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The following is a timeline of key developments of geometry:

- ca. 2000 BC – Scotland, carved stone balls exhibit a variety of symmetries including all of the symmetries of Platonic solids.
- 1800 BC – Moscow Mathematical Papyrus, findings volume of a frustum
- 1800 BC – Plimpton 322 contains the oldest reference to the Pythagorean triplets.
^{ [1] } - 1650 BC – Rhind Mathematical Papyrus, copy of a lost scroll from around 1850 BC, the scribe Ahmes presents one of the first known approximate values of π at 3.16, the first attempt at squaring the circle, earliest known use of a sort of cotangent, and knowledge of solving first order linear equations

- 800 BC – Baudhayana, author of the Baudhayana Sulba Sutra, a Vedic Sanskrit geometric text, contains quadratic equations, and calculates the square root of 2 correct to five decimal places
- ca. 600 BC – the other Vedic "Sulba Sutras" ("rule of chords" in Sanskrit) use Pythagorean triples, contain of a number of geometrical proofs, and approximate π at 3.16
- 5th century BC – Hippocrates of Chios utilizes lunes in an attempt to square the circle
- 5th century BC – Apastamba, author of the Apastamba Sulba Sutra, another Vedic Sanskrit geometric text, makes an attempt at squaring the circle and also calculates the square root of 2 correct to five decimal places
- 530 BC – Pythagoras studies propositional geometry and vibrating lyre strings; his group also discover the irrationality of the square root of two,
- 370 BC – Eudoxus states the method of exhaustion for area determination
- 300 BC – Euclid in his
*Elements*studies geometry as an axiomatic system, proves the infinitude of prime numbers and presents the Euclidean algorithm; he states the law of reflection in*Catoptrics*, and he proves the fundamental theorem of arithmetic - 260 BC – Archimedes proved that the value of π lies between 3 + 1/7 (approx. 3.1429) and 3 + 10/71 (approx. 3.1408), that the area of a circle was equal to π multiplied by the square of the radius of the circle and that the area enclosed by a parabola and a straight line is 4/3 multiplied by the area of a triangle with equal base and height. He also gave a very accurate estimate of the value of the square root of 3.
- 225 BC – Apollonius of Perga writes
*On Conic Sections*and names the ellipse, parabola, and hyperbola, - 150 BC – Jain mathematicians in India write the "Sthananga Sutra", which contains work on the theory of numbers, arithmetical operations, geometry, operations with fractions, simple equations, cubic equations, quartic equations, and permutations and combinations
- 140 BC – Hipparchus develops the bases of trigonometry.

- ca. 340 – Pappus of Alexandria states his hexagon theorem and his centroid theorem
- 500 – Aryabhata writes the "Aryabhata-Siddhanta", which first introduces the trigonometric functions and methods of calculating their approximate numerical values. It defines the concepts of sine and cosine, and also contains the earliest tables of sine and cosine values (in 3.75-degree intervals from 0 to 90 degrees)
- 7th century – Bhaskara I gives a rational approximation of the sine function
- 8th century – Virasena gives explicit rules for the Fibonacci sequence, gives the derivation of the volume of a frustum using an infinite procedure.
- 8th century – Shridhara gives the rule for finding the volume of a sphere and also the formula for solving quadratic equations
- 820 – Al-Mahani conceived the idea of reducing geometrical problems such as doubling the cube to problems in algebra.
- ca. 900 – Abu Kamil of Egypt had begun to understand what we would write in symbols as
- 975 – Al-Batani – Extended the Indian concepts of sine and cosine to other trigonometrical ratios, like tangent, secant and their inverse functions. Derived the formula: and .

- ca. 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.
- ca. 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).
- 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."
^{ [2] } - ca. 1250 – Nasir Al-Din Al-Tusi attempts to develop a form of non-Euclidean geometry.
- 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

- 17th century – Putumana Somayaji writes the "Paddhati", which presents a detailed discussion of various trigonometric series
- 1619 – Johannes Kepler discovers two of the Kepler-Poinsot polyhedra.

- 1722 – Abraham de Moivre states de Moivre's formula connecting trigonometric functions and complex numbers,
- 1733 – Giovanni Gerolamo Saccheri studies what geometry would be like if Euclid's fifth postulate were false,
- 1796 – Carl Friedrich Gauss proves that the regular 17-gon can be constructed using only a compass and straightedge
- 1797 – Caspar Wessel associates vectors with complex numbers and studies complex number operations in geometrical terms,
- 1799 – Gaspard Monge publishes Géométrie descriptive, in which he introduces descriptive geometry.

- 1806 – Louis Poinsot discovers the two remaining Kepler-Poinsot polyhedra.
- 1829 – Bolyai, Gauss, and Lobachevsky invent hyperbolic non-Euclidean geometry,
- 1837 – Pierre Wantzel proves that doubling the cube and trisecting the angle are impossible with only a compass and straightedge, as well as the full completion of the problem of constructibility of regular polygons
- 1843 – William Hamilton discovers the calculus of quaternions and deduces that they are non-commutative,
- 1854 – Bernhard Riemann introduces Riemannian geometry,
- 1854 – Arthur Cayley shows that quaternions can be used to represent rotations in four-dimensional space,
- 1858 – August Ferdinand Möbius invents the Möbius strip,
- 1870 – Felix Klein constructs an analytic geometry for Lobachevski's geometry thereby establishing its self-consistency and the logical independence of Euclid's fifth postulate,
- 1873 – Charles Hermite proves that e is transcendental,
- 1878 – Charles Hermite solves the general quintic equation by means of elliptic and modular functions
- 1882 – Ferdinand von Lindemann proves that π is transcendental and that therefore the circle cannot be squared with a compass and straightedge,
- 1882 – Felix Klein discovers the Klein bottle,
- 1899 – David Hilbert presents a set of self-consistent geometric axioms in
*Foundations of Geometry*

- 1901 – Élie Cartan develops the exterior derivative,
- 1912 – Luitzen Egbertus Jan Brouwer presents the Brouwer fixed-point theorem,
- 1916 – Einstein's theory of general relativity.
- 1930 – Casimir Kuratowski shows that the three-cottage problem has no solution,
- 1931 – Georges de Rham develops theorems in cohomology and characteristic classes,
- 1933 – Karol Borsuk and Stanislaw Ulam present the Borsuk-Ulam antipodal-point theorem,
- 1955 – H. S. M. Coxeter et al. publish the complete list of uniform polyhedron,
- 1975 – Benoit Mandelbrot, fractals theory,
- 1981 – Mikhail Gromov develops the theory of hyperbolic groups, revolutionizing both infinite group theory and global differential geometry,
- 1983 – the classification of finite simple groups, a collaborative work involving some hundred mathematicians and spanning thirty years, is completed,
- 1991 – Alain Connes and John Lott develop non-commutative geometry,
- 1998 – Thomas Callister Hales proves the Kepler conjecture,

- 2003 – Grigori Perelman proves the Poincaré conjecture,
- 2007 – a team of researchers throughout North America and Europe used networks of computers to map E8 (mathematics).
^{ [3] }

- Geometry and topology – branch of mathematics at the intersection between geometry and topology
- History of geometry – Historical development of geometry
- Timeline of ancient Greek mathematicians – Timeline and summary of ancient Greek mathematicians and their discoveries
- Timeline of mathematical logic
- Timeline of mathematics

In geometry and algebra, a real number is **constructible** if and only if, given a line segment of unit length, a line segment of length can be constructed with compass and straightedge in a finite number of steps. Equivalently, is constructible if and only if there is a closed-form expression for using only integers and the operations for addition, subtraction, multiplication, division, and square roots.

**Euclidean geometry** is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry, *Elements*. Euclid's approach consists in assuming a small set of intuitively appealing axioms (postulates) and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated earlier, Euclid was the first to organize these propositions into a logical system in which each result is *proved* from axioms and previously proved theorems.

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers (arithmetic).

The **history of mathematics** deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. From 3000 BC the Mesopotamian states of Sumer, Akkad and Assyria, followed closely by Ancient Egypt and the Levantine state of Ebla began using arithmetic, algebra and geometry for purposes of taxation, commerce, trade and also in the patterns in nature, the field of astronomy and to record time and formulate calendars.

In mathematics, the **trigonometric functions** are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.

A **triangle** is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called *vertices*, are zero-dimensional points while the sides connecting them, also called *edges*, are one-dimensional line segments. The triangle's interior is a two-dimensional region. Sometimes an arbitrary edge is chosen to be the *base*, in which case the opposite vertex is called the *apex*.

In geometry, **straightedge-and-compass construction** – also known as **ruler-and-compass construction**, **Euclidean construction**, or **classical construction** – is the construction of lengths, angles, and other geometric figures using only an idealized ruler and a pair of compasses.

In Euclidean geometry, a **cyclic quadrilateral** or **inscribed quadrilateral** is a quadrilateral whose vertices all lie on a single circle. This circle is called the *circumcircle* or *circumscribed circle*, and the vertices are said to be *concyclic*. The center of the circle and its radius are called the *circumcenter* and the *circumradius* respectively. Other names for these quadrilaterals are **concyclic quadrilateral** and **chordal quadrilateral**, the latter since the sides of the quadrilateral are chords of the circumcircle. Usually the quadrilateral is assumed to be convex, but there are also crossed cyclic quadrilaterals. The formulas and properties given below are valid in the convex case.

**Squaring the circle** is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square.

The timeline below shows the date of publication of possible major scientific breakthroughs, theories and discoveries, along with the discoverer. This article discounts mere speculation as discovery, although imperfect reasoned arguments, arguments based on elegance/simplicity, and numerically/experimentally verified conjectures qualify. The timeline begins at the Bronze Age, as it is difficult to give even estimates for the timing of events prior to this, such as of the discovery of counting, natural numbers and arithmetic.

**Indian mathematics** emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics, important contributions were made by scholars like Aryabhata, Brahmagupta, Bhaskara II, and Varāhamihira. The decimal number system in use today was first recorded in Indian mathematics. Indian mathematicians made early contributions to the study of the concept of zero as a number, negative numbers, arithmetic, and algebra. In addition, trigonometry was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there. These mathematical concepts were transmitted to the Middle East, China, and Europe and led to further developments that now form the foundations of many areas of mathematics.

* Divine Proportions: Rational Trigonometry to Universal Geometry* is a 2005 book by the mathematician Norman J. Wildberger on a proposed alternative approach to Euclidean geometry and trigonometry, called

In mathematics, **sine** and **cosine** are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opposite that angle to the length of the longest side of the triangle, and the cosine is the ratio of the length of the adjacent leg to that of the hypotenuse. For an angle , the sine and cosine functions are denoted simply as and .

Early study of triangles can be traced to the 2nd millennium BC, in Egyptian mathematics and Babylonian mathematics. Trigonometry was also prevalent in Kushite mathematics. Systematic study of trigonometric functions began in Hellenistic mathematics, reaching India as part of Hellenistic astronomy. In Indian astronomy, the study of trigonometric functions flourished in the Gupta period, especially due to Aryabhata, who discovered the sine function. During the Middle Ages, the study of trigonometry continued in Islamic mathematics, by mathematicians such as Al-Khwarizmi and Abu al-Wafa. It became an independent discipline in the Islamic world, where all six trigonometric functions were known. Translations of Arabic and Greek texts led to trigonometry being adopted as a subject in the Latin West beginning in the Renaissance with Regiomontanus. The development of modern trigonometry shifted during the western Age of Enlightenment, beginning with 17th-century mathematics and reaching its modern form with Leonhard Euler (1748).

This is a **timeline of mathematicians in ancient Greece**.

A timeline of **calculus** and **mathematical analysis**.

In trigonometry, the **law of cosines** relates the lengths of the sides of a triangle to the cosine of one of its angles. For a triangle with sides and opposite respective angles and , the law of cosines states:

This is a timeline of pure and applied mathematics history. It is divided here into three stages, corresponding to stages in the development of mathematical notation: a "rhetorical" stage in which calculations are described purely by words, a "syncopated" stage in which quantities and common algebraic operations are beginning to be represented by symbolic abbreviations, and finally a "symbolic" stage, in which comprehensive notational systems for formulas are the norm.

In mathematics, the **Pythagorean theorem** or **Pythagoras' theorem** is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

- ↑ Jones, Alexander; Proust, Christine (eds.). "Before Pythagoras: The Culture of Old Babylonian Mathematics".
*Institute for the Study of the Ancient World, New York University*. Retrieved 4 April 2023. - ↑ Arabic mathematics,
*MacTutor History of Mathematics archive*, University of St Andrews, Scotland - ↑ Thomson, Elizabeth A. (18 March 2007). "Math research team maps E8: Calculation on paper would cover Manhattan".
*MIT News*. Retrieved 19 February 2024.

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