History of mathematics

[11] Beginning in Renaissance Italy in the 15th century, new mathematical developments, interacting with new scientific discoveries, were made at an increasing pace that continues through the present day.

This includes the groundbreaking work of both Isaac Newton and Gottfried Wilhelm Leibniz in the development of infinitesimal calculus during the course of the 17th century and following discoveries of German mathematicians like Carl Friedrich Gauss and David Hilbert.

Common interpretations are that the Ishango bone shows either a tally of the earliest known demonstration of sequences of prime numbers[13][failed verification] or a six-month lunar calendar.

It has been claimed that megalithic monuments in England and Scotland, dating from the 3rd millennium BC, incorporate geometric ideas such as circles, ellipses, and Pythagorean triples in their design.

They developed a complex system of metrology from 3000 BC that was chiefly concerned with administrative/financial counting, such as grain allotments, workers, weights of silver, or even liquids, among other things.

[62] Archimedes (c. 287–212 BC) of Syracuse, widely considered the greatest mathematician of antiquity,[63] used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.

[65] He also studied the spiral bearing his name, obtained formulas for the volumes of surfaces of revolution (paraboloid, ellipsoid, hyperboloid),[64] and an ingenious method of exponentiation for expressing very large numbers.

[94] Aside from managing trade and taxes, the Romans also regularly applied mathematics to solve problems in engineering, including the erection of architecture such as bridges, road-building, and preparation for military campaigns.

[95] Arts and crafts such as Roman mosaics, inspired by previous Greek designs, created illusionist geometric patterns and rich, detailed scenes that required precise measurements for each tessera tile, the opus tessellatum pieces on average measuring eight millimeters square and the finer opus vermiculatum pieces having an average surface of four millimeters square.

[103] Perhaps relying on similar gear-work and technology found in the Antikythera mechanism, the odometer of Vitruvius featured chariot wheels measuring 4 feet (1.2 m) in diameter turning four-hundred times in one Roman mile (roughly 4590 ft/1400 m).

[106] However, the Tsinghua Bamboo Slips, containing the earliest known decimal multiplication table (although ancient Babylonians had ones with a base of 60), is dated around 305 BC and is perhaps the oldest surviving mathematical text of China.

It consists of 246 word problems involving agriculture, business, employment of geometry to figure height spans and dimension ratios for Chinese pagoda towers, engineering, surveying, and includes material on right triangles.

[135] The next significant mathematical documents from India after the Sulba Sutras are the Siddhantas, astronomical treatises from the 4th and 5th centuries AD (Gupta period) showing strong Hellenistic influence.

[137] Around 500 AD, Aryabhata wrote the Aryabhatiya, a slim volume, written in verse, intended to supplement the rules of calculation used in astronomy and mathematical mensuration, though with no feeling for logic or deductive methodology.

[149][150][151][152] The Islamic Empire established across the Middle East, Central Asia, North Africa, Iberia, and in parts of India in the 8th century made significant contributions towards mathematics.

[156] His algebra was also no longer concerned "with a series of problems to be resolved, but an exposition which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study."

Something close to a proof by mathematical induction appears in a book written by Al-Karaji around 1000 AD, who used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes.

[165] Boethius provided a place for mathematics in the curriculum in the 6th century when he coined the term quadrivium to describe the study of arithmetic, geometry, astronomy, and music.

[172] Bradwardine's analysis is an example of transferring a mathematical technique used by al-Kindi and Arnald of Villanova to quantify the nature of compound medicines to a different physical problem.

[177] Nicole Oresme at the University of Paris and the Italian Giovanni di Casali independently provided graphical demonstrations of this relationship, asserting that the area under the line depicting the constant acceleration, represented the total distance traveled.

[181][182][183] Luca Pacioli's Summa de Arithmetica, Geometria, Proportioni et Proportionalità (Italian: "Review of Arithmetic, Geometry, Ratio and Proportion") was first printed and published in Venice in 1494.

It was written primarily for, and sold mainly to, merchants who used the book as a reference text, as a source of pleasure from the mathematical puzzles it contained, and to aid the education of their sons.

Simon Stevin's De Thiende ('the art of tenths'), first published in Dutch in 1585, contained the first systematic treatment of decimal notation in Europe, which influenced all later work on the real number system.

He made numerous contributions to the study of topology, graph theory, calculus, combinatorics, and complex analysis, as evidenced by the multitude of theorems and notations named for him.

Niels Henrik Abel, a Norwegian, and Évariste Galois, a Frenchman, proved that there is no general algebraic method for solving polynomial equations of degree greater than four (Abel–Ruffini theorem).

[citation needed] Abel and Galois's investigations into the solutions of various polynomial equations laid the groundwork for further developments of group theory, and the associated fields of abstract algebra.

[citation needed] In the later 19th century, Georg Cantor established the first foundations of set theory, which enabled the rigorous treatment of the notion of infinity and has become the common language of nearly all mathematics.

An example is the classification of finite simple groups (also called the "enormous theorem"), whose proof between 1955 and 2004 required 500-odd journal articles by about 100 authors, and filling tens of thousands of pages.

[210] A group of French mathematicians, including Jean Dieudonné and André Weil, publishing under the pseudonym "Nicolas Bourbaki", attempted to exposit all of known mathematics as a coherent rigorous whole.

[citation needed] In the preceding centuries much mathematical focus was on calculus and continuous functions, but the rise of computing and communication networks led to an increasing importance of discrete concepts and the expansion of combinatorics including graph theory.