|
1706
|
Jones
introduces
the Greek letter
p
to represent the ratio of the circumference of a circle to its diameter in his
Synopsis palmariorum matheseos (A New Introduction to Mathematics).
|
|
1707
|
De Moivre uses trigonometric functions to represent complex numbers in the
form r(cos x + i sin x).
|
|
1707
|
Newton publishes
Arithmetica universalis (General Arithmetic)
which contains a collection of his results in algebra.
|
|
1713
|
Jacob Bernoulli's book
Ars conjectandi (The Art of Conjecture)
is an important work on probability. It contains the Bernoulli numbers which
appear in a discussion of the exponential series.
|
|
1715
|
Brook Taylor publishes
Methodus incrementorum directa et inversa (Direct and Indirect Methods of
Incrementation),
an important contribution to the calculus. The book discusses singular
solutions to differential equations, a change of variables formula, and a way
of relating the derivative of a function to the derivative of the inverse
function. There is also a discussion on vibrating strings.
|
|
1718
|
De Moivre publishes
The Doctrine of Chances.
The definition of statistical independence appears in this book together with
many problems with dice and other games. He also investigated mortality
statistics and the foundation of the theory of annuities.
|
|
1718
|
Jacob Bernoulli's work on the calculus of variations is published after his
death.
|
|
1719
|
Brook Taylor publishes
New principles of linear perspective.
The first edition appeared four years earlier under the title
Linear perspective.
The work gives the first general treatment of vanishing points.
|
|
1722
|
The work unfinished by Cotes on his death is published as
Harmonia mensurarum.
It deals with integration of rational functions. It contains a thorough
treatment of the calculus applied to logarithmic and circular functions.
|
|
1727
|
Euler is appointed to St Petersburg. He introduces the symbol
e
for the base of natural logarithms in a manuscript entitled
Meditation upon Experiments made recently on firing of Cannon.
The manuscript was not published until 1862.
|
|
1728
|
Grandi publishes
Flora geometrica (Geometrical Flowers).
He gives a geometrical definition of curves which resemble petals and leaves of
flowers. For example the
rhodonea curves
are so called since they look like roses while the
clelie curve
is named after the Countess Clelia Borromeo to whom he dedicated his book.
|
|
1730
|
De Moivre gives further theorems concerning his trigonometric representation of
complex numbers. He gives Stirling's formula.
|
|
1733
|
De Moivre first describes the normal distribution curve, or law of errors, in
Approximatio ad summam terminorum binomii (a+b)
n
in seriem expansi.
Gauss, in 1820, also investigated the normal distribution.
|
|
1734
|
Berkeley publishes
The analyst: or a discourse addressed to an infidel mathematician.
He argues that although the calculus led to true results, its foundations were
no more secure than those of religion.
|
|
1735
|
Euler introduces the notation
f(x).
|
|
1736
|
Euler publishes
Mechanica
which is the first mechanics textbook which is based on differential equations.
|
|
1736
|
Euler solves the topographical problem known as the "Königsberg bridges
problem". He proves mathematically that it is impossible to design a walk which
crosses each of the seven bridges exactly once.
|
|
1737
|
Simpson publishes his
Treatise on Fluxions
written as a textbook for his private students. In the book he uses infinite
series to find the definite integrals of functions.
|
|
1739
|
D'Alembert publishes
Mémoire sur le calcul intégral (Memoir on Integral Calculus).
|
|
1740
|
Maclaurin is awarded the Grand Prix of the Académie des Sciences for his work
on gravitational theory to explain the tides.
|
|
1740
|
Simpson publishes
Treatise on the Nature and Laws of Chance.
Much of this probability treatise is based on the work of de Moivre.
|
|
1742
|
Goldbach conjectures, in a letter to Euler, that every even number 4 can be
written as the sum of two primes. It is not yet known whether Goldbach's
conjecture is true.
|
|
1742
|
Maclaurin publishes
Treatise on Fluxions
which aims to provide a rigorous foundation for the calculus by appealing to
the methods of Greek geometry. It is the first systematic exposition of
Newton's methods written in reply to Berkeley's attack on the calculus for its
lack of rigorous foundations.
|
|
1743
|
D'Alembert publishes
Traité de dynamique (Treatise on Dynamics).
In this celebrated work he states his principle that the internal actions and
reactions of a system of rigid bodies in motion are in equilibrium.
|
|
1744
|
D'Alembert publishes
Traite de l'equilibre et du mouvement des fluides (Treatise on Equilibrium and
on Movement of Fluids).
He applies his principle to the equilibrium and motion of fluids.
|
|
1746
|
D'Alembert further develops the theory of complex numbers in making the first
serious attempt to prove the fundamental theorem of algebra.
|
|
1747
|
D'Alembert uses partial differential equations to study the winds in
Réflexion sur la cause générale des vents (Reflection on the General Cause of
Winds)
which receives the prize of the Prussian Academy.
|
|
1748
|
Agnesi writes
Instituzioni analitiche ad uso della giovent italiana
which is an Italian teaching text on the differential calculus. The book
contains many examples which were carefully selected to illustrate the ideas.
There is an investigation of a curve that becomes known as "the witch of
Agnesi".
|
|
1748
|
Euler publishes
Analysis Infinitorum (Analysis of the Infinite)
which is an introduction to mathematical analysis. He defines a function and
says that mathematical analysis is the study of functions. This work bases the
calculus on the theory of elementary functions rather than on geometric curves,
as had been done previously. The famous formula
e
i
p
= -1
appears for the first time in this text.
|
|
1750
|
Cramer publishes
Introduction à l'analyse des lignes courbes algébraique.
The work investigates curves. The third chapter looks at a classification of
curves and it is in this chapter that the now famous "Cramer's rule" is given.
|
|
About 1750
|
D'Alembert studies the "three-body problem" and applies calculus to celestial
mechanics. Euler, Lagrange and Laplace also work on the three-body problem.
|
|
1751
|
Euler publishes his theory of logarithms of complex numbers.
|
|
1752
|
D'Alembert discovers the Cauchy-Riemann equations while investigating
hydrodynamics.
|
|
1752
|
Euler states his theorem V - E + F = 2 for polyhedra.
|
|
1753
|
Simson notes that in the Fibonacci sequence the ratio between adjacent numbers
approaches the
golden ratio.
|
|
1754
|
Lagrange makes important discoveries on the tautochrone which would contribute
substantially to the new subject of the calculus of variations.
|
|
1755
|
Euler publishes
Institutiones calculi differentialis
which begins with a study of the calculus of finite differences.
|
|
1761
|
Lambert proves that
p
is irrational.
He publishes a more general result in 1768.
|
|
1763
|
Monge begins the study of descriptive geometry.
|
|
1764
|
Bayes publishes
An Essay Towards Solving a Problem in the Doctrine of Chances
which gives Bayes theory of probability. The work contains the important
"Bayes' theorem".
|
|
1765
|
Euler publishes
Theory of the Motions of Rigid Bodies
which lays the foundation of analytical mechanics.
|
|
1766
|
Lambert writes
Theorie der Parallellinien
which is a study of the parallel postulate. By assuming that the parallel
postulate is false, he manages to deduce a large number of results about
non-euclidean geometry.
|
|
1767
|
D'Alembert calls the problems to elementary geometry caused by failure to prove
the parallel postulate "the scandal of elementary geometry".
|
|
1769
|
Euler makes Euler's Conjecture, namely that it is impossible to exhibit three
fourth powers whose sum is a fourth power, four fifth powers whose sum is a
fifth power, and similarly for higher powers.
|
|
1770
|
Lagrange proves that any integer can be written as the sum of four squares.
|
|
1770
|
Lagrange publishes
Réflexions sur la résolution algébrique des équations
which makes a fundamental investigation of why equations of degrees up to four
can be solved by radicals. The paper is the first to consider the roots of a
equation as abstract quantities rather than numbers. He studies permutations of
the roots and this work leads to group theory.
|
|
1777
|
Euler
introduces
the symbol
i
to represent the square root of -1 in a manuscript which will not appear in
print until 1794.
|
|
1781
|
William Herschel discovers the planet Uranus.
|
|
1784
|
Legendre introduces his "Legendre polynomials" in his work
Recherches sur la figure des planètes on celestial mechanics.
|
|
1785
|
Lagrange begins work on elliptic functions and elliptic integrals.
|
|
1788
|
Lagrange publishes
Mécanique analytique (Analytical Mechanics).
It summarises all the work done in the field of mechanics since the time of
Newton and is notable for its use of the theory of differential equations. With
this work Lagrange transforms mechanics into a branch of mathematical analysis.
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