MINOR WORKS
Among the minor works of Oughtred must be ranked his booklet of forty
pages to which reference has already been made, entitled, The New
Artificial Gauging Line or Rod, London, 1633. 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 instruments. 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 “J. S.” (John Smith?).
Oughtred’s tract appeared with its own title-page, but with pagination
continued from the preceding part, as An Appendix wherein is contained a
Method of Calculating all Numbers for Watches. Written originally by that
famous Mathematician Mr. William Oughtred, and now made Publick. By J. S.
of London, Clock-maker. London, 1675.
“J. S.” says in his preface:
The method following was many years since Compiled by Mr. Oughtred for
the use of some Ingenious Gentlemen his friends, who for recreation at
the University, studied to find out the reason and Knowledge of
Watch-work, which seemed also to be a thing with which Mr. Oughtred
himself was much affected, as may in part appear by his putting out of
his own Son to the same Trade, for whose use (as I am informed) he did
compile a larger tract, but what became of it cannot be known.
Notwithstanding Oughtred’s marked activity in the design of mathematical
instruments, and his use of surveying instruments, he always spoke in
deprecating terms of their importance and their educational value. In his
epistle against Delamain he says:
The Instruments I doe not value or weigh one single penny. If I had
been ambitious of praise, or had thought them (or better then they)
worthy, at which to have taken my rise, out of my secure and quiet
obscuritie, to mount up into glory, and the knowledge of men: I could
have done it many yeares before. . . . .
Long agoe, when I was a young student of the Mathematicall Sciences, I
tryed many wayes and devices to fit my selve with some good Diall or
Instrument portable for my pocket, to finde the houre, and try other
conclusions by, and accordingly framed for that my purpose both
Quadrants, and Rings, and Cylinders, and many other composures. Yet not
to my full content and satisfaction; for either they performed but
little, or els were patched up with a diversity of lines by an
unnaturall and forced contexture. At last I . . . . found what I had
before with much studie and paines in vaine sought for.[54]
Mention has been made in the previous pages of two of his papers on
sun-dials, prepared (as he says) when he was in his twenty-third year.
The first was published in the Clavis of 1647. The second paper appeared
in his Circles of Proportion.
Both before and after the time of Oughtred much was written on sun-dials.
Such instruments were set up against the walls of prominent buildings,
much as the faces of clocks in our time. The inscriptions that were put
upon sun-dials are often very clever: “I count only the hours of
sunshine,” “Alas, how fleeting.” A sun-dial on the grounds of Merchiston
Castle, in Edinburgh, where the inventor of logarithms, John Napier,
lived for many years, bears the inscription, “Ere time be tint, tak tent
of time” (Ere time be lost, take heed of time).
Portable sun-dials were sometimes carried in pockets, as we carry
watches. Thus Shakespeare, in As You Like It, Act II, sc. vii:
“And then he drew a diall from his poke.”
Watches were first made for carrying in the pocket about 1658.
Because of this literary, scientific, 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-measurement.
Besides the accounts previously noted, there came from his pen: The
Description and Use of the double Horizontall Dyall: Whereby not onely
the hower of the day is shewne; but also the Meridian Line is found: And
most Astronomical Questions, which may be done by the Globe, are
resolved. Invented and written by W. O., London, 1636.
The “Horizontall Dyall” and “Horologicall Ring” appeared again as
appendixes to Oughtred’s translation from the French of a book on
mathematical recreations.
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, log_e 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]