The Reappearance of Analysis in Mathematics

Overview

Important mathematical developments occurred during the sixteenth century, but it was not clear how to fit them into the framework of classical mathematics, which was still used as the center of the curriculum. In particular, the work in algebra did not look like part of the system of axioms and theorems used to set out the discipline of geometry. The French mathematician François Viète (1540-1603) presented the new mathematics in a way that could make it acceptable to those who insisted on having a classical background for mathematics. Thanks to his work, algebra could thereafter be presented in a way that was both easier to explain and to extend to further new results.

Background

In the sixteenth century a number of Italian mathematicians made advances in the branch of mathematics devoted to solving equations. Perhaps the most celebrated accomplishment was that of Niccolo Fontana (1500?-1557), better known as Tartaglia, who managed to solve cubic equations, in which the variable appeared to the third power. This was a notable advance over previous techniques, and in the course of the century equations with variables to the fourth power (called quartic equations) were also solved. There was argument at the time about who had first accomplished which stages of these advances, but there was no doubt that methods for solving of equations had been immensely improved.

These new discoveries, however, did not conform to those of classical Greek mathematics represented by the Elements of Euclid (330?-260? b.c.). Even within the world of Greek mathematics some of Euclid's successors had come up with additional approaches to mathematics beyond those of the Elements. Diophantus (fl. a.d. 250) was the first mathematician to approach the business of solving equations in more than one variable systematically, but the system he used was not that of Euclid. In an effort to make the work of Diophantus at home in the world of Greek mathematics, Pappas (300?-350?) came up with the categories of zetetics and poristics as part of the machinery used by Diophantus. Zetetics involved setting up equations and poristics involved checking the truth of earlier results by use of equations.

Since little progress was made for more than a millenium in the areas of algebra in which Diophantus had worked, the terminology and divisions that Pappus used seemed to be sufficient. With the developments in the sixteenth century, however, change was called for, and the individual who made those changes to mathematics was François Viète, a French mathematician who wrote in Latin (the scholarly language of the time) under the name of Vieta. As a humanist he felt obliged to find room for the new mathematics under the heading of the Greek learning that humanism had reintroduced into Europe. Since the new mathematics did not quite fit, he had to expand the categories of Pappus to find room for the Italian advances in algebra already made, and for those likely to come in the future.

Viète introduced the new area of exegetics into the classification Pappus had created. This had to do with determining the value of the unknown in a given equation. What the techniques of Tartaglia and others had required was the substitution of one expression into another in an effort to reduce an equation to a form that could be handled. Euclid's arithmetic (stated in the Elements in geometric form) allowed for the substitution of equals for equals, but since the algebraic expressions involved unknowns, different expressions could not be recognized as being equal. Viète argued that this process of replacing one expression by another was defensible if the resulting solution could be substituted back into the original equation and tested, according to Pappus. As a result, the algebraic techniques of the Italian equation solvers could be worked into a Greek scheme, although that was not part of what the Renaissance had inherited from classical mathematics.

Viète called his new approach the ars analytice, (analytic art). This referred to the technique of solving an equation by breaking it down (from which the word "analysis" comes). The presentation of mathematics in Euclid involved the method of synthesis, building up the subject from its elements. Algebra had never followed that mode of presentation, but this was partly connected with its looking like a bag of tricks rather than a connected discipline. Viète was the first to recognize that algebra could be presented in a more coherent fashion by dividing equations into classes which could be treated as a whole. This enabled mathematicians to proceed by solving a certain class of equations and then trying to reduce other equations to examples within that class. His method of classification and discovery brought the haphazard techniques for solving cubic and quartic equations into the framework of the heritage of classical mathematics.

Impact

The presentation of algebraic results under the heading of analysis accomplished a good deal to make the solving of equations more academically respectable. The mathematical curriculum in the sixteenth century remained centered on the Elements, and, in the next century, even the work of a mathematician of the stature of Sir Isaac Newton (1642-1727) was still presented in geometrical form. Those who had accomplished the feats of solving cubic and quartic equations could be dismissed as not being mathematicians (or philosophers, in the sense of natural philosophy) if their work did not adhere to the canons of Greek mathematics. Viète's stretching of the notions of Greek mathematics put algebraic progress on the map of the academic world as well.

Much of Viète's own nomenclature for algebra was supplanted in the seventeenth century by the work of René Descartes (1596-1650). Descartes's algebra and the simultaneous work of his colleague Pierre de Fermat (1601-1665) would have been impossible without Viète taking the isolated results of his Italian predecessors and systematizing them. The influence of Descartes and Fermat has continued throughout the subsequent development of mathematics, while that of Viète has ceased to be nearly so visible. Nevertheless, both Descartes and Fermat were well aware of what they had inherited in the form of the analytic art.

There was a good deal of discussion in the nineteenth century about whether algebra was about numbers or about symbols. The issue reflects Viète's ability to transform the discussion about specific numerical examples, which he received from earlier algebraists, into classes of equations that looked as though they could be solved regardless of the kinds of numbers that the variables represented. In the course of the nineteenth century, different rules for dealing with equations had to be developed according to the nature of the numbers being considered, but the possibility for extending algebra in this way depended on having a language for the communication of algebraic results. Viète supplied that language by merging Greek and algebraic traditions into the analytic art.

THOMAS DRUCKER