# History of geometry

(Greek "γεωμετρία"; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modernGeometry mathematics , the other being the study of numbers.Classic geometry was focused in

compass and straightedge constructions. 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 andalgebraic geometry .)**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 (seeBabylonian mathematics ) from around3000 BC . Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need insurveying ,construction ,astronomy , and various crafts. Among these were some surprisingly sophisticated principles, and a modern mathematician might be hard put to derive some of them without the use ofcalculus . For example, both theEgyptians and theBabylon ians were aware of versions of thePythagorean theorem about 1500 years beforePythagoras ; the Egyptians had a correct formula for the volume of afrustum of a square pyramid; the Babylonians had atrigonometry table .cc**Egyptian geometry**The ancient Egyptians knew that they could approximate the area of a circle as follows:Ray C. Jurgensen, Alfred J. Donnelly, and Mary P. Dolciani. Editorial Advisors Andrew M. Gleason, Albert E. Meder, Jr. "Modern School Mathematics: Geometry" (Student's Edition). Houghlin Mifflin Company, Boston, 1972, p. 52. ISBN 0-395-13102-2. Teachers Edition ISBN 0-395-13103-0.]

::::Area of Circle ≈ [ (Diameter) x 8/9 ]

^{2}.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. This assumes that π is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed untilArchimedes ' approximation of 211875/67441 = 3.14163, 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 3.111...

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 afrustum of a pyramid, describing the correct formula::$V\; =\; frac\{1\}\{3\}\; h(x\_1^2\; +\; x\_1\; x\_2\; +x\_2^2).$**Babylonian geometry**The Babylonians may have known the general rules for measuring areas and volumes. 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. 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. ThePythagorean theorem was also known to the Babylonians. 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. [*Eves, Chapter 2.*]**Ancient Indian geometry (c. 3000 - 500 BC)****Harappan geometry**The geometry used in the

Indus Valley Civilization ofNorth India andPakistan from around 3000 BC was just as advanced as its contemporaries in Egypt and Mesopotamia, and mostly developed as a result of advancedurban planning , which is evident from the perfect grid pattern ofHarappa andMohenjo-daro where streets were laid out in perfect right angles. The geometry used by this earlyHarappan civilization was for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced "brick technology", which utilisedratio s. 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. 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 .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. 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 1.704 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 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. 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 religiousaltar s.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.

*Pythagorean triple s discovered algebraically.

*Statements of thePythagorean theorem and a numerical proof.

*Computations ofπ , with the closest being correct to 2 decimal places.Lagadha (circa 1350-1200 BC ) was probably the earliest known mathematician to have used geometry andtrigonometry forastronomy .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.The "

Sulba Sutras " ("Rule of Chords" inVedic Sanskrit ), which is another name for geometry, were composed between800 BC and500 BC and were appendices to theVedas giving rules for the construction of religious altars. The "Sulba Sutras" contain the first use ofirrational number s,quadratic equation s of the form a x^{2}= c and ax^{2}+ bx = c, the use of thePythagorean theorem and a list ofPythagorean triples discovered algebraically "predating" Pythagoras, geometric solutions oflinear equation s, and a number of geometrical proofs. These discoveries are mostly a result of altar construction, which also led to the first known calculations for thesquare root of 2, which were correct to a remarkable 5 decimal places.Baudhayana (circa800 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 ofπ (the closest value being 3.114), along with the first use of irrational numbers and quadratic equations of the forms ax^{2}= c and ax^{2}+ bx = c, and a computation for the square root of 2, which was correct to a remarkable five decimal places.Manava (circa750 BC ) composed the "Manava Sulba Sutra", which contains approximate constructions of circles from rectangles, and squares from circles, which give approximate values ofπ , with the closest value being 3.125.Apastamba (circa600 BC ) composed the "Apastamba Sulba Sutra", which contains the method ofsquaring 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 generallinear equation , and also contains a numerical proof of thePythagorean theorem , using an area computation. The historian Albert Burk claims this was the original proof of the theorem whichPythagoras copied on his visit to India.Fact|date=January 2008However, the legacy of geometry in India continued long after the Vedic period, with figures such as

Aryabhata (476 -550 AD).**Ancient Chinese Geometry (c. 500 BC - 500 AD)**[

Nine Chapters on the Mathematical Art ", first compiled in 179 AD, with added commentary in the 3rd century byLiu Hui .] In ancientChina , the earliest simple mathematical work stemmed back to the court records ofdivination for theShang Dynasty (c.1600 BC -1050 BC ), while the famous philosophical and cosmological work of the "I Ching " during theZhou Dynasty (1050 BC -256 BC ) had a complex arrangement of mathematicalhexagram s. However, the first definitive work (or at least oldest existent) on geometry in China was the "Mo Jing", theMohist canon of the earlyutilitarian philosopherMozi (470 BC -390 BC ). It was compiled years after his death by his later followers around the year 330 BC. 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 theQin Dynasty rulerQin Shihuang (r.221 BC -210 BC ), multitudes of written literature created before his time was purged. 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.Needham, Volume 3, 91.] Much like

Euclid 's first and third definitions andPlato '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. (As to its invisibility) there is nothing similar to it."Needham, Volume 3, 92.] Similar to theatomist s ofDemocritus , the "Mo Jing" stated that a point is the smallest unit, and cannot be cut in half, since 'nothing' cannot be halved. It stated that two lines of equal length will always finish at the same place, while providing definitions for the "comparison of lengths" and for "parallels",Needham, Volume 3, 92-93.] along with principles of space and bounded space.Needham, Volume 3, 93.] It also described the fact that planes without the quality of thickness cannot be piled up since they cannot mutually touch.Needham, Volume 3, 93-94.] The book provided definitions for circumference, diameter, and radius, along with the definition of volume.Needham, Volume 3, 94.]The

Han Dynasty (202 BC -220 AD ) period of China witnessed a new flourishing of mathematics. One of the oldest Chinese mathematical texts to presentgeometric progression s was the "Suàn shù shū " of 186 BC, during the Western Han era. The mathematician, inventor, and astronomerZhang Heng (78 -139 AD) used geometrical formulas to solve mathematical problems. Although rough estimates forpi (π ) were given in the "Zhou Li " (compiled in the 2nd century BC),Needham, Volume 3, 99.] it was Zhang Heng who was the first to make a concerted effort at creating a more accurate formula for pi. This in turn would be made more accurate by later Chinese such asZu Chongzhi (429 -500 AD). 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.Needham, Volume 3, 101.] In comparison to later works, the formula for pi given by the French mathematicianFrançois Viète (1540 -1603 ) fell halfway between Zu's approximations.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 mathematicianLiu Hui from the Kingdom ofCao Wei . 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 thePythagorean theorem . The book provided illustrated proof for the Pythagorean theorem,Needham, Volume 3, 22.] contained a written dialogue between of the earlierDuke of Zhou and Shang Gao on the properties of the right angle triangle and the Pythagorean theorem, while also referring to the astronomicalgnomon , the circle and square, as well as measurements of heights and distances.Needham, Volume 3, 21.] The editor Liu Hui listed pi as 3.141014 by using a 192 sidedpolygon , and then calculated pi as 3.14159 using a 3072 sided polygon. This was more accurate than Liu Hui's contemporaryWang Fan , a mathematician and astronomer fromEastern Wu , would render pi as 3.1555 by using^{142}⁄_{45}.Needham, Volume 3, 100.] Liu Hui also wrote of mathematicalsurveying to calculate distance measurements of depth, height, width, and surface area. 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 atetrahedral wedge.Needham, Volume 3, 98–99.] He also figured out that a wedge withtrapezoid base and both sides sloping could be made to give two tetrahedral wedges separated by a pyramid. Furthermore, Liu Hui describedCavalieri's principle on volume, as well asGaussian elimination . 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**Needham, Volume 3, 98.]

*Square

*Rectangle

*Circle

*Isosceles triangle

*Trapezium

*Rhomboid

*Trapezoid

*Double trapezium

*Segment of a circle

*Annulus (annular space between two circles)**Volumes for the**Needham, Volume 3, 98-99.]

*Parallel-piped with two square surfaces

*Parallel-piped with no square surfaces

*Pyramid

*Frustum of pyramid with square base

*Frustum of pyramid with rectangular base of unequal sides

*Cube

*Prism

*Wedge with rectangular base and both sides sloping

*Wedge with trapezoid base and both sides sloping

*Tetrahedral wedge

*Frustum of a wedge of the second type (used for applications in engineering)

*Cylinder

*Cone with circular base

*Frustum of a cone

*Sphere Continuing the geometrical legacy of ancient China, there were many later figures to come, including the famed astronomer and mathematicianShen Kuo (1031 -1095 AD),Yang Hui (1238 -1298 AD) who discoveredPascal's Triangle ,Xu Guangqi (1562 -1633 AD), and many others.**Classical Greek geometry (c. 600 – 300 BC)**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. 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.

**Thales and Pythagoras**Thales (635-543 BC) ofMiletus (now in southwestern Turkey), was the first to whom deduction in mathematics is attributed. 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 toBabylon andEgypt . 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 andirrational number s. (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." 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 markedruler or aprotractor , because these were a workman’s tools, not worthy of a scholar. This dictum led to a deep study of possiblecompass and straightedge constructions, and three classic construction problems: how to use these tools totrisect 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. 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 (seeLogic ) which was not substantially improved upon until the 19th century.**Hellenistic geometry (c. 300 BC - 500 AD)****Euclid**Euclid (c. 325-265 BC), ofAlexandria , 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 idealaxiom atic form, which came to be known asEuclidean geometry . The treatise is not a compendium of all that theHellenistic mathematicians knew at the time about geometry; Euclid himself wrote eight more advanced books on geometry. 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."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.

# Any two points can be joined by a straight line.

# Any finite straight line can be extended in a straight line.

# A circle can be drawn with any center and any radius.

# All right angles are equal to each other.

# 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 theparallel postulate ).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 aGreek 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 withIsaac Newton andCarl Friedrich Gauss ). 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 + ... 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**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. 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. 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)**The

Islam icCaliph ate (Islamic Empire ) established across theMiddle East ,North Africa ,Spain ,Portugal ,Persia and parts ofPersia , began around 640 CE.Islamic mathematics during this period was primarily algebraic rather than geometric, though there were important works on geometry. Scholarship in Europe declined and eventually theHellenistic works of antiquity were lost to them, and survived only in the Islamic centers of learning.Although the Muslim mathematicians are most famed for their work on

algebra ,number theory andnumber system s, they also made considerable contributions to geometry,trigonometry and mathematicalastronomy , and were responsible for the development ofalgebraic geometry . 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 theAlgorithm inMathematics which is the base forComputer Science (born780 ) 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. 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 (born820 ) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.Al-Karaji (born953 ) completely freed algebra from geometrical operations and replaced them with thearithmetic al type of operations which are at the core of algebra today.**Thabit family and other early geometers**Although Thabit ibn Qurra (known as Thebit in

Latin ) (born836 ) 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 number s,integral calculus , theorems inspherical trigonometry ,analytic geometry , andnon-Euclidean geometry . In astronomy Thabit was one of the first reformers of thePtolemaic system , and in mechanics he was a founder ofstatics . 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 thePythagorean theorem , which he extended fromspecial right triangles to alltriangle s in general, along with a general proof. [*Aydin Sayili (1960). "Thabit ibn Qurra's Generalization of the Pythagorean Theorem", "Isis"*]**51**(1), p. 35-37.Ibrahim ibn Sinan ibn Thabit (born908 ), who introduced a method of integration more general than that ofArchimedes , andal-Quhi (born940 ) were leading figures in a revival and continuation of Greek higher geometry in the Islamic world. These mathematicians, and in particularIbn al-Haytham , studiedoptics and investigated the optical properties of mirrors made fromconic section s.Astronomy, time-keeping and

geography provided other motivations for geometrical and trigonometrical research. For example Ibrahim ibn Sinan and his grandfatherThabit ibn Qurra both studied curves required in the construction of sundials.Abu'l-Wafa andAbu Nasr Mansur both appliedspherical geometry to astronomy.**Omar Khayyám and Sharafeddin Tusi**Omar Khayyám (born1048 ) was a Persian mathematician, astronomer, philosopher and poet who described his philosophy through poems known asquatrain s in the "Rubaiyat of Omar Khayyam ". 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 solvingcubic equation s by intersecting a parabola with a circle. In addition he discovered thebinomial expansion , and authored criticisms of Euclid's theories of parallels which made their way to England, where they contributed to the eventual development ofnon-Euclidean geometry . Omar Khayyam also combined the use of trigonometry andapproximation theory to provide methods of solving algebraic equations by geometrical means. 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 x

^{3}+ 200x = 20x^{2}+ 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.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. 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 andal-Khazin were to translate geometric problems into algebraic equations (something which was essentially impossible before the work ofMuḥammad ibn Mūsā al-Ḵwārizmī ). 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 oncontinued fraction s. 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.Persian mathematician

Sharafeddin Tusi (born1135 ) did not follow the general development that came throughal-Karaji 's school of algebra but rather followed Khayyam's application of algebra to geometry. 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.**Other contributions to non-Euclidean 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 Euclideanparallel postulate and onOmar Khayyám 's attempted proof a century earlier. 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 inRome in 1594 and was studied by European geometers. This work marked the starting point forGiovanni Girolamo Saccheri 's work on the subject, and eventually the discovery ofnon-Euclidean geometry . [*Victor J. Katz (1998). "History of Mathematics: An Introduction", p. 270-271.*]Addison-Wesley . ISBN 0321016181.A proof from Sadr al-Din's work was quoted by

John Wallis and Saccheri in the 17th and 18th centuries. They both derived their proofs of the parallel postulate from Sadr al-Din's work, while Saccheri also derived hisSaccheri quadrilateral from Sadr al-Din, who himself based it on his father's work. [*Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., "*]Encyclopedia of the History of Arabic Science ", Vol. 2, p. 447-494 [469] ,Routledge , London and New York: quote|"His book published in Rome considerably influenced the subsequent development of the theory of parallel lines. Indeed, J. Wallis (1616-1703) included a Latin translation of the proof of postulate V from this book in his own writing "On the Fifth Postulate and the Fifth Definition from Euclid's Book 6" ("De Postulato Quinto et Definitione Quinta lib. 6 Euclidis", 1663). Saccheri quited this proof in his "Euclid Cleared of all Stains" ("Euclides ab omni naevo vindicatus", 1733). It seems possible that he borrowed the idea of considering the three hypotheses about the upper angles of the 'Saccheri quadrangle' from Pseudo-Tusi. The latter inserted the exposition of this subject into his work, taking it from the writings of al-Tusi and Khayyam."The theorems of

Ibn al-Haytham (Alhazen),Omar Khayyam andNasir al-Din al-Tusi onquadrilateral s, including theLambert quadrilateral andSaccheri quadrilateral , were the first theorems onelliptical geometry andhyperbolic geometry , and along with their alternative postulates, such asPlayfair's axiom , these works marked the beginning ofnon-Euclidean geometry and had a considerable influence on its development among later European geometers, includingWitelo ,Levi ben Gerson ,Alfonso ,John Wallis , andGiovanni Girolamo Saccheri . [*Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., "*]Encyclopedia of the History of Arabic Science ", Vol. 2, p. 447-494 [470] ,Routledge , London and New York: quote|"Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the ninteenth century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European couterparts. The first European attempt to prove the postulate on parallel lines - made by Witelo, the Polish scientists of the thirteenth century, while revising Ibn al-Haytham's "Book of Optics " ("Kitab al-Manazir") - was undoubtedly prompted by Arabic sources. The proofs put forward in the fourteenth century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that "Pseudo-Tusi's Exposition of Euclid" had stimulated borth J. Wallis's and G. Saccheri's studies of the theory of parallel lines."**Geometric architecture**Recent discoveries have shown that geometrical

quasicrystal patterns were first employed in thegirih tiles found in medievalIslamic architecture dating back over five centuries ago. In 2007, ProfessorPeter Lu ofHarvard University and ProfessorPaul Steinhardt ofPrinceton University published a paper in the journal "Science" suggesting that girih tilings possessed properties consistent withself-similar fractal quasicrystalline tilings such as thePenrose tiling s, predating them by five centuries.cite journal

author = Peter J. Lu and Paul J. Steinhardt

year = 2007

title = Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture

journal = Science

volume = 315

pages = 1106–1110

url = http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007.pdf

doi = 10.1126/science.1135491] [*Supplemental figures [*]*http://www.physics.harvard.edu/~plu/publications/Science_315_1106_2007_SOM.pdf*]**The 17th century**When Europe began to emerge from its

Dark Ages , theHellenistic andIslam ic texts on geometry found in Islamic libraries were translated fromArabic intoLatin . 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.In the early 17th century, there were two important developments in geometry. The first and most important was the creation ofanalytic geometry , or geometry with coordinates andequation s, byRené Descartes (1596-1650) andPierre de Fermat (1601-1665). This was a necessary precursor to the development ofcalculus and a precise quantitative science ofphysics . The second geometric development of this period was the systematic study ofprojective geometry byGirard Desargues (1591-1661). 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 byHellenistic geometers, notably Pappus (c. 340). The greatest flowering of the field occurred withJean-Victor Poncelet (1788-1867).In the late 17th century,

calculus was developed independently and almost simultaneously byIsaac Newton (1642-1727) andGottfried Wilhelm von Leibniz (1646-1716). This was the beginning of a new field of mathematics now called analysis. 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. 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 ofnon-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, andLegendre each did excellent work on the problem in the 18th century, but still fell short of success. In the early 19th century, Gauss, Johann Bolyai, andLobatchewsky , each independently, took a different approach. 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 firstnon-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'stheory of relativity .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. 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"). 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 . 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 overfinite field s, rather than the real or complex numbers.Finite geometry itself, the study of spaces with only finitely many points, found applications incoding theory andcryptography . With the advent of the computer, new disciplines such ascomputational geometry ordigital geometry deal with geometric algorithms, discrete representations of geometric data, and so forth.**ee also***

List of geometry topics

*Important publications in geometry.

*Interactive geometry software

*History of mathematics

* "Flatland ", Book written by " A^{2}" about two andthree-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*** [

*http://agutie.homestead.com Geometry Step by Step from the Land of the Incas*] by Antonio Gutierrez.

* [*http://www.islamicarchitecture.org/art/islamic-geometry-and-floral-patterns.html Islamic Geometry*]

* Stanford Encyclopedia of Philosophy:

** [*http://plato.stanford.edu/entries/geometry-finitism/ Finitism in Geometry*]

** [*http://plato.stanford.edu/entries/geometry-19th/ Geometry in the 19th Century*]

* [*http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Arabic_mathematics.html Arabic mathematics : forgotten brilliance?*]

*Wikimedia Foundation.
2010.*

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