The Development of Analysis

¢º Caracteristic of the Eighteenth - Century Mathematics| ¢º The Early Eighteenth Century|

¢º Euler's Ages|

¢º Mathematicians of Revolution Ages|

¢º The Metric System|

¡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

¡Ý 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 and Bernoulli 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. ¡Ý 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...

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