bottom of page
of this page, ``A brief history of calculus'',
is a collection of notes I made as a prelude to
a History of Science paper that never happened.
(I did the
``Other interesting stuff'',
is just a collection of miscellaneous events in the history of math.
Citations appear in ``angle-brackets'' (e.g., <TT>).
list of sources
appears at the bottom of this page.
- c.360 B.C.
Eudoxus of Cnidus
``method of exhaustion'',
close to the
which is used by himself and
it was based on the
can be made as large as one wishes
by multiplying it by a large enough constant.
- Johannes Kepler
volumes of revolution
Nova stereometria doliorum vinarorum
(``New Measurement of the
- Bonaventura Cavalieri
- John Wallis
- Blaise Pascal,
working on the
- Isaac Newton
retires to the country to escape the
there he invents the first form of
- James Gregory
includes a geometrical version of
fundamental theorem of calculus
Geometriae pars universalis, inserviens quantitatum curvarum transmutationi & measurae
(``Geometrical Exercises and the Universal Part of
includes his method
areas under curves
De analysi per equationes numero terminorum infinitas
(``On the Analysis of
Unlimited in the Number of Their Terms''),
- Isaac Barrow
uses methods similar to
lengths of curves,
areas bounded by curves.
- Gottfried Leibniz
and the notation
he also determines the
writes two letters to
hinting at his work with
(his form of
also this year,
discovers how to
``A new method for
as well as
which is impeded neither by
and a remarkable type of
although only six pages long, few can understand it.
publishes his method of
in an issue of
-- my own translation, so might not be completely satisfactory).
- Michel Rolle
named after him.
- John Wallis
in volume two of his
- Jean Bernoulli
discovers the method known as
(a.k.a., l'Hospital's Rule);
it is known by that name because
Marquis Antoine de l'Hôpital
bought it from
and introduced it in his influential 1696 textbook
Analyse des infiniment petits pour l'Intelligence des Lignes Courbes
Curved Lines''?? -- my own translation).
- Brook Taylor
Methodus incrementorum directa et inversa
(``Direct and Inverse Incremental Method''),
in which he develops the
- Joseph-Louis Lagrange introduces the notations
f'x and y'
for the derivatives of
f(x) and y,
- Louis F. A. Arbogast introduces the symbol D
for the operation of differentiation.
- Carl Gustav Jacob Jacobi adopts the modern notation
for partial differentiation;
Adrien-Marie Legendre originally introduced it in 1786,
but immediately abandoned it.
- Bernhard Riemann defines the integral in a way
that does not require continuity.
- H. Eduard Heine, a student of Karl Weierstrass,
presents the modern ``epsilon-delta'' definition of a limit
in his Die Elemente der Functionenlehre
(``Elements of the Study of
-- my own translation).
2000 B.C. to 200 B.C.
- c.1975 B.C.
- Mesopotamian mathematicians discover how to solve
- c.1850 B.C.
- Mesopotamian mathematicians discover the so-called
approximately 12 centuries before the time of Pythagoras.
- 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).
- c.450 B.C.
- ``Achilles'' paradox of Zeno.
- c.350 B.C.
- Menaechmus discovers the conic sections:
the parabola, ellipse, and hyperbola.
- 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,
(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
some of the proofs are probably Euclid's own.
for more information.
- c.250 B.C.
- Archimedes of Syracuse and the quadrature of the parabola.
1500 A.D. to present
- The first known proof by mathematical induction
is included in Francesco Maurolico's
Arithmeticorum Libri Duo;
he proves that the sum of the first n odd integers
- Rene Descartes discovers that any simple convex polyhedron
having V vertices, E edges, and F faces
obeys the rule V - E + F = 2;
since his discovery is not published until 1860,
the theorem is named after Leonhard Euler,
who rediscovered it in 1752.
- 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.
- 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.
- 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.
- Hermann Gunther-Grassman publishes
The Study of Extensions,
which deals with multidimensional vectors;
he almost single-handedly creates modern linear algebra.
- August Ferdinand Mobius unveils his
single-sided, single-edged figure, the Mobius strip.
- 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.
- Giuseppe Peano discovers a one-dimensional, continuous curve
that passes through all the points in the interior of a square.
- Bertrand Russell discovers his ``great paradox''.
- The final edition of Giuseppe Peano's
Mathematical Formulas contains about 4,200 theorems.
- 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.
- CRC Standard Mathematical Tables, 26th edition
- William H. Beyer, ed. (CRC Press, 1981)
- The Historical Development of the Calculus
- C. H. Edwards, Jr. (Springer-Verlag, 1979)
- A History of Mathematical Notations/Volume II:
Notations Mainly in Higher Mathematics
- Florian Cajori (Open Court Publishing, 1952)
- A History of Mathematics, second edition
- Carl B. Boyer and Uta C. Merzbach (John Wiley & Sons, 1989)
- The History of the Calculus
and Its Conceptual Development
- Carl D. Boyer (Dover Publications, 1959)
- 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)
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Donald Lancon, Jr. /
Math & science
Some 20th-century scientists
High school Euclid paper
Donald Lancon, Jr.
Thu, 26 Feb 2004