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]
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.