MATH IN SIXTEEN CENTURY

General conditionsIn 15th century printing having only just been

invented and in 16th century the printed page began to prepetuate names

The world moved rapidly, and influences that bear upon the development of mathematics become more difficult to trace

Mathematical conditions: - attempt the solution of cubic eqations - a better symbolism - more rapid development of

trigonometry

Math in different countries:

The New World

The End

ITALY• Leonardo da Vinci

(1452.-1519.)

• Distinguished between curves of single and double curvature

Scipione del Ferro (1465.-1526.)

• professor of mathematics in Bologna

• In geometry he was interested in constructions depending on a single opening of the compasses

• In algebra he found a method of solving the cubic equation for the special case x3+ax=b

Giro’lamo Carda’no (1501.-1576.)

• Frist of the two prime movers in the solution of the cubic

• Man of remarkable contrast• Wrote a number of books on a

wide variety of subjects• Wrote Ars Magna,the frist great

Latin treatise devoted solely to algebra

Girolamo Cardano (1501.-1576.)• Solved all the other types

of cubics and solved the quartic equation

• He approached Tartaglia for help

• With his pupil Ludovico Ferrari had discovered the solution of the biquadric

Tartaglia (1500. – 1557.)

• Real name Nicolo Fontana• One of the greatest mathematicians of

Italy in the 16th century• He have completed the solution of cubic

equation• Wrote the best treatize on arithmetic• Published editions of Euclid and

Archimedes (1543.)• He was challenged by Antonio Maria Fior

in solving 30 mathematician problems,and he won

Lodovico Ferarri (1522.-1560.)• Was taken into Cardan’s household in Milan when he

was fifteen• professor of matematics in Bologna

Rafael Bombelli (1526.-1572.)• Wrote L’Algebra parte maggiore dell’ aritmetica divisia in tre libri• He put rules for calculating with imaginary expressions

Francesco Maurolico (1494.-1575.)• He was interested in the Greek writers,so he translated into Latin

works of Theodosius, Menelaus, Euclid, Apollonius and Archimeds• Also wrote various general works of mathematics and arithmetic

Italian geometers:

• Federigo Commandino of Urbino• Francesco Barozzi• Giambatista Benedetti• Cosimo Bartoli • Pietro Antonio Cataldi• Matteo Ricci

Minor writers: • Sivlio Belli• Petrus Bongus

Italian arithmeticians:• Girolamo and Giannantonio Tagliente• Francesco Feliciano da Lazesio• Giovanni Sfortunati• Giovani Mariani

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FRANCEJacques le Fevre d’Estaples

• Wrote an intoduction to the arithmetic of Boethius

• Work on geometry• Edited Sacrobosco’s Sphere• Description of the number

game of Rithmomachia (rithmo-arithmetic, numbers; machia-battle)

Charles de Bouelles• Wrote on geometry and the theory of

numbers• Book on perfect numbers• Work on cycloid• Wrote on regular convex and stellar

polygons

Oronce Fine• Wrote on astronomy• Works on arithmetic and geometry• Quadrature of the circle

Jean Fernel• Published a work of the

Boethian type on proportion• His computation of the length

of a degree of the meridian was so satisfactory as to entitle him to a worthy place in the history of geodesy

Claude de Boissiere• Wrote on poetry, music and astronomy• Arithmetic – related to the science of

warfare

Jean Fernel

Pierre de la Ramee• “All that Aristotle has said is

false.”• Orator and a skillful debater• Edited the “Elements” of

Euclid• Theoretical arithmetic,

geometry and optics

Francois Viéte• The greatest of all the

French mathematicians of the 16th century

• Letters represent numbers in algebra; vowels – unknowns; consonants – knowns

• Formula for sin nФ in terms of sin Ф

• Equation of the n-th degree is made up of n linear factors

• The relation between the problems of the trisection of an angle and the solution of a cubic equation

• Formulas which relate the coefficients of a polinomyal to signed sums and products of its roots

• Evaluating π by infinite

products:

Minor WritersJoannes ButteoFrancesco dal SolePierre ForcadelEstienne de la RocheJacques PeletierIan TrenchantMonte Regal Piedmontois

• Tables-products of numbers to 100 × 1000

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ENGLANDTonstall

• dedicated the book to one of greatest scolars and one of the noblest men of his generation, Sir Thomas More

• His aritmetic was not original, the material being confessedly drawn from such Italian writers as Picioli

Recordemost influential English mathematicianThe four mathematical works were written in

dialogue:1. The Ground of Artes- This was one of the most

popular arithmetics printed in the 16th century.2. The Castle of Knowledge- a work on astronomy, and

one of the first to bring the Copernican system to the attention of English readers

3. The pathewaie to knowledge-containing an abridgment of Euclid's elements

4. The whetstone of witte-containyng the extraction of Rootes: The Cossike practise, with the rule of Equation:and the woorkers of Surde Nombers.

• Title- page of Recorde’s algebra

MINOR WRITERSLeonard DiggersThomas DiggersJohn Dee

• wrote the preface to the first England translation of Euclid's Elements (picture)

Thomas MastersonThomas Blundeville

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THE NEW WORLDJuan Diez• He was of a literary turn of mind, as is shown by

three or four books which he published. One of these works was on mathematics, and this appeared in Mexico under the folowing title:

Sumario• The matematical text

consists of twenty-four pages besides the colophon

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GERMANYNature of the German Mathematics

• the mathematics of Germany was Gotic, unpolished, but strong

• Germany produced a notable group of arithmeticians, two strong algebraists

• Germany made a definite advance in geometry, in the study of higer plane curves

• the greatest influence for advance in 16th century was printing, Erasmus and Martin Luther

• it was a century of intellectual awakening and of breaking away from traditions

The Margarita Phylosophica

• the first modern encyclopedia of any note• published at Freiburg in 1503 by Gregorius

Reisch• it consists of 12 books and includes

considerable material upon arithmetic, geometry and astronomy

• there were 16 editions of this book in the 16th century

Albrecht Dürer• showed in his treaties on geometry,

fortification and human porportion his mathematical powers

Johann Stöffler• showed how the Julian calendar could

be brought into harmony with astronomical events

• predicted that the Deluge would be repeated in 1524 but he was wrong

Michael Stifel• the first German writer of

the century to devote his life to mathematics and to acquire an enviable reputation in this field

• Stifel made the error of predicting the end of the world and, when it was seen that he was wrong, he was arrested

Christoff Rudolff• worked on algebra• published three books: Coss (1525), Kunstliche rechnung

(1526) and a collection of problems (1530)

Johann Scheubel• gave the so-called Pascal Triangle a century before

Pascal wrote upon it, and extracted roots as high as the 24th by a process similar to the one which employs the Binomial Theorem

Ludolf van Ceulen• he is known for his value of , at first given to 20 and 𝜋

then to 35 decimal places

Grammateus• Real name Heinrich Scheyber• his best known work was an arithmetic in the German

language• first German writter to make free use of the signs + and

– in the treatment of algebraic expressions

Pitiscus• his trigonometry was the first satisfactory textbook

published on the subject and the first book to bear this title

THE CLASSICAL GROUP• Philip Melanchton• Joachim Camerarius• Jacobus Micyllus• Michael Neander• Guilielmus Xylander

MATHEMATICAL ASTRONOMERS

Petrus ApianusGeorg Jaochim

RhæticusChristopher Clavius

• he was engaged in the reform of the calendar under direction of Pope Gregory XIII (picture)

Johann and Andreas Schoner

Adam Riese• the greatest of all the

Rechenmeisters of this century

The only other Rechenmeister of the century to deserve special mention is Simon Jacob who wrote two commercial arithmetics

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SPAIN

• The intellectual atmosphere was not favorable to the development of mathematics, however, many Spanish scholars settled in France and Italy or at least published their works abroad

Ciruelo• The earliest Spanish mathematician of the

century was Pedro Sánchez • general work of mathematics Cursus

quattuor mathematicarum artium liberaliumOrtega

• wrote A Tractado subtilisimo d'arithmetica y de geometria which was published in 1512, in Barcelona and Lyons, being the first book on commercial computation known to have been printed in France

Joannes Martinus Blasius• an astrologer and arithmetician• published in Paris (1513) a work on computation

• Juan Perez de Moya • his Arithmetica includes calculation, applied

arithmetic, algebra, principal geometry and contains a considerable amount of interesting historical material

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NetherlandsJoachim Fortius Ringelbergius

• Astronomy, optics, arithmetic

Adriaen van Roomen• Gave the value of π to seventeen decimal places• Other works:

– Treatment of the circle by Archimedes– Spherical triangles

Adriaen Anthoniszoon• Known as Metius• Suggested 355/113 as a convenient value of π

Adriaen AdriaenszoonGiel Vander Hoecke

• Plus and minus signs as symbol operations

Gemma FrisiusValentin Menher

• Arithmetic (geometry and trigonometry)

Jodocus ClichtoveusSimon Stevin

• The theory of decimal fractions BACK

Switzerland• Henricus Loritus Glareanus

• He wrote on arithmetic, meterology and music

• Canradus Dasypodius• He had in mind the ending of all the Greek

mathematical works• He only edited Euclid’s Elements and wrote a

mathematical dictionary

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Japan• The 16th century in Japan corresponds mostly to the 13th

century in the West; it was century of preparation• Probably the chief cause which contributed to this

preparation in the field of mathematichs was the jouney to China made by one Mōri Kambei Shigeyoshi, a scholar in the servise of two of the powerful lords of Japan

• The great hero Toyotomi Hideyoshi (Taikō) sent Mōri to China to acquire and bring back mathematical knowledge (most famous mathematicion from China was K’u Ying-hsiang wrote on algebra and geometry)

• Mōri’s mission was not successful, but he bring back with him a considerable amount of material

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This presentation was made by:

The Expert

Antonija Milišić

The New Guy

Sanela Mitrović

The Supervisor

Kristina Oremuš

The Leader

Sandra Stanišić

The Joker

Irena Užar

Let’s smile:

THE END

Just to mention that

All those mathematicians didn’t die, they just losed their functions…