The Universal Characteristic

Even in his mature period one has the impression that Leibniz's knowledge of Scholastic and Renaissance philosophy was deeper than his detailed knowledge of either ancient or contemporary mathematics. This was undoubtedly true up to his second visit to London in 1676 and perhaps well beyond, and is certainly understandable. He had read almost every major work in philosophy beginning well before his teens and had also a rigorous academic training in it; by age thirteen or so he could read, he says, the scholastic philosophers with "no ordinary delight."¹ On the other hand, he was twenty six before he began in 1672 to study mathematics seriously under the tutelage of Christiaan Huygens. During the Paris years he rapidly lost his earlier near total ignorance of mathematics. By 1674 or 1675 he certainly had, as he testified, mastered the geometry of Descartes and had additionally read under Huygens direction Pascal's Lettre de A. Dettonville (1659), Gregory St. Vincent's Opus geometricum (1647), James Gregory's Geometriae pars universalis (1668), and Wallis' Arithmetica Infinitorum (1656). Leibniz also speaks of an acquaintance with Cavalieri work and a "close study of Slusius."² All this was enough to allow him to reach the research frontier in infinitesimal analysis. But to repeat some earlier judgements, it is doubtful that he ever had the kind of deep mastery of the classical geometric tradition characteristic of Huygens, Barrow or Newton, all of whom could clinch an argument by a citation from one of the ancient masters. Leibniz was certainly very gifted mathematically and his early discoveries are impressive. But considered only in isolation they are no more impressive than results found by others. We have already pointed out, for example, that his arithmetical quadrature of a circle or conic sections was just one of a family of infinite series/infinite product representations found earlier by mathematicians of the calibre of Gregory, Wallis, or Newton. His quadrature of the cycloid was similar to what had been done by Roberval or Huygens. And we certainly can find in Barrow or Huygens results at least as ingenious as his various transmutation theorems, some of which he may, in fact, have "borrowed" from Barrow.³ And in respect to problem solving or the ability to handle or synthesize involved mathematical arguments, he was probably equal to the more gifted of his contemporaries, but ranked no higher. Mathematically, he is perhaps broader than Newton since he made discoveries in several unrelated areas (some of which we discuss below), but in pure geometric power and mathematical depth he is distinctly below Newton who in the seventeenth century was in a class by himself…