The Mathematical Background of the Regulae ad Directionem Ingenii

Abstract

Descartes’s program of 1619 for reforming mathematics (see Chapter 3, § 1) was embodied in his manuscript titled Algebra. Descartes referred to this writing for the first time, as far as we know, in October of 1628 when he visited his old friend Beeckman, who was then in Dordrecht. Beeckman took notes in his Journal describing the content of his discussions with Descartes, just as he had done in 1618–1619. This document is extremely important for understanding Descartes’s mathematical thought during the period between 1619 and 1628. In it Descartes told of his nine year trek which began in 1619, the year when the two scholars bade farewell to one another. Under the margin subtitle “History of Descartes and of His Relationship with Me,” Beeckman writes as follows:M. René Descartes du Perron, who in 1618 in Breda of Brabant wrote for me the Compendium of Music, in which he revealed his view on music and which has been inserted into this Journal. He, I say, came to Dordrecht to call on me on October 8, 1628, after he had gone first to Middleburg from Holland in order to look for me there. He told me that insofar as arithmetic and geometry were concerned, he had nothing more to discover; that is, in these branches during the past nine years he had made as much progress as was possible for the human mind. He gave me perspicacious specimens of this and promised to send me his Algebra a little later from Paris, which, he said, was finished and by which not only had he arrived at a perfect knowledge of geometry but also he claimed to embrace all human knowledge. Perhaps he will come here to perfect and publish it so that together we may complete the study of what remains to be discovered in the sciences. Indeed, after travelling through Germany, France and Italy, he affirms that he has not found anybody with whom he is able to discuss from the bottom of the heart and from whom he can hope for aid in his researches besides me. So he says that everywhere there is a lack of true philosophy, which he calls the work of devotees. In truth I prefer him to all the arithmeticians and geometers whom I have ever seen and read.1