CHAPTER 8 Mathematical Method

Wallis’s emphasis in his mathematical writings was many times more about the nature of mathematical methods in themselves than it was about the solu- tion to specific problems, or the way that mathematics might be practically applied. Indeed, his whole Treatise of Algebra was designed with this aim in mind, namely, “to shew the Art it self,… and by what steps it hath arrived to the height at which now it is.”1 As explained in Chapter 5, that work was neither an algebra textbook, nor a mere history, but a history with a point, aimed at instruction for the future progress of mathematics.2 Wallis’s singular interest in method is also evident in the way he chose to present his other great publi- cation, Arithmetica Infinitorum. There he left his thinking processes transpar- ent, in hopes that it might be of use to others to see just how he attained his results. This mode of presentation was misunderstood by some. Wallis an- swered Fermat, explaining that “he doth wholly mistake the design of that Treatise; which was not so much to shew a Method of Demonstrating things already known; (which the Method that he commends, doth chiefly aim at,) as to shew a way of Investigation or finding out of things yet unknown: (Which the Ancients did studiously conceal.)”3 This opinion, that the ancient geom- eters must have had some systematic methods like algebra, only not revealed, was shared with other early promoters of algebra. Speculations of this sort help Wallis to advocate a more thoroughgoing unity between the traditional geometry of the Greeks, and the new algebraic methods. In reality, though, this

1 John Wallis, A Treatise of Algebra, Both Historical and Practical. Shewing, the Original, Progress, and Advancement thereof, from Time to Time; and by What Steps It Hath Attained to the Height at Which It Now Is. With Some Additional Treatises, I. Of the Cono-cuneus; Being a Body Representing in Part a Conus, in Part a Cuneus. II. Of Angular Sections; and Other Things Relating there-unto, and to Trigonometry. III. Of the Angle of Contact; with Other Things Appertaining to the Composition of Magnitudes, the Inceptives of Magnitudes, and the Composition of Motions, with the Results Thereof. IV. Of Combinations, Alternations, and Aliquot Parts. By John Wallis, D.D. Professor of Geometry in the University of Oxford; and a Member of the Royal Society, London (London, 1685), 117. 2 Wallis’s affection for English authors, and ongoing promotion of them, is well documented. Jacqueline Stedall, “Of Our Own Nation: John Wallis’s Account of Mathematical Learning in Medieval England,” Historia Mathematica 28 (2001): 73–122; Jacqueline A. Stedall, A Discourse Concerning Algebra: English Algebra to 1685 (Oxford, 2003), 77–87, 117–25. None of this, how- ever, exludes this pedagogical point from also being true. 3 Wallis, Treatise of Algebra, 305.

© Koninklijke Brill NV, Leiden, 2019 | doi:10.1163/9789004409149_014 170 CHAPTER 8 question of the relation between geometry classically conceived and an arith- metized geometry required significant conceptual work for mathematicians in the seventeenth century. Wallis’s lecturing and publications were influential in making the conceptual transition even though posterity has not recognized this intermediate, yet essential piece of the rise of the new mathematics. This chapter will address the centrality of arithmetic in Wallis’s thinking, both on a conceptual level, and in practice as seen in his most famous publi- cation, Arithmetica Infinitorum. In this work, we also begin to witness Wallis’s approach to mathematical argument. These two themes, that is, his concep- tion of mathematics on a philosophical level and his understanding of math- ematical demonstration, will be followed further in the context of his debate with Thomas Hobbes, in Wallis’s explanation of imaginary numbers, and in his discussion of the angle of contact. In all of these instances, it will be shown that his particular conception and solution of mathematical questions depend upon his metaphysical and epistemological principles drawn out in the pre- ceding chapters.

1 Geometry, Algebra, and Arithmetic

John Wallis’s philosophy of mathematics was already established in his own mind at the beginning of his career. In the third chapter of Mathesis Universalis, which originated in a lecture given in 1649 (see Chapter 5), he offers his opin- ion on the nature and existence of mathematical entities. He is decidedly against the Platonist view, where mathematical objects have some imaginary existence in the Platonic heaven. “Mathematicians no more suppose or affirm a line, surface, or mathematical body to exist without regard to physical body, than one supposes a natural philosopher or animal to exist because it is nei- ther man nor beast, or even somewhere to be discovered one of mankind who is neither Plato, Socrates, or some other individual person.” Rather, mathemati- cal entities have their existence in real physical bodies. But this attachment of, for instance, geometrical figures, to physical things does not necessitate some kind of corporeal discourse in order to work out their relations.

For a mathematician does not deny that his lines, surfaces, and figures inhere in physical bodies, but he carefully considers and contemplates as much these things, yet not by considering physical body. For it is one thing to abstract, another thing to deny. Indeed, the mathematician ab- stracts magnitudes from a physical body, yet does not deny it. The math- ematician no more asserts a quantity to exist without physical body, than the natural philosopher that corporeal substance exists without quantity;