18th CENTURY MATHEMATICS

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

2. • In 18th 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-Simon Laplace and Adrien-Marie
Legendre

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

5. JOSEPH LAGRANGE
• Collaborated 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 = 12 + 12 + 12 + 02;
31 = 52 + 22 + 12 + 12;
310 = 172 + 42 + 22 + 12; etc

8. 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

9. 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)

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

11. 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.

12. 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.

14. 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.

16. JOHANN BERNOULLI
• Younger brother of Bernoulli
• Teacher of Euler

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

18. 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”

19. 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

21. 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
𝑒 𝑖𝜋
= 𝑐𝑜𝑠𝑥 + 𝑖 𝑠𝑖𝑛𝑥
𝑒 𝑖𝜋 + 1 = 0

22. 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

25. 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 13⁄5, but Euler’s superior method
yielded the exact but rather unexpected result of π2⁄6

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

27. THANK YOU