William Oughtred

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

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

Brookes says in the preface:

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

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

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

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

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

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

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

The Appendix was an interesting and remarkable contribution to mathematics, for in its sixteen small pages it contains (1) the first use of the sign ×; (2) the first abbreviations, or symbols, for the sine, tangent, cosine, and cotangent; (3) the invention of the radix method of calculating logarithms; (4) the first table of hyperbolic logarithms.[55 - Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, (1915), p. 169. In this article Glaisher republishes the “Appendix” in full.]

CHAPTER IV

OUGHTRED’S INFLUENCE UPON MATHEMATICAL PROGRESS AND TEACHING

OUGHTRED AND HARRIOT

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

OUGHTRED’S PUPILS

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

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

He could not endure to see a scholar write an ill hand; he taught them all presently to mend their hands.[56 - Aubrey, op. cit., Vol. II, 1898, p. 108.]

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

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

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

OUGHTRED, THE “TODHUNTER OF THE SEVENTEENTHCENTURY”

To afford a clearer view of Oughtred as a teacher and mathematical expositor we quote some passages from various writers and from his correspondence. Anthony Wood[57 - Wood’s Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247.] gives an interesting account of how Seth Ward and Charles Scarborough went from Cambridge University to the obscure home of the country mathematician to be initiated into the mysteries of algebra:

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

Anthony Wood makes a similar statement about Thomas Henshaw:

While he remained in that coll. [University College, Oxford] which was five years.. he made an excursion for about 9 months to the famous mathematician Will. Oughtred parson of Aldbury in Surrey, by whom he was initiated in the study of mathematics, and afterwards retiring to his coll. for a time, he at length went to London, was entered a student in the Middle Temple.[58 - Wood, op. cit., Vol. II, p. 445.]

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

Amongst the mathematical rarities these times have afforded, there are none of that small number I (a late intruder into these studies) have yet viewed, which so fully demonstrates their authors’ great abilities as your Clavis, not richer in augmentations, than valuable for contraction;..

Your belief that there is in all inventions aliquid divinum, an infusion beyond human cogitations, I am confident will appear notably strengthened, if you please to afford this truth belief, that I entered upon these studies accidentally after I betook myself to the country, having never had so much aid as to be taught addition, nor the discourse of an artist (having left both Oxford and London before I knew what any proposition in geometry meant) to inform me what were the best authors.[59 - Rigaud, op. cit., Vol. I, pp. 33, 35.]

The following extracts from two letters by W. Robinson, written before the appearance of the 1647 English edition of the Clavis, express the feeling of many readers of the Clavis on its extreme conciseness and brevity of explanation:

I shall long exceedingly till I see your Clavis turned into a pick-lock; and I beseech you enlarge it, and explain it what you can, for we shall not need to fear either tautology or superfluity; you are naturally concise, and your clear judgment makes you both methodical and pithy; and your analytical way is indeed the only way…

I will once again earnestly entreat you, that you be rather diffuse in the setting forth of your English mathematical Clavis, than concise, considering that the wisest of men noted of old, and said stultorum infinitus est numerus, these arts cannot be made too easy, they are so abstruse of themselves, and men either so lazy or dull, that their fastidious wits take a loathing at the very entrance of these studies, unless it be sweetened on with plainness and facility. Brevity may well argue a learned author, that without any excess or redundance, either of matter or words, can give the very substance and essence of the thing treated of; but it seldom makes a learned scholar; and if one be capable, twenty are not; and if the master sum up in brief the pith of his own long labours and travails, it is not easy to imagine that scholars can with less labour than it cost their masters dive into the depths thereof.[60 - Rigaud, op. cit., Vol. I, pp. 16, 26.]

Here is the judgment of another of Oughtred’s friends:

… with the character I received from your and my noble friend Sir Charles Cavendish, then at Paris, of your second edition of the same piece, made me at my return into England speedily to get, and diligently peruse the same. Neither truly did I find my expectation deceived; having with admiration often considered how it was possible (even in the hardest things of geometry) to deliver so much matter in so few words, yet with such demonstrative clearness and perspicuity: and hath often put me in mind of learned Mersennus his judgment (since dead) of it, that there was more matter comprehended in that little book than in Diophantus, and all the ancients…[61 - Rigaud, op. cit., Vol. I, p. 66.]

Oughtred’s own feeling was against diffuseness in textbook writing. In his revisions of his Clavis the original character of that book was not altered. In his reply to W. Robinson, Oughtred said:

… But my art for all such mathematical inventions I have set down in my Clavis Mathematica, which therefore in my title I say is tum logisticae cum analyticae adeoque totius mathematicae quasi clavis, which if any one of a mathematical genius will carefully study, (and indeed it must be carefully studied,) he will not admire others, but himself do wonders. But I (such is my tenuity) have enough fungi vice cotis, acutum reddere quae ferrum valet, exsors ipsa secandi, or like the touchstone, which being but a stone, base and little worth, can shew the excellence and riches of gold.[62 - Ibid., Vol. I, p. 9.]

John Wallis held Oughtred’s Clavis in high regard. When in correspondence with John Collins concerning plans for a new edition, Wallis wrote in 1666-67, six years after the death of Oughtred:

… But for the goodness of the book in itself, it is that (I confess) which I look upon as a very good book, and which doth in as little room deliver as much of the fundamental and useful part of geometry (as well as of arithmetic and algebra) as any book I know; and why it should not be now acceptable I do not see. It is true, that as in other things so in mathematics, fashions will daily alter, and that which Mr. Oughtred designed by great letters may be now by others be designed by small; but a mathematician will, with the same ease and advantage, understand Ac, and a³ or aaa… And the like I judge of Mr. Oughtred’s Clavis, which I look upon (as those pieces of Vieta who first went in that way) as lasting books and classic authors in this kind; to which, notwithstanding, every day may make new additions…

But I confess, as to my own judgment, I am not for making the book bigger, because it is contrary to the design of it, being intended for a manual or contract; whereas comments, by enlarging it, do rather destroy it… But it was by him intended, in a small epitome, to give the substance of what is by others delivered in larger volumes…[63 - Rigaud, op. cit., Vol. II, p. 475.]

That there continued to be a group of students and teachers who desired a fuller exposition than is given by Oughtred is evident from the appearance, over fifty years after the first publication of the Clavis, of a booklet by Gilbert Clark, entitled Oughtredus Explicatus, London, 1682. A review of this appeared in the Acta Eruditorum (Leipzig, 1684), on p. 168, wherein Oughtred is named “clarissimus Angliae mathematicus.” John Collins wrote Wallis in 1666-67 that Clark, “who lives with Sir Justinian Isham, within seven miles of Northampton… intimates he wrote a comment on the Clavis, which lay long in the hands of a printer, by whom he was abused, meaning Leybourne.”[64 - Ibid., Vol. II, p. 471.]

We shall have occasion below to refer to Oughtred’s inability to secure a copy of a noted Italian mathematical work published a few years before. In those days the condition of the book trade in England must have been somewhat extraordinary. Dr. J. W. L. Glaisher throws some light upon this subject.[65 - J. W. L. Glaisher, “On Early Logarithmic Tables, and Their Calculators,” Philosophical Magazine, 4th Ser., Vol. XLV (1873), pp. 378, 379.] He found in the Calendar of State Papers, Domestic Series, 1637, a petition to Archbishop Laud in which it is set forth that when Hooganhuysen, a Dutchman, “heretofore complained of in the High Commission for importing books printed beyond the seas,” had been bound “not to bring in any more,” one Vlacq (the computer and publisher of logarithmic tables) “kept up the same agency and sold books in his stead… Vlacq is now preparing to go beyond the seas to avoid answering his late bringing over nine bales of books contrary to the decree of the Star Chamber.” Judgment was passed that, “Considering the ill-consequence and scandal that would arise by strangers importing and venting in this kingdom books printed beyond the seas,” certain importations be prohibited, and seized if brought over.

This want of easy intercommunication of results of scientific research in Oughtred’s time is revealed in the following letter, written by Oughtred to Robert Keylway, in 1645:

I speak this the rather, and am induced to a better confidence of your performance, by reason of a geometric-analytical art or practice found out by one Cavalieri, an Italian, of which about three years since I received information by a letter from Paris, wherein was praelibated only a small taste thereof, yet so that I divine great enlargement of the bounds of the mathematical empire will ensue. I was then very desirous to see the author’s own book while my spirits were more free and lightsome, but I could not get it in France. Since, being more stept into years, daunted and broken with the sufferings of these disastrous times, I must content myself to keep home, and not put out to any foreign discoveries.[66 - Rigaud, op. cit., Vol. I, p. 65.]

It was in 1655, when Oughtred was about eighty years old, that John Wallis, the great forerunner of Newton in Great Britain, began to publish his great researches on the arithmetic of infinites. Oughtred rejoiced over the achievements of his former pupil. In 1655, Oughtred wrote John Wallis as follows:

I have with unspeakable delight, so far as my necessary businesses, the infirmness of my health, and the greatness of my age (approaching now to an end) would permit, perused your most learned papers, of several choice arguments, which you sent me: wherein I do first with thankfulness acknowledge to God, the Father of lights, the great light he hath given you; and next I congratulate you, even with admiration, the clearness and perspicacity of your understanding and genius, who have not only gone, but also opened a way into these profoundest mysteries of art, unknown and not thought of by the ancients. With which your mysterious inventions I am the more affected, because full twenty years ago, the learned patron of learning, Sir Charles Cavendish, shewed me a paper written, wherein were some few excellent new theorems, wrought by the way, as I suppose, of Cavalieri, which I wrought over again more agreeably to my way. The paper, wherein I wrought it, I shewed to many, whereof some took copies, but my own I cannot find. I mention it for this, because I saw therein a light breaking out for the discovery of wonders to be revealed to mankind, in this last age of the world: which light I did salute as afar off, and now at a nearer distance embrace in your prosperous beginnings. Sir, that you are pleased to mention my name in your never dying papers, that is your noble favour to me, who can add nothing to your glory, but only my applause…[67 - Rigaud, op. cit., Vol. I, p. 87.]

The last sentence has reference to Wallis’ appreciative and eulogistic reference to Oughtred in the preface. It is of interest to secure the opinion of later English writers who knew Oughtred only through his books. John Locke wrote in his journal under the date, June 24, 1681, “the best algebra yet extant is Outred’s.”[68 - King’s Life of John Locke, Vol. I, London, 1830, p. 227.] John Collins, who is known in the history of mathematics chiefly through his very extensive correspondence with nearly all mathematicians of his day, was inclined to be more critical. He wrote Wallis about 1667:

It was not my intent to disparage the author, though I know many that did lightly esteem him when living, some whereof are at rest, as Mr. Foster and Mr. Gibson… You grant the author is brief, and therefore obscure, and I say it is but a collection, which, if himself knew, he had done well to have quoted his authors, whereto the reader might have repaired. You do not like those words of Vieta in his theorems, ex adjunctione plano solidi, plus quadrato quadrati, etc., and think Mr. Oughtred the first that abridged those expressions by symbols; but I dissent, and tell you ’twas done before by Cataldus, Geysius, and Camillus Gloriosus,[69 - Exercitationum Mathematicarum Decas prima, Naples, 1627, and probably Cataldus’ Transformatio Geometrica, Bonon., 1612.] who in his first decade of exercises, (not the first tract,) printed at Naples in 1627, which was four years before the first edition of the Clavis, proposeth this equation just as I here give it you, viz. 1ccc+16qcc+41qqc-2304cc-18364qc-133000qq-54505c+3728q+8064 Naequatur 4608, finds N or a root of it to be 24, and composeth the whole out of it for proof, just in Mr. Oughtred’s symbols and method. Cataldus on Vieta came out fifteen years before, and I cannot quote that, as not having it by me.

… And as for Mr. Oughtred’s method of symbols, this I say to it; it may be proper for you as a commentator to follow it, but divers I know, men of inferior rank that have good skill in algebra, that neither use nor approve it… Is not A⁵ sooner wrote than Aqc? Let A be 2, the cube of 2 is 8, which squared is 64: one of the questions between Maghet Grisio and Gloriosus is whether 64=Acc or Aqc. The Cartesian method tells you it is A⁶, and decides the doubt…[70 - Rigaud, op. cit., Vol. II, pp. 477-80.]

There is some ground for the criticisms passed by Collins. To be sure, the first edition of the Clavis is dated 1631 – six years before Descartes suggested the exponential notation which came to be adopted as the symbolism in our modern algebra. But the second edition of the Clavis, 1647, appeared ten years after Descartes’ innovation. Had Oughtred seen fit to adopt the new exponential notation in 1647, the step would have been epoch-making in the teaching of algebra in England. We have seen no indication that Oughtred was familiar with Descartes’ Géométrie of 1637.