18th ce of math emetics

8/3/2019 18th CE of Math Emetics

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The 18th CE Mathematics of

France:

The Development of Analysis


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The 18th century marks from 1701 to 1800 in

the Gregorian calendar.

Age of Reason and Change

it is known as the medieval period

1700's was the times to develop the calculus

and to expand the analysis made in 1600's.


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there were so many enlargements of thedesign trigonometry, the analytic geometry,the number theory, the equation theory, the

probability theory, the differential equation,and the analytic dynamics and also so manynew creations of the insurance statistics, thefunction of higher degree, the partialdifferential equation, the descriptivegeometry and the differential geometry.


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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 andengineering.


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BERNOULLIBROTHERS(Jacob and JohannBernoulli)

*Jacob Bernoulli

Became professor in

Basel University.

calculus of variations.

the Ars conjectandi, was

published in 1713.

Bernoulli distribution and

Bernoulli theorem of statistics

and probability theory;

died from Tuberculosis.

Jacob (1654-1705) andJohann

Bernoulli (1667-1748)


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Johann Bernoulli

he became Professor in Basel,Switzerland and at that time,

Leonard Euler was one of his student.

famous rule about 0 0 (a problem which had dogged

mathematicians since Brahmagupta's initial work on the rules for

dealing with zero back in the 7th Century).

he improved the work of his brother and he named Bernoulli

equation, He used the word 'integral' for the first time in 1690.

He greatly enriched the calculus.

asl'Hospital's rule-with the help of Marquis de l'Hospital

(1661~1704), he was able to arrange the first book in 1696 calledthe first calculus textbook.

had 3 sons namely Nicolaus (1695-1726), Daniel(1700-1782),

and Johann II (1710-1790),


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Abraham De Moivre(16671754)

French mathematical prodigy.noted for his workAnnuities

upon Lives, which played an

important role in the history of

actuarial mathematics,.In his book ofDoctrine of

Chances, which contained much

new material on the theory of

probability,and his Miscellanea

analytica.

de Moivre's formula,

(cosx+ isinx)n = cos(nx) + isin(nx),


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De Moivres formula


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Comte de Buffon(1707-1788,)

the first example of a

geometrical probability, hisfamous "needle problem" for

experimentally approximating the

value of .

he introduced differential and

integral calculus into probability

theory compiled in his book

named Sur le jeu de franc-

carreau


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Geometrical Probability involving

Calculus


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BrookTaylor(1685-1731)

English mathematician

Taylor's theorem and

theTaylor series.

The Incrementorum Directa

etInversa (1715) added a

new branch to the higher

mathematics, he designatedthe "calculus of finite

differences.


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TaylorsSeries

TaylorsTheorem


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Colin Maclaurin(1698-1746)

Scottish mathematician whomade important contributions to

geometry and algebra.

he also introduced Maclaurin

series like Taylor series. series to characterize maxima,

minima, and points of inflection

for infinitely differentiable

functions.

he study of elliptic integrals

reducing many intractable

integrals to problems of finding

arcs for hyperbolas.


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TaylorsSeriesExpansion

Taylor series expansion represents

an analytic functionf(x) as an infinite

sum of terms around the expansion

point x=a:

Maclaurin SeriesExpansion

Taylor series expansion around

x

=0 iscalled Maclaurin series expansion

Taylor series expansion requires a

function to have derivatives up to

infinite order around the expansionpoint.


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Mathematician

of Revolution Ages


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Joseph LouisLagrange (1736-

1813)

French mathematician

Mecanique analytique -

aeneral equations of motion of

a dynamical system known

today as Lagrange's equations.

Lagrange's theorem

differential equations andnumber theory.

published his work,

Miscellanea Taurinensia.


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Pierre-Simon Laplace(1749-1827)

French Mathematician and

Astronomer

Mcanique

Cleste (Celestial Mechanics)

restated the nebular

hypothesis of the origin of

the solar system .


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he started to work on differential

equations and finite differences .

His work entitled Mmoire sur la probabilit

des causes par les vnements was published

in 1774.

Spherical harmony and inductive probabi.lity


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Adrien-Marie Legendre(1752-1833)

French mathematician

he is known for Legendretransformation and ellipticfunctions

he worked on the rootsof polynomials

lments degomtrie, published in 1874

noted for his quadraticrecprocity


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Gaspard Monge(1746-1818)

French mathematician,

inventor of descriptive

geometry.

he published "Sur la thoriedes dblais et des

remblais and Trait

lmentaire de la statique

gives the ordinary differentialequation of the curves of

curvature,


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The Metric System


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The word 'meter' is from the Greek word 'metron', whichmeans 'a measure'. At the time the metric system wasbeing defined, there were two competing approaches tothe definition of the length of the meter. The initialsuggestion was that it be the length of 'a pendulumhaving a half-period of one second'. The alternativeapproach was to set it to one ten-millionth of 'the lengthof the earth's meridian along a quadrant'. Specifically,one ten-millionth of the distance from the North Pole tothe equator, along the meridian running near Dunkirk inFrance. The French Academy of Science, which was

tasked with developing the metric system, decided onthe latter approach in 1991 (on the basis that the formerapproach does not give a consistent measure, as slightvariations in the earth's gravity from one place toanother would result in different values).


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The metric system is a system of units formeasurement developed in late 18th century

France by the chemist Lavoisier to replace thedisparate systems of measures then in use with aunified, natural and universal system. In the earlymetric system there were several fundamental orbase units, the grad or grade for angles, themetre for length, the gram for weight and thelitre for capacity. These were derived from eachother via the properties of natural objects, mainlywater: 1 litre of water weighs 1 kg and measures

1 cubic decimetre (dm). Other units werederived from this, such as the Celsiustemperature scale, where water freezes as 0Cand boils as 100C.


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Measurements of length, area, volume, and

weight play an important part in the practical

applications of mathematics, Basic among the

units of these measurements is that of length,

for given a unit of length, units for the other

quantities can easily be devised.


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18th ce of math emetics

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