William Oughtred: A Great Seventeenth-Century Teacher of Mathematics

by Florian Cajori

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.