By Donald Lancon Jr

From: http://www.obkb.com/dcljr/mathemat.html

c.360 B.C.

Eudoxus of Cnidus provides a ``method of exhaustion'', close to the limiting concept of calculus, which is used by himself and later Greeks to find areas and volumes of curvilinear figures; it was based on the lemma that any non-zero quantity can be made as large as one wishes by multiplying it by a large enough constant. <HM>

1615

Johannes Kepler uses infinitesimals to calculate volumes of revolution in New Measurement of the Volume of Wine Casks. <TT>

1635

Bonaventura Cavalieri calculates volumes using infinitely small sections. <TT>

1655

John Wallis studies infinite series in Arithmetic of Infinitesimals <TT>

1658

Blaise Pascal, working on the sine function, ``almost'' discovers calculus. <TT>

1665

Isaac Newton retires to the country to escape the Great Plague in London; there he invents the first form of calculus. <TT>

1668

James Gregory includes a geometrical version of the fundamental theorem of calculus in Geometrical Exercises and the Universal Part of Geometry. <TT>

1669

Newton includes his method for finding areas under curves in his On the Analysis of Equations Unlimited in the Number of Their Terms, circulated privately. <TT>

1670

Isaac Barrow uses methods similar to calculus to draw tangents to curves, find the lengths of curves, and the areas bounded by curves. <TT>

1675

Gottfried Leibniz introduces the modern notation for integration and the
notation

1676

Newton writes two letters to Leibniz, hinting at his work with infinite
series and fluxions (his form of calculus); also this year, Leibniz discovers
how to differentiate any fractional power of

1677

Leibniz finds the quotient rule for differentiation. <TT>

1684

Leibniz publishes ``A new method for maxima and minima as well as tangents, which is impeded neither by fractional nor by irrational quantities, and a remarkable type of calculus for this''; although only six pages long, few can understand it. <TT>

1686

Leibniz publishes his method of integral calculus in an issue of Acta Eruditorum. <TT>

1691

Michel Rolle states without proof the theorem named after him. <TT>

1693

John Wallis publishes Newton's method of fluxions in volume two of his Mathematical Works. <TT>

1694

Jean Bernoulli discovers the method known as l'Hospital's Rule; it is known by that name because Marquis Antoine de l'Hospital bought it from Bernoulli and introduced it in his influential 1696 textbook Analysis of Infinitesimals. <TT>

1715

Brook Taylor introduces his famous series in Methodus Incrementorum Directa et Inversa, in which he develops the calculus of finite differences. <TT>

1797

Joseph-Louis Lagrange introduces the notations

1800

Louis F. A. Arbogast introduces the symbol

1841

Carl Gustav Jacob Jacobi adopts the modern notation for partial differentiation; Adrien-Marie Legendre originally introduced it in 1786, but immediately abandoned it. <MN/TT>

1854

Bernhard Riemann defines the integral in a way that does not require continuity. <TT>

1872

H. Eduard Heine, a student of Karl Weierstrass, presents the modern ``epsilon-delta'' definition of a limit in his Elements. <TT>

c.1975 B.C.

Mesopotamian mathematicians discover how to solve quadratic equations. <TT>

c.1850 B.C.

Mesopotamian mathematicians discover the so-called Pythagorean theorem approximately 12 centuries before the time of Pythagoras. <TT>

876 B.C.

The first known use of a symbol for zero appears in India. <TT>

c.465 B.C.

The Pythagorean Hippasus of Metapontum discovers the dodecahedron, a regular
solid whose 12 faces are regular pentagons. There are only 4 other regular
solids: the tetrahedron (4 equilateral triangles), the cube (6 squares),
the
octahedron (8 equilateral triangles), and the icosahedron (20 equilateral
triangles). <TT/CR/HM>

c.450 B.C.

``Achilles'' paradox of Zeno. <HM>

c.350 B.C.

Menaechmus discovers the conic sections: the parabola, ellipse, and hyperbola. <HM>

c.300 B.C.

The thirteen books of Euclid's Elements (the most widely read
textbook ever written) contains propositions on plane geometry, the
distributive, commutative, and associative laws of arithmetic, quadratic
equations, prime numbers (including the theorem that the set of primes is
infinite), perfect numbers, greatest common divisors, geometric series,
irrational numbers, and solid geometry, including the five regular solids.
Unlike the modern treatment of these ideas, the Greeks used an entirely
geometrical approach in mathematics, using line segments to represent all
magnitudes (numbers). Although the Elements compiles already-known
Greek mathematics, some of the proofs are probably Euclid's own. <HM>

See my Euclid paperfor more information.

c.250 B.C.

Archimedes of Syracuse and the quadrature of the parabola. <HM>

1575

The first known proof by mathematical induction is included in Francesco
Maurolico's Arithmeticorum Libri Duo; he proves that the sum of the
first

1635

Rene Descartes discovers that any simple convex polyhedron having

1679

Gottfried Leibniz introduces binary arithmetic in a letter written to Joachim Bouvet, showing that any number may be expressed by 0's and 1's only. <TT>

1781

Joseph-Louis Lagrange expresses in a letter to his mentor Jean le Rond d'Alembert his fear that no further progress can be made in mathematics; despite this dire prophesy, many of his own contributions are still to come. <TT>

1843

While strolling along the Royal Canal, Sir William Rowan Hamilton devises a mathematical system which is not commutative; he develops his idea into a system of ``quaternions'', similar to that of three-dimensional vectors. His ideas help to usher in modern abstract algebra. <TT/HM>

1844

Hermann Gunther-Grassman publishes The Study of Extensions, which deals with multidimensional vectors; he almost single-handedly creates modern linear algebra. <TT/HM>

1865

August Ferdinand Mobius unveils his single-sided, single-edged figure, the Mobius strip. <TT>

1877

Georg Cantor proves that the number of points on a line segment is the same as the number of points in the interior of a square, publication of the result is delayed for a year because mathematicians refuse to believe it. <TT>

1890

Giuseppe Peano discovers a one-dimensional, continuous curve that passes through all the points in the interior of a square. <TT>

1902

Bertrand Russell discovers his ``great paradox''. <TT>

1908

The final edition of Giuseppe Peano's Mathematical Formulas contains about 4,200 theorems. <TT>

1976

Haken, Appel, and Koch prove with the use of a computer that only four colors are required to color in any two-dimensional map in such a way that no two adjacent regions share the same color; this was conjectured in 1850 by Francis Guthrie. <TT>

<CR>

CRC Standard Mathematical Tables, 26th edition - William H. Beyer, ed. (CRC Press, 1981)

<DC>

The Historical Development of the Calculus - C. H. Edwards, Jr. (Springer-Verlag, 1979)

<MN>

A History of Mathematical Notations/Volume II: Notations Mainly in Higher Mathematics - Florian Cajori (Open Court Publishing, 1952)

<HM>

A History of Mathematics, second edition - Carl B. Boyer and Uta C. Merzbach (John Wiley & Sons, 1989)

<HC>

The History of the Calculus and Its Conceptual Development - Carl D. Boyer (Dover Publications, 1959)

<TT>

The Timetables of Science: A Chronology of the Most Important People and Events in the History of Science - Alexander Hellemans and Bryan Bunch (Simon & Schuster, 1991)