History of mathematics

Number theory:

• rational numbers, perfect numbers & prime numbers - used by Pythagoreans 6th-5thC BC
• numbers were regarded by Pythagoreans as patterns of dots which form characteristic figures, as on the sides of a dice; although we use Arabic symbols, which have no resemblance to these dot-patterns, we still call numbers "figures", ie. shapes.
• hence square numbers, triangular, cubic, pyramidal numbers, etc.
• addition of even numbers formed oblong numbers, where the ratio of the sides represented exactly the concordant intervals of the musical octave
• law of signs (ie. multiply 2 negative numbers results in a positive) - known by Babylonians & Indians in 3rdC BC
• but negative numbers rejected by Greeks & Europeans until Renaissance when used to represent debt in businesses!
• irrational numbers accepted after theory of Eudoxus (c. 370 BC)
• prime numbers:
• 300BC: Euclid  gives a proof of the Fundamental Theorem of Arithmetic: Every integer can be written as a product of primes in an essentially unique way
• Euclid also showed that if the number 2n - 1 is prime then the number 2n-1(2n - 1) is a perfect number
• 1603: Cataldi found the factors of all numbers up to 800 and also a table of all primes up to 750 (there are 132 such primes)
• early 17thC, Fermat proved a speculation of Albert Girard that every prime number of the form 4 n + 1 can be written in a unique way as the sum of two squares and was able to show how any number could be written as a sum of four squares. The number of the form 2n - 1 also attracted attention because it is easy to show that if unless n is prime these number must be composite. These are often called Mersenne numbers Mn because Mersenne studied them.
• 18thC: Euler founds Analytic Number Theory
• perfect numbers
• 100AD: Nicomachus goes on to describe certain results concerning perfect numbers, held as truths for many centuries, but were later proven fallacies.
• 1603: Cataldi finds the sixth perfect number, namely 216(217 - 1) = 8589869056 and the seventh perfect number, namely 218(219 - 1) = 137438691328
• 1732: Euler proves that the eighth perfect number was 230(231 - 1) = 2305843008139952128
• 1876:  Lucas shows that 2126(2127- 1) is indeed a perfect number
• 1883: Pervusin showed that 260(261- 1) is a perfect number
• 1911: Powers showed that 288(289 - 1) was a perfect number, then a few years later he showed that 2101- 1 is a prime and so 2100(2101- 1) is a perfect number
• today 39 perfect numbers are known, 288(289- 1) being the last to be discovered by hand calculations in 1911 (although not the largest found by hand calculations), all others being found using a computer
• complex numbers:
• in order to solve polynomial equations, Bombelli (c 1530) defined square root of negative one & then the arithmetic of the imaginary numbers
• Hamilton (1805-1865) defined complex addition, from which closure, associativity, and commutativity can be deduced
• hypercomplex numbers:
• Hamilton (c. 1843) kept the idea of ordered pairs, but he generalized from ordered pairs of reals to ordered pairs of complexes: , giving a quadruple of reals called a quaternion.
• in 1844, just one year after Hamilton finally formulated his quaternion algebra, Grassman (1809-1877) formulated general th-order hypercomplex algebras.
• 1850: primary numbers;
• 1874-99: Cantor develops set theory

Algebra:

• Babylonians use simultaneous linear equations and early matrices

• Chinese 200-100BC 1st to use matrices & use Gaussian elimination which would not become well known in the West until the early 19th Century!

• ALGEBRA comes from the title of a work written in Arabic c 825 by al-Khowarizmi, al-jabr w'al-muqâbalah, in which al-jabr means "the reunion of broken parts." When this was translated from Arabic into Latin four centuries later, the title emerged as Ludus algebrae et almucgrabalaeque.

• in 1140 Robert of Chester translated the Arabic title into Latin as Liber algebrae et almucabala.
• the earliest known use of the word algebra in English in its mathematical sense is by Robert Recorde in The Pathwaie to Knowledge in 1551
• 1629: Gerard uses brackets & other abbreviations in mathematics;
• 1631: Oughtred proposes symbol 'X' for multiplication;
• 1663: Newton discovers the binomial theorem;
• 1683: discovery of determinants in matrices simultaneously in Europe & in Japan.
• 1748: Euler's pure analytical mathematics;
• 1750: Cramer gave the general rule for n x n matrix systems in a paper Introduction to the analysis of algebraic curves
• 1770, Euler's introduction to algebra;
• ABELIAN EQUATION. Leopold Kronecker (1823-1891) introduced the term Abelsche Gleichung in an 1853 paper on algebraically soluble equations.
• 1801: The term 'determinant' was first introduced by Gauss in Disquisitiones arithmeticae (1801) while discussing quadratic forms.
• 1850: The first to use the term 'matrix' was Sylvester
• 1853-97, group theory comes of age with Burnside's Theory of groups of finite order published in 1897

Geometry:

• Pythagorus' theorem
• Euclidean geometry (c300BC):
• golden ratio
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• Ptolemy - trigonometric tables
• 1678: geometrical theorem on the nature of concurrency;
• 1826, Lobachevsky's non-Euclidean geometry;
• 1849-70: development of cubic surface geometry theorems
• Analytical geometry:
• The term ANALYTIC GEOMETRY was apparently first used (as geometria analytica) in Geometria analytica sive specimina artis analyticae, published by Samuel Horsley (1733-1806) in volume 1 of Isaaci Newtoni opera quae exstant omnia. Commentariis illustrabat Samuel Horsley (1779).
• 1908: Minkowkski's 4D geometry;

Probability:

• 1654: Pascal & Fermat state the theory of probability;

Calculus:

• 1665: Newton experiments with gravity & invents differential calculus;
• 1675: Leibniz invents integral calculus;
• 1687: Newton "Philosophiae naturalis principia mathematica";
• 1655-99: study of elliptic integrals by Wallis, Newton & then Bernoulli.
• 1715, calculus of finite differences;
• 1864, Bertrand's treatise on differential & integral calculus;
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