Francois Viète:

Father of Modern Algebraic Notation

Jennifer Orlansky

The practice of using letters rather than numbers to represent both known (but unspecified) and unknown quantities marked the beginnings of modern algebra as we know it today. Frangois Viète (Latin: Vieta), a great French mathematician, is credited with the invention of this system, and is therefore known as the "father of modern algebraic notation" [3, p. 268]. In other words, Viète's main innovation in algebra was the use of variable coefficients. Although he is best known for his "pioneering work o n symbolism" [3, p. 268], Viète also contributed much to the field of mathematics; his achievements have paved the way for the work of many mathematicians and have sparked new ideas and research that influenced the evolution of modern algebra. Viète was born in 1540 in Fontenay-le-Comte [10, p. 63]. Like many early mathematicians, Viète's primary occupation was not in the area of mathematics. He was a lawyer, like his father, and received his legal degree from the University of Poitiers in 15 60. He became a tutor for a young girl named Catherine de Parthenay, which prompted him to study astronomy, a subject in which she was interested [9, p,2]. He also served as a member of the royal privy council under King Henry III and King Henry IV, dec iphering codes during the war against Spain [10, p. 63]. Viète had a great fascination with mathematics, and used his spare time to study this area of interest, as well as cryptography. He died in Paris in 1603. A chief goal of Viète's exploration was "to join Euclidean geometry with a generalized numerical algebra" [3, p. 309]. To him, algebra was an art. He believed that his "new" algebra was really just an improvement of old ideas that had been "wrapped in g eometrical garb" [6, p. 94], as he explained in the dedication of his Isagoge to Catherine: These things which are new are wont in the beginning to be set forth rudely and formlessly and must then be polished and perfected in succeeding centuries. Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbari ans, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms... (Viète cited in [5, p. 131]. In an attempt to transform this "filth", as he called it, to "the great art," Viète wrote several works, hoping to enhance the "spoiled and defiled" system of mathematics to his standards of superiority. Various mathematicians in history influenced Viète, however, there were two primary Greek sources that served as major contributors to Viète's work: (1) Pappus' seventh book of the Mathematical Collections and (2) Diophantus' Arithmetic [7, p. 154]. In V iète's time, most people were using Arabic terms when dealing with mathematics. Since Viète felt Arabic was a barbaric language, he sought books that used Greek terms. His search resulted in Pappus' the Mathematical Collections. Pappus, "the last great mathematician of the Alexandrian school," was a Hellenistic mathematician, probably born about 340 A.D. [2, p. 49]. The Mathematical Collections is the only work of Pappus still in existence. It originally consisted of eight books, the first, and porti ons of the second, of which are now missing [2, p. 49]. In the Mathematical Collections, Pappus supplied "the geometers of his time with a succinct analysis of the most difficult mathematical works and facilitate[d] the study of them by explanatory lemma s" [2, p. 49]. Pappus distinguishes two kinds of analysis, zetetic and porisitic, terms which Viète borrowed and added to them a third stage, called rhetic or exegetic. These stages of analysis will be discussed at greater length later in the paper. The second important influence on Viète was Diophantus' Arithmetic. Diophantus "was one of the last and most fertile mathematicians of the second Alexandrian school. He flourished around 250 A.D." [2, p. 60]. Arithmetic, said to have been written in 13 books, introduced the idea of an algebraic equation expressed in algebraic symbols [2, p. 60]. Viète expanded this notion of algebraic symbols. According to Klein, some historians of mathematics: see the Diophantine science as the primitive "preliminary stage" of modern algebra. From the point of view of modern algebra only a single additional step seems necessary to perfect Diophantine logistic: the thoroughgoing substitution of "general" numeri cal expressions for the "determinate numbers," of symbolic for numerical values - a step which was, subsequent to a great deal of progress in the treatment of equations in general, finally taken by Vieta [7, p. 139]. Diophantus can be placed in the second of three categories of algebra, with respect to notation, described by Cajori. The first class is Rhetorical algebras, in which no symbols are used, everything being written out in words. Rhetorical algebra was use d in some Arabic works, the Greek works of Iamblichus and Thymaridas, and the early Italian writers. The second class is Syncopated algebras, in which, as in the first class, everything is written out in words, except that abbreviations are used for cert ain frequently recurring operations and ideas. Syncopated algebra can be found in the works of later western Arabs, Diophantus and some of the later European writers down to about the seventeenth century (with the exception of Viète and Oughtred). The t hird class is Symbolic algebras, in which all forms and operations are represented by a fully developed algebraic symbolism, as for example, x2 + 10x + 7. Symbolic algebra is the form employed by Viète, Oughtred (1574-1660), the Hindus, and the Europeans since the middle of the seventeenth century (Nesselmann, cited in [2, p. 111]. Viète combined the methods of analysis explained by Pappus with the methods used by Diophantus to create one of his most prominent works, In artem analyticem isagoge or Introduction to the Analytic Art. This book was written during the second of Viète's "highly fruitful periods of leisure and research" [3, p. 267], 1584-1589. Another well-known production of Viète was his Harmonicum Coeleste or Harmonic Construction of the Heavens, which was composed during his first period of great discoveries (1564-15 68). "In 1579 Vieta published his Canon mathematicus seu ad triangular cum appendicibus, which contains very remarkable contributions to trigonometry. It gives the first systematic elaboration in the Occident of the methods of computing plane and spheri cal triangles by the aid of the six trigonometric functions" (Braunm|hl cited in [2, p. 137]. Viète was a forerunner in many aspects of modern day algebra. He was "the first to apply algebraic transformation to trigonometry, particularly to the multisection of angles" [2, p. 138]. He established "the earliest explicit expression for ( by an infi nite number of operations" [2, p. 143]. He "employed the [use of] the Maltese cross (+) as the short-hand symbol for addition, and the (-) for subtraction" [2, p. 139]. Although the Germans introduced these characters, they were not commonly used before Viète's time. They became more popular after Viète began to use them. Among the various concepts propagated by Viète, his most well known "contribution to the development of algebra was his espousal and consistent utilization of the letters of the alphabet - he called these species - to represent both the constant and the v ariable terms in all equations" [9, p. 5]. "Diophantus seems the most likely source for Vieta's use of the word 'species'" [7, p. 321]. Viète referred to this new algebra, which used letters, as logistica speciosa, as opposed to the old algebra, logisti ca numerosa, which used numbers. "Letters had been used [previously] to represent the quantities that entered into an equation, but there had been no way of distinguishing quantities assumed to be known from those unknown quantities that were to be found " [1, p. 335]. As a solution to this dilemma, Viète designated consonants for known quantities, and used vowels for unknown quantities. The creation of this system, also known as literal notation (a notation in which letters stand in place of numbers), allowed mathematicians to solve equations in a more general form, rather than having to work out each specific equation on an individual basis. Whereas one once had to deal with each separate equation as a unique problem to be solved, one can now derive a universal way of solving all problems alike. For instance, "literal notation made it possible to build up a general theory of equations - to study not an equation like [6x2 - 5x - 1 = 0] but the quadratic equation ax2 + bx + c = 0" [1, pp.335-336]. Th is was a significant building block in the foundation of algebra as we know it today. Even though this invention was quite important, "the vowel-consonant notation of Vieta had a short existence" [1, p. 336]. After Viète's death, Descartes began using the "letters at the beginning of the alphabet...for given quantities, and those near the end (especially [z]) for the unknown. This rule was rapidly assimilated into seventeenth century practice and has survived to modern times" [1, p. 336]. On essential fact to add about his notation is that "he restricted the use of letters to positive q uantities" [6, p. 94]. Unlike today's algebra, where we use letters to denote both positive and negative numbers, Viète did not include negative quantities in his logistica speciosa. How were quantities expressed before Viète's introduction of logistica speciosa? "For unknown quantities [Diophantus] had only one symbol, ?" [2, p. 61]. He too did not recognize negative quantities. If a solution was not positive, he deemed the answer impossible and the problem to be faulty. Brahmagupta, on the other hand, called the unknown quantity yavat-tavat. "When several unknown quantities occurred, he gave, unlike Diophantus, to each a distinct name and symbol. The first unknown was designat ed by the general term 'unknown quantity.' The rest were distinguished by names of colors, as the black, blue, yellow, red or green unknown" [2, p. 93]. "The Indians were the first to recognize the existence of absolutely negative quantities" [2, p. 93] . In addition to Viète's conception of algebraic notation, he also offered some valuable insight on powers. "Prior to Viète, it was common practice to use different letters or symbols for the various powers of a quantity" [4, p. 131]. For example, althou gh Diophantus only had a single variable, he used three different symbols to represent that one quantity: he signified our x, x2 and x3 with three different symbols: ?, ?? and ??, respectively. The three dissimilar symbols make it difficult to comprehend that one is a square or cube of the other. Perhaps if we saw '?' and '??' or '? ?' we might more easily be able to decipher that '??' or '? ?' is the square of '?'. Viète, on the other hand, simplified this by using A, A quadratum and A cubum, which wa s later shortened to A, A q and A c. Even though this does not display powers as exponents as we use them today, his way makes it much easier to understand that A quadratum is the square of A and A cubum is the cube of A. Our "present system of indices - x, x2, x3, and so on," was introduced by Descartes [4, p. 131], although there are examples of very large exponents (up to 45) where Viète resorts to numbers rather than his usual abbreviations, as is with the equation put forward by Adrianus Romanus. "In 1593 the Dutch mathematician Adrianus Romanus proposed to all the mathematicians the problem of solving a certain equation of degree 45. The ambassador of the Netherlands at the court of the French king Henri IV claimed that nobody in France would be able to solve this problem" [10, p. 65]. Viète proved the ambassador wrong, and solved the equation the very next day. Branching off from powers is Viète's Law of Homogeneity. Waerden states: only magnitudes of "like genus" can be compared or added. Thus, where we would write a quadratic equation as bx2 + dx = z, Viète writes "B in A Quadratum, plus D plano in A, aequari Z solido". Namely, by Viète's Law of Homogeneity, if A (our x) and B (our b) are line segments, D must be a plane area and Z a volume. Hence he writes "D plano" and "Z solido" [10, pp.63-64]. In simpler terms, the combined powers of each term must be the same in order to add, subtract, multiply or divide them. Thus, in the example noted above, B (our b) is a line segment (or a length, side or root) with a power of 1; A Quadratum (our x2) is a plane (or a square) with a power of 2; together they comprise a power of 3 (a length and a plane). Furthermore, D plano in A (our dx) is also a length (A or x: power of 1) and a plane (D or d: power of 2) with a combined power of 3; and Z solido (our z) is a solid (or a cube) with a power of 3. The combined power of bx2 is 3, (b's power is 1, x2's power is 2), of dx is 3 (d's power is 2, x's power is 1) and z is 3. Hence, you are adding three terms, all with the same power of 3. However, this law is very confusing and it "implies a serious restriction of the algebraic formalism...Omar Khayyam managed to circumvent this restriction by introducing a unit of length e" [10, p. 64].

Viète presented several stages to solving an equation. There are 3 parts to his design: zetetic, poristic and rhetic or exegetic. The first two kinds of analysis are terms borrowed from Pappus, as stated above. Hadden defines these stages as: The zetetic finds an equation or proportion between the magnitude sought and those given, the poristic investigates the truth of a theorem from the equation of proportion thus set up, and in the exegetic the magnitude sought is produced from the equation or proportion [5, p. 131]. Viète coined this process, a three-part method for the whole analytic art, "the science of correct discovery in mathematics" [9, p. 12]. Zetetics, the first phase, is where one establishes the relationship between the known and unknown quantities and set s up an equation to illustrate that relationship. Viète proposes Rules of Zetetics, much like Euclid's "setting out" of a proposition, such as rule #1: "If it is a length that is to be found and there is an equation or proportion latent in the terms propo sed, let ["the side"] be that length" [9, p. 23]. These rules lay out the laws to follow when trying to find an appropriate equation for the sought and given magnitudes. Next, we progress to the second step, Poristics. As Viète explains, "Zetetics havi ng completed [its work], the analyst moves from hypothesis to thesis and presents the theorems derived from his discovery in the form prescribed by the art comformably to the laws" [9, pp.27-28]. In the poristic stage of analysis, we work out the problem in an attempt to discover whether or not the equation we derived in the first stage is a plausible equation to use in solving for the unknown quantity. In solving the equation, we prove that the theorem we tested is true. The third and last part in the analysis of an equation is the Rhetic or Exegetic, in which the sought quantity is found: The analyst turns geometer by executing a true construction after having worked out a solution that is analogous to the true...[and] he becomes an arithmetician, solving numerically whatever powers, either pure or affected, are exhibited. He brings forth examples of his art, either arithmetic or geometric, in accordance with the terms of the equation that he has found or of the proportion properly derived from it [9, p. 29]. The unknown magnitude is solved for, and the three stages of analysis are complete.

Francois Viète contributed a great deal to all facets of the mathematics of his day, including algebra, geometry and trigonometry. His most eminent contribution, the modern algebraic notation or logistica speciosa, is the defining point for algebra as w e know it today. From his use of vowels and consonants to indicate unknown and known quantities, to his work on analysis and powers (using the same letter as a base for different exponents: A, Aq, Ac), he has shaped the way students currently learn moder n day algebra. His efforts have enabled his successors, from Descartes down to the moderns, to further develop the field of mathematics as a powerful tool with general applications..


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