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mentioned, spain, border, demonstration, stimulated, exposition, kitab, manazir, optics, routledge, sabi, thabit, harrani, hegemony, 241, 242, 1050, 1123, tent, maker, went, khwarizmi, predecessors, provided, believed, mistakenly, 16th, hence, intersecting, conics, cubics, earlier, menaechmus, alhazan, praiseworthy, generalizing, cover, having

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lists of mathematics topics related topics descriptive geometry finite geometry flatland a book written by edwin abbott abbott about two and three dimensional space to understand the concept of four dimensions list of interactive geometry software other fields molecular geometry notes until the 19th century geometry was dominated by the assumption that all geometric constructions were euclidean in the 19th century and later this was challenged by the development of hyperbolic geometry by lobachevsky and other non euclidean geometries by gauss and others it was then realised that implicitly non euclidean geometry had appeared throughout history including the work of desargues in the 17th century all the way back to the implicit use of spherical geometry to understand the earth geodesy and to navigate the oceans since antiquity the ancient greeks had some constructions using other instruments vincenzo de risi 2015 mathematizing space the objects of geometry from antiquity to the early modern age birkhäuser pp 1 isbn 978 3 319 12102 4 archived from the original on 20 february 2021 retrieved 14 september 2019 a b tabak john 2014 geometry the language of space and form infobase publishing p xiv isbn 978 0 8160 4953 0 walter a meyer 2006 geometry and its applications elsevier isbn 978 0 08 047803 6 archived from the original on 1 september 2021 retrieved 14 september 2019 friberg jöran 1981 methods and traditions of babylonian mathematics historia mathematica 8 3 277 318 doi 10 1016 0315 0860 81 90069 0 neugebauer otto 1969 1957 chap iv egyptian mathematics and astronomy the exact sciences in antiquity 2 ed dover publications pp 71 96 isbn 978 0 486 22332 2 archived from the original on 14 august 2020 retrieved 27 february 2021 boyer 1991 egypt p 19 ossendrijver mathieu 29 january 2016 ancient babylonian astronomers calculated jupiter s position from the area under a time velocity graph science 351 6272 482 484 bibcode 2016sci 351 482o doi 10 1126 science aad8085 pmid 26823423 s2cid 206644971 depuydt leo 1 january 1998 gnomons at meroë and early trigonometry the journal of egyptian archaeology 84 171 180 doi 10 2307 3822211 jstor 3822211 slayman andrew 27 may 1998 neolithic skywatchers archaeology magazine archive archived from the original on 5 june 2011 retrieved 17 april 2011 boyer 1991 ionia and the pythagoreans p 43 eves howard an introduction to the history of mathematics saunders 1990 isbn 0 03 029558 0 kurt von fritz 1945 the discovery of incommensurability by hippasus of metapontum the annals of mathematics boston studies in the philosophy of science 240 2 211 231 doi 10 1007 978 1 4020 2640 9_11 isbn 978 90 481 5850 8 jstor 1969021 james r choike 1980 the pentagram and the discovery of an irrational number the two year college mathematics journal 11 5 312 316 doi 10 2307 3026893 jstor 3026893 boyer 1991 the age of plato and aristotle p 92 boyer 1991 euclid of alexandria p 119 boyer 1991 euclid of alexandria p 104 howard eves an introduction to the history of mathematics saunders 1990 isbn 0 03 029558 0 p 141 no work except the bible has been more widely used o connor j j robertson e f february 1996 a history of calculus university of st andrews archived from the original on 15 july 2007 retrieved 7 august 2007 staal frits 1999 greek and vedic geometry journal of indian philosophy 27 1 2 105 127 doi 10 1023 a 1004364417713 s2cid 170894641 pythagorean triples are triples of integers a b c displaystyle a b c with the property a 2 b 2 c 2 displaystyle a 2 b 2 c 2 thus 3 2 4 2 5 2 displaystyle 3 2 4 2 5 2 8 2 15 2 17 2 displaystyle 8 2 15 2 17 2 12 2 35 2 37 2 displaystyle 12 2 35 2 37 2 etc cooke 2005 p 198 the arithmetic content of the śulva sūtras consists of rules for finding pythagorean triples such as 3 4 5 5 12 13 8 15 17 and 12 35 37 it is not certain what practical use these arithmetic rules had the best conjecture is that they were part of religious ritual a hindu home was required to have three fires burning at three different altars the three altars were to be of different shapes but all three were to have the same area these conditions led to certain diophantine problems a particular case of which is the generation of pythagorean triples so as to make one square integer equal to the sum of two others hayashi 2005 p 371 a b hayashi 2003 pp 121 122 rāshid rushdī 1994 the development of arabic mathematics between arithmetic and algebra boston studies in the philosophy of science vol 156 p 35 doi 10 1007 978 94 017 3274 1 isbn 978 0 7923 2565 9 oclc 29181926 boyer 1991 the arabic hegemony pp 241 242 omar khayyam c 1050 1123 the tent maker wrote an algebra that went beyond that of al khwarizmi to include equations of third degree like his arab predecessors omar khayyam provided for quadratic equations both arithmetic and geometric solutions for general cubic equations he believed mistakenly as the 16th century later showed arithmetic solutions were impossible hence he gave only geometric solutions the scheme of using intersecting conics to solve cubics had been used earlier by menaechmus archimedes and alhazan but omar khayyam took the praiseworthy step of generalizing the method to cover all third degree equations having positive roots for equations of higher degree than three omar khayyam evidently did not envision similar geometric methods for space does not contain more than three dimensions one of the most fruitful contributions of arabic eclecticism was the tendency to close the gap between numerical and geometric algebra the decisive step in this direction came much later with descartes but omar khayyam was moving in this direction when he wrote whoever thinks algebra is a trick in obtaining unknowns has thought it in vain no attention should be paid to the fact that algebra and geometry are different in appearance algebras are geometric facts which are proved o connor john j robertson edmund f al mahani mactutor history of mathematics archive university of st andrews o connor john j robertson edmund f al sabi thabit ibn qurra al harrani mactutor history of mathematics archive university of st andrews o connor john j robertson edmund f omar khayyam mactutor history of mathematics archive university of st andrews boris a rosenfeld and adolf p youschkevitch 1996 geometry in roshdi rashed ed encyclopedia of the history of arabic science vol 2 pp 447 494 470 routledge london and new york 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 19th 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 this 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 investigations of their european counterparts the first european attempt to prove the postulate on parallel lines made by witelo the polish scientists of the 13th 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 14th 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 both j wallis s and g saccheri s studies of the theory of parallel lines a b carl b boyer 2012 history of analytic geometry courier corporation isbn 978 0 486 15451 0 archived 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