18th Century Math

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

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 'calculus 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.

The Early Eighteenth Century

The Bernoulli Family : One of the most distinguished families in the history of mathematics and science is the Bernoulli family of Switzerland, which, from the late seventeenth century on, produceed an unusual number of capable mathematicians and scientists.  The most famous were Jakob bernoulli (1654~1750) and Johann Bernoulli(1667~1748) among them.
They were among the first mathematicians to realize the surprising power of the calculus and to apply the tool to a great diversity of problems.
And was thus one of the first mathematicians to work in the calculus of variations.  He was also one of the early students of mathematical probability ; his book in this field, the Ars conjectandi, was posthumously published in 1713.  Several things in mathematics now bear Jakob Bernoulli's name.  Among these are the Bernoulli distribution andBernoulli theorem of statistics and probabillty theory; the Bernoulli equation, met by every student of a first course in differential equations.  He used the word 'integral' for the first time in 1690.     Johann Bernoulli was an even more prolific contributor to mathematics than was his brother Jakob.   He greatly enriched the calculus and was very influential in making the power of the new subject appreciated in continental Europe.  As we have seen, it was his material that the Marquis de l'Hospital (1661~1704), under a curious financial agreement with Johann, assembled in 1696 into the first calculus textbook.
In this way, the familiar method of evaluating the indeterminate form 0/0 became incorrectly known, in later calculus texts, as l'Hospital's rule.
Johann Bernoulli had three sons, Nicolaus (1695-1726), Daniel(1700-1782), and Johann II (1710-1790), all of whom won renown as eighteenth century mathematicians and scientists.
He was the most famous of Johann's three sons, and devoted most of his energies to probability, astronomy, physics, and hydrodynamics.

De Moivre and Probability : In the eighteenth century, the pioneering ideas of Fermat, Pascal, and Huygens in probability theory were considerably elaborated, and the theory made rapid advances, with the result that the Ars conjectandi of Jakob Bernoulli was followed by further treatments of the subject.   Important among those contributing to probability theory was Abraham De Moivre (1667-1754), a French Hugenot who moved to the more congenial political climate of London after the revocation of the Edict of Nantes in 1685.  He earned his living in England by private tutoring and he became an intimate friend of Isaac Newton.
De Moivre is particularly noted for his work Annuities upon Lives, which played an important role in the history of actuarial mathematics, his Doctrine of Chances, which contained much new material on the theory of probability, and his Miscellanea analytica, which contributed to recurrent series, probability, and analytic trigonometry.  De Moivre is credited with the first treatment of the probability integral,

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.

Comte de Buffon(1707-1788, France) : The insurance business made great strides in the eighteenth century, and a number of mathematicians were attracted to the underlying probability theory.
Comte de Buffon (1707-1788), gave in 1777 the first example of a geometrical probability, his famous "needle problem" for experimentally approximating the value of ¥ً.

Taylor and Maclaurin : Every student of the calculus is familiar with the name of the Englishman Brook Taylor(1685-1731) and the name of the Scotsman Colin Maclaurin(1698-1746), through the very useful Taylor's expansion and Maclaurin's expansion of a function.  It was in 1715 that Taylor published (with no consideration of convergence) his well-known expansion theorem,

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's Ages

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:


for functional notation,


for the base of natural logarithms,


for the sides of triangle ABC,


for the semiperimeter of triangle ABC,


for the inradius of triangle ABC,


for the circumradius of triangle ABC,

for the summation sign,


for the imaginary unit,

    His contributions to mathematics are as follows:
  • Classification of functions
  • Suggesting that arbitrary function is expressed by a straight line or contrary to this, arbitrary curve is expressed by a function. So the center of mathematics moves from geometry to algebra.
  • Euler's formula : e ix=cos x + i sin x etc.
  • Basic theory of graph :konigsberg bridge problem
  • Regulation of the way of marking
  • expansion of function (sinx, cosx, log(1+x to the indefinite series.
  • Telling the difference between exponential and trigonometric function.
  • Raising the known problem which Goldbach guessed.
  • Definition of ix and function.

Mathematicians of Revolution Ages

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 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.

Laplace and Legendre :He published two monumental works, <Traite de mecanique celeste, five volumes, 1799-1825> and <Theorie analytique des probabilites, 1812>.  HIs name is connected with the so-called 'Laplace tramsform' and with the 'Laplace expansion' of a determinant.
For Laplace, mathematics was merely a kit of tools used to explain nature.  To Lagrange, mathematics was a sublime art and was its own excuse for being.
Adrien-Marie Legendre(1752-1833) is known in the history of elementary mathematics principally for his very popular , in which he attempted a pedagogical improvement of Euclid's by considerably rearranging and simplifying many of the propositions.   This work was very favorably received in America and became the prototype of the geometry textbooks in this country.
Legendre's name is today connected with the second order differential equation

(1-x2)y- 2xy+ n(n+1)y = 0

which is of considerable importance in applied mathematics.

Monge : In addition to creating descriptive geometry, Monge is considered as the father of differential geometry.
His method, which was one of cleverly representing three-dimensional objects by appropriate projections on the two-dimensional plane, was adopted by the military and classified as top-secret.  It later became widely taught as 'descriptive geometry'.
His lectures there inspired a large following of able geometers, among whom were Charles Dupin(1784-1873) and Jean Victor Poncelet(1788-1867), the former a contributor to the field of differential geometry, and the latter to that of projective geometry.
Unlike the theree L's (Lagrange, Laplace, and Legendre), who remaind aloof from the French Revolution, Monge and carnot supported it and played active roles in revolutionary matters.

The Metric System

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 themetric 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.

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