In 18th century mathematics is already a modern science Mathematics begins to develop very fast because of introducing it to schools Therefore everyone.

In 18th century mathematics is In 18th century mathematics is already a modern science already a modern science

Mathematics begins to develop very Mathematics begins to develop very fast because of introducing it to fast because of introducing it to schoolsschools

Therefore everyone have a chance to Therefore everyone have a chance to learn the basic learnings of learn the basic learnings of mathematicsmathematics

Thanks to that, large number of new Thanks to that, large number of new mathematicians appear on stage mathematicians appear on stage

There are many new ideas, solutions There are many new ideas, solutions to old mathematical to old mathematical problems,researches which lead to problems,researches which lead to creating new fields of mathematics.creating new fields of mathematics.

Old fields of mathematics are also Old fields of mathematics are also expanding.expanding.

FAMOUS FAMOUS MATHEMATICIANSMATHEMATICIANS

LEONHARD EULERLEONHARD EULER

Leonhard Paul EulerLeonhard Paul Euler(1707-1783)(1707-1783)

He was a Swiss mathematicianHe was a Swiss mathematician Johann Bernoulli made the biggest Johann Bernoulli made the biggest

influence on Leonhardinfluence on Leonhard 1727 he went to St Petersburg where he 1727 he went to St Petersburg where he

worked in the mathematics department worked in the mathematics department and became in 1731 the head of this and became in 1731 the head of this departmentdepartment

1741 went in Berlin and worked in Berlin 1741 went in Berlin and worked in Berlin Academy for 25 years and after that he Academy for 25 years and after that he returned in St Ptersburg where he spent returned in St Ptersburg where he spent the rest of his life.the rest of his life.

Euler worked in almost all areas of mathematics: Euler worked in almost all areas of mathematics: geometrygeometry, , calculuscalculus, , trigonometrytrigonometry, , algebraalgebra,,applied applied mathematics, graph theorymathematics, graph theory and and number theorynumber theory, , as well as , as well as , lunar theory, opticslunar theory, optics and other areas of and other areas of physphysicsics. .

He introduced several notational conventions in He introduced several notational conventions in mathematicsmathematics

Concept of a function as we use today was Concept of a function as we use today was introduced by him;he was the first mathematician introduced by him;he was the first mathematician to write to write f(x) f(x) to denote functionto denote function

He also introduced the modern notation for the He also introduced the modern notation for the trigonometric functionstrigonometric functions, the letter , the letter ee for the base for the base of the of the natural logarithmnatural logarithm (now also known as (now also known as Euler’s numberEuler’s number), the Greek letter ), the Greek letter ΣΣ for for summations and the letter summations and the letter ii to denote the to denote the imaginary unitimaginary unit

He wrote 45 books an over 700 He wrote 45 books an over 700 theses.theses.

His main book is His main book is Introduction in Introduction in Analisyis of the Infinite.Analisyis of the Infinite.

AnalysisAnalysis

He discovered ways to express He discovered ways to express various logarithmic functions using various logarithmic functions using power series, and he successfully power series, and he successfully defined logarithms for negative and defined logarithms for negative and complex numberscomplex numbers

He also defined the exponential He also defined the exponential function for complex numbers, and function for complex numbers, and discovered its relation to the discovered its relation to the trigonometric functionstrigonometric functions

EULER’ S FORMULAEULER’ S FORMULA

For anyFor any real number x real number x,,

Euler’s formulaEuler’s formula states states

that the that the complexcomplex

exponentialexponential function function

ssatisfiesatisfies

Number theoryNumber theory

He contributed significantly to the He contributed significantly to the theory of theory of perfect numbersperfect numbers, which had , which had fascinated mathematicians since fascinated mathematicians since Euclid.Euclid.

His His prime number theoremprime number theorem and and the the law of law of quadratic reciprocity quadratic reciprocity are are regarded as fundamental theorems regarded as fundamental theorems of number theoryof number theory..

GeometryGeometry Euler (1765) showed that in any Euler (1765) showed that in any

triangle, the orthocenter, triangle, the orthocenter, circumcenter, centroid, and nine-circumcenter, centroid, and nine-point center are point center are collinear.collinear.

Because of that the line which Because of that the line which connects the points above is connects the points above is called called Euler line.Euler line.

Seven bridges of KonigsbergSeven bridges of Konigsberg

Seven bridges of KonigsbergSeven bridges of Konigsberg

Seven bridges of KonigsbergSeven bridges of Konigsberg

Seven bridges of KonigsbergSeven bridges of Konigsberg

This was old mathematical problem.This was old mathematical problem. The problem The problem waswas to decide whether to decide whether

it is possible to follow a path that it is possible to follow a path that crosses each bridge exactly once and crosses each bridge exactly once and returns to the starting point. returns to the starting point.

1736 Euler solved this problem, and 1736 Euler solved this problem, and prooved that it is not possible.prooved that it is not possible.

This solution is considered to be the This solution is considered to be the first theorem of first theorem of graph theorygraph theory

Euler was very importnat for further Euler was very importnat for further development of mathematicsdevelopment of mathematics

Next quotation tells enough about his Next quotation tells enough about his importance:importance:

““Lisez Euler, lisez Euler, c'est notre Lisez Euler, lisez Euler, c'est notre maître à tous maître à tous ”(Read Euler, read ”(Read Euler, read Euler, he is the master of us all.)Euler, he is the master of us all.)

Pierre-Simon Pierre-Simon LaplaceLaplace

GABRIEL CRAMERGABRIEL CRAMER

GABRIEL CRAMERGABRIEL CRAMER(1704-1752)(1704-1752)

Swiss mathematicianSwiss mathematician He give the solution of St. Peterburg He give the solution of St. Peterburg

paradoxparadox He worked on analysis and He worked on analysis and

determinantsdeterminants He is the most famous by his rule He is the most famous by his rule

(Cramer’s rule) which gives a (Cramer’s rule) which gives a solution of a system of linear solution of a system of linear equations using determinants.equations using determinants.

THOMAS SIMPSONTHOMAS SIMPSON

He received little formal education and He received little formal education and taught himself mathematics while he was taught himself mathematics while he was working like a weaver.working like a weaver.

Soon he became one of the most Soon he became one of the most distinguished members of the English distinguished members of the English schoolschool

SimpsonSimpson isis best remembered for his work best remembered for his work on interpolation and numerical methods of on interpolation and numerical methods of integration.integration.

He wrote books He wrote books Algebra, Geometry, Algebra, Geometry, Trigonometry, Fluxions, Laws of ChanceTrigonometry, Fluxions, Laws of Chance, , and othersand others

THOMAS SIMPSONTHOMAS SIMPSON(1710-1761)(1710-1761)

JEAN LE ROND D’ALAMBERTJEAN LE ROND D’ALAMBERT

He dealt with problems of dinamics He dealt with problems of dinamics and fluids and especially with and fluids and especially with problem of vibrating string which problem of vibrating string which leads to solving partial diferential leads to solving partial diferential equationsequations

During his second part of life, he was During his second part of life, he was mainly occupied with the great mainly occupied with the great French encyclopediaFrench encyclopedia

JEAN LE ROND D’ALAMBERTJEAN LE ROND D’ALAMBERT(1717-1783)(1717-1783)

For this he wrote the introduction, For this he wrote the introduction, and numerous philosophical and and numerous philosophical and mathematical articles; the best are mathematical articles; the best are those on geometry and on those on geometry and on probabilities.probabilities.

JOSEPH LOUIS LANGRANGEJOSEPH LOUIS LANGRANGE

He didn’t show any intersts for He didn’t show any intersts for mathematics untill his 17.mathematics untill his 17.

From his 17, he alone threw himself From his 17, he alone threw himself into mathematical studiesinto mathematical studies

Already with 19, he wrote a letter to Already with 19, he wrote a letter to Euler in which he solved the Euler in which he solved the isoperimetrical problem which for isoperimetrical problem which for more than half a century had been a more than half a century had been a subject of discussion.subject of discussion.

JOSEPH LOUIS LANGRANGEJOSEPH LOUIS LANGRANGE(1736-1813)(1736-1813)

Lagrange established a society known Lagrange established a society known as Turing Academy, and published as Turing Academy, and published Miscellanea TaurinesiaMiscellanea Taurinesia, his work in , his work in which he corrects mistakes made by which he corrects mistakes made by some of great mathematicianssome of great mathematicians

He was studing problems of analytical He was studing problems of analytical geometry, algebra, theory of numbers, geometry, algebra, theory of numbers, differential eqations, mechanics, differential eqations, mechanics, astronomy, and many other...astronomy, and many other...

Napoleon named Lagrange to the Napoleon named Lagrange to the Legion of Honour and Legion of Honour and made him the made him the Count of the Empire in 1808.Count of the Empire in 1808.

On 3 April 1813 he was awarded the Grand On 3 April 1813 he was awarded the Grand Croix of the Ordre Impérial de la Réunion. Croix of the Ordre Impérial de la Réunion. He died a week later.He died a week later.

PIERRE SIMON LAPLACEPIERRE SIMON LAPLACE

French mathematician and astronomerFrench mathematician and astronomer His most known works are His most known works are Traite de Traite de

mecanique celeste mecanique celeste andand Theory analytique Theory analytique des probabiliteisdes probabiliteis

His name is also connected with the His name is also connected with the “Laplace transform” and with the “Laplace “Laplace transform” and with the “Laplace ex pansion” of a determintex pansion” of a determint

He is He is one of the first scientists to postulate one of the first scientists to postulate the existence of the existence of black holesblack holes..

He is one of only seventy-two people to He is one of only seventy-two people to have their have their name engraved on Eiffel Towername engraved on Eiffel Tower..

PIERRE-SIMON LAPLACEPIERRE-SIMON LAPLACE(1749-1827)(1749-1827)

It is also interesting to say the It is also interesting to say the difference between Laplace and difference between Laplace and LagrangeLagrange

For Laplace, mathematics was merely For Laplace, mathematics was merely a kit of tools used to explain naturea kit of tools used to explain nature

To Lagrange, mathematics was a To Lagrange, mathematics was a sublime art and was its own excuse sublime art and was its own excuse for being for being

He is remembered He is remembered as one of the as one of the greatest scientists greatest scientists of all time, of all time, sometimes referred sometimes referred to as a to as a French French Newton Newton or or Newton Newton of Franceof France

He became a He became a countcount of the of the First French First French EmpireEmpire in 1806 and in 1806 and was named a was named a marquismarquis in 1817 in 1817

GASPARD MONGEGASPARD MONGE

French mathematician also known as French mathematician also known as Comte de PéluseComte de Péluse

Monge Monge is considered the father of is considered the father of differential geometry because of his differential geometry because of his work work Application de l'analyse à la Application de l'analyse à la géométrie géométrie where he introduced the where he introduced the concept of lines of curvature of a concept of lines of curvature of a surface in 3-space. surface in 3-space.

GASPARD MONGE(1746-1818)

His method, which was one of His method, which was one of cleverly representing 3-dimensional cleverly representing 3-dimensional objects by appropriate projections 2-objects by appropriate projections 2-dimensional plane, was adopted by dimensional plane, was adopted by the military and classified as top the military and classified as top secret secret

ADRIEN – MARIE LEGENDREADRIEN – MARIE LEGENDRE

He made important contributions to He made important contributions to statisticsstatistics, , number theorynumber theory, , abstract abstract algebraalgebra and and mathematical analysismathematical analysis..

Legendre is known in the history of Legendre is known in the history of elementary methematics principially elementary methematics principially for his very popular for his very popular Elements de Elements de geometriegeometrie

He gave a simple proof that He gave a simple proof that ππ(pi)(pi) is is irrational as well as the first proof irrational as well as the first proof that that π2π2(pi squared)(pi squared) is irrational. is irrational.

ADRIEN – MARIE LEGENDREADRIEN – MARIE LEGENDRE(1752-1833)(1752-1833)

JJEANEAN B BAPTISTEAPTISTE J JOSEPHOSEPH FFOURIEROURIER

French mathematician,French mathematician, physicist physicist and historianand historian

He He studied the mathematical studied the mathematical theory of heat conduction.theory of heat conduction.

JJEANEAN B BAPTISTEAPTISTE J JOSEPHOSEPH F FOURIEROURIER(1768-1830)(1768-1830)

Fourier established the Fourier established the partial differential partial differential equation governing heat equation governing heat diffusion and solved in by diffusion and solved in by using infinite series of using infinite series of trigonometric functionstrigonometric functions

JJOHANNOHANN C CARLARL F FRIEDRICHRIEDRICH G GAUSSAUSS

JJOHANNOHANN C CARLARL F FRIEDRICHRIEDRICH G GAUSSAUSS((1777 – 18551777 – 1855))

He He worked in a wide variety of fields worked in a wide variety of fields in both mathematics and physics in both mathematics and physics incuding number theory, analysis, incuding number theory, analysis, differential geometry, geodesy, differential geometry, geodesy, magnetism, astronomy and optics.magnetism, astronomy and optics.

““Mathematics is the queen of the Mathematics is the queen of the sciences and number theory is the sciences and number theory is the queen of mathematics.queen of mathematics.””

AUGUSTIN LOUIS CAUCHYAUGUSTIN LOUIS CAUCHY

Cauchy started the project of Cauchy started the project of formulating and proving the teorems formulating and proving the teorems of calculus in a rigorous manner and of calculus in a rigorous manner and was thus an early pioneer of analysiswas thus an early pioneer of analysis

He also gave several important He also gave several important theorems in complex analysis and theorems in complex analysis and initiated the study of permutation initiated the study of permutation groupsgroups

AUGUSTIN LOUIS CAUCHYAUGUSTIN LOUIS CAUCHY(1789-1857)(1789-1857)

He also researched in convergence He also researched in convergence and divergence of infinite series, and divergence of infinite series, differential equations, determinants, differential equations, determinants, probability and mathematical probability and mathematical physics.physics.

He was first to prove Taylor’s He was first to prove Taylor’s theorem, he brought a whole new theorem, he brought a whole new set of teorems and definitions, he set of teorems and definitions, he dealed with mechanics, optics, dealed with mechanics, optics, elasticity and many other problemselasticity and many other problems

His last words were:His last words were:

““Men pass away, but their Men pass away, but their deeds abide.deeds abide.””

Anela BocorAnela Bocor

Mateja JelušićMateja Jelušić

Ivan JelićIvan Jelić

Vojislav ĐuračkovićVojislav Đuračković

Boris DokićBoris Dokić