Viète

Viète

François Viète was born in Fontenay-le-Comte, in Poitou in 1540. He studied law in Poitiers, following his father’s profession. Viète was a Huguenot and his life was much affected by the religious wars that afflicted France in the late sixteenth century. He managed to steer a steady course in these turbulent times, serving kings on both sides of the religious divide, the Catholic, Henri III (1551-1589), and his Protestant successor, Henri IV (1553-1610). In the war with Philip II of Spain (1527-1598), that emerged after relative peace had been restored in France, Viète used his mathematical talent to crack Spanish code to the advantage of France.

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François Viète, De aequationum recognitione et emendatione tractatus duo (Paris, 1615), title page.

Although Viète worked also in geometry and trigonometry, it is for his pioneering breakthrough in algebra that he is best known, especially in the development of notation, laying the foundation for what we use today. In his perception of the development of mathematical ideas, he preferred to give credit to the work of Greek, rather than Arabic/Islamic, mathematicians. Stedall (2008, p. 47) summarises his point of view as follows:

‘Viète’s most important contribution to mathematics was his recognition that geometric relationships could be expressed and explored through algebraic equations, leading to a powerful fusion of two previously distinct legacies: classical Greek geometry and Islamic algebra. The central technique of algebra, taught in many sixteenth-century texts as the “Rule of Algebra”, was one that should assign a symbol or letter to an unknown quantity and then, bearing in mind the requirements of the problem, manipulate it alongside known quantities to produce an equation. For Viète, never content with a simple idea unless he could clothe it in a Greek term, this was the classical method of “analysis” in which, he claimed, one assumes that what one is seeking is somehow known and then sets up the relationships or equations it must satisfy. Thus Viète saw the application of algebra to geometric quantities as the restored art of analysis.’

The Edward Worth Library has a copy of his De aequationum recognitione et emendatione tractatus duo (Paris, 1615). (Two treatises on the understanding and amendment of equations).

Although some earlier mathematicians, such as Diophantus in the third century, al-Khwārizmī in the ninth and Cardano in the sixteenth, had a good grasp of the generality of algebra, they did not combine the mathematical concepts with an efficient notation to express the power of algebra fluently. For example, when Cardano discovered the general formula for the solution of cubic equations, he expressed it as a procedure applied to specific equations, such as x³ + 6x = 20. Moreover, he stated the problem in terms of words (‘a cube and 6 things are equal to 20’) rather than symbols. Viète might have expressed this problem more generally as: A cubus + B plano in A, æquetur C solido, and then would have solved for A in terms of B and C. He adopted the convention of using capital letters for both (unknown) variables and (known) constants, choosing vowels for variables and consonants for constants. He could not accept adding terms having different dimensions, such as a ‘volume’, x³, to a ‘length’, 6x. He insisted that equations should be homogeneous, so that, if A is a length, B will need to be an area (‘plano’) for A³+BA to make sense (as a volume); moreover, the C on the right-hand side of this equation is a volume (‘solido’).

De aequationum recognitione et emendation contains a large number of theorems, written in Latin. Although Viète used symbols for variables and constants, the substantial textual content makes his work difficult to access for today’s reader. Let us conclude with one short theorem from this work.

François Viète, De aequationum recognitione et emendatione tractatus duo (Paris, 1615), p. 19.

The statement of the theorem is essentially: if A² = Z (an area), and A – B = E, then we can conclude that E² + 2BE = Z – B². Here, we should think of A and E as variables, while Z and B are constants. This simple theorem shows how the substitution y = x – b allows us to express the quadratic equation, x² = c, in terms of a quadratic equation in y: y² + 2by = c – b².

Sources

Encyclopædia Britannica, entry for Mathematics in the 17th and 18th centuries (2016).

Hartshorne, Robin, François Viète, (University of California at Berkeley).

O’Connor, J.J. and Robertson E.F., François Viète, MacTutor History of Mathematics, (University of St Andrews, Scotland, February 2000).

Stedhall, Jacqueline, Mathematics Emerging. A Sourcebook 1540-1900 (Oxford University Press, 2008).

Text: Maurice OReilly and Conor McElduff.