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 (late sixteenth - the early seventeenth centuries).

Contents

General ideas

Until 1591, the year in which François Viète published his In artem Analyticem Isagoge, the formalization of algebra was limited to the introduction of a single letter. We find this fundamental innovation in Johannes Hispalensis and Jordanus Nemorarius at the end of the twelfth century, and, later, in some other mathematicians like Regiomontanus (see below). However, this notation was not followed by the mathematicians of Middle-Ages and Renaissance who have the greatest difficulties in handling formal equations.

The path followed by François Viète was the beginning of the construction of the current formalism. It is a global project which provides, in modern terms, both an axiomatic of literal calculations and a method to invent mathematics. This new algebra was a fundamental contribution to our modern algebra, as illustrated by B. Lefebvre in 1890:

This digital algebra, where the unknown is designated only by a letter or a word (coss, res, radix, corn), and the known are represented by numbers, has persisted through the ages until the time of Vieta.... Aristotle, Euclid, Archimedes, Pappus often think of letters; Johannes Hispalensis, Fibonacci, sometimes; Jordanus Nemorarius frequently, and others, such as Pacioli, Stifel, Regiomontanus, Peletier, Butéon stated and demonstrated mathematical theorems with letters expressing quantities or variables. Is this already literal Algebra ? - No. But to submit those letters to calculation, like literal quantities; to play with virtual letters as digits; to perform transformations of algebraic expressions, to solve equations with literal coefficients, in a word to begin symbolic calculation, that is the subject of literal algebra or the science of formulas.[1]

Admittedly, the literal form proposed by Viète is not ours. In his mind, the numbers are decimals or square roots. Only positive solutions are considered, and so on. In fact, he builds not a general algebra of polynomials, but a collection of algebras for homogeneous polynomials, where indeterminates are letters designating length, area, volume and hypervolumes.

However, through this alternative way, Viète managed to give us the first effective symbolization of algebraic equations.

Viète published his work at his own expense. He taught his method to a few students, who published his works and used his methods and notations. Among the most famous, we find: Aleaume Jacques, Marino Ghetaldi, Jean de Beaugrand and Alexander Anderson. Thereafter, Le Sieur de Vaulezard, Claude Hardy, Pierre de Fermat and Thomas Harriot will perfect his language.

When in 1637, René Descartes illustrated his method by a treatise on geometry, the philosopher concluded this revolution. By removing the constraints of homogeneity introduced by Viète, Descartes gave algebra its present form. But that is another story.

Chronology of publications

In 1591, the description of this program in In artem Analyticem Isagoge republished in 1624 by Jean de Beaugrand

In 1593, the publication of the Zététic or Zeteticorum Libri quinque, a series of Diophantine problems determined by the method developed in Isagoge.

The same year two geometrical Exegetics: Effectionum Geometricarum Canonica recensio and an example of the new algebra: Variorum de rebus Mathematicis responsorum , Libri Septem

In 1600, an Exegetic based on digital numbers De numerosâ potestatum ad Exegesim resolutione released by Marino Ghetaldi

In 1600, examples of use of the new algebra in Viète Francisci Vietae Apollonius Gallus seu Apollonii Pergaei περι επαιρων Geometri

Theses works were formed while Viète was still living. Let's see, now, the publication of his students:

In 1600,Confutatio problematis ab Henrico Monantholio ... proposiiti. Quo conatus est demonstrare octavam partem diametri circuli aequalem esse lateri polygoni aequilateri & aequianguli eidem circulo inscripti, cuius perimeter ad diametrum rationem habet triplam sesquioctavam... in Paris at David Clerc, by Jacques Aleaume.

In 1607, a geometrical 'Exegetic': Apollonius Gallus published by Marino Ghetaldi

In 1615, two treaties of symbolic logistic: De recognitione Æquationum et De Emendatione Æquationum Tractatus Secundus, published by Alexander Anderson, who said he had much to do to enable them to be published, passages missing entirely, of others are simply listed and paper everywhere soiled and torn.

The same examples of use of the new algebra in Ad Angularium Sectionum Analyticem Theoremata, developing the theory of equations.

In 1624 a 'Zététic' (or symbolic logistic), Ad Logisticem Speciosam. Reprinted in 1631 and published twice by Jean de Beaugrand.

This edition will continue through the translations of Le Sieur de Vaulezard, of Antoine Vasset (about 1630), and through the complete edition of the works of Viète (except Harmonicum Celestae) by Frans van Schooten in 1646.

The Isagoge

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

This work is available available via gallica[2], written in Latin, and announcing that it will be the first volume of a work divided into ten parts:

  • In artem Analyticem Isagoge
  • Ad Logiticem speciosam Nota priores
  • Zeteticorum libri quinque
  • De numerosa potestatum ad exegesim resolution
  • De recognitione Aequationum
  • Ad logiticem speciosam nota posteriores
  • Effectionum geometricarum Canonica recensio
  • Supllementum Geometria
  • Analytica angularium sectionum in tres partes
  • Varorium de rebus Mathematicis responsorum

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

Chapter I: Introduction

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.

  • The Zetetic is the art of translating a problem into an equation and the art of handling this equation to put it in a canonical form which gives rise to an interpretation in terms of proportions.
  • The Poristic is the examination of the truth (by the means of ordinary theorems).
  • The Exegetic, is the determination (the exhibition, says Antoine Vasset), of geometric solution or numerical solution, obtained from the general propositions of the Poristic.

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.

Chapter II About symbols, equalities, and proportions

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

  • paragraphs 1 to 6 cover the properties of equality:

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

  • paragraphs 7 to 11 cover the properties of laws (addition, product, etc. with fractions).
  • paragraphs 15 and 16 explain when fractions are equal or not.

Chapter III: Des lege homogeneorum

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.

Chapter IV De praeceptis logistices speciosae

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

Then, secondly, he defines products and quotients of homogeneous quantities. He then notes

\frac{A \text{ plano}}{B} \text{ subducere } \frac {Z \text{ quadratum}}{G}\text{ residua erit }\frac {A \text{ planum in } G - Z \text{ quadrato in } B}{B\text{ in }G}

what we note now

\frac{A }{B} - \frac {Z }{G} =  \frac {A G - Z  B}{B  G}

without attempting to mark the homogeneity factor.

Chapter V: Laws of the Zététic

In this chapter we have the foundations of the formulation of equations and particularly in paragraph 5 of this chapter, the idea that some letters should be reserved for known quantities (data) and other letters to unknown quantities (incertitus). Viète designates the first quantities by consonants and the others by vowels. Then, after a few propositions, the book ends with two short chapters that describe how, in practice, it is necessary to conduct the analysis of a problem, its resolution and geometrical checking.

Chapter VI: The theorems of Poristic

Chapter VII: About Rhétic (or Exegetic)

Chapter VIII: Epilogue

In this final part Viète defines some notations, including the first and second roots (in other words square and cube roots).

Variations of 1631

The manuscript published by Vasset doesn't contain the definition of Poristic and Exegetic (ch VI and VII), but some results on the development of the binomial (to level 6) and general theorems of Poristic: which way to insert a medium proportional you want between two lengths.

This means, for instance, that the sequence

A^6,A^5B,A^4B^2,A^3B^3,A^2B^4,AB^5,B^6\,

is geometric.

Reflecting Viète, Vasset writes also:

A − B cubus cubus aequabitur A cubo-cubus − 6 A quadrato-cubus in B + 15 A quad.quad. in B quad. − 20 A cubus in B cubum + 15A quadratum in B quad.-quad − 6 A B quad.-cub. + B cubus-cubus

which we denote now by:

(A-B)^6 = A^6-6A^5B+15A^4B^2-20A^3B^3+15A^2B^4-6AB^5+B^6.\,

He then gave the rule for forming binomial coefficients (already known by Stiffel and Tartaglia), noting that he obtains the coefficients of the development, by addition in the development of the previous power, of the first and second coefficient, of the second and third, and so on.

Francois Viete

Francois Viete, French mathematician
Born 1540
Fontenay-le-Comte, Poitou
Died 23 December 1603
Paris, France
Fields algebra
Known for first notation of new algebra
Influences Pierre de Fermat
Influenced by Ramus

Zététic, Poristic and Exegetic

The five books of the Zetetic.

In 1593, Viet publishes Zeteticorum libri quinque, which complements and enriches the new algebra. Zététique comes from the Greek zêtêin: try to enter the reason of things.

This book consists of five books containing ten problems where Viète researchs quantities which he knows through their sum or difference and through their quotient or product. There are also equations of degree 2 and 3, and partitions of numbers into squares. It ends, as an example of symbolic logistic, with a Diophantine problem.

These five books develop the method proposed in Isagoge.

The new algebra is presented as a new language to formalize the calculation but also as the instrument to solve new problems. These books are a test, where Viète addresses issues raised by Diophantus like his elders did, but, also, particularly in Book III, new issues, without equivalent in Diophantus.

It includes among other things, the following nice problem:

Dato adgregato extremarum, et adgregato mediarum in serie quatuor continue roportionalium, invenire continue proportionales.

And some arithmetical issues solved with triangles whose lengths are, in fact, the real and imaginary parts of the product of two complex numbers (which Viète seems not to realise).

The exegetical geometric Canon

In 1593, in Effectionum Geometricarum Canonica recensio or review of geometric constructions, Viète begins by demonstrating the relationship between geometric constructions and algebraic equations. His purpose is the graphical resolution of quadratic equations. There are also solutions of geometric problems, leading to quadratic equations and treated algebraically.

The poristique, The Supplementum

In 1593, in the Supplementum Geometriae, Viète gives a more complete characterization of the Poristic. They include: the trisection of the angle, the construction of the regular heptagon; solving cubic equations or quadratic-quadrato and their equivalence to the problem of trisection.

In 1592 - 1593, Viète gives in Variorum de Rebus in Mathematics Responsorum, Liber VIII, an answer to the problems raised by Scaliger. It deals with problems of duplication of the cube and trisection of the angle. What he called an irrational problem.

The digital exegetic De Numerosa

In 1600, in De Numerosa Potestatum ad Exegesim resolutione, Viète sets the goal of solving equations of any degree with the help of radicals. This belief will be definitely disappointed by Niels Abel in 1828. He provides a method to approximate the roots of an equation. This method will influence Newton and the rule of Newton-Raphson owes him a lot. Raphson Joseph stated it in 1690.

Posthumous publications

In 1615, in De Recognitione œquationum published by Anderson, twenty chapters fairly repetitive, there are the relationship between roots and coefficients, equations in which the unknown is given by relation between his first and his cubic power, how to make off the second term of an equation (or Ferrari's method for solving equations of degree 4) and the equivalence between equations of the third degree and knowledge of the first of four quantities continually proportional, and of the difference between the second and fourth, in order to find the second. Viète still gave in this book some ways to lower the degree of an equation.

In De emendatione œquationum, published the same year by Anderson, and composed of fourteen chapters, one finds the names given by Viète to some algebraic operations: The isomerism to remove the denominators of equations without introducing coefficient to the highest power of the unknown; the Paraplérosinal Climaction to reduce an equation of fourth degree equation by taking the second equation as an intermediate of the third; a method similar to that of Ferrari (Viète rediscovers it). There is also (as duplicate hypostases) the resolution of equations of the third degree (Cardan had published, in 1545, the verification that the formulas were correct, but empirically).

Finally, this book describes the decomposition of a polynomial with as a number of solutions equal to its degree.[3]

The contributions of the new algebra

The isagoge: a work which was a milestone

The introduction of literal notation for the parameters in the algebraic equations and the willingness to make algebra free of medieval procedures, providing an effective formalization, are central to the project of Francis Viète.

To do this, he describes the rules of multiplication, addition, subtraction, etc., operating, not on numbers but on the sizes (length, area, volume, etc.). The need to maintain the homogeneity of the formulas obliges Viète to give different names to these operations. The notation ducere in is inherited from the Italians and their Arab predecessors and operationalizes the idea that led the side to the side b to form the rectangle ab.

We have seen that the main idea (note by letters an unknown number), had already been introduced by Jordanus Nemorarius about the thirteenth century, by Michael Stifel and Regiomontanus in Germany in the sixteenth century, but, here, all measurable quantities, kwnown or unknown, may become subject of an equation; sa that, François Viète, developing his ideas in a systematic way and separating the alphabet between known and unknown parameters, offers in this introduction to the art of the analysis the first symbolic algebra work never done.[4]

Moreover, the results reported by this method are varied and numerous. From the determination of positive solutions of equations of degree 2, 3 and 4 to the statement of the relationship between coefficients and roots, the formation of binomial coefficients in the development of a polynomial into a product of factors, and so on. So that, this innovation, considered one of the largest in the history of mathematics, really opens the way for the development of modern algebra

Historical critics

Viète made good Algebra with excellent geometry. However, in his desire to spend his new label under the Diophantine's aegis, Viète was taken to preserve the language of the elders. Moreover, he has no symbol to rate the multiplication, roots or equality. For example

\frac{S\text{ in }A\text{ planum }+\text{ Rbis in }A\text{ planum}}{R} \text{ aequabitur }B\text{ plano}

written today

 \frac{S A+ 2RA}{R} = B.

Although effective, this new algebra maintained a requirement of homogeneity which is very heavy and condemns the constant reference to the meaning of geometrical parameters involved. A second Algebraic revolution will be done in the next generation with William Oughtred, Thomas Harriot, Pierre de Fermat, and finally René Descartes. Nevertheless, Viètes gave us for the first time the ability to work efficiently on letters. And, for that, he must be honored as one of the fathers of Algebra.

See also

Sources

References

  1. ^ (French) B.Lefebvre Cours d'introduction à l'algèbre élémentaire
  2. ^ (French) François Viète In artem analyticem Isagoge, Meteyer publisher, in Tours, (1591)
  3. ^ Ronald Calinger Vita mathematica Mathematical Association of America
  4. ^ H. J. M. Bos: Redefining geometrical exactness: Descartes' transformation [1]

Bibliography

  • D. J. Struik A Source Book in Mathematics, 1200-1800; Harvard University Press, 1969 p 73-81 available here.

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  • (Latin) The integral corpus (excluding Harmonicon) was published by Frans van Schooten, professor at Leyde as Francisci Vietæ. Opera mathematica, in unum volumen congesta ac recognita, opera atque studio Francisci a Schooten, Officine de Bonaventure et Abraham Elzevier, Leyde, 1646; Online Text.

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  • Robin Hartshorne; [2].

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