William Oughtred

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 - A. von Braunmühl, Geschichte der Trigonometrie, 2. Teil, Leipzig, 1903, pp. 42, 91.] 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.

SOLUTION OF NUMERICAL EQUATIONS

In the solution of numerical equations Oughtred does not mention the sources from which he drew, but the method is substantially that of the great French algebraist Vieta, as explained in a publication which appeared in 1600 in Paris under the title, De numerosa potestatum purarum atque adfectarum ad exegesin resolutione tractatus. In view of the fact that Vieta’s process has been described inaccurately by leading modern historians including H. Hankel[45 - H. Hankel, Geschichte der Mathematik in Alterthum und Mittelalter, Leipzig, 1874, pp. 369, 370.] and M. Cantor,[46 - M. Cantor, Vorlesungen über Geschichte der Mathematik, II, 1900, pp. 640, 641.] it may be worth while to go into some detail.[47 - This matter has been discussed in a paper by F. Cajori, “A History of the Arithmetical Methods of Approximation, etc., Colorado College Publication, General Series No. 51, 1910, pp. 182-84. Later this subject was again treated by G. Eneström in Bibliotheca mathematica, 3. Folge, Vol. XI, 1911, pp. 234, 235.] By them it is made to appear as identical with the procedure given later by Newton. The two are not the same. The difference lies in the divisor used. What is now called “Newton’s method” is Newton’s method as modified by Joseph Raphson.[48 - See F. Cajori, op. cit., p. 193.] The Newton-Raphson method of approximation to the roots of an equation f(x)=0 is usually given the form a-[f(a)/f´(a)], where a is an approximate value of the required root. It will be seen that the divisor is f´(a). Vieta’s divisor is different; it is

|f(a+s₁) -f(a)|-s₁ⁿ,

where f(x) is the left of the equation f(x)=k, n is the degree of equation, and s₁ is a unit of the denomination of the digit next to be found. Thus in x³+420000x=247651713, it can be shown that 417 is approximately a root; suppose that a has been taken to be 400, then s₁=10; but if, at the next step of approximation, a is taken to be 410, then s₁=1. In this example, taking a=400, Vieta’s divisor would have been 9120000; Newton’s divisor would have been 900000.

A comparison of Vieta’s method with the Newton-Raphson method reveals the fact that Vieta’s divisor is more reliable, but labors under the very great disadvantage of requiring a much larger amount of computation. The latter divisor is accurate enough and easier to compute. Altogether the Newton-Raphson process marks a decided advance over that of Vieta.

As already stated, it is the method of Vieta that Oughtred explains. The Englishman’s exposition is an improvement on that of Vieta, printed forty years earlier. Nevertheless, Oughtred’s explanation is far from easy to follow. The theory of equations was at that time still in its primitive stage of development. Algebraic notation was not sufficiently developed to enable the argument to be condensed into a form easily surveyed. So complicated does Vieta’s process of approximation appear that M. Cantor failed to recognize that Vieta possessed a uniform mode of procedure. But when one has in mind the general expression for Vieta’s divisor which we gave above, one will recognize that there was marked uniformity in Vieta’s approximations.

Oughtred allows himself twenty-eight sections in which to explain the process and at the close cannot forbear remarking that 28 is a “perfect” number (being equal to the sum of its divisors, 1, 2, 4, 7, 14).

The early part of his exposition shows how an equation may be transformed so as to make its roots 10, 100, 1000, or 10

times smaller. This simplifies the task of “locating a root”; that is, of finding between what integers the root lies.

Taking one of Oughtred’s equations, x⁴-72x³+238600x=8725815, upon dividing 72x³ by 10, 238600x by 1000, and 8725815 by 10,000, we obtain x⁴-7·2x³+238·6x=872·5. Dividing both sides by x, we obtain x³+238·6-7·2x²=x)872·5. Letting x=4, we have 64+238·6-115·2=187·4.

But 4)872·5(218·1; 4 is too small. Next let x=5, we have 125+238·6-180=183·6.

But 5)872·5(174·5; 5 is too large. We take the lesser value, x=4, or in the original equation, x=40. This method may be used to find the second digit in the root. Oughtred divides both sides of the equation by x², and obtains x²+x)238600-72x=x²)8725815. He tries x=47 and x=48, and finds that x=47.

He explains also how the last computation may be done by logarithms. Thereby he established for himself the record of being the first to use logarithms in the solution of affected equations.

As an illustration of Oughtred’s method of approximation after the root sought has been located, we have chosen for brevity a cubic in preference to a quartic. We selected the equation x³+420000x=247651713. By the process explained above a root is found to lie between x=400 and x=500. From this point on, the approximation as given by Oughtred is as shown on p. 43 (#Page_43).

In further explanation of this process, observe that the given equation is of the form Lc+CqL=Dc, where Lc is our x, Cq=420000, Dc=247651713. In the first step of approximation, let L=A+E, where A=400 and E is, as yet, undetermined. We have

Lc=(A+E)³=A³+3A²E+3AE²+E³

and

CqL=420000(A+E).

Subtract from 247651713 the sum of the known terms A³ (his Ac) and 420000 A (his CqA). This sum is 232000000 the remainder is 15651713.

Next, he evaluates the coefficients of E in 3A²E and 420000E, also 3A, the coefficient of E². He obtains 3A²=480000, 3A=1200, Cq=420000. He interprets 3A² and Cq as tens, 3A as hundreds. Accordingly, he obtains as their sum 9120000, which is the divisor for finding the second digit in the approximation. Observe that this divisor is the value of |f(a+s₁) -f(a)|-s₁ⁿ in our general expression, where a=400, s₁=10, n=3, f(x)=x³+420000x.

Dividing the remainder 15651713 by 9120000, he obtains the integer 1 in ten’s place; thus E=10, approximately. He now computes the terms 3A²E, 3AE² and E³ to be, respectively, 4800000, 120000, 1000. Their sum is 9121000. Subtracting it from the previous remainder, 15651713, leaves the new remainder, 6530713.

From here on each step is a repetition of the preceding step. The new A is 410, the new E is to be determined. We have now in closer approximation, L=A+E. This time we do not subtract A³ and CqA, because this subtraction is already affected by the preceding work.

We find the second trial divisor by computing the sum of 3A², 3A and Cq; that is, the sum of 504300, 1230, 420000, which is 925530. Again, this divisor can be computed by our general expression for divisors, by taking a=410, s₁=1, n=3.

Dividing 6530713 by 925530 yields the integer 7. Thus E=7. Computing 3A²E, 3AE², E³ and subtracting their sum, the remainder is 0. Hence 417 is an exact root of the given equation.

Since the extraction of a cube root is merely the solution of a pure cubic equation, x³=n, the process given above may be utilized in finding cube roots. This is precisely what Oughtred does in chap. xiv of his Clavis. If the foregoing computation is modified by putting Cq=0, the process will yield the approximate cube root of 247651713.

Oughtred solves 16 examples by the process of approximation here explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In all cases he finds only one or two real roots. Of the roots sought, five are irrational, the remaining are rational and are computed to their exact values. Three of the computed roots have 2 figures each, 9 roots have 3 figures each, 4 roots have 4 figures each. While no attempt is made to secure all the roots – methods of computing complex roots were invented much later – he computes roots of equations which involve large coefficients and some of them are of a degree as high as the fifth. In view of the fact that many editions of the Clavis were issued, one impression as late as 1702, it contributed probably more than any other book to the popularization of Vieta’s method in England.

Before Oughtred, Thomas Harriot and William Milbourn are the only Englishmen known to have solved numerical equations of higher degrees. Milbourn published nothing. Harriot slightly modified Vieta’s process by simplifying somewhat the formation of the trial divisor. This method of approximation was the best in existence in Europe until the publication by Wallis in 1685 of Newton’s method of approximation.

It should be stated that, before the time of Newton, the best method of approximation to the roots of numerical equations existed, not in Europe, but in China. As early as the thirteenth century the Chinese possessed a method which is almost identical with what is known today as “Horner’s method.”

LOGARITHMS

Oughtred’s treatment of logarithms is quite in accordance with the more recent practice.[49 - See William Oughtred’s Key of the Mathematicks, London, 1694, pp. 173-75, tract, “Of the Resolution of the Affected Equations,” or any edition of the Clavis after the first.] He explains the finding of the “index” (our “characteristic”); he states that “the sum of two Logarithms is the Logarithm of the Product of their Valors; and their difference is the Logarithm of the Quotient,” that “the Logarithm of the side [436] drawn upon the Index number [2] of dimensions of any Potestas is the logarithm of the same Potestas” [436²], that “the logarithm of any Potestas [436²] divided by the number of its dimensions [2] affordeth the Logarithm of its Root [436].” These statements of Oughtred occur for the first time in the Key of the Mathematicks of 1647; the Clavis of 1631 contains no treatment of logarithms.

If the characteristic of a logarithm is negative, Oughtred indicates this fact by placing the – above the characteristic. He separates the characteristic and mantissa by a comma, but still uses the sign |_ to indicate decimal fractions. He uses the contraction “log.”

INVENTION OF THE SLIDE RULE; CONTROVERSY ON PRIORITY OF INVENTION

Oughtred’s most original line of scientific activity is the one least known to the present generation. Augustus De Morgan, in speaking of Oughtred, who was sometimes called “Oughtred Aetonensis,” remarks: “He is an animal of extinct race, an Eton mathematician. Few Eton men, even of the minority which knows what a sliding rule is, are aware that the inventor was of their own school and college.”[50 - A. De Morgan, op. cit., p. 451; 2d ed., Vol. II, p. 303.] The invention of the slide rule has, until recently,[51 - See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909, pp. 7-14, Addenda, p. ii.] been a matter of dispute; it has been erroneously ascribed to Edmund Gunter, Edmund Wingate, Seth Partridge, and others. We have been able to establish that William Oughtred was the first inventor of slide rules, though not the first to publish thereon. We shall see that Oughtred invented slide rules about 1622, but the descriptions of his instruments were not put into print before 1632 and 1633. Meanwhile one of his own pupils, Richard Delamain, who probably invented the circular slide rule independently, published a description in 1630, at London, in a pamphlet of 32 pages entitled Grammelogia; or the Mathematicall Ring. In editions of this pamphlet which appeared during the following three or four years, various parts were added on, and some parts of the first and second editions eliminated. Thus Delamain antedates Oughtred two years in the publication of a description of a circular slide rule. But Oughtred had invented also a rectilinear slide rule, a description of which appeared in 1633. To the invention of this Oughtred has a clear title. A bitter controversy sprang up between Delamain on one hand, and Oughtred and some of his pupils on the other, on the priority and independence of invention of the circular slide rule. Few inventors and scientific men are so fortunate as to escape contests. The reader needs only to recall the disputes which have arisen, involving the researches of Sir Isaac Newton and Leibniz on the differential and integral calculus, of Thomas Harriot and René Descartes relating to the theory of equations, of Robert Mayer, Hermann von Helmholtz, and Joule on the principle of the conservation of energy, or of Robert Morse, Joseph Henry, Gauss and Weber, and others on the telegraph, to see that questions of priority and independence are not uncommon. The controversy between Oughtred and Delamain embittered Oughtred’s life for many years. He refers to it in print on more than one occasion. We shall confine ourselves at present to the statement that it is by no means clear that Delamain stole the invention from Oughtred; Delamain was probably an independent inventor. Moreover, it is highly probable that the controversy would never have arisen, had not some of Oughtred’s pupils urged and forced him into it. William Forster stated in the preface to the Circles of Proportion of 1632 that while he had been carefully preparing the manuscript for the press, “another to whom the Author [Oughtred] in a louing confidence discouered this intent, using more hast then good speed, went about to preocupate.” It was this passage which started the conflagration. Another pupil, W. Robinson, wrote to Oughtred, when the latter was preparing his Apologeticall Epistle as a reply to Delamain’s countercharges: “Good sir, let me be beholden to you for your Apology whensoever it comes forth, and (if I speak not too late) let me entreat you, whip ignorance well on the blind side, and we may turn him round, and see what part of him is free.”[52 - Rigaud, op. cit., Vol. I, p. 12.] As stated previously, Oughtred’s circular slide rule was described by him in his Circles of Proportion, London, 1632, which was translated from Oughtred’s Latin manuscript and then seen through the press by his pupil, William Forster. In 1633 appeared An Addition vnto the Vse of the Instrvment called the Circles of Proportion which contained at the end “The Declaration of the two Rulers for Calculation,” giving a description of Oughtred’s rectilinear slide rule. This Addition was bound with the Circles of Proportion as one volume. About the same time Oughtred described a modified form of the rectilinear slide rule, to be used in London for gauging.[53 - The New Artificial Gauging Line or Rod: together with rules concerning the use thereof: Invented and written by William Oughtred, London, 1633.]

CHAPTER III

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 - W. Oughtred, Apologeticall Epistle, p. 13.]

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.