The Analytic Art

This is a translation of the ‘Opera Mathematica de Viète’, compiled by van Schooten (1646). Specifically, this Dover edition is an unabridged reprint of Witmer’s book, originally published by the Kent State University Press in 1983. The title ‘The Analytic Art’ refers to Viète’s ‘Isagoge in Artem Analyticem’ that was published in 1591 and forms the first chapter. Together with the remaining eight chapters, this corresponds to the ‘nine studies’ referred to in the subtitle above. The introduction contains a short biography of Viète (1540-1603) and briefly places his work within a historical context (he came one generation after Cardano and one generation before Descartes and Fermat ).

The translation of Viète’s Latin prose into English certainly conveys the quaintness of the 16^th century vernacular, but the mathematics is presented mainly in modern form, which means that the original notation has largely been lost in translation. For example, where Viète would write:

Confectarium: Itaque si A cubus + B plano 3 in A, aequetur Z solido 2,
& aequetur D cubo.
Ergo ,  sit A de qua quaeritur.

Witmer gives this syncopated version on page 287:

Corollary: So, if A³+3B^pA⁸⁰=2Z ^s and ,
A, the unknown, is (B^p-D²)/D

Comparing this to the original, I was surprised by that term A⁸⁰, until I realised it referred to footnote 80 at the bottom of page 287!

This prompts the observation that, in Witmer’s book, although there are various footnotes, there is no commentary on Viète’s work, and readers have to deduce the meaning of terms such as B^ppp. Another clue to its actual meaning is obtained from the translation of the same piece of Viète’s writing provided by Cajori [1 ]:

Conclusion:

If therefore x³+3B²x=2Z³, and , then 
is x, as required.

Hence, Cajori reads B-plano-plano-plani and B^ppp as both representing B⁶, each being (B²)³, and the terms Z-solido-solido and Z^ss represent Z⁶, since each is equivalent to (Z³)², whilst D cubo obviously refers to D³.

Viète was the first to make systematic use of letters to denote coefficients, and this enabled him to deal with equations expressed in general form, as opposed to the Italian algebraists, who mainly solved particular equations. But why did he resort to the use of such strange terminology? The answer certainly isn’t obvious from Witmer’s footnotes, but it can be extracted from the general context of Viète’s mathematical thinking. For a start, he rejected the word ‘algebra’ because of its association with Arabic mathematics, which he regarded as non-rigorous. Instead, he preferred the terminology and processes of Greek mathematics, due to the traditions of its supposed logical rigor. In fact, he declares his allegiance to the ‘analysis’ of Plato and Theon in the very first sentence of this translation.

Moreover, in the introduction to the Analytic Art, he gives his ‘Fundamental Rules of Equations and Proportions’ in a form analogous to the axioms of Euclid, and he describes his work on algebra under three categories, the first two of which he adapted from Pappus:

• zetetics, by which one sets up an equation or proportion
• poristics, by which the truth of a stated theorem is tested by means of an equation or proportion
• exegetics, by which the value of the unknown term in a given equation or proportion is determined

As with Greek geometricians, Viète used capital letters to denote lines and surfaces, so he thought in terms of elementary spatial concepts. Naturally, he adopted an associated terminology to describe the various magnitudes appearing in his equations. Thus, where Witmer gives us ‘x times x⁵ yields x⁶’, Viète would say ‘a breadth times a plano-solid produces a solido-solid’. Moreover, he uses the term ‘proportional’ to describe his equations due to his adherence to homogeneity of dimension. So he would deal with an equation like A⁴ + B³A = B³Z in which each term is of order 4. Conversely, an equation such as A² +A = 2Z would not conform to any geometric sense because A² represents a plane, while the other terms are one-dimensional.

Interestingly, although the ‘=’ sign first appeared in 1557, Viète didn’t ever use it and expressed that relationship verbally (aequetur).

Despite his heavy reliance upon spatial terminology, Viète avoided use of geometric methods and the first part of this book (Isagoge in Artem Analyticem, 1591) represented the first effective effort to establish algebra as a subject independent of geometry. Of course, by improving on Viéte’s algebra, and implicitly laying the foundations for the coordinatisation of dependent variables, Descartes subsequently completed the role reversal, whereby geometry has since been mainly algebraic.

Viète’s algebra occupies the bulk of this book, which he describes in two parts as ‘Five Books of the Zetetica’ and ‘Two Treatises on the Understanding and Amendment of Equations’. The Zetetica contains about eighty individual zetetics (problems), of which an example from Book I would be: ‘Given the product of the roots and their ratio, find the two roots’ And from Book III: ‘To find numerically two squares with a given difference between them’. As for the two treatises on the ‘amendment’ of equations, they represent an advance in the study of the formal manipulation of equations and methods for solving them. For example, chapter VI of the second treatise is called ‘How Biquadratic Equations are Reduced to Quadratics by Means of Cubes of a Plane Root, or on Completing the Power’. An example his method of substitution, is found on p. 449 of Smith [2] and is expressed in the original as:

Si A quad. + B2 in A, aequetur Z plano. A+B esto E. Igitur E quad. Aequabitur Z plano+ B quad.

Translated by Witmer as: If A²+2BA=Z², let A+B=E, then E²= Z^p+B²

The penultimate chapter covers Viète’s writing on numerical methods for the extraction of roots whilst the very last chapter, which seemingly concerns the geometry of the circle, really focuses upon some of Viete’s notable work on trigonometry. For instance, Witmer translates ‘AB quadratum ad AD in AC minus CB in DB, ut AB ad AE’, as

In modern notation this becomes cos(A+B) = cosB.cosA – sinB.sinA

Overall, I was fascinated by Witmer’s book, but I reserve judgement on the accuracy of his translation, since there is no equivalent English version for purposes of comparison. Nonetheless, many of the translated excerpts contained in some of the general histories are in agreement with the equivalent bits done by Witmer, who is deserving of belated thanks for the completion of the long and difficult task of translating and transposing Viète’s most important work into a single volume. As a result, the work of that great French algebraist is now accessible to all those interested in the history of mathematics. This book is therefore recommended for inclusion in the libraries of all institutions offering such courses.

References 

[1] A History of Mathematical Notations, Florian Cajori (Dover, 1993)

[2] History of Mathematics, D.E. Smith (Dover, 1958)

Peter Ruane has now escaped the bureaucratic confines of higher education, where he spent a working life training primary and secondary mathematics teachers.