William Oughtred

Z=A+E and A>E;

he lets also X=A-E. From these relations he obtains identities which, in modern notation, are ¼Z²-AE=(½Z-E)²=¼X². Now, if we know Z and AE, we can find ½X. Then ½(Z+X)=A, and ½(Z-X)=E, and

A=½Z+√¼Z²-AE.

Having established these preliminaries, he proceeds thus:

Datis igitur linea inaequaliter secta Z (10), & rectangulo sub segmentis AE (21) qui gnomon est: datur semidifferentia segmentorum ½X: & per consequens ipsa segmenta. Nam ponatur alterutrum segmentum A: alterum erit Z-A: Rectangulum auctem est ZA-Aq=AE. Et quia dantur Z & AE: estque ¼Zq-AE=¼Xq: & per 5c. 18, ½Z+½X=A: & ½Z-½X=E: Aequatio sic resoluetur: ½Z±√q:¼Zq-AE:=A

Itaque proposita equatione, in qua sunt tres species aequaliter in ordine tabellae adscendentes, altissima autem species ponitur negata: Magnitudo data coefficiens mediam speciem est linea bisecanda: & magnitudo absoluta data, ad quam sit aequatio, est rectangulum sub segmentis inaequalibus, sine gnomon: vt ZA-Aq=AE: in numeris autem 10l-lq=21: Estque A, vel 1l, alterutrum segmentum inaequale. Inuenitur autem sic:

Dimidiata coefficiens median speciem est

(5); cuius quadratum est

(25): ex hoc tolle AE (21) absolutum: eritque

(4) quadratum semidifferentiae segmentorum: latus huius quadratum (2) est semidifferentia: quam si addas ad

(5) semissem coefficientis, sive lineae bisecandae, erit maius segment.; sin detrahas, erit minus segment: Dico

We translate the Latin passage, using the modern exponential notation and parentheses, as follows:

Given therefore an unequally divided line Z (10), and a rectangle beneath the segments AE (21) which is a gnomon. Half the difference of the segments ½X is given, and consequently the segment itself. For, if one of the two segments is placed equal to A, the other will be Z-A. Moreover, the rectangle is ZA-A²=AE. And because Z and AE are given, and there is ¼Z²-AE=¼X², and by 5c.18, ½Z+½X=A, and ½Z-½X=E, the equation will be solved thus: ½Z±√(¼Z²-AE)=A

And so an equation having been proposed in which three species (terms) are in equally ascending powers, the highest species, moreover, being negative, the given magnitude which constitutes the middle species is the line to be bisected. And the given absolute magnitude to which it is equal is the rectangle beneath the unequal segments, without gnomon. As ZA-A²=AE, or in numbers, 10x-x²=21. And A or x is one of the two unequal segments. It may be found thus:

The half of the middle species is

(5), its square is

(25). From it subtract the absolute term AE (21), and

(4) will be the square of half the difference of the segments. The square root of this,

(2), is half the difference. If you add it to half the coefficient

(5), or half the line to be bisected, the longer segment is obtained; if you subtract it, the smaller segment is obtained. I say:

The quadratic equation Aq+ZA=AE receives similar treatment. This and the preceding equation, ZA-Aq=AE, constitute together a solution of the general quadratic equation, x²+ax=b, provided that E or Z are not restricted to positive values, but admit of being either positive or negative, a case not adequately treated by Oughtred. Imaginary numbers and imaginary roots receive no consideration whatever.

A notation suggested by Vieta and favored by Girard made vowels stand for unknowns and consonants for knowns. This conventionality was adopted by Oughtred in parts of his algebra, but not throughout. Near the beginning he used Q to designate the unknown, though usually this letter stood with him for the “square” of the expression after it.[34 - We have noticed the representation of known quantities by consonants and the unknown by vowels in Wingate’s Arithmetick made easie, edited by John Kersey, London, 1650, algebra, p. 382; and in the second part, section 19, of Jonas Moore’s Arithmetick in two parts, London, 1660, Moore suggests as an alternative the use of z, y, x, etc., for the unknowns. The practice of representing unknowns by vowels did not spread widely in England.]

It is of some interest that Oughtred used

to signify the ratio of the circumference to the diameter of a circle. Very probably this notation is the forerunner of the π=3.14159.. used in 1706 by William Jones. Oughtred first used

in the 1647 edition of the Clavis mathematicae. In the 1652 edition he says, “Si in circulo sit 7.22::δ·π::113.355: erit δ·π::2 R.P: periph.” This notation was adopted by Isaac Barrow, who used it extensively. David Gregory[35 - Philosophical Transactions, Vol. XIX, No. 231, London, p. 652.] used

in 1697, and De Moivre[36 - Ibid., Vol. XIX, p. 56.] used

about 1697, to designate the ratio of the circumference to the radius.

We quote the description of the Clavis that was given by Oughtred’s greatest pupil, John Wallis. It contains additional information of interest to us. Wallis devotes chap. xv of his Treatise of Algebra, London, 1685, pp. 67-69, to Mr. Oughtred and his Clavis, saying:

Mr. William Oughtred (our Country-man) in his Clavis Mathematicae, (or Key of Mathematicks,) first published in the Year 1631, follows Vieta (as he did Diophantus) in the use of the Cossick Denominations; omitting (as he had done) the names of Sursolids, and contenting himself with those of Square and Cube, and the Compounds of these.

But he doth abridge Vieta’s Characters or Species, using only the letters q, c, &c. which in Vieta are expressed (at length) by Quadrate, Cube, &c. For though when Vieta first introduced this way of Specious Arithmetick, it was more necessary (the thing being new,) to express it in words at length: Yet when the thing was once received in practise, Mr. Oughtred (who affected brevity, and to deliver what he taught as briefly as might be, and reduce all to a short view,) contented himself with single Letters instead of those words.

Thus what Vieta would have written

would with him be thus expressed

And the better to distinguish upon the first view, what quantities were Known, and what Unknown, he doth (usually) denote the Known to Consonants, and the Unknown by Vowels; as Vieta (for the same reason) had done before him.

He doth also (to very great advantage) make use of several Ligatures, or Compendious Notes, to signify Summs, Differences, and Rectangles of several Quantities. As for instance, Of two Quantities A (the Greater), and E (the Lesser), the Sum he calls Z, the Difference X, the Rectangle AE…

Which being of (almost) a constant signification with him throughout, do save a great circumlocution of words, (each Letter serving instead of a Definition;) and are also made use of (with very great advantage) to discover the true nature of divers intricate Operations, arising from the various compositions of such Parts, Sums, Differences, and Rectangles; (of which there is great plenty in his Clavis, Cap. 11, 16, 18, 19. and elsewhere,) which without such Ligatures, or Compendious Notes, would not be easily discovered or apprehended…

I know there are who find fault with his Clavis, as too obscure, because so short, but without cause; for his words be always full, but not Redundant, and need only a little attention in the Reader to weigh the force of every word, and the Syntax of it;.. And this, when once apprehended, is much more easily retained, than if it were expressed with the prolixity of some other Writers; where a Reader must first be at the pains to weed out a great deal of superfluous Language, that he may have a short prospect of what is material; which is here contracted for him in a short Synopsis…

Mr. Oughtred in his Clavis, contents himself (for the most part) with the solution of Quadratick Equations, without proceeding (or very sparingly) to Cubick Equations, and those of Higher Powers; having designed that Work for an Introduction into Algebra so far, leaving the Discussion of Superior Equations for another work… He contents himself likewise in Resolving Equations, to take notice of the Affirmative or Positive Roots; omitting the Negative or Ablative Roots, and such as are called Imaginary or Impossible Roots. And of those which, he calls Ambiguous Equations, (as having more Affirmative Roots than one,) he doth not (that I remember) any where take notice of more than Two Affirmative Roots: (Because in Quadratick Equations, which are those he handleth, there are indeed no more.) Whereas yet in Cubick Equations, there may be Three, and in those of Higher Powers, yet more. Which Vieta was well aware of, and mentioneth in some of his Writings; and of which Mr. Oughtred could not be ignorant.

“CIRCLES OF PROPORTION” AND “TRIGONOMETRIE”

Oughtred wrote and had published three important mathematical books, the Clavis, the Circles of Proportion,[37 - There are two title-pages to the edition of 1632. The first title-page is as follows: The Circles of Proportion and The Horizontall Instrument. Both invented, and the vses of both Written in Latine by Mr. W. O. Translated into English: and set forth for the publique benefit by William Forster. London. Printed for Elias Allen maker of these and all other mathematical Instruments, and are to be sold at his shop over against St. Clements church with out Temple-barr. 1632. T. Cecill Sculp.In 1633 there was added the following, with a separate title-page: An addition vnto the Vse of the Instrvment called the Circles of Proportion… London, 1633, this being followed by Oughtred’s To the English Gentrie etc. In the British Museum there is a copy of another impression of the Circles of Proportion, dated 1639, with the Addition vnto the Vse of the Instrument etc., bearing the original date, 1633, and with the epistle, To the English Gentrie, etc., inserted immediately after Forster’s dedication, instead of at the end of the volume.] and a Trigonometrie.[38 - The complete title of the English edition is as follows: Trigonometrie, or, The manner of calculating the Sides and Angles of Triangles, by the Mathematical Canon, demonstrated. By William Oughtred Etonens. And published by Richard Stokes Fellow of Kings Colledge in Cambridge, and Arthur Haughton Gentleman. London, Printed by R. and W. Leybourn, for Thomas Johnson at the Golden Key in St. Pauls Church-yard. M.DC.LVII.] This last appeared in the year 1657 at London, in both Latin and English.

It is claimed that the trigonometry was “neither finished nor published by himself, but collected out of his scattered papers; and though he connived at the printing it, yet imperfectly done, as appears by his MSS.; and one of the printed Books, corrected by his own Hand.”[39 - Jer. Collier, The Great Historical, Geographical, Genealogical and Poetical Dictionary, Vol. II, London, 1701, art. “Oughtred.”] Doubtless more accurate on this point is a letter of Richard Stokes who saw the book through the press:

I have procured your Trigonometry to be written over in a fair hand, which when finished I will send to you, to know if it be according to your mind; for I intend (since you were pleased to give your assent) to endeavour to print it with Mr. Briggs his Tables, and so soon as I can get the Prutenic Tables I will turn those of the sun and moon, and send them to you.[40 - Rigaud op. cit., Vol. I, p. 82.]

In the preface to the Latin edition Stokes writes:

Since this trigonometry was written for private use without the intention of having it published, it pleased the Reverend Author, before allowing it to go to press, to expunge some things, to change other things and even to make some additions and insert more lucid methods of exposition.

This much is certain, the Trigonometry bears the impress characteristic of Oughtred. Like all his mathematical writings, the book was very condensed. Aside from the tables, the text covered only 36 pages. Plane and spherical triangles were taken up together. The treatise is known in the history of trigonometry as among the very earliest works to adopt a condensed symbolism so that equations involving trigonometric functions could be easily taken in by the eye. In the work of 1657, contractions are given as follows: s=sine, t=tangent, se=secant, s co=cosine (sine complement), t co=cotangent, se co=cosecant, log=logarithm, Z cru=sum of the sides of a rectangle or right angle, X cru=difference of these sides. It has been generally overlooked by historians that Oughtred used the abbreviations of trigonometric functions, named above, a quarter of a century earlier, in his Circles of Proportion, 1632, 1633. Moreover, he used sometimes also the abbreviations which are current at the present time, namely sin=sine, tan=tangent, sec=secant. We know that the Circles of Proportion existed in manuscript many years before they were published. The symbol sv for sinus versus occurs in the Clavis of 1631. The great importance of well-chosen symbols needs no emphasis to readers of the present day. With reference to Oughtred’s trigonometric symbols. Augustus De Morgan said:

This is so very important a step, simple as it is, that Euler is justly held to have greatly advanced trigonometry by its introduction. Nobody that we know of has noticed that Oughtred was master of the improvement, and willing to have taught it, if people would have learnt.[41 - A. De Morgan, Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago, 1915, Vol. II, p. 303.]

We find, however, that even Oughtred cannot be given the whole credit in this matter. By or before 1631 several other writers used abbreviations of the trigonometric functions. As early as 1624 the contractions sin for sine and tan for tangent appear on the drawing representing Gunter’s scale, but Gunter did not use them in his books, except in the drawing of his scale.[42 - E. Gunter, Description and Use of the Sector, the Crosse-staffe and other Instruments, London, 1624, second book, p. 31.] A closer competitor for the honor of first using these trigonometric abbreviations is Richard Norwood in his Trigonometrie, London, 1631, where s stands for sine, t for tangent, sc for sine complement (cosine), tc for tangent complement (cotangent), and sec for secant. Norwood was a teacher of mathematics in London and a well-known writer of books on navigation. Aside from the abbreviations just cited Norwood did not use nearly as much symbolism in his mathematics as did Oughtred.

Mention should be made of trigonometric symbols used even earlier than any of the preceding, in “An Appendix to the Logarithmes, shewing the practise of the Calculation of Triangles, etc.,” printed in Edward Wright’s edition of Napier’s A Description of the Admirable Table of Logarithmes, London, 1618. We referred to this “Appendix” in tracing the origin of the sign ×. It contains, on p. 4, the following passage: “For the Logarithme of an arch or an angle I set before (s), for the antilogarithme or compliment thereof (s*) and for the Differential (t).” In further explanation of this rather unsatisfactory passage, the author (Oughtred?) says, “As for example: sB+BC=CA. that is, the Logarithme of an angle B. at the Base of a plane right-angled triangle, increased by the addition of the Logarithm of BC, the hypothenuse thereof, is equall to the Logarithme of CA the cathetus.”

Here “logarithme of an angle B” evidently means “log sin B,” just as with Napier, “Logarithms of the arcs” signifies really “Logarithms of the sines of the angles.” In Napier’s table, the numbers in the column marked “Differentiae” signify log sine minus log cosine of an angle; that is, the logarithms of the tangents. This explains the contraction (t) in the “Appendix.” The conclusion of all this is that as early as 1618 the signs s, s*, t were used for sine, cosine, and tangent, respectively.

John Speidell, in his Breefe Treatise of Sphaericall Triangles, London, 1627, uses Si. for sine, T. and Tan for tangent, Se. for secant, Si. Co. for cosine, Se. Co. for cosecant, T. Co. for cotangent.

The innovation of designating the sides and angles of a triangle by A, B, C, and a, b, c, so that A was opposite a, B opposite b, and C opposite c, is attributed to Leonard Euler (1753), but was first used by Richard Rawlinson of Queen’s College, Oxford, sometimes after 1655 and before 1668. Oughtred did not use Rawlinson’s notation.[43 - F. Cajori, “On the History of a Notation in Trigonometry,” Nature, Vol. XCIV, 1915, pp. 642, 643.]