New algebra

The new algebra or symbolic analysis is a formalization of algebra promoted by François Viète in 1591 and by his successors after 1603. It marks the beginning of the algebraic formalization in the late sixteenth and the early seventeenth centuries.

In artem analyticem Isagoge (1591) is the program of this large axiomatic project.

This work is available via gallica, written in Latin, and announcing that it will be the first volume of a work divided into ten parts:

It provides a new approach to writing algebra and begins with the famous dedication to the Melusinide princess Catherine de Parthenay.

In the first part of his Isagoge, Viète provides definitions of his symbolic analysis, and gives, in a rhythmic movement, the definitions of Zetetic, Poristic, and Exegetic, for the purpose of writing the science of inventing Mathematics. He gives, concurrently, an axiomatic for calculation on the quantities (known and unknown) and a program, which provides heuristic rules.

In this introduction, Viète requires three steps to solve algebraic or geometrical problems: formalization, general resolution, special resolution. He adds that, contrary to the former analysts, his method will act on the resolution of symbols (non iam in numeris sed sub specie)... which is the major input. He also predicts that after his works, training in Zetetic will be done through the analysis of symbols and not by the numbers.

Viète continues, in this second part, to describe the symbols and gives axiomatic rules:

Transitivity of equality, conservation sum, subtraction, product, and division

Viète, then, continues to give the law of homogeneity, and distinguishes the symbols according to their powers, where 1 is the side (or root), 2 square, cube 3, and so on. Factors and powers are of complementary homogeneity; he notes them:

1. Length, 2 Plane, Solid 3, and 4 Plane / Plane 5 Plane / Solid 6 Solid / Solid, etc. as if he had the intuition that a geometry can be deployed beyond the ordinary dimension 3.

In this fourth chapter, Viète gives the rules of a calculus of symbols, i.e. the axioms of addition, product, etc. Symbols designate types of comparable dimension.

Firstly, his attention is focused on addition and subtraction of quantities of the same order, with rules such as A − (B + D) = A − B − D or A − (B − D) = A − B + D

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