Franciscus Vieta

Christopher Ploe

Term paper for History of Mathematics, Rutgers, Spring 1999

Today when one thinks of algebra, one immediately thinks of equations and variables. The notation we use today allows us to write not only specific equations to solve, but also a general form for many equations. The development of the algebraic notation we use today started in the sixteenth century. One of the first mathematicians to have an impact on the development of this algebraic notation was Franois Vi te. The contribution Vi te made towards algebraic notation has earned him the name "The father of modern algebraic notation" (Struik, p. 268).

François Viéte (1540-1603), a Frenchman who wrote under the Latinized name Franciscus Vieta, was not a mathematician, at least by vocation. He studied law at the University of Poitiers (Struik, p. 267), and went on to become legal counselor to the parliament of Brittany at Rennes (Boyer, HM, p. 333). Vieta later became a member of the king's council, serving under Henry III and Henry IV (Struik, p. 267). Vieta did however spend his free time in mathematical studies; and he was able to make significant contributions to mathematics in the areas of arithmetic, algebra, trigonometry, and geometry (Eves, p. 277). In arithmetic Vieta is remembered for his plea to use decimal fractions, rather than sexagesimal. In his Canon- mathematicus of 1579, he wrote: Sexagesimals and sixties are to be used sparingly or never in mathematics, and thousandths and thousands, hundredths and hundreds, tenths and tens, and similar progressions, ascending and descending, are to be used frequently or exclusively (Boyer, MT, p. 124). In his work he wrote numbers with the integral portion in boldface, which would look like 54,493,675,78; but at times he would also print numbers with a vertical line, that is: 54,493|675,78 (Boyer, HM, p. 334).

The use of the decimal point as we know it today is not attributed to Vieta, and did not even become a popular notation until it was used by John Napier in the early 1600's (Boyer, HM, p. 334).

Vieta made, without a doubt, more significant contributions in algebra (Boyer, HM, p. 334). The views he had in algebra are close to the way we look at algebra today. Up until this time mathematicians would use letters to stand for known and unknown quantities, but had no real way of writing both in a general form of an equation because it would not be known which quantities were which. Here Vieta introduced a simple system for doing so, he would let vowels stand for unknown quantities while he would use a consonant to stand for a given value, like a known distance or magnitude (Eves 278). This was the first time in the history of algebra that someone made a clear distinction between the use of unknowns and the concepts of parameters (Boyer, HM, p. 335). Although this use of symbols was innovative, it is still different from our current use of symbols. However it led towards the development of modern algebra. Today we let letters in the beginning of the alphabet like A, B, C stand for known values while we use x, y, z to stand for the unknown quantities we want to solve for in our equations. This way of using letters was introduced by Descartes in 1637 (Eves 278). As far as other similarities to the symbolic algebra we know today, Vieta produced little change here. His thought may have been more modern than previous work done; he however did not use much symbolic notation. Vieta would write things like A cubus, to stand for AAA, as others before had written (Eves 278). The idea is similar to our symbolic representation A3. Albeit the symbols were not the same, Vieta was thinking clearly, and others would follow continuing to make slow changes in the symbols leading us towards the algebraic notation we employ today.

Vieta tried to create a new science that would combine the rigor of the geometry of the ancients with the operativeness of algebra. This analytic art, as he called it, was to be powerful enough to leave no problem unsolved: nullum non problema solvere (Bashmakova and Smirnova, p. 260). Vieta introduced the language of formulas into mathematics. Vieta used literal notations for parameters as well as for the unknowns. By doing this Vieta was able to write equations and mathematical identities in general form. This was the first fundamentally new step in notation taken since Diophantus (Bashmakova and Smirnova 260). Up until Vieta's time the mathematics that existed was one without formulas. But it was not until Descartes later perfected the work done by Vieta, which led to the modern algebraic notation, that we are familiar with today (Bashmakova and Smirnova 261).

Vieta also made contributions towards trigonometry. Vieta thought of trigonometry as an independent branch of mathematics (Boyer, HM, p. 338). Vieta in 1579 prepared extensive tables of the six functions for angles to the nearest minute (Boyer, HM, p. 338). In Variorum de rebus mathematicis, Vieta makes a statement that is equivalent to the law of tangents (Boyer, HM, p. 339):

At this time in Europe different types of trigonometric identities were beginning to appear, this was leading more emphasis on the analytic functional relationships, as opposed to the emphasis that was formally placed on the computational solutions using triangles. By considering the lengths of the sides of the triangle ABC in the following diagram:

Vieta obtained the formula , which passes to a more familiar form via the substitution a = (x+y)/2, b= (x-y)/2 , from which the modern forms of the addition formulas can be derived by simple manipulations. (Cf. Boyer, HM, p. 339.)

Vieta also investigated multiple angle formulas, and in particular the link between angle trisection and the solution of a cubic equation (Boyer, HM, p. 341). Vieta was not looking at trigonometry as an independent branch of mathematics; rather, he felt that trigonometry could be used to help solve algebraic problems that had resisted the older methods of algebra. In applying trigonometry to algebra, Vieta was one of the first to begin to look at the mathematics in a unified way, not just as separate independent studies, as it had been viewed in the past (Boyer, HM, p. 341).

However one weakness of Vieta's methods was his inability to understand that there could be negative roots, and also the existence of imaginary numbers. On the other hand some of the work that he did with the operations over right triangles has proven, surprisingly, to be directly related to the multiplication and division of complex numbers (Glushkov 127).

Stanislav Glushkov's work on this topic appeared in Historia Mathematica in 1977. In his paper Glushkov looks at some of the operations Vieta performed on right triangles. In Vieta's work he only considered positive magnitudes of sides, so A-B, and B-A (here A, B represent side lengths of right triangles) were the same to Vieta, as he would only use the absolute value, but in his paper Glushkov modified this, choosing only the positive orientations, and showed that this makes the correspondence to complex numbers completely correct (Glushkov 128). In that paper, Vieta's two theorems characterizing his operations on right triangles are considered, and the following conclusions are arrived at:

  1. Theorem I simply means that the arguments of complex numbers are subtracted in division (Glushkov 132).
  2. Theorem II states that the arguments of complex numbers are added in multiplication (Glushkov 133).
In order to express Vieta's operations in terms of the geometrical interpretation of complex numbers, one must construct the triangles in the complex plane, with the acute angle of the triangle at the origin, and the base lying on the positive x-axis. Then the argument of the complex number is the acute angle at the origin. Now when the complex numbers are multiplied the two angles, or arguments, are added to form the new angle of the product of the complex numbers, represented in the complex plane. For the division of complex numbers, one would follow the same procedure, subtracting the angles. In this paper Glushkov also looks at Vieta's multiple angle formula, and lastly looks at how by viewing Vieta's work in the way that Glushkov has throughout the paper, we can even see that Vieta had investigated what would later become know as De Moivre's theorem. To see this, Glushkov had to write two of Vieta's equations together in one formula, and still the equation was in a different form than the one De Moivre formulated later in 1707 (Glushkov 135). This does not actually establish that Vieta discovered De Moivre's theorem, but it does show that he had an equivalent approach to the subject (Glushkov 135).

For someone who by trade was not a mathematician, and who only worked on solving problems in his leisure time (much like Fermat later on), Vieta contributed greatly towards the development of modern algebra. Vieta's pioneering work, the development of a set of literal notations that enabled him to write both parameters and unknowns in the same equation, became the building blocks for the notation that we use today. Although, Vieta did not create the modern algebraic notation we know today. He did make a significant step, which enabled others to build upon his work and perfect the ideas he had leading to today's algebraic notation. Vieta may not be well know for his work, nevertheless, one can easily see how important his work was and why Vieta is often called the father of modern algebraic notation.