Table of Geometry, from the 1728 Cyclopaedia. Cyclopaedia or A Universal Dictionary of Arts and Sciences ( folio, 2 vols

Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position Greek (el ελληνική γλώσσα or simply el ελληνικά — "Hellenic" is an Indo-European language, spoken today by 15-22 million people mainly Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and

Classic geometry was focused in compass and straightedge constructions. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles As they are the composition of five elemental constructions over a set of elements, as an algebra over an axiomatic system, the barrier between algebra and geometry began to fade out.

In modern times, geometric concepts have been generalized to a high level of abstraction and complexity, and have been subjected to the methods of calculus and abstract algebra, so that many modern branches of the field are barely recognizable as the descendants of early geometry. (See areas of mathematics and algebraic geometry. Here is a list of areas of modern Mathematics, with a brief explanation of their scope and links to other parts of this encyclopedia set out in a systematic way Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with )

Early geometry

The earliest recorded beginnings of geometry can be traced to cavemen, who discovered obtuse triangles in the ancient Indus Valley (see Harappan Mathematics), and ancient Babylonia (see Babylonian mathematics) from around 3000 BC. A caveman is a popular Stock character based upon Stereotyped concepts of the way in which early prehistoric Humans or Homininans may have looked and The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of The 30th century BC is a Century which lasted from the year 3000 BC to 2901 BC Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying, construction, astronomy, and various crafts. Surveying is the technique and science of accurately determining the terrestrial or three-dimensional space Position of points and the distances and angles between In the fields of Architecture and Civil engineering, construction is a process that consists of the Building or assembling of Infrastructure Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use of calculus. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras; the Egyptians had a correct formula for the volume of a frustum of a square pyramid; the Babylonians had a trigonometry table. This article is about the contemporary North African ethnic group Babylon was a City-state of ancient Mesopotamia, the remains of which can be found in present-day Al Hillah, Babil Province, Iraq In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. Elements special cases and related concepts Each plane section is a base of the frustum Exact constant expressions for Trigonometric expressions are sometimes useful mainly for simplifying solutions into radical forms which allow further simplification

Egyptian geometry

Main article: Egyptian mathematics

The ancient Egyptians knew that they could approximate the area of a circle as follows:^[1]

Area of Circle ≈ [ (Diameter) x 8/9 ]². Egyptian mathematics refers to the style and methods of Mathematics performed in Ancient Egypt. ^[1]

Problem 50 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. Ahmes (c 1680 BC-c 1620 BC (more accurately Ahmose) was an Egyptian scribe who lived during the Second Intermediate Period. This assumes that π is 4×(8/9)² (or 3. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 160493. . . ), with an error of slightly over 0. 63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital 125, within 0. 53 percent), but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3. Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer 14163, which had an error of just over 1 in 10,000). Interestingly, Ahmes knew of the modern 22/7 as an approximation for pi, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for pi for computing his hekat volume found in a cylinder.

Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 111. . .

The two problems together indicate a range of values for Pi between 3. 11 and 3. 16.

Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula:

Babylonian geometry

Main article: Babylonian mathematics

The Babylonians may have known the general rules for measuring areas and volumes. The Moscow Mathematical Papyrus is also called the Golenischev Mathematical Papyrus, after its first owner Egyptologist Vladimir Goleniščev. Elements special cases and related concepts Each plane section is a base of the frustum Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (ancient Iraq) from the days of the early Sumerians to the fall of They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time. ^[2]

Ancient Indian geometry (c. 3000 - 500 BC)

See also: Indian mathematics

Harappan geometry

The geometry used in the Indus Valley Civilization of North India and Pakistan from around 3000 BC was just as advanced as its contemporaries in Egypt and Mesopotamia, and mostly developed as a result of advanced urban planning, which is evident from the perfect grid pattern of Harappa and Mohenjo-daro where streets were laid out in perfect right angles. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin Geography Northern India lies mainly on continental India and a very small part of it lies on the Indian peninsula Pakistan () officially the Islamic Republic of Pakistan, is a country located in South Asia, Southwest Asia, Middle East and Harappa ( Urdu:, Hindi: हड़प्पा) is a City in Punjab, northeast Pakistan, about 35km (22 miles southwest Mohenjo-daro (موئن جودڑو موئن جو دڙو मोहन जोदड़ो Mound of the Dead was one of the largest city-settlements of the Indus Valley Civilization The geometry used by this early Harappan civilization was for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced brick technology, which utilised ratios. The Indus Valley Civilization (Mature period 2600&ndash1900 BCE abbreviated IVC, was an ancient Civilization that flourished in the Indus River basin A ratio is an expression which compares quantities relative to each other The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding. Brick sizes were in a perfect ratio of 4:2:1. Decimal weights were based on ratios of 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with each unit weighing approximately 28 grams, similar to the English ounce or Greek uncia.

Many of the weights uncovered have been produced in definite geometrical shapes (cuboid, barrel, cone, and cylinder to name a few) which present knowledge of basic geometry, including the circle. In anatomy the Cuboid bone is a bone in the foot See also Hyperrectangle Oblong A barrel or cask is a hollow cylindrical container traditionally made of Wood Staves and bound with Iron Hoops The A cone is a three-dimensional Geometric shape that tapers smoothly from a flat round base to a point called the apex or vertex A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis This culture also produced artistic designs of a mathematical nature and there is evidence on carvings that these people could draw concentric and intersecting circles and triangles.

Further to the use of circles in decorative design there is indication of the use of bullock carts, the wheels of which may have had a metallic band wrapped round the rim. Some historians believe this points to the possession of knowledge of the ratio of the length of the circumference of the circle and its diameter, and thus values of π. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems

In Lothal, a thick ring-like shell object found with four slits each in two margins served as a compass to measure angles on plane surfaces or in horizon in multiples of 40–360 degrees. Lothal ( Gujarātī: લોથલ ˈloːtʰəl Eng Mound of the Dead was one of the most prominent cities of the ancient Indus valley civilization. A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as Such shell instruments were probably invented to measure 8–12 whole sections of the horizon and sky, explaining the slits on the lower and upper margins. Archaeologists consider this as evidence the Lothal experts had achieved something 2,000 years before the Greeks are credited with doing: an 8–12 fold division of horizon and sky, as well as an instrument to measure angles and perhaps the position of stars, and for navigation purposes. Lothal contributes one of three measurement scales that are integrated and linear (others found in Harappa and Mohenjodaro). An ivory scale from Lothal has the smallest-known decimal divisions in Indus civilization. The scale is 6 mm thick, 15 mm broad and the available length is 128 mm, but only 27 graduations are visible over 146 mm, the distance between graduation lines being 1. The Millimetre ( American spelling: millimeter, symbol mm) is a unit of Length in the Metric system, equal to 704 mm (the small size indicate use for finer purposes). The sum total of ten graduations from Lothal is approximate to the angula in the Arthashastra. The Arthashastra ( IAST: Arthaśāstra) is a Treatise on statecraft, economic policy and Military strategy which The Lothal craftsmen took care to ensure durability and accuracy of stone weights by blunting edges before polishing. The Lothal weight of 12. 184 gm is almost equal to the Egyptian Oedet of 13. 792 gm.

Vedic geometry

During the Vedic period of Indian mathematics (c. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. 1500-500 BC), many rules and developments of geometry are found in Vedic works as a result of the mathematics required for the construction of religious altars. This article discusses the historical religious practices in the Vedic time period see Hinduism and Indian religions for details An altar is any structure upon which Sacrifices or other offerings are made for religious purposes or some other sacred place where ceremonies take place

As a result of the mathematics required for the construction of these altars, many rules and developments of geometry are found in Vedic works. These include:

• Use of geometric shapes, including triangles, rectangles, squares, trapezia and circles.
• Equivalence through numbers and area.
• Squaring the circle and vice versa. Squaring the circle is a problem proposed by ancient Geometers.
• Pythagorean triples discovered algebraically. A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 = 
• Statements of the Pythagorean theorem and a numerical proof. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry
• Computations of π, with the closest being correct to 2 decimal places. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems

Lagadha (circa 1350-1200 BC) was probably the earliest known mathematician to have used geometry and trigonometry for astronomy. The Vedanga Jyotisha, is an Indian text on Jyotisha ( Indian astronomy) redacted by Lagadha (लगध Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study

Yajnavalkya (9th century BC) composed the Shatapatha Brahmana, which contains geometric aspects, including several computations of π, with the closest being correct to 2 decimal places (the most accurate value of π up to that time), and gives a rule implying knowledge of the Pythagorean theorem. Sage Yajnavalkya ( याज्ञवल्क्य) of Mithila was a legendary sage of Vedic India, credited with the authorship of the The 9th century BC started the first day of 900 BC and ended the last day of 801 BC The Shatapatha Brahmana (sa शतपथ ब्राह्मण śatapatha brāhmaṇa, " Brahmana of one-hundred paths" abbreviated ŚB

The Sulba Sutras ("Rule of Chords" in Vedic Sanskrit), which is another name for geometry, were composed between 800 BC and 500 BC and were appendices to the Vedas giving rules for the construction of religious altars. The Shulba Sutras or Śulbasūtras ( Sanskrit śulba: "string cord rope" are Sutra texts belonging to the Vedic Sanskrit is an ancient Indian language, the language of the Vedas, the oldest Shruti texts of Hinduism. "Veda" redirects here For other uses see Veda (disambiguation. The Sulba Sutras contain the first use of irrational numbers, quadratic equations of the form a x² = c and ax² + bx = c, the use of the Pythagorean theorem and a list of Pythagorean triples discovered algebraically predating Pythagoras, geometric solutions of linear equations, and a number of geometrical proofs. In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry A Pythagorean triple consists of three positive Integers a, b, and c, such that a 2 +  b 2 =  A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable These discoveries are mostly a result of altar construction, which also led to the first known calculations for the square root of 2, which were correct to a remarkable 5 decimal places. In Mathematics, a square root of a number x is a number r such that r 2 = x, or in words a number r whose

Baudhayana (circa 800 BC) composed the Baudhayana Sulba Sutra, which contains a statement of the Pythagorean theorem, geometric solutions of a linear equation in a single unknown, several approximations of π (the closest value being 3. Baudhāyana, (fl ca 800 BCE was an Indian mathematician whowas most likely also a priest IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 114), along with the first use of irrational numbers and quadratic equations of the forms ax² = c and ax² + bx = c, and a computation for the square root of 2, which was correct to a remarkable five decimal places.

Manava (circa 750 BC) composed the Manava Sulba Sutra, which contains approximate constructions of circles from rectangles, and squares from circles, which give approximate values of π, with the closest value being 3. Manava (c 750 BC - 690 BC) is the author of the Indian geometric text of Sulba Sutras. Events and trends 756 BC — Founding of Cyzicus. 755 BC — Ashur-nirari V succeeds Ashur-Dan III as king of Assyria IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems 125.

Apastamba (circa 600 BC) composed the Apastamba Sulba Sutra, which contains the method of squaring the circle, considers the problem of dividing a segment into 7 equal parts, calculates the square root of 2 correct to five decimal places, solves the general linear equation, and also contains a numerical proof of the Pythagorean theorem, using an area computation. Squaring the circle is a problem proposed by ancient Geometers. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The historian Albert Burk claims this was the original proof of the theorem which Pythagoras copied on his visit to India. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor.

However, the legacy of geometry in India continued long after the Vedic period, with figures such as Aryabhata (476-550 AD). Āryabhaṭa ( Devanāgarī: आर्यभट (AD 476 &ndash 550 is the first in the line of great mathematician-astronomers from the classical age of Indian mathematics Events By place Western Roman Empire September 4 — Romulus Augustus, the last Emperor of the Western Roman Empire Events By place Byzantine Empire January 16 — Gothic War (535–552: The Ostrogoths, under King Totila

Ancient Chinese Geometry (c. 500 BC - 500 AD)

See also: Chinese mathematics

The Nine Chapters on the Mathematical Art, first compiled in 179 AD, with added commentary in the 3rd century by Liu Hui. Mathematics in China emerged independently by the 11th century BC The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom.

In ancient China, the earliest simple mathematical work stemmed back to the court records of divination for the Shang Dynasty (c. China ( Wade-Giles ( Mandarin) Chung¹kuo² is a cultural region, an ancient Civilization, and depending on perspective a National Divination (from Latin divinare "to be inspired by a god" related to Divine, Diva and Deus) is the attempt of ascertaining The Shang Dynasty ( Chinese: 商[[wiktionary 朝|朝]] or Yin Dynasty ( 殷[[wiktionary 代|代]] was according to traditional sources the 1600 BC-1050 BC), while the famous philosophical and cosmological work of the I Ching during the Zhou Dynasty (1050 BC-256 BC) had a complex arrangement of mathematical hexagrams. The I Ching ( Wade-Giles) or “Yì Jīng” ( Pinyin) also called “Classic of Changes” or “Book of Changes” is one of the oldest of the The Zhou Dynasty ( POJ: Chiu Tiau 1122 BC to 256 BC was preceded by the Shang Dynasty and followed by the Qin Dynasty in China. Events By place Roman Republic Rome aims for a quick end to hostilities in the First Punic War and decides to invade the For symbols used in the I Ching, see Hexagram (I Ching For a Jewish symbol see Star of David. However, the first definitive work (or at least oldest existent) on geometry in China was the Mo Jing, the Mohist canon of the early utilitarian philosopher Mozi (470 BC-390 BC). Mohism or Moism ( was a Chinese philosophy developed by the followers of Mozi (also referred to as Mo Di 470 &ndashc Utilitarianism is the idea that the moral worth of an action is solely determined by its contribution to overall Utility, that is its contribution to happiness Mozi ( Lat as Micius, ca 470 BCE&ndashca 391 BCE was a Philosopher who lived in China during the Hundred Schools of Thought Events By Place Greece Suspected of plotting to seize power in Sparta by instigating a Helot uprising Pausanias takes Events By place Roman Republic July 18 - Brennus, a chieftain of the Senones of the Adriatic coast of It was compiled years after his death by his later followers around the year 330 BC. ^[3] Although the Mo Jing is the oldest existent book on geometry in China, there is the possibility that even older written material exists. However, due to the infamous Burning of the Books in the political maneauver by the Qin Dynasty ruler Qin Shihuang (r. Burning of the books and burial of the scholars ( is a phrase that refers to a policy and a sequence of events in the Qin Dynasty of China, between the period of Not to be confused with the Qing Dynasty, the last dynasty of China Qin Shi Huang ( (259 BC – September 10 210 BC personal name Yíng Zhèng, was king of the Chinese State of Qin from 247 BCE to 221 BCE (during the 221 BC-210 BC), multitudes of written literature created before his time was purged. Events By place Carthage The Carthaginian general Hasdrubal is murdered by a Celtic assassin while campaigning to increase Events By place Roman Republic Following the death of his father Publius Cornelius Scipio, and his uncle Gnaeus Cornelius Scipio In addition, the Mo Jing presents geometrical concepts in mathematics that are perhaps too advanced not to have had a previous geometrical base or mathematic background to work upon.

The Mo Jing described various aspects of many fields associated with physical science, and provided a small wealth of information on mathematics as well. It provided an 'atomic' definition of the geometric point, stating that a line is separated into parts, and the part which has no remaining parts (i. e. cannot be divided into smaller parts) and thus forms the extreme end of a line is a point. ^[3] Much like Euclid's first and third definitions and Plato's 'beginning of a line', the Mo Jing stated that "a point may stand at the end (of a line) or at its beginning like a head-presentation in childbirth. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry Biography Early life Birth and family Plato was born in Athens Greece (As to its invisibility) there is nothing similar to it. "^[4] Similar to the atomists of Democritus, the Mo Jing stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved. In Natural philosophy, atomism is the theory that all the objects in the universe are composed of very small indestructible building blocks - Atoms Or stated in Democritus ( Greek:) was a pre-Socratic Greek Materialist Philosopher (born at Abdera in Thrace ca ^[4] It stated that two lines of equal length will always finish at the same place,^[4] while providing definitions for the comparison of lengths and for parallels,^[5] along with principles of space and bounded space. ^[6] It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch. ^[7] The book provided definitions for circumference, diameter, and radius, along with the definition of volume. ^[8]

The Sea Island Mathematical Manual, Liu Hui, 3rd century. The Sea Island Mathematical Manual or Haidao suanjing (海岛算经 was written by the Chinese mathematican Liu Hui of the Three Kingdoms Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom.

The Han Dynasty (202 BC-220 AD) period of China witnessed a new flourishing of mathematics. The Han Dynasty ( 206 BC–220 AD followed the Qin Dynasty and preceded the Three Kingdoms in China. Events By place Carthage Accused of treason by the Carthaginians after being defeated by the Romans at the Battle of the Events By Place Roman Empire The Goths invade Asia Minor and the Balkans. One of the oldest Chinese mathematical texts to present geometric progressions was the Suàn shù shū of 186 BC, during the Western Han era. In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found The mathematician, inventor, and astronomer Zhang Heng (78-139 AD) used geometrical formulas to solve mathematical problems. Zhang Heng ( (CE 78–139 was an astronomer, mathematician, inventor, geographer, cartographer, artist, poet Year 78 was a Common year starting on Thursday (link will display the full calendar of the Julian calendar. Although rough estimates for pi (π) were given in the Zhou Li (compiled in the 2nd century BC),^[9] it was Zhang Heng who was the first to make a concerted effort at creating a more accurate formula for pi. IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems IMPORTANT NOTICE Please note that Wikipedia is not a database to store the millions of digits of π please refrain from adding those to Wikipedia as it could cause technical problems The Rites of Zhou ( also known as Zhouguan (Offices of Zhou is one of three ancient ritual texts listed among the classics of Confucianism. This in turn would be made more accurate by later Chinese such as Zu Chongzhi (429-500 AD). Zu Chongzhi ( 429–500 Courtesy name Wenyuan (文遠 was a prominent Chinese mathematician and astronomer during the Liu Events By Place Western Roman Empire Vandals under Geiseric cross from the Iberian Peninsula Events By Place Europe Possible date for the Battle of Mons Badonicus: Romano-British and Celts defeat an Anglo-Saxon Zhang Heng approximated pi as 730/232 (or approx 3. 1466), although he used another formula of pi in finding a spherical volume, using the square root of 10 (or approx 3. 162) instead. Zu Chongzhi's best approximation was between 3. 1415926 and 3. 1415927, with ³⁵⁵⁄₁₁₃ (密率, Milü, detailed approximation) and ²²⁄₇ (约率, Yuelü, rough approximation) being the other notable approximation. \pi \approx 3141\ 592\ 653\dots\\frac{355}{113} \approx 3141\ 592\ 920\dots\The best rational Proofs of the famous mathematical result that the Rational number 22⁄7 is greater than π date back to antiquity ^[10] In comparison to later works, the formula for pi given by the French mathematician François Viète (1540-1603) fell halfway between Zu's approximations. François Viète (or Vieta) seigneur de la Bigotière ( 1540 - February 13, 1603) generally known as Franciscus Vieta,

The Nine Chapters on the Mathematical Art, the title of which first appeared by 179 AD on a bronze inscription, was edited and commented on by the 3rd century mathematician Liu Hui from the Kingdom of Cao Wei. The Nine Chapters on the Mathematical Art ( is a Chinese Mathematics book composed by several generations of scholars from the 10th&ndash2nd century BC and Liu Hui ( fl 3rd century) was a Chinese Mathematician who lived in the Wei Kingdom. Cao Wei ( was one of the empires that competed for control of China during the Three Kingdoms period This book included many problems where geometry was applied, such as finding surface areas for squares and circles, the volumes of solids in various three dimensional shapes, and included the use of the Pythagorean theorem. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry The book provided illustrated proof for the Pythagorean theorem,^[11] contained a written dialogue between of the earlier Duke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomical gnomon, the circle and square, as well as measurements of heights and distances. God of Dreams Duke of Zhou is also known as the 'God of Dreams' The gnomon is the part of a Sundial that casts the Shadow. Gnomon (γνώμων is an Ancient Greek word meaning "indicator" "one who ^[12] The editor Liu Hui listed pi as 3. 141014 by using a 192 sided polygon, and then calculated pi as 3. In Geometry a polygon (ˈpɒlɨɡɒn ˈpɒliɡɒn is traditionally a plane figure that is bounded by a closed path or circuit 14159 using a 3072 sided polygon. This was more accurate than Liu Hui's contemporary Wang Fan, a mathematician and astronomer from Eastern Wu, would render pi as 3. Eastern Wu ( Chinese: 東吳 Pinyin: Dōng Wú also known as Sun Wu ( Traditional Chinese: 孫吳 pinyin Sūn Wú refers to a 1555 by using ¹⁴²⁄₄₅. ^[13] Liu Hui also wrote of mathematical surveying to calculate distance measurements of depth, height, width, and surface area. Surveying is the technique and science of accurately determining the terrestrial or three-dimensional space Position of points and the distances and angles between In terms of solid geometry, he figured out that a wedge with rectangular base and both sides sloping could be broken down into a pyramid and a tetrahedral wedge. A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. ^[14] He also figured out that a wedge with trapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one ^[14] Furthermore, Liu Hui described Cavalieri's principle on volume, as well as Gaussian elimination. Bonaventura Francesco Cavalieri (in Latin, Cavalerius) ( 1598 - November 30, 1647) was an Italian mathematician In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix From the Nine Chapters, it listed the following geometrical formulas that were known by the time of the Former Han Dynasty (202 BCE–9 CE).

Areas for the^[15]

Volumes for the^[14]

Continuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematician Shen Kuo (1031-1095 AD), Yang Hui (1238-1298 AD) who discovered Pascal's Triangle, Xu Guangqi (1562-1633 AD), and many others. A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line This article is about mathematics For Rhomboid muscles in anatomy see Rhomboid major muscle and Rhomboid minor muscle. A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one Elements special cases and related concepts Each plane section is a base of the frustum General right and uniform prisms A right prism is a prism in which the joining edges and faces are perpendicular to the base faces A tetrahedron (plural tetrahedra) is a Polyhedron composed of four triangular faces three of which meet at each vertex. Shen Kuo or Shen Kua ( (1031&ndash1095 style name Cunzhong and pseudonym Mengqi Weng, was a Polymathic Chinese Yang Hui ( ca 1238–1298 Courtesy name Qianguang (谦光 was a Chinese Mathematician from Qiantang (modern Hangzhou \begin{matrix}&&&&&1\\&&&&1&&1\\&&&1&&2&&1\\&&1&&3&&3&&1\\&1&&4&&6&&4&&1\end{matrix Xu Guangqi ( 1562–1633 Courtesy name Zixian (子先 was a Chinese bureaucrat agricultural scientist astronomer and mathematician in the Ming Dynasty

Classical Greek geometry (c. 600 – 300 BC)

See also: Greek mathematics

For the ancient Greek mathematicians, geometry was the crown jewel of their sciences, reaching a completeness and perfection of methodology that no other branch of their knowledge had attained. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Greece (Ελλάδα transliterated: Elláda, historically, Ellás,) officially the Hellenic Republic (Ελληνική Δημοκρατία Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century They expanded the range of geometry to many new kinds of figures, curves, surfaces, and solids; they changed its methodology from trial-and-error to logical deduction; they recognized that geometry studies "eternal forms", or abstractions, of which physical objects are only approximations; and they developed the idea of an "axiomatic theory", which, for more than 2000 years, was regarded to be the ideal paradigm for all scientific theories. Plato 's Theory of Forms asserts that Forms (or Ideas) and not the material world of change known to us through sensation, possess In Mathematics, an axiomatic system is any set of Axioms from which some or all axioms can be used in conjunction to logically derive Theorems

Thales and Pythagoras

Pythagorean theorem: a² + b² = c²

Thales (635-543 BC) of Miletus (now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Thales of Miletus According to Bertrand Russell, "Philosophy begins with Thales Miletus (mī lē' təs ( Ancient Greek: Μίλητος literally Transliterated Milētos, Latin Miletus) was an Ancient There are five geometric propositions for which he wrote deductive proofs, though his proofs have not survived. Pythagoras (582-496 BC) of Ionia, and later, Italy, then colonized by Greeks, may have been a student of Thales, and traveled to Babylon and Egypt. "Pythagoras of Samos" redirects here For the Samian statuary of the same name see Pythagoras (sculptor. Babylon was a City-state of ancient Mesopotamia, the remains of which can be found in present-day Al Hillah, Babil Province, Iraq This article is about the country of Egypt For a topic outline on this subject see List of basic Egypt topics. The theorem that bears his name may not have been his discovery, but he was probably one of the first to give a deductive proof of it. He gathered a group of students around him to study mathematics, music, and philosophy, and together they discovered most of what high school students learn today in their geometry courses. In addition, they made the profound discovery of incommensurable lengths and irrational numbers. In Mathematics, two non- Zero Real numbers a and b are said to be commensurable Iff a / b In Mathematics, an irrational number is any Real number that is not a Rational number — that is it is a number which cannot be expressed as a fraction (There is no evidence that Thales provided any deductive proofs, and in fact, deductive mathematical proofs did not appear until after Parmemides. At best, all that we can say about Thales is that he introduced various geometric theorems to the Greeks. The idea that mathematics was from its inception deductive is false. At the time of Thales, mathematics was inductive. This means that Thales would have "provided" empirical and direct proofs, but not deductive proofs. )

Plato

Plato (427-347 BC), the philosopher most esteemed by the Greeks, had inscribed above the entrance to his famous school, "Let none ignorant of geometry enter here. Biography Early life Birth and family Plato was born in Athens Greece " Though he was not a mathematician himself, his views on mathematics had great influence. Mathematicians thus accepted his belief that geometry should use no tools but compass and straightedge – never measuring instruments such as a marked ruler or a protractor, because these were a workman’s tools, not worthy of a scholar. A ruler, or rule, is an instrument used in Geometry, Technical drawing and engineering/building to measure distances and/or to rule straight In Geometry, a protractor is a circular or semicircular tool for measuring an Angle or a Circle. This dictum led to a deep study of possible compass and straightedge constructions, and three classic construction problems: how to use these tools to trisect an angle, to construct a cube twice the volume of a given cube, and to construct a square equal in area to a given circle. Pentagon constructgif|thumb|right|Construction of a regular pentagon]] Compass-and-straightedge or ruler-and-compass construction is the construction of lengths or Angles The problem of trisecting the angle is a classic problem of Compass and straightedge constructions of ancient Greek mathematics. The proofs of the impossibility of these constructions, finally achieved in the 19th century, led to important principles regarding the deep structure of the real number system. Aristotle (384-322 BC), Plato’s greatest pupil, wrote a treatise on methods of reasoning used in deductive proofs (see Logic) which was not substantially improved upon until the 19th century. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Logic is the study of the principles of valid demonstration and Inference.

Hellenistic geometry (c. 300 BC - 500 AD)

See also: Greek mathematics

Euclid

Woman teaching geometry. Greek mathematics, as that term is used in this article is the Mathematics written in Greek, developed from the 6th century BC to the 5th century Illustration at the beginning of a medieval translation of Euclid's Elements, (c. In Mathematics, the elements or members of a set (or more generally a class) are all those objects which when collected together make up the 1310)

Euclid (c. Euclid ( Greek:.) fl 300 BC also known as Euclid of Alexandria, is often referred to as the Father of Geometry 325-265 BC), of Alexandria, probably a student of one of Plato’s students, wrote a treatise in 13 books (chapters), titled The Elements of Geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. Alexandria ( Egyptian Arabic: اسكندريه Eskendereyya; Standard Arabic: ar الإسكندرية Al-Iskandariyya; Ἀλεξάνδρεια Euclid's Elements ( Greek:) is a mathematical and geometric Treatise consisting of 13 books written by the Greek In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. The treatise is not a compendium of all that the Hellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. We know from other references that Euclid’s was not the first elementary geometry textbook, but it was so much superior that the others fell into disuse and were lost. He was brought to the university at Alexandria by Ptolemy I, King of Egypt. For the astronomer see Ptolemy; for others named "Ptolemy" or "Ptolemaeus" see Ptolemy (disambiguation.

The Elements began with definitions of terms, fundamental geometric principles (called axioms or postulates), and general quantitative principles (called common notions) from which all the rest of geometry could be logically deduced. Following are his five axioms, somewhat paraphrased to make the English easier to read.

1. Any two points can be joined by a straight line.
2. Any finite straight line can be extended in a straight line.
3. A circle can be drawn with any center and any radius.
4. All right angles are equal to each other.
5. If two straight lines in a plane are crossed by another straight line (called the transversal), and the interior angles between the two lines and the transversal lying on one side of the transversal add up to less than two right angles, then on that side of the transversal, the two lines extended will intersect (also called the parallel postulate). In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive

It was soon observed, and no doubt Euclid himself knew, that his fifth axiom could be replaced by the shorter statement “Given a line and a point not on the line, there is only one line through the given point and in the same plane with the given line that does not intersect the given line. ” This is called Playfair’s Axiom, after the British teacher who proposed to make the replacement in all the school textbooks.

The axioms, according to Plato, should be simple and self-evident principles, so clearly true that they need no proof. Euclid’s first four axioms meet this criterion, but the fifth, even if replaced by Playfair’s Axiom, is not simple, and most would say not self-evident like the first four. The fifth resembled more the theorems that Euclid proved from the axioms. Furthermore, Euclid developed a substantial part of his theory of triangles without using the Fifth Axiom. The speculation arose, probably during Euclid’s lifetime, that the Fifth Axiom can and should be proved as a theorem from the first four, and thus is unnecessary as an axiom. Thus began many centuries of attempts to prove the Fifth Axiom, and the question was not settled until the 19th century.

Archimedes

Archimedes (287-212 BC), of Syracuse, Sicily, when it was a Greek city-state, is often considered to be the greatest of the Greek mathematicians, and occasionally even named as one of the three greatest of all time (along with Isaac Newton and Carl Friedrich Gauss). Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer Syracuse (Siracusa Sicilian: Sarausa, Classical Greek: / transliterated Syrakousai) is a historic City in Sicily ( Italian and Sicilian: Sicilia) is an autonomous region of Italy. A polis ( πόλις, pronunciation, in English-- plural poleis ( πόλεις, pronunciation, in English --is a City, a Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Had he not been a mathematician, he would still be remembered as a great physicist, engineer, and inventor. In his mathematics, he developed methods very similar to the coordinate systems of analytic geometry, and the limiting process of integral calculus. The only element lacking for the creation of these fields was an efficient algebraic notation in which to express his concepts.

Archimedes had followed Eudoxian methods to write out geometric solutions. One solution to the area and volume of a parabola used unit fractions, a form of rigorous arithmetic notation that was created by Egyptians 1,700 years earlier. A unit fraction link between Archimedes' method of slicing the parabola into small pieces, creating the first form of calculus, as given by the proof (noted by E. J. Dijksterhuis)

4A/3 = A + A/3 + A/12

and, its 1/4th geometric infinite series form

4A/3 = A + A/4 + A/16 + A/64 + . Eduard Jan Dijksterhuis ( 28 October 1892, Tilburg - 18 May 1965, De Bilt) was a Dutch Historian of science. . . A/(4n) + . . .

The Moscow Mathematical Papyrus, dating to 2,000 BCE also sliced the area of a truncated pyramid, exactly finding its area, as Archimedes later applied by following the Eudoxian 1/4th geometric series, and proving his result by unit fraction arithmetic.

After Archimedes

Geometry was connected to the divine for most medieval scholars. In the Middle Ages, Science progressed dramatically from the time of antiquity in areas as diverse as Astronomy, Medicine, and Mathematics The compass in this 13th Century manuscript is a symbol of God's act of Creation. A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as

After Archimedes, Hellenistic mathematics began to decline. There were a few minor stars yet to come, but the golden age of geometry was over. Proclus (410-485), author of Commentary on the First Book of Euclid, was one of the last important players in Hellenistic geometry. Proclus Lycaeus ( February 8, c 411 &ndash April 17, 485) called "The Successor" or "Diadochos" ( Greek Próklos He was a competent geometer, but more importantly, he was a superb commentator on the works that preceded him. Much of that work did not survive to modern times, and is known to us only through his commentary. The Roman Republic and Empire that succeeded and absorbed the Greek city-states produced excellent engineers, but no mathematicians of note.

The great Library of Alexandria was later burned. The Royal Library of Alexandria or Ancient Library of Alexandria in Alexandria, Egypt, was once the largest library in the ancient world There is a growing consensus among historians that the Library of Alexandria likely suffered from several destructive events, but that the destruction of Alexandria's pagan temples in the late 4th century was probably the most severe and final one. The evidence for that destruction is the most definitive and secure. Caesar's invasion may well have led to the loss of some 40,000-70,000 scrolls in a warehouse adjacent to the port (as Luciano Canfora argues, they were likely copies produced by the Library intended for export), but it is unlikely to have affected the Library or Museum, given that there is ample evidence that both existed later.

Civil wars, decreasing investments in maintenance and acquisition of new scrolls and generally declining interest in non-religious pursuits likely contributed to a reduction in the body of material available in the Library, especially in the fourth century. The Serapeum was certainly destroyed by Theophilus in 391, and the Museum and Library may have fallen victim to the same campaign.

Islamic geometry (c. 700 - 1500)

See also: Islamic mathematics

Page from the Al-Jabr wa-al-Muqabilah

The Islamic Caliphate (Islamic Empire) established across the Middle East, North Africa, Spain, Portugal, Persia and parts of Persia, began around 640 CE. Al-Kitāb al-mukhtaṣar fī hīsāb al-ğabr wa’l-muqābala ( Arabic for "The Compendious Book on Calculation by Completion and Balancing" For other meanings including people named 'Islam' see Islam (disambiguation. The Caliph is the Head of state in a Caliphate, and the title for the leader of the Islamic Ummah, an Islamic community ruled by the Shari'ah The Middle East is a Subcontinent with no clear boundaries often used as a synonym to Near East, in opposition to Far East. North Africa or Northern Africa is the Northernmost Region of the African Continent, separated by the Sahara from Sub-Saharan Spain () or the Kingdom of Spain (Reino de España is a country located mostly in southwestern Europe on the Iberian Peninsula. Portugal, officially the Portuguese Republic (República Portuguesa is a country on the Iberian Peninsula. The Persian Empire was a series of Iranian empires that ruled over the Iranian plateau, the original Persian homeland and beyond in Western Asia The Persian Empire was a series of Iranian empires that ruled over the Iranian plateau, the original Persian homeland and beyond in Western Asia Events By Place Europe Tulga succeeds his father Suinthila as king of the Visigoths. Islamic mathematics during this period was primarily algebraic rather than geometric, though there were important works on geometry. Scholarship in Europe declined and eventually the Hellenistic works of antiquity were lost to them, and survived only in the Islamic centers of learning. This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. Classical antiquity (also the classical era or classical period) is a broad term for a long period of cultural History centered on the Mediterranean

Although the Muslim mathematicians are most famed for their work on algebra, number theory and number systems, they also made considerable contributions to geometry, trigonometry and mathematical astronomy, and were responsible for the development of algebraic geometry. Algebra is a branch of Mathematics concerning the study of structure, relation, and Quantity. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes In Mathematics, a number system is a set of Numbers (in the broadest sense of the word together with one or more operations such as Addition Circle-trig6svg|300px|thumb|right|All of the Trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Astronomy (from the Greek words astron (ἄστρον "star" and nomos (νόμος "law" is the scientific study Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Geometrical magnitudes were treated as "algebraic objects" by most Muslim mathematicians however.

The successors of Muḥammad ibn Mūsā al-Ḵwārizmī who was Persian Scholar, mathematician and Astronomer who invented the Algorithm in Mathematics which is the base for Computer Science (born 780) undertook a systematic application of arithmetic to algebra, algebra to arithmetic, both to trigonometry, algebra to the Euclidean theory of numbers, algebra to geometry, and geometry to algebra. In Mathematics, Computing, Linguistics and related subjects an algorithm is a sequence of finite instructions often used for Calculation Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Events By Place Byzantine Empire Constantine VI becomes Byzantine Emperor with Irene as guardian This was how the creation of polynomial algebra, combinatorial analysis, numerical analysis, the numerical solution of equations, the new elementary theory of numbers, and the geometric construction of equations arose.

Al-Mahani (born 820) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Abu-Abdullah Muhammad ibn Īsa Māhānī, was a Persian mathematician and astronomer from Mahan, Kerman, Persia. Events By Place Asia Tahir, the son of a slave is rewarded with the governorship of Khurasan for supporting the Caliphate Al-Karaji (born 953) completely freed algebra from geometrical operations and replaced them with the arithmetical type of operations which are at the core of algebra today. (or) (c 953 in Karaj or Karkh &ndash c 1029 was a 10th century Persian Muslim mathematician and engineer. Events Europe First documented mention of the village of Aach in Rhineland-Palatinate, Germany. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone

Thabit ibn Qurra

Although Thabit ibn Qurra (known as Thebit in Latin) (born 836) contributed to a number of areas in mathematics, where he played an important role in preparing the way for such important mathematical discoveries as the extension of the concept of number to (positive) real numbers, integral calculus, theorems in spherical trigonometry, analytic geometry, and non-Euclidean geometry. (836 in Harran, Mesopotamia &ndash February 18, 901 in Baghdad) was an Arab astronomer, mathematician Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. Events By Place Asia Abbasid caliph Al-Mutasim establishes a new capital at Samarra, Iraq. In Mathematics, the real numbers may be described informally in several different ways The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Spherical trigonometry is a part of Spherical geometry that deals with Polygons (especially Triangles on the Sphere and explains how to find relations Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry In astronomy Thabit was one of the first reformers of the Ptolemaic system, and in mechanics he was a founder of statics. In Astronomy, the geocentric model of the Universe is the superseded theory that the Earth is the center of the universe and other Statics is the branch of Mechanics concerned with the analysis of loads ( Force, torque/moment) on Physical systems in Static equilibrium

An important geometrical aspect of Thabit's work was his book on the composition of ratios. In this book, Thabit deals with arithmetical operations applied to ratios of geometrical quantities. The Greeks had dealt with geometric quantities but had not thought of them in the same way as numbers to which the usual rules of arithmetic could be applied. By introducing arithmetical operations on quantities previously regarded as geometric and non-numerical, Thabit started a trend which led eventually to the generalisation of the number concept.

In some respects, Thabit is critical of the ideas of Plato and Aristotle, particularly regarding motion. It would seem that here his ideas are based on an acceptance of using arguments concerning motion in his geometrical arguments.

Another important contribution Thabit made to geometry was his generalization of the Pythagorean theorem, which he extended from special right triangles to all triangles in general, along with a general proof. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position In Mathematics, the Pythagorean theorem ( American English) or Pythagoras' theorem ( British English) is a relation in Euclidean geometry Two types of special right triangles appear commonly in geometry the "angle based" and the "side based" (or Pythagorean Triangles The former are characterised A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line In Mathematics, a proof is a convincing demonstration (within the accepted standards of the field that some Mathematical statement is necessarily true ^[16]

Omar Khayyám

Omar Khayyám (born 1048) was a Persian mathematician,Astronomer and Philosopher who described his Philosophy by poets known as Robaeiyat Omar Khayyam's Quatrains. For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین layout and formatting it should ensure no clashes with the top of the infobox Along with his fame as a poet, he was also famous during his lifetime as a mathematician, well known for inventing the general method of solving cubic equations by intersecting a parabola with a circle. This article discusses cubic equations in one variable For a discussion of cubic equations in two variables see Elliptic curve. In addition he discovered the binomial expansion, and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development of non-Euclidean geometry. In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry Omar Khayyam also combined the use of trigonometry and approximation theory to provide methods of solving algebraic equations by geometrical means. In Mathematics, approximation theory is concerned with how functions can best be approximated with simpler functions, and with Quantitatively He was mostly responsible for the development of algebraic geometry.

In a paper written by Khayyam before his famous algebra text Treatise on Demonstration of Problems of Algebra, he considers the problem: Find a point on a quadrant of a circle in such manner that when a normal is dropped from the point to one of the bounding radii, the ratio of the normal's length to that of the radius equals the ratio of the segments determined by the foot of the normal. Khayyam shows that this problem is equivalent to solving a second problem: Find a right triangle having the property that the hypotenuse equals the sum of one leg plus the altitude on the hypotenuse. This problem in turn led Khayyam to solve the cubic equation x³ + 200x = 20x² + 2000 and he found a positive root of this cubic by considering the intersection of a rectangular hyperbola and a circle. An approximate numerical solution was then found by interpolation in trigonometric tables. Perhaps even more remarkable is the fact that Khayyam states that the solution of this cubic requires the use of conic sections and that it cannot be solved by compass and straightedge, a result which would not be proved for another 750 years.

An engraving by Albrecht Dürer featuring Mashallah, from the title page of the De scientia motus orbis (Latin version with engraving, 1504). Albrecht Dürer (ˈalbʀɛçt ˈdyʀɐ ( May 21, 1471 &ndash April 6, 1528) was a German painter, Printmaker Masha'allah ibn Atharī (c740-d815 AD was an eighth century Persian Jewish astrologer and astronomer from the city of Basra (now located in As in many medieval illustrations, the compass here is an icon of religion as well as science, in reference to God as the architect of creation

His Treatise on Demonstration of Problems of Algebra contained a complete classification of cubic equations with geometric solutions found by means of intersecting conic sections. A compass or pair of compasses is a Technical drawing instrument that can be used for inscribing Circles or arcs They can also be used as In fact Khayyam gives an interesting historical account in which he claims that the Greeks had left nothing on the theory of cubic equations. Indeed, as Khayyam writes, the contributions by earlier writers such as al-Mahani and al-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work of Muḥammad ibn Mūsā al-Ḵwārizmī). Abu Jafar Muhammad ibn al-Hasan Al-Khazini (900-971 was a Persian astronomer and mathematician from Khorasan. However, Khayyam himself seems to have been the first to conceive a general theory of cubic equations.

In Commentaries on the difficult postulates of Euclid's book Khayyam made a contribution to non-Euclidean geometry, although this was not his intention. In trying to prove the parallel postulate he accidentally proved properties of figures in non-Euclidean geometries. Khayyam also gave important results on ratios in this book, extending Euclid's work to include the multiplication of ratios. The importance of Khayyam's contribution is that he examined both Euclid's definition of equality of ratios (which was that first proposed by Eudoxus) and the definition of equality of ratios as proposed by earlier Islamic mathematicians such as al-Mahani which was based on continued fractions. In Mathematics, a continued fraction is an expression such as x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{\ddots\}}}} Khayyam proved that the two definitions are equivalent. He also posed the question of whether a ratio can be regarded as a number but leaves the question unanswered.

Other geometers

Ibrahim ibn Sinan (born 908), who introduced a method of integration more general than that of Archimedes, and al-Quhi (born 940) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. Ibrahim ibn Sinan ibn Thabit ibn Qurra (908 Baghdad – 946 Baghdad was an Arab Mathematician and Astronomer who studied Geometry Events By Place Asia The Battle of Belach Mugna is fought Zhu Wen kills the last Tang Dynasty emperor The European Space Agency 's INTErnational Gamma-Ray Astrophysics Laboratory ( INTEGRAL) is detecting some of the most energetic radiation that comes from space Archimedes of Syracuse ( Greek:) ( c. 287 BC – c 212 BC was a Greek mathematician, Physicist, Engineer (sometimes) was a Persian mathematician, physicist and astronomer. Events By Place Asia Saadia Gaon compiles his Siddur (Jewish prayer book in Iraq. These mathematicians, and in particular Ibn al-Haytham, studied optics and investigated the optical properties of mirrors made from conic sections. TemplateInfobox Muslim scholars --> ( Arabic: ابو علی، حسن بن حسن بن هيثم Latinized In Mathematics, a conic section (or just conic) is a Curve obtained by intersecting a cone (more precisely a circular Conical surface

Astronomy, time-keeping and geography provided other motivations for geometrical and trigonometrical research. Geography (from Greek γεωγραφία - geografia) is the study of the Earth and its lands features inhabitants and phenomena For example Ibrahim ibn Sinan and his grandfather Thabit ibn Qurra both studied curves required in the construction of sundials. (836 in Harran, Mesopotamia &ndash February 18, 901 in Baghdad) was an Arab astronomer, mathematician Abu'l-Wafa and Abu Nasr Mansur both applied spherical geometry to astronomy. Abu Nasr Mansur ibn Ali ibn Iraq (c 960 - 1036 was a was a Persian Muslim mathematician. Spherical geometry is the Geometry of the two- Dimensional surface of a Sphere.

Persian mathematician Sharafeddin Tusi (born 1135) did not follow the general development that came through al-Karaji's school of algebra but rather followed Khayyam's application of algebra to geometry. (1135 - 1213 was a Persian mathematician and astronomer of the Islamic Golden Age (during the Middle Ages) (or) (c 953 in Karaj or Karkh &ndash c 1029 was a 10th century Persian Muslim mathematician and engineer. He 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 study of algebraic geometry.

In 1250, Nasīr al-Dīn al-Tūsī, in his Al-risala al-shafiya'an al-shakk fi'l-khutut al-mutawaziya (Discussion Which Removes Doubt about Parallel Lines), wrote detailed critiques of the Euclidean parallel postulate and on Omar Khayyám's attempted proof a century earlier. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive For the Thoroughbred racehorse see Omar Khayyam (horse Ghiyās od-Dīn Abol-Fath Omār ibn Ebrāhīm Khayyām Neyshābūri (غیاث الدین Nasir al-Din attempted to derive a contradiction of the parallel postulate. His son, Sadr al-Din wrote a book on the subject in 1298, based on Nasir al-Din's later thoughts, which presented an argument for a hypothesis equivalent to the parallel postulate. Sadr al-Din's work was published in Rome in 1594 and was studied by European geometers. Rome ( Roma ˈroma Roma is the capital city of Italy and Lazio, and is Italy's largest and most populous city with more than 2 This work marked the starting point for Giovanni Girolamo Saccheri's work on the subject, and eventually the discovery of non-Euclidean geometry. Giovanni Girolamo Saccheri ( September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry ^[17]

The 17th century

When Europe began to emerge from its Dark Ages, the Hellenistic and Islamic texts on geometry found in Islamic libraries were translated from Arabic into Latin. This article is about the phrase "Dark Age(s" as a characterization of the Early Middle Ages in Western Europe This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. For other meanings including people named 'Islam' see Islam (disambiguation. Arabic (ar الْعَرَبيّة (informally ar عَرَبيْ) in terms of the number of speakers is the largest living member of the Semitic language Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. The rigorous deductive methods of geometry found in Euclid’s Elements of Geometry were relearned, and further development of geometry in the styles of both Euclid (Euclidean geometry) and Khayyam (algebraic geometry) continued, resulting in an abundance of new theorems and concepts, many of them very profound and elegant. Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with

Discourse on Method by René Descartes

In the early 17th century, there were two important developments in geometry. Organization How to think correctly The Method of Science Morals Maxims deduced from this Method Proof of God and the Soul Physics the heart The first and most important was the creation of analytic geometry, or geometry with coordinates and equations, by René Descartes (1596-1650) and Pierre de Fermat (1601-1665). Analytic geometry, also called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of Geometry In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point An equation is a mathematical statement, in symbols, that two things are exactly the same (or equivalent Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the This was a necessary precursor to the development of calculus and a precise quantitative science of physics. Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591-1661). Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Girard Desargues ( February 21 or March 2, 1591 -October 1661 was a French Mathematician and engineer who is considered one of Projective geometry is the study of geometry without measurement, just the study of how points align with each other. There had been some early work in this area by Hellenistic geometers, notably Pappus (c. This article focuses on the cultural aspects of the Hellenistic age for the historical aspects see Hellenistic period. 340). The greatest flowering of the field occurred with Jean-Victor Poncelet (1788-1867). Jean-Victor Poncelet ( July 1, 1788 &ndash December 22, 1867) was a French Engineer and Mathematician who served

In the late 17th century, calculus was developed independently and almost simultaneously by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716). Calculus ( Latin, calculus, a small stone used for counting is a branch of Mathematics that includes the study of limits, Derivatives Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements This was the beginning of a new field of mathematics now called analysis. Analysis has its beginnings in the rigorous formulation of Calculus. Though not itself a branch of geometry, it is applicable to geometry, and it solved two families of problems that had long been almost intractable: finding tangent lines to odd curves, and finding areas enclosed by those curves. The methods of calculus reduced these problems mostly to straightforward matters of computation.

The 18th and 19th centuries

Non-Euclidean geometry

The old problem of proving Euclid’s Fifth Postulate, the "Parallel Postulate", from his first four postulates had never been forgotten. In Geometry, the parallel postulate, also called Euclid 's fifth postulate since it is the fifth postulate in Euclid's ''Elements'', is a distinctive Beginning not long after Euclid, many attempted demonstrations were given, but all were later found to be faulty, through allowing into the reasoning some principle which itself had not been proved from the first four postulates. Though Omar Khayyám was also unsuccessful in proving the parallel postulate, his criticisms of Euclid's theories of parallels and his proof of properties of figures in non-Euclidean geometries contributed to the eventual development of non-Euclidean geometry. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry By 1700 a great deal had been discovered about what can be proved from the first four, and what the pitfalls were in attempting to prove the fifth. Saccheri, Lambert, and Legendre each did excellent work on the problem in the 18th century, but still fell short of success. Giovanni Girolamo Saccheri ( September 5, 1667 - October 25, 1733) was an Italian Jesuit priest and mathematician Johann Heinrich Lambert ( August 26, 1728 &ndash September 25 1777) was a Swiss Mathematician, Physicist and Adrien-Marie Legendre ( September 18 1752 – January 10 1833) was a French Mathematician. In the early 19th century, Gauss, Johann Bolyai, and Lobatchewsky, each independently, took a different approach. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German János Bolyai ( December 15, 1802 – January 27, 1860) was a Hungarian Mathematician, known for his work in Non-Euclidean Nikolai Ivanovich Lobachevsky (Никола́й Ива́нович Лобаче́вский ( December 1 1792 &ndash February 24 1856 ( N Beginning to suspect that it was impossible to prove the Parallel Postulate, they set out to develop a self-consistent geometry in which that postulate was false. In this they were successful, thus creating the first non-Euclidean geometry. In mathematics non-Euclidean geometry describes how this all works--> hyperbolic and Elliptic geometry, which are contrasted with Euclidean geometry By 1854, Bernhard Riemann, a student of Gauss, had applied methods of calculus in a ground-breaking study of the intrinsic (self-contained) geometry of all smooth surfaces, and thereby found a different non-Euclidean geometry. This work of Riemann later became fundamental for Einstein's theory of relativity. Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism.

William Blake's "Newton" is a demonstration of his opposition to the 'single-vision' of scientific materialism; here, Isaac Newton is shown as 'divine geometer' (1795)

It remained to be proved mathematically that the non-Euclidean geometry was just as self-consistent as Euclidean geometry, and this was first accomplished by Beltrami in 1868. William Blake (28 November 1757 – 12 August 1827 was an English poet, painter, and Printmaker. Philosophical naturalism has been described in various ways In its broadest and strongest sense naturalism is the metaphysical position that "nature is all there is Sir Isaac Newton, FRS (ˈnjuːtən 4 January 1643 31 March 1727) Biography Early years See also Isaac Newton's early life and achievements Eugenio Beltrami ( 16 November, 1835 - 4 June, 1899) was an Italian mathematician notable for his work on Non-Euclidean geometry With this, non-Euclidean geometry was established on an equal mathematical footing with Euclidean geometry.

While it was now known that different geometric theories were mathematically possible, the question remained, "Which one of these theories is correct for our physical space?" The mathematical work revealed that this question must be answered by physical experimentation, not mathematical reasoning, and uncovered the reason why the experimentation must involve immense (interstellar, not earth-bound) distances. With the development of relativity theory in physics, this question became vastly more complicated.

Introduction of mathematical rigor

All the work related to the Parallel Postulate revealed that it was quite difficult for a geometer to separate his logical reasoning from his intuitive understanding of physical space, and, moreover, revealed the critical importance of doing so. Careful examination had uncovered some logical inadequacies in Euclid's reasoning, and some unstated geometric principles to which Euclid sometimes appealed. This critique paralleled the crisis occurring in calculus and analysis regarding the meaning of infinite processes such as convergence and continuity. In geometry, there was a clear need for a new set of axioms, which would be complete, and which in no way relied on pictures we draw or on our intuition of space. Such axioms were given by David Hilbert in 1894 in his dissertation Grundlagen der Geometrie (Foundations of Geometry). David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most Some other complete sets of axioms had been given a few years earlier, but did not match Hilbert's in economy, elegance, and similarity to Euclid's axioms.

Analysis situs, or topology

In the mid-18th century, it became apparent that certain progressions of mathematical reasoning recurred when similar ideas were studied on the number line, in two dimensions, and in three dimensions. Thus the general concept of a metric space was created so that the reasoning could be done in more generality, and then applied to special cases. This method of studying calculus- and analysis-related concepts came to be known as analysis situs, and later as topology. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of The important topics in this field were properties of more general figures, such as connectedness and boundaries, rather than properties like straightness, and precise equality of length and angle measurements, which had been the focus of Euclidean and non-Euclidean geometry. Topology soon became a separate field of major importance, rather than a sub-field of geometry or analysis.

The 20th century

Developments in algebraic geometry included the study of curves and surfaces over finite fields, rather than the real or complex numbers. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements Finite geometry itself, the study of spaces with only finitely many points, found applications in coding theory and cryptography. A finite geometry is any geometric system that has only a finite number of points. Coding theory is one of the most important and direct applications of Information theory. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" With the advent of the computer, new disciplines such as computational geometry or digital geometry deal with geometric algorithms, discrete representations of geometric data, and so forth. Computational geometry is a branch of Computer science devoted to the study of algorithms which can be stated in terms of Geometry. Digital geometry deals with discrete sets (usually discrete point sets considered to be digitized models or Images of objects of the

See also

• List of geometry topics
• Important publications in geometry. This is list of Geometry topics, by Wikipedia page Geometric shape covers standard terms for plane shapes List of mathematical shapes Algebra Theory of equations Hisab
• Interactive geometry software
• History of mathematics
• Flatland, Book written by " A² " about two and three-dimensional space, to understand the concept of four dimensions

Notes

References

• Needham, Joseph (1986). Science and Civilization in China: Volume 3, Mathematics and the Sciences of the Heavens and the Earth. Taipei: Caves Books Ltd.

External links

• Geometry Step by Step from the Land of the Incas by Antonio Gutierrez.
• Islamic Geometry
• Stanford Encyclopedia of Philosophy:
  • Finitism in Geometry
  • Geometry in the 19th Century
• Arabic mathematics : forgotten brilliance?