¡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
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
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
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
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
¡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 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,
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.
¡Ý 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 ¥ð.
¡Ý 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
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.
¡ß 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
First of all, we owe to Euler the conventionalization of the following
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 : ei¥ö=cos¥ö£«isin¥öetc.
- 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 ¥Ã and©¬function.
¡ß Mathematicians of Revolution Ages
¡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
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.
¡Ý 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
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 <Elements de
geometrie>, in which he attempted a pedagogical improvement of
Euclid's <Elements> 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
(1-x2)y¡È- 2xy¡Ç+ n(n+1)y = 0
which is of considerable importance in applied mathematics.
¡Ý 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.
¡ß 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 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
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