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VI. The Eighteenth - Century Mathematics of France :
The Development of Analysis

¡ß Caracteristic of the Eighteenth - Century Mathematics

1700's was the times to develop the calculus and to expand the analysis made in 1600's.
In this century, there were so many enlargements of the design trigonometry, the analytic geometry, the number theory, the equation theory, the probability theory, the differential equation, and the analytic dynamics and also so many new creations of the insurance statistics, the function of higher degree, the partial differential equation, the descriptive geometry and the differential geometry.
1700's was the times that Bernoulli family in Swiss and mathematicians in France were active.
The Euler's creative talent like the active of Bernoulli family renovated analysis.
Lagrange, a Frenchman living in Italy, made 'calulus of variation' with Euler.   D'Alembert was interested in the basic of analysis and Lambert wrote the paper about oparallel postulate.
Laplace making a great contribution towards analysis and Monge creating descriptive geometry were the people of this times.
The French republican government succeeding the French Revolution choose the metric system of weights and measures in 1799.

¡ß The Early Eighteenth Century

so important in the study of statistics.

Known by De Moivre's name and found in every theory of equations textbook, was familiar to De Moivre for the case where n is a positive integer.  This formula has become the keystone of analytic trigonometry.

Recognition of the full importance of Taylor's series awaited until 1755, when Euler applied them in his differential calculus, and still later, when Lagrange used the series with a remainder as the foundation of his theory of functions.
Maclaurin was one of the ablest mathematicians of the eighteenth century.   The so-called Maclaurin expansion is nothing but the case where a = 0 in the Taylor expansion above.

Euler was a voluminous writer on mathematics, indeed, far and away the most prolific writer in the history of the subject ; his name is attached to every branch of the study.  It is of interest that his amazing productivity was not in the least impaired when, shortly after his return to the St. Petersburg Academy, he had the misfortune to become totally blind.
Euler's studies ranged over theory of numbers, algebra, theory of series algebraic analysis, theory of probability and dynamics.  He also wrote 45 book and 700 theses.
Euler's contributions to mathematics are too numerous to expound completely here.
First of all, we owe to Euler the conventionalization of the following notations:

 f(x) for functional notation, e for the base of natural logarithms, a,b,c for the sides of triangle ABC, s for the semiperimeter of triangle ABC, r for the inradius of triangle ABC, R for the circumradius of triangle ABC, ¢² for the summation sign, i for the imaginary unit,

¡Ý Lagrange :  The two greatest mathematicians of the eighteenth century were Euler and Joseph Louis Lagrange(1736-1813).  Whereas Euler wrote with a profusion of detail and a free employment of intuition, Largrange wrote concisely and with attempted rigor.
His monumental <Mecanique analytique> contains the aeneral equations of motion of a dynamical system known today as Lagrange's equations.
In fact, He made the important theorem of group theory that states that the order of a subgroup of a finite group G is a factor of the order of G is called
Lagrange's theorem.
Lagrange's work had a very deep influence on later mathematical research, for he was the earliest first-rank mathematician to recognize the thoroughly unsatisfactory state of the foundations of analysis and accordingly to attempt a rigorization of the calculus.

(1-x2)y¡È- 2xy¡Ç+ n(n+1)y = 0

which is of considerable importance in applied mathematics.

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.   One of the important accomplishments of the eighteenth century was the construction of the metric system, designed to replace the world's vast welter of chaotic and unscientific systems of weights and measures by one that is orderly, uniform, scientific, exact, and simple.
In view of the accuracy with which Legendre and others had measured the length of a terrestrial meridian, the committee finally agreed to take the meter to be the tenmillionth part of the meridional distance from the North Pole to the equator(1797).
But today the standard meter is more accurately defined as 1,650,763.73 wavelengths of the orange-red light from the isotope krypton-86, measured in a vacuum.