Timeline of Mathematics - 1000–1500

1000–1500

c. 1000 — Abū Sahl al-Qūhī (Kuhi) solves equations higher than the second degree.
c. 1000 — Abu-Mahmud al-Khujandi first states a special case of Fermat's Last Theorem.
c. 1000 — Law of sines is discovered by Muslim mathematicians, but it is uncertain who discovers it first between Abu-Mahmud al-Khujandi, Abu Nasr Mansur, and Abu al-Wafa.
c. 1000 — Pope Sylvester II introduces the abacus using the Hindu-Arabic numeral system to Europe.
1000 — Al-Karaji writes a book containing the first known proofs by mathematical induction. He used it to prove the binomial theorem, Pascal's triangle, and the sum of integral cubes. He was “the first who introduced the theory of algebraic calculus.”
c. 1000 — Ibn Tahir al-Baghdadi studied a slight variant of Thabit ibn Qurra's theorem on amicable numbers, and he also made improvements on the decimal system.
1020 — Abul Wáfa — Gave this famous formula: sin (α + β) = sin α cos β + sin β cos α. Also discussed the quadrature of the parabola and the volume of the paraboloid.
1021 — Ibn al-Haytham formulated and solved “Alhazen's problem” geometrically.
1030 — Ali Ahmad Nasawi writes a treatise on the decimal and sexagesimal number systems. His arithmetic explains the division of fractions and the extraction of square and cubic roots (square root of 57,342; cubic root of 3, 652, 296) in an almost modern manner.
1070 — Omar Khayyám begins to write Treatise on Demonstration of Problems of Algebra and classifies cubic equations.
c. 1100 — Omar Khayyám “gave a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections.” He became the first to find general geometric solutions of cubic equations and laid the foundations for the development of analytic geometry and non-Euclidean geometry. He also extracted roots using the decimal system (Hindu-Arabic numeral system).
12th century — Indian numerals have been modified by Arab mathematicians to form the modern Hindu-Arabic numeral system (used universally in the modern world)
12th century — the Hindu-Arabic numeral system reaches Europe through the Arabs
12th century — Bhaskara Acharya writes the Lilavati, which covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations
12th century — Bhāskara II (Bhaskara Acharya) writes the “Bijaganita” (“Algebra”), which is the first text to recognize that a positive number has two square roots
12th century — Bhaskara Acharya conceives differential calculus, and also develops Rolle's theorem, Pell's equation, a proof for the Pythagorean Theorem, proves that division by zero is infinity, computes π to 5 decimal places, and calculates the time taken for the earth to orbit the sun to 9 decimal places
1130 — Al-Samawal gave a definition of algebra: “ with operating on unknowns using all the arithmetical tools, in the same way as the arithmetician operates on the known.”
1135 — Sharafeddin Tusi followed al-Khayyam's application of algebra to geometry, and wrote a treatise on cubic equations which “represents an essential contribution to another algebra which aimed to study curves by means of equations, thus inaugurating the beginning of algebraic geometry.”
1202 — Leonardo Fibonacci demonstrates the utility of Hindu-Arabic numerals in his Liber Abaci (Book of the Abacus).
1247 — Qin Jiushao publishes Shùshū Jiǔzhāng (“Mathematical Treatise in Nine Sections”).
1260 — Al-Farisi gave a new proof of Thabit ibn Qurra's theorem, introducing important new ideas concerning factorization and combinatorial methods. He also gave the pair of amicable numbers 17296 and 18416 which have also been joint attributed to Fermat as well as Thabit ibn Qurra.
c. 1250 — Nasir Al-Din Al-Tusi attempts to develop a form of non-Euclidean geometry.
1303 — Zhu Shijie publishes Precious Mirror of the Four Elements, which contains an ancient method of arranging binomial coefficients in a triangle.
14th century — Madhava is considered the father of mathematical analysis, who also worked on the power series for p and for sine and cosine functions, and along with other Kerala school mathematicians, founded the important concepts of Calculus
14th century — Parameshvara, a Kerala school mathematician, presents a series form of the sine function that is equivalent to its Taylor series expansion, states the mean value theorem of differential calculus, and is also the first mathematician to give the radius of circle with inscribed cyclic quadrilateral
1400 — Madhava of Sangamagrama|Madhava discovers the series expansion for the inverse-tangent function, the infinite series for arctan and sin, and many methods for calculating the circumference of the circle, and uses them to compute π correct to 11 decimal places
c. 1400 — Ghiyath al-Kashi “contributed to the development of decimal fractions not only for approximating algebraic numbers, but also for real numbers such as π. His contribution to decimal fractions is so major that for many years he was considered as their inventor. Although not the first to do so, al-Kashi gave an algorithm for calculating nth roots which is a special case of the methods given many centuries later by Ruffini and Horner.” He is also the first to use the decimal point notation in arithmetic and Arabic numerals. His works include The Key of arithmetics, Discoveries in mathematics, The Decimal point, and The benefits of the zero. The contents of the Benefits of the Zero are an introduction followed by five essays: “On whole number arithmetic”, “On fractional arithmetic”, “On astrology”, “On areas”, and “On finding the unknowns ”. He also wrote the Thesis on the sine and the chord and Thesis on finding the first degree sine.
15th century — Ibn al-Banna and al-Qalasadi introduced symbolic notation for algebra and for mathematics in general.
15th century — Nilakantha Somayaji, a Kerala school mathematician, writes the “Aryabhatiya Bhasya”, which contains work on infinite-series expansions, problems of algebra, and spherical geometry
1424 — Ghiyath al-Kashi computes π to sixteen decimal places using inscribed and circumscribed polygons.
1427 — Al-Kashi completes The Key to Arithmetic containing work of great depth on decimal fractions. It applies arithmetical and algebraic methods to the solution of various problems, including several geometric ones.
1478 — An anonymous author writes the Treviso Arithmetic.
1494 — Luca Pacioli writes "Summa de arithmetica, geometria, proportioni et proportionalità"; introduces primitive symbolic algebra using "co" (cosa) for the unknown.