18th CENTURY MATHEMATICS

18th CENTURY MATHEMATICS

1. 18 TH CENTURY MATHEMATICS PREPARED BY: HEIDEMAE R. REMATA MST-MATH

2. In 18 th Century Mathematics is already a Modern Science Mathematics begins to develop very fast because of introducing it to schools A 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

3. Most of the late 17th Century and a good part of the early 18th were taken up by the work of disciples of  Newton and  Leibniz, who applied their ideas on calculus to solving a variety of problems in physics, astronomy and engineering. Dominated by French Mathematicians despite the popularity of Euler and Bernoulli Most developments were attributed to the “three L’s” – Joseph Lagrange, Pierre- S imon Laplace and Adrien-Marie Legendre

4. FAMOUS MATHEMATICIANS JOSEPH LAGRANGE PIERRE-SIMON LAPLACE ADRIEN-MARIE LEGENDRE THE BERNOULLIS LEONHARD PAUL EULER

5. JOSEPH LAGRANGE C ollaborated with  Euler in an important joint work on the calculus of variation, but he also contributed to differential equations and number theory Credited with originating the theory of groups which states that the number of elements of every sub-group of a finite group divides evenly into the number of elements of the original finite group .

6. JOSEPH LAGRANGE He is credited with the four-square theorem, that any natural number can be represented as the sum of four squares EXAMPLES: 3 = 1 2  + 1 2  + 1 2  + 0 2 ; 31 = 5 2  + 2 2  + 1 2  + 1 2 ; 310 = 17 2  + 4 2  + 2 2  + 1 2 ; etc

7. PIERRE-SIMON LAPLACE Referred to as the “French Newton” Mainly on differential equations and finite differences, mathematical and philosophical concepts of probability and statistics Developed his own version of the so-called Bayesian interpretation of probability independently of Thomas Bayes

8. PIERRE-SIMON LAPLACE His monumental work “Celestial Mechanics” translated the geometric study of classical mechanics to one based on calculus, opening up a much broader range of problems Maintained that there should be a set of scientific laws that would allow us to predict everything about the universe and how it works (complete science determinism)

9. ADRIEN-MARIE LEGENDRE Statistics, number theory, abstract algebra and mathematical analysis in the late 18 th and early 19 th centuries Inspired the creation, and almost universal adoption of the metric system of measures and weights

10. ADRIEN-MARIE LEGENDRE His “Elements of Geometry”, a re-working of Euclid’s book, became the leading geometry textbook for almost 100 years, and his extremely accurate measurement of the terrestrial meridian inspired the creation, and almost universal adoption, of the metric system of measures and weights.

11. THE BERNOUILLIS The first Mathematicians to not only study and understand infinitesimal calculus but to apply it to various problems They were largely responsible for further developing  Leibniz’s infinitesimal calculus - particularly through the generalization and extension of calculus known as the "calculus of variations" - as well as  Pascal and  Fermat’s probability and number theory.

12. JACOB BERNOULLI Older brother of Johann Bernoulli The Art of Conjecture book Discovered the approximate value of the irrational number e while exploring the compound interest in loans.

13. JOHANN BERNOULLI Younger brother of Bernoulli Teacher of Euler

14. DANIEL BERNOULLI Son of Johann Bernoulli Well known for his work on fluid mechanics especially Bernoulli’s Principle

15. LEONHARD PAUL EULER Born in Basel, Switzerland Geometry, calculus, trigonometry, algebra, Number theory, Optics, Astronomy, Cartography, Mechanics, Weighs and Measures and even the theory of Music Published 900 books His main book is “Introduction in Analysis of the Infinite”

16. LEONHARD PAUL EULER A new method for solving quartic equations The Prime Number Theorem Proofs (and in some cases disproofs ) of some of Fermat’s theorem and conjectures The calculus of variations, including its best known result, the Euler-Lagrange equation The integration of Leibniz’s differential calculus with Newton’s Method of Fluxions into a form of calculus we would recognize today

17. EULER’S FORMULA Combined several symbols together in an amazing feat of mathematical alchemy to produce one o the most beautiful of all mathematical equation – Euler’s Identity For any real number x, the complex exponential function satisfies  

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

19. LEONHARD PAUL EULER The discovery that initially sealed Euler’s reputation was announced in 1735 and concerned the calculation of infinite sums. It was called the Basel problem after the Bernoulli’s had tried and failed to solve it, and asked what was the precise sum of the of the reciprocals of the squares of all the natural numbers to infinity i.e.  1 ⁄ 1 2  +  1 ⁄ 2 2  +  1 ⁄ 3 2  +  1 ⁄ 4 2  ... (a zeta function using a zeta constant of 2). Euler’s friend Daniel Bernoulli had estimated the sum to be about 1 3 ⁄ 5 , but Euler’s superior method yielded the exact but rather unexpected result of π 2 ⁄ 6

20. REFERENCES https://www.storyofmathematics.com

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