p4 #cfh1 the paper having been first published in #afhh2 on p4 #dai is given ,,ge3,,ag,,la3,,ml1 the paper having been first published in #afhi4 ,bernhardi ,nieuwentiit1 ,considerationes circa analyseos ad quantitates infinit parvas applicatae principia1 ,amsterdam1 #afid1 p4 #bj1 and ,analysis infinitorum1 ,amsterdam1 #afie1 on p4 #bgf1 have x3c33s3r4 ,paul ,halcken's ,deliciae mathematicae1 ,hamburg1 #agai1 gives a3b33c3d4 ,johannis ,baptistae ,caraccioli1 ,geometria algebraica universa1 ,rome1 #agei1 p4 #gi1 has a4b33c4d4 ,delle corde ouverto fibre elastiche schediasmi fisico-matematici del conte ,giordano ,riccati1 ,bologna1 #agfg1 p4 #fe1 gives ,p3b33r3ds4 8,produzioni mathematiche0 del ,conte ,giulio ,carlo de ,fagnano1 ,vol4 ,i1 ,pesario1 #agej1 p4 #aic1 has a4b33c4d4 ,l4 ,mascheroni1 ,gomtrie du compas1 translated by ,a4 ,m4 ,carette1 ,paris1 #agih1 p4 #ahh1 gives 7#c73b337b73,lp4 ,danielis ,melandri and ,paulli ,frisi1 ,de theoria lunae commentarii1 ,parma1 #agfi1 p4 #ac1 has a3b33c3d4 ,vicentio ,riccato and ,hieronymo ,saladino1 ,institutiones analyticae1 ,vol4 ,i1 ,bologna1 #agfe1 p4 #dg1 gives x3a33m3nm4 ,r4 ,g4 ,boscovich1 ,opera pertinentia ad opticam et astronomiam1 ,bassani1 #aghe1 p4 #dji1 uses a3b33c3d4 ,jacob ,bernoulli1 ,ars ,conjectandi1 ,basel1 #agac1 has n-r4n-#a33;c4d4 ,pavlini ,chelvicii1 ,institutiones analyticae1 editio post tertiam ,romanam prima in ,germania1 ,vienna1 #agfa1 p4 #b1 a4b33c4d4 ,christiani ,wolfii1 ,elementa matheseos universae1 ,vol4 ,,iii1 ,geneva1 #agce1 p4 #fc1 has ,,ab3,,ae#a3q4 ,johann ,bernoulli1 ,opera omnia1 ,vol4 ,i1 ,lausanne and ,geneva1 #agdb1 p4 #dc1 has a3bc3d4 ,d4 ,c4 ,walmesley1 ,analyse des mesures des rapports et des angles1 ,paris1 #agdi1 uses extensively a4b33c4d1 later a3b33c3d4 ,g4 ,w4 ,krafft1 ,institutiones geometriae sublimoris1 ,tbingen1 #agec1 p4 #aid1 has a3bc3d4 ,j4 ,h4 ,lambert1 ,photometria1 #agfj1 p4 #ajd1 has ,c3 ,,bc3,,mh4 ,meccanica sublime del ,dott4 ,domenico ,bartaloni1 ,naples1 #agfe1 has a3b33c3d4 ,occasionally ratio is not designated by a4b1 nor by a3b1 but by a1 b1 as for instance in ,a4 de ,moivre's ,doctrine of ,chance1 ,london1 #agef1 p4 #cd1 where he writes a1 b33#a1 q4 ,a further variation in the designation of ratio is found in ,james ,atkinson's ,epitome of the ,art of ,navigation1 ,london1 #agah1 p4 #bd1 namely1 #c44b33gb44dh4 ,curious notations are given in ,rich4 ,balam's ,algebra1 ,london1 #afec4 #aaf;,chr4 ,clavii ,operum mathematicorum tomus secundus1 ,mayence1 #afaa1 ,algebra1 p4 #ci4 #aag;,invention nouvelle en l'algbre1 by ,albert ,girard1 ,amsterdam1 #afbi1 p4 #ag4 #aah,la gomtrie et pratique gnrale d'icelle1 par ,i4 ,errard de ,bar-le-,duc1 ,ingnieur ordinaire de sa ,majest1 #c;d ed41 revised by ,d4 ,h4 ,p4 ,e4 ,m41 ,paris1 #afai1 p4 #baf4 #aai,novae geometriae clavis algebra1 authore ,p4 ,jacobo de ,billy1 ,paris1 #afdc1 p4 #aeg2 also an ,abridgement of the ,precepts of ,algebra4 ,written in ,french by ,james de ,billy1 ,london1 #afei1 p4 #cdf4 #abj,miscellanies3 or ,mathematical ,lucubrations1 of ,mr4 ,samuel ,foster1 ,sometime publike ,professor of ,astronomie in ,gresham ,colledge in ,london1 ,london1 #afei1 p4 #g4 #aba,quarterly ,jour4 of ,pure and ,applied ,math41 ,vol4 ,,xlvi 7,london1 #aiae71 p4 #aia4 #abb,pietro ,cossali1 ,origine1 trasporto in ,italia primi progressi in essa dell' algebra1 ,vol4 ,i1 ,parmense1 #agig1 p4 #eb4 #abc;,in ,is4 ,bullialdi astronomiae philolaicae fundamenta inquisitio brevis1 ,auctore ,setho ,wardo1 ,oxford1 #afec1 p4 #a4 #abd;,john ,wallis1 ,algebra1 ,london1 #afhe1 p4 #cba1 and in some of his other works4 ,he makes greater use of ,harriot's symbols4 #abe;,euclidis data1 #afeg1 p4 #a2 also ,euclidis elementorum libris ,,xv1 ,london1 #afei1 p4 #a4 #abf;,john ,kersey1 ,algebra1 ,london1 #afgc1 p4 #cba4 #abg;,e4 ,wells1 ,elementa arithmeticae numerosae et speciosae1 ,oxford1 #afih1 p4 #adb4 #abh;,cocker's ,decimal ,arithmetick1 perused by ,john ,hawkins1 ,london1 #afie 7preface dated #afhd71 p4 #bgh4 #abi,th4 ,baker1 ,the ,geometrical ,key1 ,london1 #afhd1 p4 #ae4 #acj,richard ,sault1 ,a ,new ,treatise of ,algebra1 ,london 7no date74 #aca,richard ,rawlinson in a pamphlet without date1 issued sometime between #afee and #affh1 containing trigonometric formulas4 ,there is a copy in the ,british ,museum4 #acb;,f4 ,dulaurens1 ,specima mathematica1 ,paris1 #affg1 p4 #a4 #acc;,j4 ,milnes1 ,sectionum conicarum elementa1 ,oxford1 #agjb1 p4 #db4 #acd;,cheyne1 ,philosophical ,principles of ,natural ,religion1 ,london1 #agje1 p4 #ee4 #ace;,j4 ,craig1 ,de calculo fluentium1 ,london1 #agah1 p4 #hf4 #acf;,jo4 ,wilson1 ,trigonometry1 #b;d ed41 ,edinburgh1 #agbd1 p4 v4 #acg;,commercium ,epistolicum1 #agab1 p4 #bj4 #ach;,c4 ,le ,paige1 8,sur l'origine de certains signes d'opration10 ,annales de la socit scientifique de ,bruxelles1 #afth year1 #ahia-ib1 ,part ,,ii1 pp4 #gi-hb4 #aci;,gravelaar1 8,over den oorsprong van ons maalteeken 7710 ,wiskundig ,tijdschrift1 #fth year4 ,we have not had access to this article4 #adj;,h4 ,bosmans1 op4 cit41 p4 #dj4 #ada;,claudii ,ptolemaei 4 4 4 4 annotationes1 ,ble1 #aeea4 ,this reference is taken from the ,encyclopdie des sciences mathmatiques1 ,tome ,i1 ,vol4 ,i1 ,fasc4 #a1 p4 #dj4 #adb;,due ,correction for ,mr4 ,hobbes4 ,or ,schoole ,discipline1 for not saying his ,lessons right4 ,in answer to his ,six ,lessons1 directed to the ,professors of ,mathematicks4 ,by the ,professor of ,geometry4 ,oxford1 #afef1 pp4 #g1 #dg1 #ej4 #adc,oughtred1 ,apologeticall ,epistle1 p4 #bg4 #add;,j4 ,edleston1 ,correspondence of ,sir ,isaac ,newton and ,professor ,cotes1 ,london1 #ahej1 pp4 #bgi-ib4 ,,index ,adam1 ,charles1 #ga ,agnesi1 ,maria ,g41 #gg ,alexander1 ,j41 #gf ,allen1 ,e41 #ce ,analysis1 #ai1 #bj ,apollonius of ,perga1 #bj1 #he ,archimedes1 #ah1 #bj1 #he ,aristotle1 #fi ,ashmole1 ,e41 #ac ,atkinson1 ,j41 #gi ,atwood1 #ef ,aubrey1 #c1 #g1 #h1 #ab-af1 #eh1 #ei ,austin1 #eh ,baker1 ,t41 #hb ,balam1 ,r41 #gi ,bar-le-,duc1 de1 #hj ,barrow1 ,s41 #a1 #cb1 #gc1 #gd1 #hj1 #ha ,bartaloni1 ,d41 #gi ,beman1 ,w4 ,w41 #gd1 #ge ,bernoulli1 ,jakob1 #gh-hj ,bernoulli1 ,john1 #gi1 #hj ,billingsley's ,euclid1 #ae ,billion1 #bj ,billy1 ,jacobo de1 #hj ,binomial formula1 #be1 #bi ,bliss1 ,p41 #fj ,boscovich1 ,r4 ,g41 #gh ,bosmans1 ,h41 #gb1 #hc ,boyle1 ,r41 #a1 #fi ,braunmhl1 von1 #ci ,brearly1 ,w41 #ei ,briggs1 #f1 #cf1 #ee ,brookes1 ,christopher1 #g1 #ec1 #ei ,cajori1 ,f41 #bg1 #ci1 #dj1 #dg ,cantor1 ,m41 #dj1 #da ,caraccioli1 ,j4 ,b41 #gh ,cardan1 #ga ,carr1 #gg ,carrete1 ,n4 ,m41 #gh ,caryll1 ,c41 #g ,cataldi1 #fg ,cavalieri1 #fe1 #ff ,cavendish1 ,charles1 #ag1 #fb1 #ff ,charles ,i1 #i1 #fj ,chelvicius1 ,p41 #gi ,cheyne1 ,g41 #hb ,circles of ,proportion1 #ce1 #cg1 #dh1 #di1 #ea1 #ei1 #hg1 #hh ,clairaut1 #gg ,clark1 ,a41 #c ,clark1 ,g41 #fc ,clarke1 ,f4 ,l41 #c ,clavis mathematicae1 #a1 #e1 #aj1 #ad1 #ag-ce1 #de1 #df1 #ea1 #eg-fc1 #fh-gc1 #ha1 #he1 #hg ,clavius1 #bf1 #hj ,clerc1 le1 #gg ,cobb1 ,s41 #gf ,cocker1 #gf1 #hb ,collins1 ,john1 #ae1 #ai1 #fc1 #fd1 #fg1 #fh1 #gf1 #hb ,colson1 ,j41 #gg ,conchoid1 #ab ,conic sections1 #aa1 #ec ,cossali1 ,p41 #ha ,cotes1 ,r41 #a1 #he ,craig1 ,j41 #gf1 #hb ,cross1 symbol of multiplication1 #bg1 #ch1 #ee1 #ef1 #hb1 #hc ,cubic equations1 #bh1 #cd1 #db1 #de ,decimal fractions1 notation of1 #ba ,degree1 centesimal division1 #ci ,delamain1 ,r41 #d1 #i1 #aj1 #aa1 #dg1 #dh1 #ea1 #fj1 #hd1 #hh1 #hi1 #ia1 #ic1 #id ,de ,moivre1 #cb1 #gi ,de ,morgan1 ,a41 #e1 #af1 #cg1 #df1 #dg1 #ed ,descartes1 ,r41 #a1 #be1 #dg1 #eg1 #fh-gb1 #hj ,dibuadius1 #gi ,difference1 symbol for1 #bg1 #ha ,diophantus1 #fc1 #he ,division1 abbreviated1 #ba1 #bc1 #bd ,dulaurens1 ,f41 #gd1 #hb ,earl of ,arundel1 #aj1 #ac1 #ae1 #ag ,edleston1 ,j41 #ie ,enestrm1 ,g41 #dj ,equations1 solution of1 #ah1 #bh1 #bi1 #ca1 #cd1 #ci-de1 #hg ,errard de ,bar-le-,duc1 #hj ,eton ,college1 #c1 #d ,euclid1 #a1 #ae1 #ah1 #bj1 #be1 #bg1 #bh1 #gi1 #ha1 #hc1 #he ,euler1 ,l41 #cg1 #ci ,ewart1 #ei ,exponents1 #be1 #bh1 #bi ,fagnano1 de1 #gh ,flower1 #ef ,fontenelle1 de1 #gg ,forster1 ,w41 #ce1 #dh1 #ei1 #hh ,foster1 ,s41 #bg1 #fg1 #fi1 #gc1 #hj1 #hi ,frisi1 ,p41 #gh ,gascoigne1 #ei1 #fa ,gauss1 ,c4 ,f41 #dh ,geysius1 #fg ,ghetaldi1 #gj1 #ga ,gibson1 #fg ,girard1 ,a41 #cb1 #hj ,glaisher1 ,j4 ,w4 ,l41 #ed-ef1 #fd1 #hj ,glorioso1 #fg1 #fh ,grammelogia1 #d1 #dg1 #hi ,gravelaar1 #hc ,greater than1 symbol for1 #ha ,greatrex1 ,r41 #ae ,gregory1 ,d41 #cb ,gregory1 ,j41 #bg1 #gf ,gresham ,college1 #a1 #f1 #bg1 #ei1 #fa1 #hj ,guisne1 #gg ,gunter1 ,e41 #cg1 #dg1 #hf ,gunter's scale1 #cg ,halcken1 ,p41 #gh ,hales1 ,j41 #g ,halley1 ,e41 #a1 #ah1 #fi ,hankel1 ,h41 #dj ,harper1 ,t41 #ah ,harriot1 ,t41 #de1 #dg1 #eg1 #eh1 #fi-ga1 #ha ,harris1 ,j41 #gf ,hartlib1 #fi ,haughton1 ,a41 #ce1 #ei ,hawkins1 ,j41 #gf1 #hb ,hearn1 #ef ,helmholtz1 #dh ,henry1 ,j41 #dh ,henry van ,etten1 #eb1 #ec ,henshaw1 ,t41 #h1 #eh1 #fa ,hobbes1 #gc1 #hf ,hollar1 #ad ,holsatus1 #ac ,hooganhuysen1 #fd ,hooke1 ,rb41 #a ,horner's method1 #de ,horology1 #ah1 #ej ,horrox1 ,j41 #d ,hospital1 de l'1 #gd1 #gg ,howard1 ,th4 ,see ,earl of ,arundel4 ,howard1 ,w41 #ag1 #ah1 #ei ,hutchinson1 ,a41 #f ,invisible college1 #a ,joule1 #dh ,kepler1 ,j41 #f ,kersey1 ,j41 #cb1 #gc1 #hb ,keylway1 ,r41 #fe ,king1 #fg ,kings ,college1 ,cambridge1 #c1 #ce ,krafft1 ,g4 ,w41 #gi ,lambert1 ,j4 ,h41 #gi ,lamy1 ,r4 ,p4 ,b41 #gd ,laud1 ,archbishop1 #fe ,leake1 ,w41 #ec ,le ,clerc1 #gg ,leech1 ,w41 #ei ,le ,fvre1 #gg ,leibniz1 #dg1 #gh1 #hj ,leonelli1 #ef ,le ,paige1 de1 #hc ,less than1 symbol for1 #ha ,leurechon1 #eb ,leybourn1 #ce1 #fd1 #gf ,lichfield1 ,mrs41 #ai ,lilly1 ,w41 #h1 #i ,locke1 ,j41 #fg ,logarithms1 #f1 #ba1 #bg1 #bh1 #ch1 #ci1 #db1 #df1 #ed-ef1 #fe1 #ib1 #ic2 natural1 #ee2 radix method of computing1 #ee1 #ef ,lower1 ,w41 #eh ,ludolph ,ceulen1 #gi ,manning1 #ef ,manning1 ,o41 #g1 #h1 #ac-ae ,mascheroni1 ,l41 #gh ,mayer1 ,r41 #dg ,melandri1 ,d41 #gh ,mercator1 ,n41 #ac ,mersenne1 #fc ,milbourn1 ,w41 #de ,million1 #bj ,milnes1 ,j41 #hb ,moivre1 de1 #cb1 #gi ,moore1 ,jonas1 #cb1 #ed1 #eh1 #gc ,moreland1 ,s41 #gj ,morse1 ,r41 #dh ,multiplication1 abbreviated1 #ba1 #bb1 #bd2 symbol for1 #bg1 #hb1 #hc ,mydorge1 #ed ,napier1 ,j41 #f1 #g1 #ba1 #bg1 #ch1 #ci1 #eb1 #ed1 #eg1 #ei ,napier's analogies1 #ci ,newton1 ,sir ,isaac1 #a1 #be1 #bi1 #dj1 #da1 #de1 #dg1 #ei1 #fe1 #hf1 #ib-ie ,nichols1 ,j41 #f1 #ad ,nicolas1 ,r4 ,p4 ,p41 #gd ,nieuwentiit1 ,b41 #gh ,norwood1 ,r41 #cg1 #ch1 #hj ,opuscula mathematica hactenus inedita1 #af1 #ba1 #ge ,orchard1 #ef ,oughtredus explicatus1 #fd ,ozanam1 #gd 1 symbol for1 #cb ,paige1 ,c4 de1 #hc ,pardies1 #gf ,parentheses1 #bf1 #gi1 #hj ,partridge1 ,s41 #dg ,peano1 #hf ,perfect number1 #da ,pitiscus1 #ae ,planisphere1 #ec1 #ib1 #ic ,prestet1 ,j41 #gd ,price1 #aa ,proportion1 notation for1 #bf1 #bg1 #gc-gi ,protheroe1 #eh ,ptolemy1 #hc ,quadratic equation1 #bi1 #ca1 #cd ,radix method1 #ee1 #ef ,rahn1 #bg ,raphson1 ,j41 #dj1 #da1 #gf ,ratio1 notation of1 #ba1 #gc-hj ,rawlinson1 ,r41 #ci1 #hb ,regula falsa1 #ah ,regular solids1 #ah ,riccati1 ,g41 #gh ,riccati1 ,v41 #gh ,rigaud1 #g1 #ab1 #ac1 #ai1 #dh1 #fa-ff1 #fh ,robillard1 #gg ,robinson1 ,w41 #ac1 #dh1 #ei1 #fb1 #fc ,rooke1 ,l41 #ei1 #fa ,saladini1 ,h41 #gh ,sanders1 ,w41 #gf ,sault1 ,r41 #hb ,scarborough1 ,charles1 #af1 #ed1 #eh1 #fj ,schooten1 ,van1 #a ,schreshensuchs1 ,o41 #hc ,scratch method1 #bc ,shakespeare1 #eb ,shelley1 ,g41 #gf ,shipley1 ,a4 ,e41 #a ,shuttleworth1 #ei ,slide rule1 #i1 #df-di1 #ej1 #fj1 #hh1 #ic ,smethwyck1 #eh ,smith1 ,j41 #ej ,snellius1 ,w41 #gi ,solids1 regular1 #ah ,speidell1 ,john1 #ch1 #ee ,spherical triangles1 #ec1 #ed1 #ic ,stokes1 ,r41 #ce1 #cf1 #eh ,sudell1 #ei ,sun dials1 #e1 #i1 #ej1 #ea1 #eb1 #fj1 #ib ,tannery1 ,p41 #ga ,todhunter1 #fj ,torporley1 #eh ,triangles1 spherical1 #ec1 #ed1 #ic ,trigonometria1 #ba1 #cf1 #ee1 #ge ,trigonometric functions1 symbols for1 #cf1 #cg1 #ee1 #ef ,trigonometrie1 #ba1 #ce1 #ci ,trisection of angles1 #bh ,twysden1 #ei1 #fh1 #fi1 #gc ,varignon1 #gg ,vieta1 #a1 #b1 #be1 #cb1 #cc1 #ce1 #ci-da1 #de1 #fc1 #fg1 #gj1 #ga ,vlack1 #fe ,von ,braunmhl1 #ci ,wadham ,college1 #e1 #ec ,wallis1 ,john1 #a1 #ai1 #bg1 #cc1 #de1 #eg-ei1 #fc1 #fd1 #ff-gd1 #gi-ha1 #hf ,walmesley1 ,d4 ,c41 #gi ,ward1 ,bishop1 #ac ,ward1 ,john1 #gf ,ward1 ,seth1 #ee1 #eh1 #fj1 #fh1 #gc1 #gd1 #ha ,watch-making1 #ah1 #ej ,weber1 ,w4 ,e41 #dh ,weddle1 #ef ,wells1 ,e41 #gf1 #hb ,wharton1 #fj ,whitlock1 ,b41 #h1 #i ,wilson1 ,j41 #gg1 #hb ,wing1 ,v41 #gc1 #ge ,wingate1 ,e41 #cb1 #dg1 #gc ,wolf1 ,christian1 #gi ,wood1 ,a41 #fj1 #fa ,wood1 ,r41 #ah1 #ei ,wren1 ,christopher1 #e1 #eh1 #ei1 #gf ,wright1 ,e41 #f1 #bg1 #ch1 #ed ,wright1 ,s41 #ed ,transcriber's ,notes ,a handful of typos1 mostly misplaced punctuation1 were silently corrected4 ,,html and ,,utf text versions make heavy use of mathematical symbols3 particularly superscripts1 subscripts1 and combining characters4 ,some viewers may require user assistance to find fonts containing these characters4 ,the text versions miss much of the formatting1 especially in mathematical formulas3 --,several arithmetic examples must be viewed in a monospaced font 7which recognizes combining characters7 to be legible4 --,formulas under the horizontal line of a square root symbol are parenthesized4 --,subscripts are preceded by 804 --,superscripts are preceded by 804 --,italics1 used primarily in formulas and bibliographical entries1 are not indicated in the text4 --,italics in the index are delimited by 804 --,underlines are not indicated 7in particular1 in the fractional part of a decimal number in ,oughtred's notation74 --,the idiosyncratic 8greater than0 and 8less than0 symbols are indicated as symbol in the ,,ascii version4 --,overdots1 underdots1 and slashmarks around digits 7in the long division example7 are not indicated in the ,,ascii version4 --,greek letters are spelled out within curly brackets in the ,,ascii version4