“CIRCLES OF PROPORTION” AND “TRIGONOMETRIE”
Oughtred wrote and had published three important mathematical books, the Clavis, the Circles of Proportion,[37] and a Trigonometrie.[38] This last appeared in the year 1657 at London, in both Latin and English.
It is claimed that the trigonometry was “neither finished nor published by himself, but collected out of his scattered papers; and though he connived at the printing it, yet imperfectly done, as appears by his MSS.; and one of the printed Books, corrected by his own Hand.”[39] Doubtless more accurate on this point is a letter of Richard Stokes who saw the book through the press:
I have procured your Trigonometry to be written over in a fair hand, which when finished I will send to you, to know if it be according to your mind; for I intend (since you were pleased to give your assent) to endeavour to print it with Mr. Briggs his Tables, and so soon as I can get the Prutenic Tables I will turn those of the sun and moon, and send them to you.[40]
In the preface to the Latin edition Stokes writes:
Since this trigonometry was written for private use without the intention of having it published, it pleased the Reverend Author, before allowing it to go to press, to expunge some things, to change other things and even to make some additions and insert more lucid methods of exposition.
This much is certain, the Trigonometry bears the impress characteristic of Oughtred. Like all his mathematical writings, the book was very condensed. Aside from the tables, the text covered only 36 pages. Plane and spherical triangles were taken up together. The treatise is known in the history of trigonometry as among the very earliest works to adopt a condensed symbolism so that equations involving trigonometric functions could be easily taken in by the eye. In the work of 1657, contractions are given as follows: s=sine, t=tangent, se=secant, s co=cosine (sine complement), t co=cotangent, se co=cosecant, log=logarithm, Z cru=sum of the sides of a rectangle or right angle, X cru=difference of these sides. It has been generally overlooked by historians that Oughtred used the abbreviations of trigonometric functions, named above, a quarter of a century earlier, in his Circles of Proportion, 1632, 1633. Moreover, he used sometimes also the abbreviations which are current at the present time, namely sin=sine, tan=tangent, sec=secant. We know that the Circles of Proportion existed in manuscript many years before they were published. The symbol sv for sinus versus occurs in the Clavis of 1631. The great importance of well-chosen symbols needs no emphasis to readers of the present day. With reference to Oughtred’s trigonometric symbols. Augustus De Morgan said:
This is so very important a step, simple as it is, that Euler is justly held to have greatly advanced trigonometry by its introduction. Nobody that we know of has noticed that Oughtred was master of the improvement, and willing to have taught it, if people would have learnt.[41]
We find, however, that even Oughtred cannot be given the whole credit in this matter. By or before 1631 several other writers used abbreviations of the trigonometric functions. As early as 1624 the contractions sin for sine and tan for tangent appear on the drawing representing Gunter’s scale, but Gunter did not use them in his books, except in the drawing of his scale.[42] A closer competitor for the honor of first using these trigonometric abbreviations is Richard Norwood in his Trigonometrie, London, 1631, where s stands for sine, t for tangent, sc for sine complement (cosine), tc for tangent complement (cotangent), and sec for secant. Norwood was a teacher of mathematics in London and a well-known writer of books on navigation. Aside from the abbreviations just cited Norwood did not use nearly as much symbolism in his mathematics as did Oughtred.
Mention should be made of trigonometric symbols used even earlier than any of the preceding, in “An Appendix to the Logarithmes, shewing the practise of the Calculation of Triangles, etc.,” printed in Edward Wright’s edition of Napier’s A Description of the Admirable Table of Logarithmes, London, 1618. We referred to this “Appendix” in tracing the origin of the sign ×. It contains, on p. 4, the following passage: “For the Logarithme of an arch or an angle I set before (s), for the antilogarithme or compliment thereof (s*) and for the Differential (t).” In further explanation of this rather unsatisfactory passage, the author (Oughtred?) says, “As for example: sB+BC=CA. that is, the Logarithme of an angle B. at the Base of a plane right-angled triangle, increased by the addition of the Logarithm of BC, the hypothenuse thereof, is equall to the Logarithme of CA the cathetus.”
Here “logarithme of an angle B” evidently means “log sin B,” just as with Napier, “Logarithms of the arcs” signifies really “Logarithms of the sines of the angles.” In Napier’s table, the numbers in the column marked “Differentiae” signify log sine minus log cosine of an angle; that is, the logarithms of the tangents. This explains the contraction (t) in the “Appendix.” The conclusion of all this is that as early as 1618 the signs s, s*, t were used for sine, cosine, and tangent, respectively.
John Speidell, in his Breefe Treatise of Sphaericall Triangles, London, 1627, uses Si. for sine, T. and Tan for tangent, Se. for secant, Si. Co. for cosine, Se. Co. for cosecant, T. Co. for cotangent.
The innovation of designating the sides and angles of a triangle by A, B, C, and a, b, c, so that A was opposite a, B opposite b, and C opposite c, is attributed to Leonard Euler (1753), but was first used by Richard Rawlinson of Queen’s College, Oxford, sometimes after 1655 and before 1668. Oughtred did not use Rawlinson’s notation.[43]
In trigonometry English writers of the first half of the seventeenth century used contractions more freely than their continental contemporaries; even more freely, indeed, than English writers of a later period. Von Braunmühl, the great historian of trigonometry, gives Oughtred much praise for his trigonometry, and points out that half a century later the army of writers on trigonometry had hardly yet reached the standard set by Oughtred’s analysis.[44] Oughtred must be credited also with the first complete proof that was given to the first two of “Napier’s analogies.” His trigonometry contains seven-place tables of sines, tangents, and secants, and six-place tables of logarithmic sines and tangents; also seven-place logarithmic tables of numbers. At the time of Oughtred there was some agitation in favor of a wider introduction of decimal systems. This movement is reflected in those tables which contain the centesimal division of the degree, a practice which is urged for general adoption in our own day, particularly by the French.