New Algebra
The new algebra, logistical or specious analysis, is a project to formalize the algebraic notation directed by Francois Vieta and his successors. The founding act is the publication in Mettayer Jamet (in 1591 ) of artem Analyticem Isagoge. His appearance led end XVI - early XVII century , the gradual abandonment of the rhetoric algebra. This formalism has evolved according to the authors, especially under the leadership of Anderson , of Vaulezard of James Hume of Godscroft and Pierre Hrigone. When in 1637 , Rene Descartes illustrates his method by a treatise on geometry, the philosopher concludes this revolution , it provides the current algebra its literal form (or nearly so).
Algebra before Isagoge
Until 1591 the formalization of the algebraic language was limited to the introduction of a letter or two , designating one or two unknown quantities. We find this fundamental innovation in Jordan Nemorarius on the border of XII - XIII century , but this symbolism, already known to the Greeks , does not progress until Jacques Peletier du Mans, Jean Borrel and Guillaume Gosselin. This notation, which is not used on an ongoing basis by medieval mathematicians disappears even at the dawn of the Renaissance, where we use more abbreviations. The first translations of Johannes Hispalense until Nicolas Chuquet or Regiomontanus , you can not really speak of literal algebra. As noted by F. Russo :
"The symbols for unknown quantities is found already among the Greeks, also in the Middle Ages, especially in Leonardo of Pisa and Jordanus Nemorarius but these symbols are not really engaged in a" surgical technique. " As they remain in a static state. The symbols of unspecified quantities are not found before Vieta algebraists in the sixteenth century. This was the first revives the ancient and medieval tradition, but by combining it with a surgical technique that will give him all her fertility. "
Mathematicians of the sixteenth century also experiencing severe difficulties in handling formal polynomial equations and the same letter is sometimes used along with the unknown and the square root, as in Peletier (in 1554 ), Jean Borrel (in 1559 ) and Gosselin (in 1577 ). The arithmetic remains rhetoric and it is customary to justify geometric problem solving algebraic . The efforts of the German school focusing more on the structuring of transactions on their formalization, implementation of the symbolic notation is done in a scattered manner.
The teaching of Pierre de la Ramee and resolutions of digital systems exposed by his students , however, will prepare a radical. Pierre de la Ramee , said Ramus, restored the place of mathematics in the university . Logician, literate Hellenizing Latinized and Hebrew, he opposed the thought of Aristotle and built his own system of binary logic, where consistency plays a fundamental role. His "students," Gosselin Guillaume and Jacques Peletier du Mans introduced the first formal notation for the unknowns of digital systems of two linear equations. However, their algebra remains at most syncopated.
In Italy, Francesco Maurolico , master Federico Commandino and Clavius published in 1575 , the year of his death, some proposals involving letters with their product, rated "A in B" and called "C plano" respecting the homogeneity of formulas. The influence of his works on Vieta remains unknown , it should be noted, however, the similarity of ideas Maurolico (aka Marule) and its concerns, and geometric cosmographic with those of Viet. However, writing the father Francesco Messina remains marginal in his work, and devoid of theorizing.
In this context, the publication of Isagoge by Francois Vieta inaugurates the beginning of a new era and contemporary Announces algebraic formalization .
The birth of algebra
The Isagoge can also be read as heralding a global project , . And he immediately offered, in modern terms, as an axiomatic literal and calculations to invent a method of mathematics. In this new algebra is also a fundamental contribution to the construction of the current algebra literal, as shown in B. Lefebvre in 1890 :
"This digital algebra, where the unknown is only designated by a letter or a word (coss, res, radix, corn) and where known are represented by numbers, persisted through the ages until the time of Vieta. However, long before and even Vieta any time , has emerged the use of letters not only to represent the unknown of a problem, but even to refer to the following reasoning quantities or objects is determined to be indefinite. Aristotle, Euclid, Archimedes, Pappus reason often on bills, John of Seville , Leonardo of Pisa sometimes; Jordanus Saxon frequently, and others, such as Pacioli , Stifel , Regiomontanus, Peletier, Buton, some more, the others less so, state and demonstrate mathematical theorem on letters, which express the quantities determined or undetermined. Is it not already Algebra literal? - No. But these letters refer to the calculation, these quantities literal; appear on these virtual letters calculations we can enforce only on numbers, perform transformations of algebraic expressions, solving equations with literal coefficients, in a word begin the calculation symbols, that is the subject of literal algebra or science formulas. "
Publishing this work at its expense and sometimes confidential manner , Vieta had few students, who completed his publications and used its methods of reasoning and its ratings. We find the list below. Among the most famous, known Nathanael Tarporley , Jacques Aleaume , Ghetaldi Marin , Jean de Beaugrand and Alexander Anderson. Subsequently, Vasset Antoine , Sieur Vaulezard , James Hume of Godscroft, Christmas Duret the ditrent and served as its language. Finally found its ratings from Adrien Romain , Thomas Harriot , Albert Girard , Pierre de Fermat , Blaise Pascal , Schooten Frans van , Christiaan Huygens and Isaac Newton.
It should be noted however, that the literal form given by Vieta's not really ours: In Vieta, only positive solutions are considered and deeper way, it does not build a polynomial algebra in n indeterminates, but a graded algebra , a collection of homogeneous polynomials which denote undetermined length, area, volume, hypervolumes.
|
However, by this circuitous route, Vieta manages to give the first effective symbolization of algebraic equations. One must add the other, he is particularly aware to make a break in his dedication to the Isagoge Catherine de Parthenay he says in effect
"Anything new is usually present at its origin rude and shapeless to be polished and perfected in the following centuries. The art I produce today is a new art, or at least so degraded by time, so sullied and soiled by barbarians, I felt it important to give it a completely new shape, and after stripped of all his erroneous propositions, so it should detain any stain, and she did not feel the age, imagine and produce new words which the ears are unaccustomed to this, it will be difficult as many people don ' are not in at the very threshold terrified and offended "
Chronology of publications
We can date the different steps of this process :
In 1591 : the program description in In artem Analyticem Isagoge.
In 1591 : The Zeteticorum libri quinque. Tours Mettayer, 24 folio, which make up the five books of Zttique. This is a series of diophantine problems, solved by the method developed in Isagoge.
Between 1591 and 1593: two exegetical geometric Effectionum Geometricarum Canonica recensio that links between algebraic expressions and equations of the second degree and some geometric problems and an example of using the new algebra: Variorum of rebus responsorum Mathematics, Libri Septem (Eighth Book of varied answers) Tours Mettayer , 1593, 49 foil, about the challenges of Joseph Justus Scaliger , in which he discusses the problems of angle trisection
In 1600 : an exegetical digital numbers potestatum Exegesim Resolutiones ad, released by Marino Ghetaldi
In 1600 : examples of use of the new algebra of Vieta in Francisci Vietae Apollonius Gallus seu Apollonii Pergaei Geometri
Such works are identified during the lifetime of Vieta. It continues with the publication of his students:
In 1600 : Confutatio problematic ab Henrico Monantholio ... proposiiti. Conatus quo is demonstrare octavam partem esse diameter circuli aequalem latere polygoni aequilateri & aequianguli Eidem circulo registration, ad cuius perimeter diametrum rationem habet triplam sesquioctavam ... in Paris with David Clerc, by Jacques Aleaume.
In 1607 : an exegetical geometric Apollonius Gallus, released by Marino Ghetaldi
In 1612 : Supplementum Apollonii Galli Marino Ghetaldi. In 1615 , two treaties Logistics specious and De De Recognitiones quationum Emendatione quationum Tractatus Secundus, published by Alexander Anderson who, he said, had much to do to put them in condition to be published, passages missing entirely, of others are simply listed and soiled and torn paper everywhere.
The same gives examples of using the new algebra in Ad Angularium Sectionum Analyticem theorematic, developing the theory of equations.
In 1624 : A Zttique or logistical specious Ad Logisticem Speciosam, reissued in 1631 and published two times a Jean Beaugrand.
In 1629 , Albert Girard publishes its own new algebra.
This movement publishing works of Vieta and comments appropriators ratings so more and more personal, will continue through the translations of Antoine Vasset and Sieur de Vaulezard to 1630 and again in 1631 by a Algebra easy by James Godscroft Hume , Scottish mathematician and then as a Apollonii Pergaei tactionum geometria, Francisco Vieta restituta.
In 1634 the dissemination of specious algebra or speculative gains momentum across the Curriculum Mathematicus Hrigone Peter , who performs in Volume V of his course a first extension of the work of Vieta.
Although in 1637 with the publication of his "Geometry", Rene Descartes openly challenges the constraints of homogeneity and verification (geometric or digital), by the new algebra, it still faces several decades of popularity. His edition continues in 1644 , for the algebra of Vieta Christmas Duret and finally, by publishing the complete works of Vieta (Harmonicum Celeste excepted) in 1646 , by Frans van Schooten , professor at the University Leiden (in press Elzevirs), assisted in his work by Jacques Golius and the Father Mersenne.
We limit ourselves here solely for editions that can be attributed to students and direct heirs. Other books being published in this writing, particularly in Italy, where it lasted nearly a century for some authors.
The Isagoge
Its correct name is In artem analyticem Isagoge (1591) and is the declared program of this vast project axiomatic.
The book, 18 pages, available at Gallica , is written in Latin by Francois Vieta. It promises to be the first in a series divided into ten parts:
- In artem Analyticem Isagoge
- Ad Logiticem speciosam Note priors
- Zeteticorum libri quinque
- From numerosa potestatum ad exegesim resolution
- From Recognitiones Aequationum
- Ad logiticem speciosam noted posteriores
- Effectionum geometricarum Canonica recensio
- Supplementum Geometria
- Analytica angularium sectionum in very special
- Varorium rebus of Mathematics responsorum
He alone provides a new approach to writing algebraic and opens the famous dedication to the princess mlusinide Catherine de Parthenay which is given a translation by Frederick Ritter .
Chapter I: Introduction
In the first part provides definitions of Vieta's specious analysis. It decomposes in a ternary movement: Zttique, Poristique, exegetical. It aims to provide well for the Doctrine invent Mathematics.
- Zttique the equation is the setting of the problem and the handling of this equation and put it under a canonical form that gives rise to an interpretation in terms of proportions.
- The Poristique is examining the truth of propositions through the ordinary theorems.
- The exegete is to determine, say Vasset exhibition, solutions, numerical and geometric, obtained from general propositions of poristique.
This is the introduction, concomitantly, an axiomatic calculations on the quantities (known and unknown) and a program intended to provide heuristic rules, which requires three steps to solving an algebraic problem or geometric formalization, General Resolution, Special Resolution.
Vieta adds that, contrary to former analysts, his method is to act on the resolution of symbols (not numeris sed iam in sub specie), which is a significant aspect. He further predicts that after him, the training will Zttique by the analysis of symbols and not by numbers.
Chapter II Symbols and aequalitum proportionum
Vieta continues in this second part, describe the symbols used in the equations and proportions, and it gives axiomatic rules:
- 1 to 6 on the properties of equality:
Transitivity of equality, keeping in sum, subtraction, product, and division,
- July to November on the properties of laws on fractions.
- 15 and 16 on equal fractions.
Chapter III: homogeneorum lege ...
Vieta then continued giving the laws of homogeneity, thus distinguishing symbols according to their powers, with 1 being the side (or root), 2 square, 3 cube, etc.. This factor requires that these powers are complementary to homogeneity, it notes
1. Length, 2 Plane Solid 3, then 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 3-dimensional.
Chapter IV Of prcepti logistics speciosae
Vieta in this fourth chapter provides the precepts of specious logic, that is to say the axioms of addition, product, etc., symbols denoting quantities of a similar nature.
Initially, his focus is on adding quantities of the same order, their subtraction, giving rules such as A - (B + D) = A - B - D or A - (B - D) = A - B + D
Then, in a second time on the naming of products homogeneous quantities, on the naming of quotients. He then notes
what we now note 
without attempting to mark the homogeneity factor.
Chapter V: Laws of Zttique
This chapter is enclosed foundations for the formulation of equations and particularly in paragraph 5 of this chapter, the idea that he should reserve some letters to known quantities (datas) and other unknown quantities (incertitus) , Vieta designating a first version, the first by vowels and consonants seconds.
Then follow a few suggestions.
The book ends with two short chapters that describe how in practice it is appropriate to conduct problem analysis, resolution, and geometric verification.
Chapter VI: theorems review poristique
Vieta develops the idea that once the modeling completed by the art of zetetic, the mathematician's theorems produced by his invention and the rules of syntax, as has been established since antiquity with Apollonius and Theon Archimedes.
Chapter VII: Rhaetian or exegetical
Exegetical dividing into two parts, one digital and geometric Vieta explicit in this chapter the need to transform the 'formula' generally obtained at the end of the review poristique a numerical result or a geometrical construction. The mathematician must, as appropriate, be arithmetician, showing that the roots can extract and calculate their assignments or surveyor and draw a figure the true result . It specifies that the result obtained on the letters is true, but a truth of another kind that does not specify.
Chapter VIII: Epilogue
In this last part, Vieta also defines some notations and summarizes twenty-nine steps of reasoning, he also defines the roots of order 1 and 2 (in fact square and cubic in the current nomenclature). At the end of this chapter he announces that, by this method, we can solve the problem of all problems, ie leave no issue unresolved or No nullum probeblum solvere
Variants of 1631
The manuscript contains published by Vasset up the definition of poristique and exegetical, some results on the binomial expansion (grade 6) and general theorems poristique including how to insert both average proportional they like between two lengths.
This means that the sequence A 6 A 5 B A 4 B 2, A 3 B 3 A 2 B 4 A B 5, B 6 is geometric.
Reflecting Vieta, Vasset wrote:
A-B Cubus Cubus Cubus aequabitur Cubo-A - 6 A quadrato cubus in B-15 A quad.quad. in B quad. -20 To cubus in B + 15A cubum quadratum quad. in B-AB-6 quad-quad. cub. + B-cubus cubus
in place of (A - B) 6 = A 6-6 A 5 A 4 B + 15 B 2 to 20 A 3 b 3 + 15 A 2 B 4-6 A B 5 + B 6.
He then gave the rule for forming the binomial coefficients (already known to Tartaglia and Stiffel), noting that to form the expansion coefficients, simply add in the power development preceding the first and second coefficient, the second and third, etc.. This practice gives Pascal's triangle.
The translation of Ritter (1867)
Engineer of bridges and roads serving in Fontenay le Comte , Frdric Ritter has spent his life studying the works of Vieta. He wrote a biography and a translation of his work that remain largely untapped. In 1868, Count Baldassare Boncompagni publish in its Bulletin the first 50 pages from the book, which has no less than 9 pounds. Ritter to book his new translation of the algebra of Vieta as publishes Frans van Schooten in 1646. The variants are significant and although the original plan and that of Vasset are respected, this translation takes a fresh look at the project editors Vieta.
In addition, Ritter and publish in this newsletter Boncompagni translating priors Note, first set-formulas of algebra specious-real introduction to the books of Zttique. It is this work that we detail here.
- The Treaty is divided into two parts, one algebraic and one geometric. Permires the proposals are quite innocent and prepare the future work:
- Proposition I: Three sizes are data to find a fourth proportional.
- Proposal II: Two sizes are given, find a third proportional, fourth, fifth, and other quantities continually proportional further orders, to infinity.
- Proposal III: Find a mean proportional between two given squares.
- Proposal IV: Find two means continually proportional between two given cubic
- Proposal V: Between two given sides, find any number of means continually proportional. What Viete gives as a solution: A square-cube, square-square A by B, A cubic B squared, A squared per cubic B, A with B and B-square carrc square cube.
- Proposal VI: The sum of two quantities add their difference. What meets the theorem: The sum of two quantities added to their difference is equal to twice the largest.
(The difference being taken in absolute value)
- Proposal VII: the sum of two quantities subtract their difference. What meets the theorem: The sum of two quantities minus their difference is equal to twice the smallest.
- Proposition VIII: When the same quantity is reduced by unequal amounts, subtract any of the other differences. What meets the theorem: If a quantity is reduced by unequal amounts, the difference remains the same is that the difference amounts subtracted.
- The approach is similar to the following proposals:
- Proposal IX: Where the same size is increased unequal amounts, subtract any amounts of the other.
- Proposition X: Where the same size is increased and decreased by unequal amounts, subtract from each other.
The binomial theorem
- Then come the proposals in the binomial expansion.
- Proposition XI: Form a pure power of a binomial root. Vieta developments here gives the square of A + B of the cube, square, square, and so on until the cubo-cube, a rule recognizing training monomials and identity of playing left right development binomial.
- Proposition XII: square of the sum of ribs, add the square of their difference. What meets the theorem: The square of the sum of the sides plus the square of their difference is equal to twice the sum of the squares.
- Proposal XIII: From the square of the sum of the two coasts subtract the square of their difference. What meets the theorem: The square of the sum of two sides minus the square of their difference is equal to four times the product plan for these sides. Then an important point, noted in a contemporary language
The remarkable identities
- Come some remarkable identities:
- Proposal XIV: Multiply the difference between two sides by their sum.
- Proposition XV: The cube of the sum of two sides add the cube of their difference.
- Proposal XVI: From the cube of the sum of both sides subtract the cube of their difference.
- Proposition XVII: Multiply the difference between two sides by three partial plans, which make up the square of the sum of the same sides, these planes being taken only once. This translates into theorem gives (x - y) (x 2 + xy + y 2) = x 3 - y 3
- Proposition XVIII: Multiply the sum of two sides by three partial plans, which make up the square of the difference of the same side, these planes being taken only once. This translates into theorem gives (x + y) (x 2 - xy + y 2) = x 3 + y 3
- Proposition XIX: Multiply the difference between two sides by four strong partials, which make up the cube of the sum of the same sides, these solids are taken only once. This translates into theorem gives (x - y) (x 3 + x 2 y + y x 2 + y 3) = x 4 - y 4
- Proposal XX: Multiply the sum of two coasts by four solid partial, which make up the cube of the difference of the same side, these solids are taken only once. This translates into theorem gives (x + y) (x 3 - x 2 y + y x 2 - y 3) = x 4 - y 4
- The following proposals,
- Proposal giving XXI (x - y) (x 4 + x 3 y + x 2 + y 2 and x 3 + y 4) = x 5 - y 5 and XXII similar proposal and the two following the end of grade 6 This first collection of basic formulas. Vieta generalizes the form of two theorems. It gives the second:
- "The product of the sum of two sides by the homogeneous terms, which make up the power of the difference of the same side, these terms are taken only once, is equal to the sum or difference of powers of the next higher order , that is to say the sum, if the number of homogeneous terms is odd, and the difference, if the number of homogeneous terms is even. "
Preparations for solving equations
- dd> Proposals from the basis of solving the equations outlined by Vieta in Numbers potestatis. They are extremely repetitive, given the impossibility for Vieta give meaning to a length oriented (this work will be completed until the nineteenth century by Hermann Grassmann with the creation of the exterior algebra ). Proposals to XXXIII XXV learn to form, for example the square affected by the addition of the sub side, produced by a coefficient in lateral length is suitably chosen to say, according to modern writing to see that x 2 + 2 xy + y 2 + d x + d y = (x + y) (x + y + d) = (x + y) 2 + d (x + y) and so on until the expansion of ( y + x) 5 (x + y + d). Proposals XXXIV to XXXVI resume with previous subtraction, XXXVII Proposals to merge IXL subtraction and addition in the same spirit. Proposals XL to XLIV repeat the same development of (x + y) - (x + y) n for n between 2 and 6.
Complex and triangles
- In the last part, Vieta and Ritter attack representation figures (triangles) of a number of classical problems.
- Proposition XLV: With two roots form a triangle data. This amounts to verify geometrically that (x 2 + y 2) 2 = (x 2 - y 2) 2 + (2 x y) 2.
- Proposal XLVI: With two triangles form a third triangle. This amounts to show that for x, y, z, t, we have (x y z + t) 2 + (x t - y z) 2 = (x 2 + y 2) (z 2 + t 2), identity Which makes sense if read as the product of the squares of modules of two complex numbers. It is usually attributed to Lagrange. Vieta and found geometrically, the principles underlying the calculation of the imaginary, initiated 50 years earlier by Scipione del Ferro. But the link between these geometric forms and their complex equivalents will not be truly established that the nineteenth century by Gauss.
- Vieta subsequently resolves the following issues:
- Proposition XLVII: With two similar right triangles form a third triangle as the square of the hypotenuse equals the third the sum of the squares of the hypotenuse of the first and the second hypotenuse.
- Proposition XLVII: With two equal and equiangular triangles, form a third triangle.
- Proposition XLVIII: With the triangle angle simple, and the triangle of the double angle form a triangle. The third triangle will be called "triple triangle angle.
- He plays thereafter (until Proposition LI) with the angle triple, quadruple the corner, then moves to the higher order, and so on to infinity, finding geometrically (and without saying so explicitly, since Vieta does not recognize the existence of imaginary) the real and imaginary parts of (x + I y) n is obtained by alternating pairs of coefficients that has developed in previous parts.
- Follow other proposals, which form the apex of this art, which can be interpreted in terms of complex geometry and the solution appears here :
- Proposal LII. : Create a right triangle with the sum of two roots and their difference.
- Proposition LIII. With the base of a triangle given the sum of its hypotenuse and the perpendicular, forming a triangle.
- Proposal LIV: Less of a right triangle right triangles of equal height, such as the triangle of equal height formed by their juxtaposition (which will cathetus hypotenuses of these triangles, and whose base is the sum of their bases) will the angle at the top right.
Zttique, Poristique and exegetical
As suggested by Maximilian Mary in his lectures at Polytechnic The five books of the Zttique. In the wake of Isagoge, Vieta publishes Zeteticorum libri quinque, which complements and enriches the new algebra. Zttique comes from the Greek ztin: seeking to penetrate the reason of things . This book consists of five books containing ten research problems with known quantities of the sum or difference and the quotient or product, equations of degree 2 and 3 and partition of numbers into squares. There is an example of the logistical and symbolic ends with a problem of Diophantus. 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 pose and solve new problems. These books are a test bed where Vieta addresses issues raised by Diophantus in the manner of old but also, especially in Book III, new issues, unparalleled in Diophantus . It includes among other things, the pretty problem: Dato adgregato extremarum and adgregato mediarum in series quartet continues roportionalium, Invenire Proportional continues. And some issues resolved arithmetic using triangles pointing by their lengths the real and imaginary parts of the product of two complex numbers. We give below a summary of the issues outlined in some of these books, we give them translated into modern language from the original and translations of Vaulezard, those of Frederick Ritter in his notes , . . and John Griswold's statement that in fact in his thesis of 1968 . In 1593 , in Effectionum Geometricarum Canonica recensio review or canonical geometric constructions, Vieta begins by demonstrating the relationship between geometric constructions and algebraic equations. Its object is the graphic resolution of quadratic equations, there are also the solutions of problems of geometry of the second degree, treated algebraically. In 1593 , the Supplementum Geometriae, Vieta gives a more complete characterization of poristique. They include: the trisection of the angle, the construction of the regular heptagon; solving cubic equations or quadratic and quadrato-equivalence problems trisection angle. In one thousand five hundred and ninety-two - one thousand five hundred ninety-three in Variorum of Mathematics Responsorum Rebus, Liber VIII, Vieta gives an answer to the problems raised by Scaliger. There is again dealing with problems of duplication of the cube and trisection of the angle. What he calls an irrational problem. In 1600 , in De Numeros Potestatum ad Exegesim Resolutiones, Vieta, aided by Marin Ghetaldi , binds to initial goal of solving equations of any degree using radicals. This belief will definitely disappointed by Niels Abel in 1828. It gives off a method of approximation of roots of an equation. This method will influence the rule of Newton and Newton-Raphson owes him much. Raphson Joseph stated it in 1690. In 1615 , in De Recognitiones quationum published by Anderson, there are some twenty chapters fairly repetitive, the relationship between roots and coefficients, equations in which the stranger enters her power cube and his first, The way to eliminate the second term of an equation (or method of Ferrari for solving equations of degree 4) and the equivalence between the equations of the third degree and knowing the first of four variables, proportional, and the difference between the second and fourth, find this second. Vieta in this book still gives a few ways to lower the degree of an equation. Maximilien Marie , summarizes a few pages The contributions of the new algebra The introduction of literal notation for the parameters in the algebraic equations and algebra will generate intricate rules and procedures medieval formalization by providing efficient, are at the heart of the proposed Francois Vieta. But to establish this formalization, we had to do much more Vieta and The idea to write letters by a stranger has already been introduced by Euclid , and after him by Diophantus , however, this is geometry or abbreviations in a language where every letter conceals a number or a number and without any explicit calculation rule permits it to follow a formal calculation reduced to the status of mere notation. Moreover, any algebra based on ancient figures obtained rescission (called Gnomons ), which replace the algebraic formalization, but make it difficult to solve equations of high degree . Arab mathematicians, who returned a large part of the requirements of Greek geometers, gave the first self-algebra. However, the same language barriers does not allow them to set those rules in the form of literal form comparable to our modern algebra and symbolism appears in the current research, limited to abbreviations . The rating of the unknown by a (single) letter was taken by Jordanus Nemorarius in XII century , then by the computers medieval. Neither Michael Stifel or Regiomontanus in Germany in the sixteenth century , nor the Lyon Nicolas Chuquet , yet very ahead of its time, nor Luca Pacioli , nor Bombelli and even less Cardano , however, have given their ratings on their general will have with Vieta. Until Isagoge all measurable quantities can take action together in an equation in the form of unknowns and parameters. To realize his project of formalization issues geometric Vieta had therefore describe new rules multiplications, additions, subtraction, etc., operating, not on numbers but on the sizes (length, area, volume, etc.). Vieta adopted as the basic principle of Greek geometers that one can not add, subtract and take the report until homogeneous magnitudes . The need to maintain the homogeneity of the deduced formulas such as giving different names to these operations. For example, the multiplication of quantities (ducere in), notation inherited from the Italians and their Arab predecessors operationalizes the idea that leads to the side a side b to form the rectangle ab. Their removal is noted the sign =, which at the time, has not yet taken its current meaning of equality. The idea to operate on symbols (species) may be partly explained by his legal studies , the word designating species, so all their customers in the jargon of lawyers (the first occupation of the protected Parthenay ) . The idea to write in vowels and consonants, some reserved for unknown to the other known quantities (or parameters) the design points, angles, or the lengths of the sides of a geometric figure, had already been explored in antiquity and the sixteenth century by Ramus (Vieta which speaks as a man of very subtle: logikotatos ); Gosselin Guillaume and Jacques Peletier du Mans had evolved such ratings decisively between 1560 and 1570 , giving effective means of working with two unknowns. However, their algebra, like that of Pedro Nunez , remained essentially digital. In the current state of knowledge, Isagoge is the first book appeared when the foundations of the current formalism (The manuscripts of Thomas Harriot , published for the first time in 2007 , suggest that the English mathematician had developed (about the same time) the beginnings of a literal calculation, but was kept secret ). Provided the contribution of Isagoge not limited to this significant invention: the results announced by this method are varied and numerous. They range from determining positive solutions of equations of degree 2, 3 and 4 to the statement of relationships between coefficients and roots of a polynomial, in English, called the Vieta formula , the formulas of binomial, and training of binomial coefficients, which are taken over by Pascal and Newton, the appearance of the first infinite product, recognizing the link between trisection of angle and third-degree equation, and this innovation, considered one the most important in the history of mathematics, really opens the way for the development of modern algebra . But there's more. Because the need to base a calculation on the symbolic variables respecting the homogeneity (requirement of former renewed by Ramus ), Vieta forced to explain how to conduct the operations on the same quantities. And this stress has made the Isagoge a real program of reasoning. Thus, it would be profoundly reducing the project to summarize his idea to write down the known quantities and unknown letters. The book offers the first truly axiomatic algebraic, and is gripped by what means any of the modern presentation of Vieta , who develops his ideas systematically, offers made in this introduction to the art of the Analysis, not only a first symbolic algebra work completed, but also a first shaping the art of reasoning algebraically similar to what was done Euclid for geometry . In the words of Maximilian Mary , Vieta was good with excellent Algebra Geometry. However, in its desire to uphold the homogeneous form of writing Diophantine, : Similarly, the choice to share the alphabet vowel (unknown) and consonants (parameters), although directly inherited Ramus , did not reveal happier. If it marks the revival of studies of Semitic languages in the sixteenth century , it will not stand the choices made by Descartes to book unknown to the last letters of the alphabet. Furthermore, writing provides a symbol that forge reduced to rate multiplication (in), roots or equality (aeq or aequabitur), powers (even if we find in the writings of Anderson qc, quadcubo or to be quadrato-cubus he criticizes already forgotten in its resolutions negative solutions (less than nothing) and complex (he calls wrapped). But the same year, James Hume of notes already Godscroft However, when Vieta wrote: we see the seeds that this is an effective writing, allowing replacements and mechanical transformations. Eventually, the whole way of thinking about mathematics and the world that it will be modified . As assure Boole : After Vieta, the French Pierre de Fermat and Roberval , the Dutch Snell , the English Thomas Harriot and Newton and Christian Huygens will use notations Vieta . Later, Leibniz , who appreciated his inheritance tries to do what Vieta analysis was done for equations but after 1649 and the reissue of The Geometry of Descartes, his fame is eclipsed by that of the philosopher Hague , which was extensively renovated this formalism, and which the next century will issue a common misconception, the sole authorship of the algebraic formalization . Also consult: The poristique, The Supplementum
The exegetical digital, Random Number
Posthumous publications
The Isagoge: a landmark work
Barriers to innovation
Sources of Innovation
The art of reasoning
The hazards of posterity
), But has not completed the character that will give him the great age and its requirements of homogeneity to condemn the constant reference to the geometric meaning of the parameters involved This second revolution in the art will be done at the algebraic generation with the following publications of Alexander Anderson , of William Oughtred , the Thomas Harriot , of Duret Christmas , then after them, Pierre de Fermat , of Rene Descartes and Frans van Schooten. The incompleteness of these ratings is also feeling very early on when Albert Girard honors Vieta : 
in place of A b u s u c preparing the synthesis of the views of Vieta and Harriot. 
Sources
References
Bibliography
Loading ...