Abstract
This chapter discusses some of the more contemporary developments that might have influenced the progress of early modern ideas about number and magnitude. Since the introduction of mathematical symbolism inspired a broadening of number concepts and the creation of the notion of abstract magnitude, several aspects of the new algebraic methods will be explored. First, we will examine the German cossic notation that was utilised by Robert Recorde, and still included in later algebraic works such as Wallis’ Treatise. We will also consider the early proto-algebraic techniques of the Italian abacus school writers, since their texts had much in common with Recorde’s and they provide us with a demonstration of the changing nature of the problems that practitioners were trying to solve. Second, we shall investigate some of the mathematical contributions of Simon Stevin. His notion of decimal expansion was familiar to Harriot, Napier, Briggs, Barrow, and Wallis and his views about the nature of numbers were diametrically opposed to the Greeks. Third, we will explore the impact of Viète’s seminal work that seems to have influenced almost all the practitioners in this study. Descartes’ contributions will also be investigated, as Wallis and Barrow closely studied his writings. Although some English mathematicians preferred to attribute Descartes’ advances to Harriot, they were nonetheless familiar with his work.
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References
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Frank J. Swetz, Capitalism and Arithmetic (La Salle: Opencourt, 1987), p.13.
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van Egmond p. 231. The answer is given as: Do it thus: you see first that each of them is to have according to his capital and because one has 2 hundred and the other 3 hundred we add then together, 2 and 3, which makes 5, and 5 will be the divisor. Now we say 2 times 200 makes 400, divided by 5 whence comes 80 and 80 lire will have that which put in 200 and the other should have the remainder of the 200 lire which is 120 lire. Now we say, if they had stayed 3 years and they had these 200 lire of earnings, each would have of it 100 lire. Now you see how much of the 200 lire that one would have above 80 which he is to have by the rule which gives 20 lire because we say in 36 months he would have 20 lire. Now how much would he have had of it in 20? Therefore multiply 20 times 20 which makes 400; divide by 36 whence comes 11 lire 1/9 of a lire, add it to the above 80 and there are 91 lire 1/9 and that man has 91 lire 1/9 of a lire of that 200 lire and the other would have the remainder of the 200 lire, which is 108 lire and 8/9 of one lire. ...
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These rules were based on a principle recognised in the modern theory of arithmetic as the congruence relation of modular arithmetic. For every a, b ∈ N, [a] + [b] = [a + b] and [a] * [b] = [a * b] where [a] = a (mod n). This is true because for every m ∈ N, there exists n, q, r ∈ N, such that m = n* q + r.
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“Cossike” was the preferred spelling in England in this period.
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Stevin, p. 495. “Nombre est cela, par lequel s’explique la quantité de chascune chose.”
Stevin, p. 495. “Que l’unite est nombre.”
Stevin, p. 501. “QUE NOMBRE N’EST POINCT QUANTITE DISCONTINUE”
Stevin, p. 501. The translation is by Struik, the emphasis is mine.
Stevin, p. 505. “Nombre entier unité, ou composée multitude d’unitez.”
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See Helena M. Pycior’s, Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra though the Commentaries on Newton’s Universal Arithmetik, chapter two “Setting the Scene: The Foundations of Early Modem Algebra” (Cambridge University Press, 1997) for an excellent discussion of both Cardano and Viète.
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Paolo Mancosu, “Descartes’s Géométrie and Revolutions in Mathematics,” in Revolutions in Mathematics, ed. Donald Gilles (Oxford: Clarendon Press, 1992) contains an extensive summary of the highlights of this text. See also, Paolo Mancosu’s Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century (Oxford: Oxford University Press, 1996), pp. 65–91.
H. J. M. Bos, “On the representation of Curves in Descartes’ Géométrie,” AHES 24 (1981) p.278.
Bos, Redefining Geometrical Exactness, p. 296. I am following Bos’ interpretation of Descatres’ text here.
The exception might be Napier; it is difficult to tell if he was acquainted with the Greek ideas but he was certainly influenced by certain contemporary ideas, such as Stevin’s.
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Neal, K. (2002). The Contemporary Influences. In: From Discrete to Continuous. Australasian Studies in History and Philosophy of Science, vol 16. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0077-1_3
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