Lecture 23. Viete and Modern AlgebraicNotation

Viete’s formula is well-known: for a quadratic equation 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 with roots 𝑥1

and 𝑥2, it has

𝑥1 + 𝑥2 = − 𝑏

𝑎, 𝑥1𝑥2 =

𝑐

𝑎.

Viete and his life Francois Viete (or Vieta)(1540-1603), a French mathematician, wasborn in Fontenay-le-Comte, a town in what is now the Vendee department of France. Hisfather, Etienne, was lawyer and his mother, Marguerite Dupont, was well connected to rulingcircles in France. Vieta was educated by the Franciscans in Fontenay and at the Universityof Poitiers. He received his bachelor’s degree in law in 1560 and then returned to Fontenayto commence practice.

Figure 23.1 Francois Viete (1540-1603).

For the rest of his life he was engaged mainly in law or related judicial and court services,doing mathematics only in periods of leisure.

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Viete really had mathematical talent. In 1593. Adriaen van Roomen 1 posted a publicchallenge to solve the equation of the 45th-degree equation: 2

45𝑥− 3795𝑥3 + 95634𝑥5 − ...− 12300𝑥39 + 945𝑥41 − 45𝑥43 + 𝑥45 = 𝐴.

Van Roomen proposed special cases, e.g., 𝐴 =

√2−

√2 +

√2 +

√3, which was suggested

by consideration of regular polygons. Viete saw immediately that this equation resultedfrom the expansion of 𝑠𝑖𝑛 45𝜃 in powers of 𝑠𝑖𝑛 𝜃, and he was able to give 23 solutions (hedid not recognize other negative solutions).

Viete later became a member of the king’s council, serving under Henry III, acting as anadvisor and negotiator for King Henry III of France. At that stage he was banished throughthe efforts of political rivals, but he returned to court in 1589 when Henry III moved his seatof government from Paris to Tours. After the assassination of Henry III in 1589, he servedHenry IV until 1602.

Albegraic notation The European algebraists of the sixteenth century had achievedabout as much as possible following the Islamic algebra of the Middle Ages. Viete was thefirst mathematicians to have an impact on the development of the algebraic notation in thehistory of mathematics.

Viete’s main achievement were in the improvement of the theory of equations, where hewas among the first to represent numbers by letters. He introduced the systematic algebraicnotation in his book In artem analyticam isagoge, published in 1591, in which he introducedletters to represent unknowns. He used letters as symbols for quantities, and he used vowelsfor the unknowns and consonants for known quantities. The convention where letters nearthe beginning of the alphabet, such as 𝑎, 𝑏, 𝑐, represent known quantities while letters nearthe end, such as 𝑥, 𝑦, 𝑧, represent unknown quantities was introduced later by Descartes.

Viete adopted the German form + and - for addition and subtraction in our presentmeaning, although sometimes he still used words. For division, he used the fraction bar, forexample, 𝐴𝐵/𝐶2 was written as

𝐴 𝑖𝑛𝐵

𝐶 𝑞𝑢𝑎𝑑𝑟𝑎𝑡𝑢𝑚

where Viete used the word “in” for “multiplication,” and used the word “quadratum” for“square.”

1Adriaen van Roomen (1561 - 4 May 1615), was a Belgian mathematician.2cf. D.J. Struik, A concise History of Mathematics, the 4th edition, Dover Publications, Inc., New York,

1987, p.86. Also,

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While Viete had come only part way toward modern symbolism, his pioneering workbecame the building blocks for the notation that we use today, which allowed other math-ematicians to build modern algebra upon his work. Therefore, Viete has earned him thename “the father of modern algebraic notation.”

Of course there were someone before him who used symbols different from numerals,such as the letters of the alphabet, to denote the quantities of arithmetic. But what Vieteachieved was to make the custom popular.

In the Five Books of Zetetics (1591), Viete used his symbolic methods to study a largenumber of algebraic problems, ancient or comtemporate. For example 3, for the problem:

Given the difference between two numbers and their sum, to find the numbers.

Viete’s solution: Let 𝐵 be the difference, 𝐷 the sum, and 𝐴 the smallest of the two numbers.Then he noted that 𝐴+𝐵 is the greater number. Then

𝐴+ (𝐴+𝐵) = 𝐷.

Hence 𝐴 = 12𝐷− 1

2𝐵. Another number is 𝐸 = 𝐴+𝐵 = 1

2𝐷+ 1

2𝐵. Having written down the

solution in symbols, Viete restated the answer in words:

“Half the sum of the numbers minus half the difference equals the least num-ber, plus the difference, the greater.”

He concluded with an example: If 𝐵 is 40 and 𝐷 is 100, then 𝐴 is 30 and 𝐸 is 70. Thisis a typical style for Viete’s work. Although he had introduced symbolic methods, he oftenrestated his answers in words. He may do this to convince skeptical readers to show theusefulness of his symbolic method.

Viete imporved on Archimedes and found 𝜋 in 9 decimals; shortly afterward 𝜋 wascomputed in 35 decimals by Ludolph van Coolen. Vete also expressed 𝜋 as an infiniteproduct (1593), in our notation: 4

2

𝜋= 𝑐𝑜𝑠

𝜋

4𝑐𝑜𝑠

𝜋

8𝑐𝑜𝑠

𝜋

16𝑐𝑜𝑠

𝜋

32...

3Victor J. Katz, A History of Mathematics - an introduction, 3rd edition, Addison -Wesley, 2009, 411-412.4D. Sruik, A Concise History of Mathematics, Dover Publications, INC., 1987, p.88.

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The improvement in technique was a result of the improvement of notation. The in-troduction of Hindu-Arabic numerals is one example; Leibniz’s notation for the calculus isanother one. Viete’s improvement in notation was followed, a generation later, by Descar-tee’s application of algebra to geometry, and by our present notation.

Figure 23.2 Image how hard to read a page from a mathematical book without notation.

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