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.