New algebra
Encyclopedia
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).
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:
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.
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.
This work is available available via gallica, written in Latin, and announcing that it will be the first volume of a work divided into ten parts:
It provides a new approach to writing algebra and begins with the famous dedication to the Melusinide
princess Catherine de Parthenay.
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.
Transitivity of equality, conservation sum, subtraction, product, and division
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.
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
what we note now
without attempting to mark the homogeneity factor.
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.
This means, for instance, that the sequence
is geometric.
Reflecting Viète, Vasset writes also:
which we denote now by:
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.
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).
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.
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.
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.
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
For example
written today
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.
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
promoted by François Viète
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...
in 1591 and by his successors (after 1603). It marks the beginning of the algebraic formalization (late sixteenth - the early seventeenth centuries).
General ideas
Until 1591, the year in which François VièteFrançois Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...
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
Jordanus Nemorarius
Jordanus de Nemore was a thirteenth-century European mathematician who wrote treatises on at least 6 different important mathematical subjects: the science of weights; “algorismi” treatises on practical arithmetic; pure arithmetic; algebra; geometry; and stereographic projection. Most of these...
at the end of the twelfth century, and, later, in some other mathematicians like Regiomontanus
Regiomontanus
Johannes Müller von Königsberg , today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator and instrument maker....
(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
François Viète
François Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...
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:
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
Jean de Beaugrand
Jean de Beaugrand was the foremost French lineographer of the seventeenth century. Though born in Mulhouse, de Beaugrand moved to Paris in 1581. He also worked as a mathematician and published works on geostatics...
and Alexander Anderson
Alexander Anderson
Alexander Anderson is the name of:* Alexander Anderson , Scottish mathematician* Alexander Anderson , American illustrator* Alexander O...
. Thereafter, Le Sieur de Vaulezard, Claude Hardy, Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
and Thomas Harriot
Thomas Harriot
Thomas Harriot was an English astronomer, mathematician, ethnographer, and translator. Some sources give his surname as Harriott or Hariot or Heriot. He is sometimes credited with the introduction of the potato to Great Britain and Ireland...
will perfect his language.
When in 1637, René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
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 BeaugrandIn 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
Alexander Anderson
Alexander Anderson is the name of:* Alexander Anderson , Scottish mathematician* Alexander Anderson , American illustrator* Alexander O...
, 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
Frans van Schooten
Franciscus van Schooten was a Dutch mathematician who is most known for popularizing the analytic geometry of René Descartes.-Life:...
in 1646.
The Isagoge
In artem analyticem Isagoge (1591) is the program of this large axiomatic project.This work is available available via gallica, 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
Melusine
Melusine is a figure of European legends and folklore, a feminine spirit of fresh waters in sacred springs and rivers.She is usually depicted as a woman who is a serpent or fish from the waist down...
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
what we note now
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 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
is geometric.
Reflecting Viète, Vasset writes also:
which we denote now by:
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.
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.
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.
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
written today
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
William Oughtred
William Oughtred was an English mathematician.After John Napier invented logarithms, and Edmund Gunter created the logarithmic scales upon which slide rules are based, it was Oughtred who first used two such scales sliding by one another to perform direct multiplication and division; and he is...
, Thomas Harriot
Thomas Harriot
Thomas Harriot was an English astronomer, mathematician, ethnographer, and translator. Some sources give his surname as Harriott or Hariot or Heriot. He is sometimes credited with the introduction of the potato to Great Britain and Ireland...
, Pierre de Fermat
Pierre de Fermat
Pierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...
, and finally René Descartes
René Descartes
René Descartes ; was a French philosopher and writer who spent most of his adult life in the Dutch Republic. He has been dubbed the 'Father of Modern Philosophy', and much subsequent Western philosophy is a response to his writings, which are studied closely to this day...
. 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
- Johannes Hispalensis
- Jordanus NemorariusJordanus NemorariusJordanus de Nemore was a thirteenth-century European mathematician who wrote treatises on at least 6 different important mathematical subjects: the science of weights; “algorismi” treatises on practical arithmetic; pure arithmetic; algebra; geometry; and stereographic projection. Most of these...
- RegiomontanusRegiomontanusJohannes Müller von Königsberg , today best known by his Latin toponym Regiomontanus, was a German mathematician, astronomer, astrologer, translator and instrument maker....
- Michael StifelMichael StifelMichael Stifel or Styfel was a German monk and mathematician. He was an Augustinian who became an early supporter of Martin Luther. Stifel was later appointed professor of mathematics at Jena University...
- ParthenayParthenayParthenay is an ancient fortified town and commune in the Poitou-Charentes region in western France, sited on a rocky spur that is surrounded on two sides by the River Thouet...
- Marino Ghetaldi
- Alexander AndersonAlexander AndersonAlexander Anderson is the name of:* Alexander Anderson , Scottish mathematician* Alexander Anderson , American illustrator* Alexander O...
- Jean de BeaugrandJean de BeaugrandJean de Beaugrand was the foremost French lineographer of the seventeenth century. Though born in Mulhouse, de Beaugrand moved to Paris in 1581. He also worked as a mathematician and published works on geostatics...
- François VièteFrançois VièteFrançois Viète , Seigneur de la Bigotière, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to its innovative use of letters as parameters in equations...
- Simon StevinSimon StevinSimon Stevin was a Flemish mathematician and military engineer. He was active in a great many areas of science and engineering, both theoretical and practical...
- Thomas HarriotThomas HarriotThomas Harriot was an English astronomer, mathematician, ethnographer, and translator. Some sources give his surname as Harriott or Hariot or Heriot. He is sometimes credited with the introduction of the potato to Great Britain and Ireland...
- Pierre de FermatPierre de FermatPierre de Fermat was a French lawyer at the Parlement of Toulouse, France, and an amateur mathematician who is given credit for early developments that led to infinitesimal calculus, including his adequality...