MATHEMATICS IN 18th century

In 18th century mathematics is already a modern science Mathematics begins to develop very fast because of introducing it to schools Therefore everyone have a chance to learn the basic learnings of mathematics

Thanks to that, large number of new mathematicians appear on stage There are many new ideas, solutions to old mathematical problems,researches which lead to creating new fields of mathematics. Old fields of mathematics are also expanding.

FAMOUS MATHEMATICIANS

LEONHARD EULER

Leonhard Paul Euler (1707-1783) He was a Swiss mathematician Johann Bernoulli made the biggest influence on Leonhard 1727 he went to St Petersburg where he worked in the mathematics department and became in 1731 the head of this department 1741 went in Berlin and worked in Berlin Academy for 25 years and after that he returned in St Ptersburg where he spent the rest of his life.

Euler worked in almost all areas of mathematics: geometry, calculus, trigonometry, algebra,applied mathematics, graph theory and number theory, as well as , lunar theory, optics and other areas of physics. He introduced several notational conventions in mathematics Concept of a function as we use today was introduced by him;he was the first mathematician to write f(x) to denote function He also introduced the modern notation for the trigonometric functions, the letter e for the base of the natural logarithm (now also known as Euler’s number), the Greek letter Σ for summations and the letter i to denote the imaginary unit

He wrote 45 books an over 700 theses. His main book is Introduction in Analisyis of the Infinite.

Analysis He discovered ways to express various logarithmic functions using power series, and he successfully defined logarithms for negative and complex numbers He also defined the exponential function for complex numbers, and discovered its relation to the trigonometric functions

EULER’ S FORMULA For any real number x, Euler’s formula states that the complex exponential function satisfies

Number theory He contributed significantly to the theory of perfect numbers, which had fascinated mathematicians since Euclid. His prime number theorem and the law of quadratic reciprocity are regarded as fundamental theorems of number theory.

Geometry Euler (1765) showed that in any triangle, the orthocenter, circumcenter, centroid, and nine- point center are collinear. Because of that the line which connects the points above is called Euler line.

Seven bridges of Konigsberg

Seven bridges of Konigsberg

Seven bridges of Konigsberg

Seven bridges of Konigsberg This was old mathematical problem. The problem was to decide whether it is possible to follow a path that crosses each bridge exactly once and returns to the starting point. 1736 Euler solved this problem, and prooved that it is not possible. This solution is considered to be the first theorem of graph theory

Euler was very importnat for further development of mathematics Next quotation tells enough about his importance: “Lisez Euler, lisez Euler, c'est notre maître à tous ”(Read Euler, read Euler, he is the master of us all.) Pierre-Simon Laplace

GABRIEL CRAMER

GABRIEL CRAMER (1704-1752) Swiss mathematician He give the solution of St. Peterburg paradox He worked on analysis and determinants He is the most famous by his rule (Cramer’s rule) which gives a solution of a system of linear equations using determinants.

THOMAS SIMPSON

THOMAS SIMPSON (1710-1761) He received little formal education and taught himself mathematics while he was working like a weaver. Soon he became one of the most distinguished members of the English school Simpson is best remembered for his work on interpolation and numerical methods of integration. He wrote books Algebra, Geometry, Trigonometry, Fluxions, Laws of Chance, and others

JEAN LE ROND D’ALAMBERT

JEAN LE ROND D’ALAMBERT (1717-1783) He dealt with problems of dynamics and fluids and especially with problem of vibrating string which leads to solving partial diferential equations During his second part of life, he was mainly occupied with the great French encyclopedia

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

JOSEPH LOUIS LANGRANGE

JOSEPH LOUIS LANGRANGE (1736-1813) He didn’t show any intersts for mathematics untill his 17. From his 17, he alone threw himself into mathematical studies Already at 19, he wrote a letter to Euler in which he solved the isoperimetrical problem which for more than half a century had been a subject of discussion.

Lagrange established a society known as Turing Academy, and published Miscellanea Taurinesia, his work in which he corrects mistakes made by some of great mathematicians He was studing problems of analytical geometry, algebra, theory of numbers, differential eqations, mechanics, astronomy, and many other... Napoleon named Lagrange to the Legion of Honour and made him the Count of the Empire in 1808.

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

PIERRE SIMON LAPLACE

PIERRE-SIMON LAPLACE (1749-1827) French mathematician and astronomer His most known works are Traite de mecanique celeste and Theory analytique des probabiliteis His name is also connected with the “Laplace transform” and with the “Laplace ex pansion” of a determint He is one of the first scientists to postulate the existence of black holes. He is one of only seventy-two people to have their name engraved on Eiffel Tower.

It is also interesting to say the difference between Laplace and Lagrange For Laplace, mathematics was merely a kit of tools used to explain nature To Lagrange, mathematics was a sublime art

He is remembered as one of the greatest scientists of all time, sometimes referred to as a French Newton or Newton of France He became a count of the First French Empire in 1806 and was named a marquis in 1817

GASPARD MONGE

GASPARD MONGE (1746-1818) French mathematician also known as Comte de Péluse Monge is considered the father of differential geometry because of his work Application de l'analyse à la géométrie where he introduced the concept of lines of curvature of a surface in 3-space.

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

ADRIEN – MARIE LEGENDRE

ADRIEN – MARIE LEGENDRE (1752-1833) He made important contributions to statistics, number theory, abstract algebra and mathematical analysis. Legendre is known in the history of elementary methematics principially for his very popular Elements de geometrie He gave a simple proof that π(pi) is irrational as well as the first proof that π2(pi squared) is irrational.

JEAN BAPTISTE JOSEPH FOURIER

JEAN BAPTISTE JOSEPH FOURIER (1768-1830) French mathematician, physicist and historian He studied the mathematical theory of heat conduction.

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

JOHANN CARL FRIEDRICH GAUSS

JOHANN CARL FRIEDRICH GAUSS (1777 – 1855) He worked in a wide variety of fields in both mathematics and physics incuding number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. “Mathematics is the queen of the sciences and number theory is the queen of mathematics.”

AUGUSTIN LOUIS CAUCHY

AUGUSTIN LOUIS CAUCHY (1789-1857) Cauchy started the project of formulating and proving the teorems of calculus in a rigorous manner and was thus an early pioneer of analysis He also gave several important theorems in complex analysis and initiated the study of permutation groups

He also researched in convergence and divergence of infinite series, differential equations, determinants, probability and mathematical physics. He was first to prove Taylor’s theorem, he brought a whole new set of teorems and definitions, he dealed with mechanics, optics, elasticity and many other problems

“Men pass away, but their deeds abide.” His last words were: “Men pass away, but their deeds abide.”