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The year preceding Oughtred’s death Mr. John Twysden expressed himself as follows in the preface to his Miscellanies:

It remains that I should adde something touching the beginning, and use of these Sciences… I shall only, to their honours, name some of our own Nation yet living, who have happily laboured upon both stages. That succeeding ages may understand that in this of ours, there yet remained some who were neither ignorant of these Arts, as if they had held them vain, nor condemn them as superfluous. Amongst them all let Mr. William Oughtred, of Aeton, be named in the first place, a Person of venerable grey haires, and exemplary piety, who indeed exceeds all praise we can bestow upon him. Who by an easie method, and admirable Key, hath unlocked the hidden things of geometry. Who by an accurate Trigonometry and furniture of Instruments, hath inriched, as well geometry, as Astronomy. Let D. John Wallis, and D. Seth Ward, succeed in the next place, both famous Persons, and Doctors in Divinity, the one of geometry, the other of astronomy, Savilian Professors in the University of Oxford.⁷¹

The astronomer Edmund Halley, in his preface to the 1694 English edition of the Clavis, speaks of this book as one of “so established a reputation, that it were needless to say anything thereof,” though “the concise Brevity of the author is such, as in many places to need Explication, to render it Intelligible to the less knowing Mathematical matters.”

In closing this part of our monograph, we quote the testimony of Robert Boyle, the experimental physicist, as given May 8, 1647, in a letter to Mr. Hartlib:

The Englishing of, and additions to Oughtred’s Clavis mathematica does much content me, I having formerly spent much study on the original of that algebra, which I have long since esteemed a much more instructive way of logic, than that of Aristotle.⁷²

WAS DESCARTES INDEBTED TO OUGHTRED?

This question first arose in the seventeenth century, when John Wallis, of Oxford, in his Algebra (the English edition of 1685, and more particularly the Latin edition of 1693), raised the issue of Descartes’ indebtedness to the English scientists, Thomas Harriot and William Oughtred. In discussing matters of priority between Harriot and Descartes, relating to the theory of equations, Wallis is generally held to have shown marked partiality to Harriot. Less attention has been given by historians of mathematics to Descartes’ indebtedness to Oughtred. Yet this question is of importance in tracing Oughtred’s influence upon his time.

On January 8, 1688-89, Samuel Morland addressed a letter of inquiry to John Wallis, containing a passage which we translate from the Latin:

Some time ago I read in the elegant and truly precious book that you have written on Algebra, about Descartes, this philosopher so extolled above all for having arrived at a very perfect system by his own powers, without the aid of others, this Descartes, I say, who has received in geometry very great light from our Oughtred and our Harriot, and has followed their track though he carefully suppressed their names. I stated this in a conversation with a professor in Utrecht (where I reside at present). He requested me to indicate to him the page-numbers in the two authors which justified this accusation. I admitted that I could not do so. The Géométrie of Descartes is not sufficiently familiar to me, although with Oughtred I am fairly familiar. I pray you therefore that you will assume this burden. Give me at least those references to passages of the two authors from the comparison of which the plagiarism by Descartes is the most striking.⁷³

Following Morland’s letter in the De algebra tractatus, is printed Wallis’ reply, dated March 12, 1688 (“Stilo Angliae”), which is, in part, as follows:

I nowhere give him the name of a plagiarist; I would not appear so impolite. However this I say, the major part of his algebra (if not all) is found before him in other authors (notably in our Harriot) whom he does not designate by name. That algebra may be applied to geometry, and that it is in fact so applied, is nothing new. Passing the ancients in silence, we state that this has been done by Vieta, Ghetaldi, Oughtred and others, before Descartes. They have resolved by algebra and specious arithmetic [literal arithmetic] many geometrical problems… But the question is not as to application of algebra to geometry (a thing quite old), but of the Cartesian algebra considered by itself.

Wallis then indicates in the 1659 edition of Descartes’ Géométrie where the subjects treated on the first six pages are found in the writings of earlier algebraists, particularly of Harriot and Oughtred. For example, what is found on the first page of Descartes, relating to addition, subtraction, multiplication, division, and root extraction, is declared by Wallis to be drawn from Vieta, Ghetaldi, and Oughtred.

It is true that Descartes makes no mention of modern writers, except once of Cardan. But it was not the purpose of Descartes to write a history of algebra. To be sure, references to such of his immediate predecessors as he had read would not have been out of place. Nevertheless, Wallis fails to show that Descartes made illegitimate use of anything he may have seen in Harriot or Oughtred.

The first inquiry to be made is, Did Descartes possess copies of the books of Harriot and Oughtred? It is only in recent time that this question has been answered as to Harriot. As to Oughtred, it is still unanswered. It is now known that Descartes had seen Harriot’s Artis analyticae praxis (1631). Descartes wrote a letter to Constantin Huygens in which he states that he is sending Harriot’s book.⁷⁴

An able discussion of the question, what effect, if any, Oughtred’s Clavis mathematicae of 1631 had upon Descartes’⁷⁵ Géométrie of 1637, is given by H. Bosmans in a recent article. According to Bosmans no evidence has been found that Descartes possessed a copy of Oughtred’s book, or that he had examined it. Bosmans believes nevertheless that Descartes was influenced by the Clavis, either directly or indirectly. He says:

If Descartes did not read it carefully, which is not proved, he was none the less well informed with regard to it. No one denies his intimate knowledge of the intellectual movement of his time. The Clavis mathematica enjoyed a rapid success. It is impossible that, at least indirectly, he did not know the more original ideas which it contained. Far from belittling Descartes, as I much desire to repeat, this rather makes him the greater.⁷⁶

We ourselves would hardly go as far as does Bosmans. Unless Descartes actually examined a copy of Oughtred it is not likely that he was influenced by Oughtred in appreciable degree. Book reviews were quite unknown in those days. No evidence has yet been adduced to show that Descartes obtained a knowledge of Oughtred by correspondence. A most striking feature about Oughtred’s Clavis is its notation. No trace of the Englishman’s symbolism has been pointed out in Descartes’ Géométrie of 1637. Only six years intervened between the publication of the Clavis and the Géométrie. It took longer than this period for the Clavis to show evidence of its influence upon mathematical books published in England; it is not probable that abroad the contact was more immediate than at home. Our study of seventeenth-century algebra has led us to the conviction that Oughtred deserves a higher place in the development of this science than is usually accorded to him; but that it took several decennia for his influence fully to develop.

THE SPREAD OF OUGHTRED’S NOTATIONS

An idea of Oughtred’s influence upon mathematical thought and teaching can be obtained from the spread of his symbolism. This study indicates that the adoption was not immediate. The earliest use that we have been able to find of Oughtred’s notation for proportion, A.B::C.D, occurs nineteen years after the Clavis mathematicae of 1631. In 1650 John Kersey brought out in London an edition of Edmund Wingates’ Arithmetique made easie, in which this notation is used. After this date publications employing it became frequent, some of them being the productions of pupils of Oughtred. We have seen it in Vincent Wing (1651),⁷⁷ Seth Ward (1653),⁷⁸ John Wallis (1655),⁷⁹ in “R. B.,” a schoolmaster in Suffolk,⁸⁰ Samuel Foster (1659),⁸¹ Jonas Moore (1660),⁸² and Isaac Barrow (1657).⁸³ In the latter part of the seventeenth century Oughtred’s notation, A.B::C.D, became the prevalent, though not universal, notation in Great Britain. A tremendous impetus to their adoption was given by Seth Ward, Isaac Barrow, and particularly by John Wallis, who was rising to international eminence as a mathematician.

In France we have noticed Oughtred’s notation for proportion in Franciscus Dulaurens (1667),⁸⁴ J. Prestet (1675),⁸⁵ R. P. Bernard Lamy (1684),⁸⁶ Ozanam (1691),⁸⁷ De l’Hospital (1696),⁸⁸ R. P. Petro Nicolas (1697).⁸⁹

In the Netherlands we have noticed it in R. P. Bernard Lamy (1680),⁹⁰ and in an anonymous work of 1690.⁹¹ In German and Italian works of the seventeenth century we have not seen Oughtred’s notation for proportion.

In England a modified notation soon sprang up in which ratio was indicated by two dots instead of a single dot, thus A:B::C:D. The reason for the change lies probably in the inclination to use the single dot to designate decimal fractions. W. W. Beman pointed out that this modified symbolism (:) for ratio is found as early as 1657 in the end of the trigonometric and logarithmic tables that were bound with Oughtred’s Trigonometria.⁹² It is not probable, however, that this notation was used by Oughtred himself. The Trigonometria proper has Oughtred’s A.B::C.D throughout. Moreover, in the English edition of this trigonometry, which appeared the same year, 1657, but subsequent to the Latin edition, the passages which contained the colon as the symbol for ratio, when not omitted, are recast, and the regular Oughtredian notation is introduced. In Oughtred’s posthumous work, Opuscula mathematica hactenus inedita, 1677, the colon appears quite often but is most likely due to the editor of the book.

We have noticed that the notation A:B::C:D antedates the year 1657. Vincent Wing, the astronomer, published in 1651 in London the Harmonicon coeleste, in which is found not only Oughtred’s notation A.B::C.D but also the modified form of it given above. The two are used interchangeably. His later works, the Logistica astronomica (1656), Doctrina spherica (1655), and Doctrina theorica, published in one volume in London, all use the symbols A:B::C:D exclusively. The author of a book entitled, An Idea of Arithmetick at first designed for the use of the Free Schoole at Thurlow in Suffolk.. by R. B., Schoolmaster there, London, 1655, writes A:a::C:c, though part of the time he uses Oughtred’s unmodified notation.

We can best indicate the trend in England by indicating the authors of the seventeenth century whom we have found using the notation A:B::C:D and the authors of the eighteenth century whom we have found using A.B::C.D. The former notation was the less common during the seventeenth but the more common during the eighteenth century. We have observed the symbols A:B::C:D (besides the authors already named) in John Collins (1659),⁹³ James Gregory (1663),⁹⁴ Christopher Wren (1668-69),⁹⁵ William Leybourn (1673),⁹⁶ William Sanders (1686),⁹⁷ John Hawkins (1684),⁹⁸ Joseph Raphson (1697),⁹⁹ E. Wells (1698),¹⁰⁰ and John Ward (1698).¹⁰¹

Of English eighteenth-century authors the following still clung to the notation A.B::C.D: John Harris’ translation of F. Ignatius Gaston Pardies (1701),¹⁰² George Shelley (1704),¹⁰³ Sam Cobb (1709),¹⁰⁴ J. Collins in Commercium Epistolicum (1712), John Craig (1718),¹⁰⁵ Jo. Wilson (1724).¹⁰⁶ The latest use of A.B::C.D which has come to our notice is in the translation of the Analytical Institutions of Maria G. Agnesi, made by John Colson sometime before 1760, but which was not published until 1801. During the seventeenth century the notation A:B::C:D acquired almost complete ascendancy in England.

In France Oughtred’s unmodified notation A.B::C.D, having been adopted later, was also discarded later than in England. An approximate idea of the situation appears from the following data. The notation A.B::C.D was used by M. Carré (1700),¹⁰⁷ M. Guisnée (1705),¹⁰⁸ M. de Fontenelle (1727),¹⁰⁹ M. Varignon (1725),¹¹⁰ M. Robillard (1753),¹¹¹ M. Sebastien le Clerc (1764),¹¹² Clairaut (1731),¹¹³ M. L’Hospital (1781).¹¹⁴

In Italy Oughtred’s modified notation a, b::c, d was used by Maria G. Agnesi in her Instituzioni analitiche, Milano, 1748. The notation a:b::c:d found entrance the latter part of the eighteenth century. In Germany the symbolism a:b=c:d, suggested by Leibniz, found wider acceptance.¹¹⁵

It is evident from the data presented that Oughtred proposed his notation for ratio and proportion at a time when the need of a specific notation began to be generally felt, that his symbol for ratio a.b was temporarily adopted in England and France but gave way in the eighteenth century to the symbol a:b, that Oughtred’s symbol for proportion :: found almost universal adoption in England and France and was widely used in Italy, the Netherlands, the United States, and to some extent in Germany; it has survived to the present time but is now being gradually displaced by the sign of equality =.

Oughtred’s notation to express aggregation of terms has received little attention from historians but is nevertheless interesting. His books, as well as those of John Wallis, are full of parentheses but they are not used as symbols of aggregation in algebra; they are simply marks of punctuation for parenthetical clauses. We have seen that Oughtred writes (a+b)² and √a+b thus, Q:a+b:, √:a+b:, or Q:a+b, √:a+b, using on rarer occasions a single dot in place of the colon. This notation did not originate with Oughtred, but, in slightly modified form, occurs in writings from the Netherlands. In 1603 C. Dibvadii in geometriam Evclidis demonstratio numeralis, Leyden, contains many expressions of this sort, √·136+√2048, signifying √(136+√2048). The dot is used to indicate that the root of the binomial (not of 136 alone) is called for. This notation is used extensively in Ludolphi à Cevlen de circulo, Leyden, 1619, and in Willebrordi Snellii De circuli dimensione, Leyden, 1621. In place of the single dot Oughtred used the colon (:), probably to avoid confusion with his notation for ratio. To avoid further possibility of uncertainty he usually placed the colon both before and after the algebraic expression under aggregation. This notation was adopted by John Wallis and Isaac Barrow. It is found in the writings of Descartes. Together with Vieta’s horizontal bar, placed over two or more terms, it constituted the means used almost universally for denoting aggregation of terms in algebra. Before Oughtred the use of parentheses had been suggested by Clavius¹¹⁶ and Girard.¹¹⁷ The latter wrote, for instance, √(2+√3). While parentheses never became popular in algebra before the time of Leibniz and the Bernoullis they were by no means lost sight of. We are able to point to the following authors who made use of them: I. Errard de Bar-le-Duc (1619),¹¹⁸ Jacobo de Billy (1643),¹¹⁹ one of whose books containing this notation was translated into English, and also the posthumous works of Samuel Foster.¹²⁰ J. W. L. Glaisher points out that parentheses were used by Norwood in his Trigonometrie (1631), p. 30.¹²¹

The symbol for the arithmetical difference between two numbers, ~, is usually attributed to John Wallis, but it occurs in Oughtred’s Clavis mathematicae of 1652, in the tract on Elementi decimi Euclidis declaratio, at an earlier date than in any of Wallis’ books. As Wallis assisted in putting this edition through the press it is possible, though not probable, that the symbol was inserted by him. Were the symbol Wallis’, Oughtred would doubtless have referred to its origin in the preface. During the eighteenth century the symbol found its way into foreign texts even in far-off Italy.¹²² It is one of three symbols presumably invented by Oughtred and which are still used at the present time. The others are × and ::.

The curious and ill-chosen symbols, for “greater than,” and for “less than,” were certain to succumb in their struggle for existence against Harriot’s admirably chosen > and <. Yet such was the reputation of Oughtred that his symbols were used in England quite extensively during the seventeenth and the beginning of the eighteenth century. Considerable confusion has existed among algebraists and also among historians as to what Oughtred’s symbols really were. Particularly is this true of the sign for “less than” which is frequently written . Oughtred’s symbols, or these symbols turned about in some way, have been used by Seth Ward,¹²³ John Wallis,¹²⁴ Isaac Barrow,¹²⁵ John Kersey,¹²⁶ E. Wells,¹²⁷ John Hawkins,¹²⁸ Tho. Baker,¹²⁹ Richard Sault,¹³⁰ Richard Rawlinson,¹³¹ Franciscus Dulaurens,¹³² James Milnes,¹³³ George Cheyne,¹³⁴ John Craig,¹³⁵ Jo. Wilson,¹³⁶ and J. Collins.¹³⁷

General acceptance has been accorded to Oughtred’s symbol ×. The first printed appearance of this symbol for multiplication in 1618 in the form of the letter x hardly explains its real origin. The author of the “Appendix” (be he Oughtred or someone else) may not have used the letter x at all, but may have written the cross ×, called the St. Andrew’s cross, while the printer, in the absence of any type accurately representing that cross, may have substituted the letter x in its place. The hypothesis that the symbol × of multiplication owes its origin to the old habit of using directed bars to indicate that two numbers are to be combined, as for instance in the multiplication of 23 and 34, thus,

has been advanced by two writers, C. Le Paige¹³⁸ and Gravelaar.¹³⁹ Bosmans is more inclined to the belief that Oughtred adopted the symbol somewhat arbitrarily, much as he did the numerous symbols in his Elementi decimi Euclidis declaratio.¹⁴⁰

Le Paige’s and Gravelaar’s theory finds some support in the fact that the cross ×, without the two additional vertical lines shown above, occurs in a commentary published by Oswald Schreshensuchs¹⁴¹ in 1551, where the sign is written between two factors placed one above the other.

CHAPTER V
OUGHTRED’S IDEAS ON THE TEACHING OF MATHEMATICS

GENERAL STATEMENT

Nowhere has Oughtred given a full and systematic exposition of his views on mathematical teaching. Nevertheless, he had very pronounced and clear-cut ideas on the subject. That a man who was not a teacher by profession should have mature views on teaching is most interesting. We gather his ideas from the quality of the books he published, from his prefaces, and from passages in his controversial writing against Delamain. As we proceed to give quotations unfolding Oughtred’s views, we shall observe that three points receive special emphasis: (1) an appeal to the eye through suitable symbolism; (2) emphasis upon rigorous thinking; (3) the postponement of the use of mathematical instruments until after the logical foundations of a subject have been thoroughly mastered.

The importance of these tenets is immensely reinforced by the conditions of the hour. This voice from the past speaks wisdom to specialists of today. Recent methods of determining educational values and the modern cult of utilitarianism have led some experts to extraordinary conclusions. Laboratory methods of testing, by the narrowness of their range, often mislead. Thus far they have been inferior to the word of a man of experience, insight, and conviction.