2201:(perhaps for the last letter of arithmos). It is instead a collection of some 150 problems, all worked out in terms of specific numerical examples, although perhaps generality of method was intended. There is no postulation development, nor is an effort made to find all possible solutions. In the case of quadratic equations with two positive roots, only the larger is give, and negative roots are not recognized. No clear-cut distinction is made between determinate and indeterminate problems, and even for the latter for which the number of solutions generally is unlimited, only a single answer is given. Diophantus solved problems involving several unknown numbers by skillfully expressing all unknown quantities, where possible, in terms of only one of them."
2262:
was composed. Also, because of the two-dimensional character of the Arabic notation, it would have been written and read visually, independent of real or imagined speech. It thus fits nicely into
Nesselmann's "symbolic" category. The rhetorical version of the same work, on the other hand, was categorized as being "rhetorical". These two ways of writing algebra do not reflect two stages of the development of algebra but are different ways of expressing the same ideas. Second, Nesselmann was unaware of the conceptual differences between premodern and modern algebra, and thus, he could not have appreciated the leap made in the time of Viète and Descartes that included a radical shift in how notation was interpreted.
3720:
2153:, "Revival and Decline of Greek Mathematics" pp. 180-182) "In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent. But whereas Babylonian mathematicians had been concerned primarily with
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25:
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2032:, "Revival and Decline of Greek Mathematics" p. 178) "Uncertainty about the life of Diophantus is so great that we do not know definitely in which century he lived. Generally he is assumed to have flourished about A.D. 250, but dates a century or more earlier or later are sometimes suggested If this conundrum is historically accurate, Diophantus lived to be eighty-four-years old. The chief Diophantine work known to us is the
795:
1837:
1944:
of
Diophantus (ca. A.D. 250) are extant in Greek. The remaining books were believed to be lost, until the recent discovery of a medieval Arabic translation of four of the remaining books in a manuscript in the Shrine Library in Meshed in Iran (see the catalogue . The manuscript was discovered in 1968
2261:
There are two major flaws with this trichotomy. First, the language written in books is not always the language in which problems were worked out. In Arabic, problems were often solved in notation on a dust-board or some other temporary surface, and then for inclusion in a book a rhetorical version
367:
is the earliest extant work present that solve arithmetic problems by algebra. Diophantus however did not invent the method of algebra, which existed before him. Algebra was practiced and diffused orally by practitioners, with
Diophantus picking up technique to solve problems in arithmetic.
1606:
However the distinction between "rhetorical algebra", "syncopated algebra" and "symbolic algebra" is considered outdated by
Jeffrey Oaks and Jean Christianidis. The problems were solved on dust-board using some notation, while in books solution were written in "rhetorical style".
1397:
Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of
Diophantus's syncopated equation into a modern symbolic equation would be the following:
375:
is linear combination of some variables, raised to integer powers, which behaves under multiplication, addition, and subtraction. Algebra of
Diophantus, similar to medieval arabic algebra is aggregation of objects of different types with no operations present
702:
1616:
2214:, "Revival and Decline of Greek Mathematics" p. 178) "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."
275:. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it was not settled until
509:. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So for example, what would be written in modern notation as
497:
Diophantus does not give classification of equations in six types like Al-Khwarizmi in extant parts of
Arithmetica. He does says that he would give solution to three terms equations later, so this part of work is possibly just lost
241:
In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form
694:
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790:{\displaystyle \mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;}
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574:
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1226:
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there is a systematic use of abbreviations for powers of numbers and for relationships and operations. An unknown number is represented by a symbol resembling the Greek letter
480:
505:, Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as
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1071:
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1832:{\displaystyle {\begin{alignedat}{4}\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)&=(ac+db)^{2}+(bc-ad)^{2}\\&=(ad+bc)^{2}+(ac-bd)^{2}\\\end{alignedat}}}
269:
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1172:
2277:, "Europe in the Middle Ages" p. 257) "The book makes frequent use of the identities which had appeared in Diophantus and had been widely used by the Arabs."
357:
mathematician who lived circa 250 AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written
2443:. Cum comm. C(laude) G(aspar) Bacheti et observationibus P(ierre) de Fermat. Acc. doctrinae analyticae inventum novum, coll. ex variis eiu. Tolosae 1670,
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was originally written in thirteen books, but the Greek manuscripts that survived to the present contain no more than six books. In 1968,
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317:) because the librarian was apparently not able to read the main line of the cover page where Diophantus’s name appears in geometric
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1990:) "Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine
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141:
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68:
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object of one kind with 25 object of second kind which lack 9 objects of third kind with no operation present".
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in Meshed (Iran) in a copy from 1198 AD. It was not catalogued under the name of
Diophantus (but under that of
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where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:
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Similar to medieval Arabic algebra
Diophantus uses three stages to solution of a problem by Algebra:
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Books IV to VII of
Diophantus' Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā
2011:) "Abu'l-Wefa was a capable algebraist as well as a trigonometer. He commented on al-Khwarizmi's
1078:
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in northeastern Iran. The four books are thought to have been translated from Greek to Arabic by
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361:, a treatise that was originally thirteen books but of which only the first six have survived.
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8:
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3220:
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82:
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2099:"Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria"
2066:"Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria"
1847:
1157:
372:
345:
235:
2397:
Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century
2036:, a treatise originally in thirteen books, only the first six of which have survived."
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1904:
689:{\displaystyle \left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5,}
491:
2) An equation is simplified to a standard form( al-jabr and al-muqābala in arabic)
3599:
3470:
3387:
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3079:
2815:
2444:
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1599:{\displaystyle \left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5}
2441:
Diophanti Alexandrini Arithmeticorum libri 6, et De numeris multangulis liber unus
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Diophantus Alexandrinus, Pierre de Fermat, Claude Gaspard Bachet de Meziriac,
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cannot be the sum of two squares. Diophantus also appears to know that
188:
153:
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2740:
2690:
2585:
2575:
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2165:
of Diophantus (such as we have it) is almost entirely devoted to the
207:
2315:
The Arithmetica of Diophantus: a complete translation and commentary
24:
2892:
2670:
2515:
2257:
The Arithmetica of Diophantus A Complete Translation and Commentary
2137:
The Arithmetica of Diophantus A Complete Translation and Commentary
2122:
The Arithmetica of Diophantus A Complete Translation and Commentary
2098:
2048:
The Arithmetica of Diophantus A Complete Translation and Commentary
127:
1929:"Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica"
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2650:
2378:
Diophantus of Alexandria: A Study in the History of Greek Algebra
1022:
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965:
831:{\displaystyle \sigma \;\,\mathrm {M} {\overline {\varepsilon }}}
298:
203:
450:
inverse Powers, 25 Powers lacking 9 units", or "a collection of
2760:
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2015:
and translated from Greek one of the last great classics, the
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318:
171:
133:
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927:
2356:
Unknown Quantity: A Real And Imaginary History of Algebra
2237:
2235:
2233:
1021:(uppercase: Ϝ, lowercase: ϝ) in the 6th position between
1959:"Diophantus of Alexandria : a Text and its History"
696:
would be written in Diophantus's syncopated notation as
1482:{\displaystyle {x^{3}}1{x}10-{x^{2}}2{x^{0}}1={x^{0}}5}
297:
at the shrine of Imam Rezā in the holy Islamic city of
2230:
1075:
represents the subtraction of everything that follows
461:
390:
273:
every number can be written as the sum of four squares
2187:
1994:
became familiar before the end of the tenth century."
1619:
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1404:
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1207:
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1138:
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1081:
1059:
979:
941:
903:
861:
807:
705:
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248:
226:. The method for solving these equations is known as
1382:{\displaystyle \mathrm {K} ^{\upsilon }\mathrm {K} }
202:) in the 3rd century AD. It is a collection of 130
446:
in modern notation is written by Diophantus as "6 4
206:problems giving numerical solutions of determinate
49:. Unsourced material may be challenged and removed.
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488:1) An unknown is named and an equation is set up
3738:
2395:Katz, Victor J.; Parshall, Karen Hunger (2014).
2227:, "Mathematics in the Roman Empire" pp. 167-168)
2007:, "Revival and Decline of Greek Mathematics" p.
1986:, "Revival and Decline of Greek Mathematics" p.
1873:"Diophantus of Alexandria (Greek mathematician)"
439:{\displaystyle 6{\tfrac {1}{4}}x^{-1}+25x^{2}-9}
2418:. New York Heidelberg Berlin: Springer-Verlag.
2298:(Second ed.). John Wiley & Sons, Inc.
2022:
1342:{\displaystyle \Delta \mathrm {K} ^{\upsilon }}
379:For example, the Laurent polynomial written as
2313:Christianidis, Jean; Oaks, Jeffrey A. (2023).
2484:
2470:
2045:
1197:this may be thought of as "the first power")
2394:
1128:the zeroth power (that is, a constant term)
305:(820–912). Norbert Schappacher has written:
2255:Oaks, Jeffrey; Christianidis, Jean (2023).
2135:Oaks, Jeffrey; Christianidis, Jean (2021).
2120:Oaks, Jeffrey; Christianidis, Jean (2023).
2097:Oaks, Jeffrey; Christianidis, Jean (2013).
2064:Oaks, Jeffrey; Christianidis, Jean (2013).
1956:
841:where the symbols represent the following:
222:Equations in the book are presently called
132:Cover of the 1621 edition, translated into
2477:
2463:
2350:
2334:The History of Mathematics: A Brief Course
2317:. Abingdon, Oxon New York, NY: Routledge.
2242:
2161:equations as far as the third degree, the
1304:{\displaystyle \Delta ^{\upsilon }\Delta }
955:{\displaystyle {\overline {\varepsilon }}}
811:
786:
769:
747:
742:
728:
126:
2177:. Throughout the six surviving books of
2081:
1923:
1903:. Vol. 1. Salem Press. p. 362.
968:is the 5th letter of the Greek alphabet)
930:is the 2nd letter of the Greek alphabet)
812:
785:
748:
743:
109:Learn how and when to remove this message
1892:
1890:
1262:{\displaystyle \mathrm {K} ^{\upsilon }}
2413:
1154:the unknown quantity (because a number
293:found four previously unknown books of
3739:
3506:Latin translations of the 12th century
1940:Only six of the thirteen books of the
1896:
3236:Straightedge and compass construction
2458:
2372:
2331:
2290:
2274:
2224:
2211:
2204:
2150:
2029:
2004:
1983:
1887:
875:{\displaystyle {\overline {\alpha }}}
569:{\displaystyle x^{3}-2x^{2}+10x-1=5,}
339:
3201:Incircle and excircles of a triangle
2245:, "The Father of Algebra" pp. 35-36)
2059:
2057:
2046:Oaks, Jeffrey; Christianidis, Jean.
993:{\displaystyle {\overline {\iota }}}
917:{\displaystyle {\overline {\beta }}}
47:adding citations to reliable sources
18:
1957:Schappacher, Norbert (April 2005).
1221:{\displaystyle \Delta ^{\upsilon }}
330:mathematicians in the Islamic world
210:(those with a unique solution) and
13:
1613:also makes use of the identities:
1375:
1364:
1329:
1324:
1298:
1289:
1249:
1209:
1174:raised to the first power is just
1112:
814:
771:
750:
708:
14:
3768:
2433:
2054:
1013:but it was the 10th letter of an
494:3) Simplified equation is solved
142:Claude Gaspard Bachet de Méziriac
16:Ancient Greek text on mathematics
3752:Ancient Greek mathematical works
3718:
3705:
475:{\displaystyle 6{\tfrac {1}{4}}}
23:
2267:
2248:
2217:
2128:
2113:
2090:
279:proved it using results due to
34:needs additional citations for
3538:A History of Greek Mathematics
3051:The Quadrature of the Parabola
2399:. Princeton University Press.
2039:
1997:
1976:
1950:
1917:
1897:Magill, Frank N., ed. (1998).
1865:
1853:Muhammad ibn Mūsā al-Khwārizmī
1816:
1797:
1785:
1766:
1747:
1728:
1716:
1697:
1015:ancient archaic Greek alphabet
309:resurfaced around 1971 in the
1:
2284:
1900:Dictionary of World Biography
1230:the second power, from Greek
3319:Intersecting secants theorem
2169:solution of equations, both
1858:
1271:the third power, from Greek
1236:, meaning strength or power
1119:{\displaystyle \mathrm {M} }
985:
947:
909:
867:
823:
780:
764:
737:
723:
7:
3314:Intersecting chords theorem
3181:Doctrine of proportionality
1841:
1088:{\displaystyle \pitchfork }
1066:{\displaystyle \pitchfork }
336:translated it into Arabic.
10:
3773:
3010:On the Sphere and Cylinder
2963:On the Sizes and Distances
1273:
1232:
1097:
1043:
1035:
798:
576:which can be rewritten as
343:
332:in the tenth century when
217:
172:
3712:Ancient Greece portal
3701:
3651:
3529:
3516:Philosophy of mathematics
3486:
3479:
3453:
3431:Ptolemy's table of chords
3375:
3357:
3256:
3249:
3105:
3067:
2884:
2492:
2486:Ancient Greek mathematics
2414:Sesiano, Jacques (2011).
1875:. Encyclopædia Britannica
1006:is the 9th letter of the
888:is the 1st letter of the
149:
125:
3383:Aristarchus's inequality
2956:On Conoids and Spheroids
2296:A History of Mathematics
2083:10.1016/j.hm.2012.09.001
3491:Ancient Greek astronomy
3304:Inscribed angle theorem
3294:Greek geometric algebra
2949:Measurement of a Circle
212:indeterminate equations
3725:Mathematics portal
3511:Non-Euclidean geometry
3466:Mouseion of Alexandria
3339:Tangent-secant theorem
3289:Geometric mean theorem
3274:Exterior angle theorem
3269:Angle bisector theorem
2973:On Sizes and Distances
2380:. Martino Fine Books.
2358:. Joseph Henry Press.
2336:. Wiley-Interscience.
2195:
2194:{\displaystyle \zeta }
1833:
1600:
1483:
1383:
1343:
1305:
1263:
1222:
1191:
1168:
1146:
1145:{\displaystyle \zeta }
1120:
1089:
1067:
994:
956:
918:
876:
832:
791:
690:
570:
476:
440:
323:
265:
200: 284/298 AD
196: 200/214 AD
3413:Pappus's area theorem
3349:Theorem of the gnomon
3226:Quadratrix of Hippias
3149:Circles of Apollonius
3097:Problem of Apollonius
3075:Constructible numbers
2899:Archimedes Palimpsest
2332:Cooke, Roger (1997).
2196:
1834:
1601:
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1344:
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571:
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307:
277:Joseph Louis Lagrange
266:
224:Diophantine equations
3629:prehistoric counting
3426:Ptolemy's inequality
3367:Apollonius's theorem
3206:Method of exhaustion
3176:Diophantine equation
3166:Circumscribed circle
2983:On the Moving Sphere
2374:Heath, Sir Thomas L.
2185:
2103:Historia Mathematica
2070:Historia Mathematica
1617:
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1359:
1321:
1285:
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1041:"equals" (short for
1017:that had the letter
977:
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901:
859:
805:
703:
580:
513:
454:
383:
371:In modern algebra a
264:{\displaystyle 4n+3}
246:
228:Diophantine analysis
43:improve this article
3715: •
3521:Neusis construction
3441:Spiral of Theodorus
3334:Pythagorean theorem
3279:Euclidean algorithm
3221:Lune of Hippocrates
3090:Squaring the circle
2846:Theon of Alexandria
2521:Aristaeus the Elder
2449:10.3931/e-rara-9423
850:What it represents
236:quadratic equations
122:
3757:History of algebra
3408:Menelaus's theorem
3398:Irrational numbers
3211:Parallel postulate
3186:Euclidean geometry
3154:Apollonian circles
2696:Isidore of Miletus
2259:. pp. 78–79.
2191:
1848:Diophantus II.VIII
1829:
1827:
1596:
1479:
1379:
1339:
1301:
1259:
1218:
1190:{\displaystyle x,}
1187:
1164:
1142:
1116:
1085:
1063:
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952:
914:
872:
828:
787:
686:
566:
507:syncopated algebra
472:
470:
436:
399:
373:Laurent polynomial
346:Syncopated algebra
340:Syncopated algebra
311:Astan Quds Library
261:
121:
3747:3rd-century books
3732:
3731:
3697:
3696:
3449:
3448:
3436:Ptolemy's theorem
3309:Intercept theorem
3159:Apollonian gasket
3085:Doubling the cube
3058:The Sand Reckoner
2406:978-0-691-14905-9
2387:978-1-57898-754-2
2139:. pp. 53–66.
2124:. pp. 51–52.
1925:Hogendijk, Jan P.
1395:
1394:
1313:the fourth power
1277:, meaning a cube
1167:{\displaystyle x}
988:
950:
912:
870:
826:
783:
767:
740:
726:
469:
398:
234:problems lead to
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118:
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93:
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3723:
3722:
3710:
3709:
3708:
3484:
3483:
3471:Platonic Academy
3418:Problem II.8 of
3388:Crossbar theorem
3344:Thales's theorem
3284:Euclid's theorem
3254:
3253:
3171:Commensurability
3132:Axiomatic system
3080:Angle trisection
3045:
3035:
2997:
2987:
2977:
2967:
2943:
2933:
2916:
2479:
2472:
2465:
2456:
2455:
2429:
2410:
2391:
2369:
2352:Derbyshire, John
2347:
2328:
2309:
2278:
2271:
2265:
2264:
2252:
2246:
2239:
2228:
2221:
2215:
2208:
2202:
2200:
2198:
2197:
2192:
2147:
2141:
2140:
2132:
2126:
2125:
2117:
2111:
2110:
2094:
2088:
2087:
2085:
2061:
2052:
2051:
2043:
2037:
2026:
2020:
2001:
1995:
1980:
1974:
1973:
1971:
1969:
1963:
1954:
1948:
1947:
1937:
1935:
1921:
1915:
1914:
1894:
1885:
1884:
1882:
1880:
1869:
1838:
1836:
1835:
1830:
1828:
1824:
1823:
1793:
1792:
1759:
1755:
1754:
1724:
1723:
1689:
1685:
1684:
1683:
1671:
1670:
1656:
1652:
1651:
1650:
1638:
1637:
1605:
1603:
1602:
1597:
1592:
1591:
1590:
1577:
1573:
1569:
1568:
1567:
1551:
1550:
1549:
1531:
1527:
1523:
1512:
1511:
1510:
1488:
1486:
1485:
1480:
1475:
1474:
1473:
1457:
1456:
1455:
1442:
1441:
1440:
1424:
1416:
1415:
1414:
1391:the sixth power
1388:
1386:
1385:
1380:
1378:
1373:
1372:
1367:
1351:the fifth power
1348:
1346:
1345:
1340:
1338:
1337:
1332:
1310:
1308:
1307:
1302:
1297:
1296:
1276:
1275:
1268:
1266:
1265:
1260:
1258:
1257:
1252:
1235:
1234:
1227:
1225:
1224:
1219:
1217:
1216:
1196:
1194:
1193:
1188:
1173:
1171:
1170:
1165:
1151:
1149:
1148:
1143:
1125:
1123:
1122:
1117:
1115:
1100:
1099:
1094:
1092:
1091:
1086:
1072:
1070:
1069:
1064:
1048:
1047:
1038:
1037:
999:
997:
996:
991:
989:
981:
961:
959:
958:
953:
951:
943:
923:
921:
920:
915:
913:
905:
881:
879:
878:
873:
871:
863:
844:
843:
837:
835:
834:
829:
827:
819:
817:
801:
800:
796:
794:
793:
788:
784:
776:
774:
768:
760:
758:
757:
741:
733:
727:
719:
717:
716:
711:
695:
693:
692:
687:
679:
678:
677:
664:
660:
656:
655:
654:
638:
637:
636:
618:
614:
610:
599:
598:
597:
575:
573:
572:
567:
541:
540:
525:
524:
481:
479:
478:
473:
471:
462:
449:
445:
443:
442:
437:
429:
428:
413:
412:
400:
391:
328:became known to
319:Kufi calligraphy
270:
268:
267:
262:
201:
197:
175:
174:
130:
123:
120:
114:
107:
103:
100:
94:
92:
51:
27:
19:
3772:
3771:
3767:
3766:
3765:
3763:
3762:
3761:
3737:
3736:
3733:
3728:
3717:
3706:
3704:
3693:
3659:Arabian/Islamic
3647:
3636:numeral systems
3525:
3475:
3445:
3393:Heron's formula
3371:
3353:
3245:
3241:Triangle center
3231:Regular polygon
3108:and definitions
3107:
3101:
3063:
3043:
3033:
2995:
2985:
2975:
2965:
2941:
2931:
2914:
2880:
2851:Theon of Smyrna
2496:
2488:
2483:
2436:
2426:
2407:
2388:
2366:
2344:
2325:
2306:
2287:
2282:
2281:
2272:
2268:
2253:
2249:
2243:Derbyshire 2006
2240:
2231:
2222:
2218:
2209:
2205:
2186:
2183:
2182:
2148:
2144:
2133:
2129:
2118:
2114:
2095:
2091:
2062:
2055:
2044:
2040:
2027:
2023:
2019:of Diophantus."
2002:
1998:
1981:
1977:
1967:
1965:
1961:
1955:
1951:
1933:
1931:
1922:
1918:
1911:
1895:
1888:
1878:
1876:
1871:
1870:
1866:
1861:
1844:
1826:
1825:
1819:
1815:
1788:
1784:
1757:
1756:
1750:
1746:
1719:
1715:
1690:
1679:
1675:
1666:
1662:
1661:
1657:
1646:
1642:
1633:
1629:
1628:
1624:
1620:
1618:
1615:
1614:
1586:
1582:
1581:
1563:
1559:
1558:
1545:
1541:
1540:
1539:
1535:
1519:
1506:
1502:
1501:
1500:
1496:
1494:
1491:
1490:
1469:
1465:
1464:
1451:
1447:
1446:
1436:
1432:
1431:
1420:
1410:
1406:
1405:
1403:
1400:
1399:
1374:
1368:
1363:
1362:
1360:
1357:
1356:
1333:
1328:
1327:
1322:
1319:
1318:
1292:
1288:
1286:
1283:
1282:
1253:
1248:
1247:
1245:
1242:
1241:
1212:
1208:
1206:
1203:
1202:
1179:
1176:
1175:
1159:
1156:
1155:
1137:
1134:
1133:
1111:
1109:
1106:
1105:
1080:
1077:
1076:
1058:
1055:
1054:
980:
978:
975:
974:
942:
940:
937:
936:
904:
902:
899:
898:
862:
860:
857:
856:
818:
813:
806:
803:
802:
775:
770:
759:
753:
749:
732:
718:
712:
707:
706:
704:
701:
700:
673:
669:
668:
650:
646:
645:
632:
628:
627:
626:
622:
606:
593:
589:
588:
587:
583:
581:
578:
577:
536:
532:
520:
516:
514:
511:
510:
460:
455:
452:
451:
447:
424:
420:
405:
401:
389:
384:
381:
380:
348:
342:
247:
244:
243:
230:. Most of the
220:
199:
195:
184:written by the
145:
115:
104:
98:
95:
52:
50:
40:
28:
17:
12:
11:
5:
3770:
3760:
3759:
3754:
3749:
3730:
3729:
3702:
3699:
3698:
3695:
3694:
3692:
3691:
3686:
3681:
3676:
3671:
3666:
3661:
3655:
3653:
3652:Other cultures
3649:
3648:
3646:
3645:
3644:
3643:
3633:
3632:
3631:
3621:
3620:
3619:
3609:
3608:
3607:
3597:
3596:
3595:
3585:
3584:
3583:
3573:
3572:
3571:
3561:
3560:
3559:
3549:
3548:
3547:
3533:
3531:
3527:
3526:
3524:
3523:
3518:
3513:
3508:
3503:
3501:Greek numerals
3498:
3496:Attic numerals
3493:
3487:
3481:
3477:
3476:
3474:
3473:
3468:
3463:
3457:
3455:
3451:
3450:
3447:
3446:
3444:
3443:
3438:
3433:
3428:
3423:
3415:
3410:
3405:
3400:
3395:
3390:
3385:
3379:
3377:
3373:
3372:
3370:
3369:
3363:
3361:
3355:
3354:
3352:
3351:
3346:
3341:
3336:
3331:
3326:
3324:Law of cosines
3321:
3316:
3311:
3306:
3301:
3296:
3291:
3286:
3281:
3276:
3271:
3265:
3263:
3251:
3247:
3246:
3244:
3243:
3238:
3233:
3228:
3223:
3218:
3216:Platonic solid
3213:
3208:
3203:
3198:
3196:Greek numerals
3193:
3188:
3183:
3178:
3173:
3168:
3163:
3162:
3161:
3156:
3146:
3141:
3140:
3139:
3129:
3128:
3127:
3122:
3111:
3109:
3103:
3102:
3100:
3099:
3094:
3093:
3092:
3087:
3082:
3071:
3069:
3065:
3064:
3062:
3061:
3054:
3047:
3037:
3027:
3024:Planisphaerium
3020:
3013:
3006:
2999:
2989:
2979:
2969:
2959:
2952:
2945:
2935:
2925:
2918:
2908:
2901:
2896:
2888:
2886:
2882:
2881:
2879:
2878:
2873:
2868:
2863:
2858:
2853:
2848:
2843:
2838:
2833:
2828:
2823:
2818:
2813:
2808:
2803:
2798:
2793:
2788:
2783:
2778:
2773:
2768:
2763:
2758:
2753:
2748:
2743:
2738:
2733:
2728:
2723:
2718:
2713:
2708:
2703:
2698:
2693:
2688:
2683:
2678:
2673:
2668:
2663:
2658:
2653:
2648:
2643:
2638:
2633:
2628:
2623:
2618:
2613:
2608:
2603:
2598:
2593:
2588:
2583:
2578:
2573:
2568:
2563:
2558:
2553:
2548:
2543:
2538:
2533:
2528:
2523:
2518:
2513:
2508:
2502:
2500:
2494:Mathematicians
2490:
2489:
2482:
2481:
2474:
2467:
2459:
2453:
2452:
2435:
2434:External links
2432:
2431:
2430:
2424:
2411:
2405:
2392:
2386:
2370:
2364:
2348:
2342:
2329:
2323:
2310:
2304:
2292:Boyer, Carl B.
2286:
2283:
2280:
2279:
2266:
2247:
2229:
2216:
2203:
2190:
2142:
2127:
2112:
2089:
2076:(2): 158–160.
2053:
2038:
2021:
1996:
1975:
1949:
1945:by F. Sezgin).
1916:
1909:
1886:
1863:
1862:
1860:
1857:
1856:
1855:
1850:
1843:
1840:
1822:
1818:
1814:
1811:
1808:
1805:
1802:
1799:
1796:
1791:
1787:
1783:
1780:
1777:
1774:
1771:
1768:
1765:
1762:
1760:
1758:
1753:
1749:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1722:
1718:
1714:
1711:
1708:
1705:
1702:
1699:
1696:
1693:
1691:
1688:
1682:
1678:
1674:
1669:
1665:
1660:
1655:
1649:
1645:
1641:
1636:
1632:
1627:
1623:
1622:
1595:
1589:
1585:
1580:
1576:
1572:
1566:
1562:
1557:
1554:
1548:
1544:
1538:
1534:
1530:
1526:
1522:
1518:
1515:
1509:
1505:
1499:
1478:
1472:
1468:
1463:
1460:
1454:
1450:
1445:
1439:
1435:
1430:
1427:
1423:
1419:
1413:
1409:
1393:
1392:
1389:
1377:
1371:
1366:
1353:
1352:
1349:
1336:
1331:
1326:
1315:
1314:
1311:
1300:
1295:
1291:
1279:
1278:
1269:
1256:
1251:
1238:
1237:
1228:
1215:
1211:
1199:
1198:
1186:
1183:
1163:
1152:
1141:
1130:
1129:
1126:
1114:
1102:
1101:
1084:
1073:
1062:
1051:
1050:
1039:
1031:
1030:
1011:Greek alphabet
1010:
1000:
987:
984:
970:
969:
962:
949:
946:
932:
931:
924:
911:
908:
894:
893:
890:Greek alphabet
882:
869:
866:
852:
851:
848:
839:
838:
825:
822:
816:
810:
782:
779:
773:
766:
763:
756:
752:
746:
739:
736:
731:
725:
722:
715:
710:
685:
682:
676:
672:
667:
663:
659:
653:
649:
644:
641:
635:
631:
625:
621:
617:
613:
609:
605:
602:
596:
592:
586:
565:
562:
559:
556:
553:
550:
547:
544:
539:
535:
531:
528:
523:
519:
508:
468:
465:
459:
435:
432:
427:
423:
419:
416:
411:
408:
404:
397:
394:
388:
344:Main article:
341:
338:
315:Qusta ibn Luqa
303:Qusta ibn Luqa
281:Leonhard Euler
260:
257:
254:
251:
219:
216:
157:
156:
151:
147:
146:
131:
117:
116:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3769:
3758:
3755:
3753:
3750:
3748:
3745:
3744:
3742:
3735:
3727:
3726:
3721:
3714:
3713:
3700:
3690:
3687:
3685:
3682:
3680:
3677:
3675:
3672:
3670:
3667:
3665:
3662:
3660:
3657:
3656:
3654:
3650:
3642:
3639:
3638:
3637:
3634:
3630:
3627:
3626:
3625:
3622:
3618:
3615:
3614:
3613:
3610:
3606:
3603:
3602:
3601:
3598:
3594:
3591:
3590:
3589:
3586:
3582:
3579:
3578:
3577:
3574:
3570:
3567:
3566:
3565:
3562:
3558:
3555:
3554:
3553:
3550:
3546:
3542:
3541:
3540:
3539:
3535:
3534:
3532:
3528:
3522:
3519:
3517:
3514:
3512:
3509:
3507:
3504:
3502:
3499:
3497:
3494:
3492:
3489:
3488:
3485:
3482:
3478:
3472:
3469:
3467:
3464:
3462:
3459:
3458:
3456:
3452:
3442:
3439:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3421:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3389:
3386:
3384:
3381:
3380:
3378:
3374:
3368:
3365:
3364:
3362:
3360:
3356:
3350:
3347:
3345:
3342:
3340:
3337:
3335:
3332:
3330:
3329:Pons asinorum
3327:
3325:
3322:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3300:
3299:Hinge theorem
3297:
3295:
3292:
3290:
3287:
3285:
3282:
3280:
3277:
3275:
3272:
3270:
3267:
3266:
3264:
3262:
3261:
3255:
3252:
3248:
3242:
3239:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3202:
3199:
3197:
3194:
3192:
3189:
3187:
3184:
3182:
3179:
3177:
3174:
3172:
3169:
3167:
3164:
3160:
3157:
3155:
3152:
3151:
3150:
3147:
3145:
3142:
3138:
3135:
3134:
3133:
3130:
3126:
3123:
3121:
3118:
3117:
3116:
3113:
3112:
3110:
3104:
3098:
3095:
3091:
3088:
3086:
3083:
3081:
3078:
3077:
3076:
3073:
3072:
3070:
3066:
3060:
3059:
3055:
3053:
3052:
3048:
3046:
3042:
3038:
3036:
3032:
3028:
3026:
3025:
3021:
3019:
3018:
3014:
3012:
3011:
3007:
3005:
3004:
3000:
2998:
2994:
2990:
2988:
2984:
2980:
2978:
2974:
2970:
2968:
2966:(Aristarchus)
2964:
2960:
2958:
2957:
2953:
2951:
2950:
2946:
2944:
2940:
2936:
2934:
2930:
2926:
2924:
2923:
2919:
2917:
2913:
2909:
2907:
2906:
2902:
2900:
2897:
2895:
2894:
2890:
2889:
2887:
2883:
2877:
2874:
2872:
2871:Zeno of Sidon
2869:
2867:
2864:
2862:
2859:
2857:
2854:
2852:
2849:
2847:
2844:
2842:
2839:
2837:
2834:
2832:
2829:
2827:
2824:
2822:
2819:
2817:
2814:
2812:
2809:
2807:
2804:
2802:
2799:
2797:
2794:
2792:
2789:
2787:
2784:
2782:
2779:
2777:
2774:
2772:
2769:
2767:
2764:
2762:
2759:
2757:
2754:
2752:
2749:
2747:
2744:
2742:
2739:
2737:
2734:
2732:
2729:
2727:
2724:
2722:
2719:
2717:
2714:
2712:
2709:
2707:
2704:
2702:
2699:
2697:
2694:
2692:
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2050:. p. 80.
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60: –
59:
58:"Arithmetica"
55:
54:Find sources:
48:
44:
38:
37:
32:This article
30:
26:
21:
20:
3734:
3716:
3703:
3545:Thomas Heath
3536:
3419:
3403:Law of sines
3259:
3191:Golden ratio
3056:
3049:
3040:
3034:(Theodosius)
3030:
3022:
3015:
3008:
3001:
2992:
2982:
2976:(Hipparchus)
2972:
2962:
2954:
2947:
2938:
2928:
2920:
2915:(Apollonius)
2911:
2904:
2903:
2891:
2866:Zeno of Elea
2626:Eratosthenes
2616:Dionysodorus
2440:
2415:
2396:
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2041:
2033:
2024:
2016:
2012:
1999:
1991:
1978:
1966:. Retrieved
1964:. p. 18
1952:
1941:
1939:
1932:. Retrieved
1919:
1899:
1877:. Retrieved
1867:
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105:
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86:
79:
72:
65:
53:
41:Please help
36:verification
33:
3612:mathematics
3420:Arithmetica
3017:Ostomachion
2986:(Autolycus)
2905:Arithmetica
2681:Hippocrates
2611:Dinostratus
2596:Dicaearchus
2526:Aristarchus
2179:Arithmetica
2171:determinate
2163:Arithmetica
2159:determinate
2155:approximate
2034:Arithmetica
2017:Arithmetica
1992:Arithmetica
1942:Arithmetica
1611:Arithmetica
503:Arithmetica
365:Arithmetica
359:Arithmetica
355:Hellenistic
326:Arithmetica
295:Arithmetica
291:Fuat Sezgin
287:Arithmetica
232:Arithmetica
182:mathematics
163:Arithmetica
3741:Categories
3664:Babylonian
3564:arithmetic
3530:History of
3359:Apollonius
3044:(Menelaus)
3003:On Spirals
2922:Catoptrics
2861:Xenocrates
2856:Thymaridas
2841:Theodosius
2826:Theaetetus
2806:Simplicius
2796:Pythagoras
2781:Posidonius
2766:Philonides
2726:Nicomachus
2721:Metrodorus
2711:Menaechmus
2666:Hipparchus
2656:Heliodorus
2606:Diophantus
2591:Democritus
2571:Chrysippus
2541:Archimedes
2536:Apollonius
2506:Anaxagoras
2498:(timeline)
2425:1461381762
2324:1138046353
2285:References
2275:Boyer 1991
2225:Cooke 1997
2212:Boyer 1991
2151:Boyer 1991
2030:Boyer 1991
2005:Boyer 1991
1984:Boyer 1991
351:Diophantus
334:Abu'l-Wefa
198: – c.
189:Diophantus
173:Ἀριθμητικά
154:Diophantus
69:newspapers
3125:Inscribed
2885:Treatises
2876:Zenodorus
2836:Theodorus
2811:Sosigenes
2756:Philolaus
2741:Oenopides
2736:Nicoteles
2731:Nicomedes
2691:Hypsicles
2586:Ctesibius
2576:Cleomedes
2561:Callippus
2546:Autolycus
2531:Aristotle
2511:Anthemius
2189:ζ
1968:9 October
1859:Citations
1807:−
1738:−
1533:−
1429:−
1370:υ
1335:υ
1325:Δ
1299:Δ
1294:υ
1290:Δ
1255:υ
1214:υ
1210:Δ
1140:ζ
1083:⋔
1061:⋔
986:¯
983:ι
948:¯
945:ε
910:¯
907:β
868:¯
865:α
824:¯
821:ε
809:σ
781:¯
778:α
765:¯
762:β
755:υ
751:Δ
745:⋔
738:¯
735:ι
730:ζ
724:¯
721:α
714:υ
620:−
552:−
527:−
431:−
407:−
208:equations
204:algebraic
99:July 2010
3689:Japanese
3674:Egyptian
3617:timeline
3605:timeline
3593:timeline
3588:geometry
3581:timeline
3576:calculus
3569:timeline
3557:timeline
3260:Elements
3106:Concepts
3068:Problems
3041:Spherics
3031:Spherics
2996:(Euclid)
2942:(Euclid)
2939:Elements
2932:(Euclid)
2893:Almagest
2801:Serenus
2776:Porphyry
2716:Menelaus
2671:Hippasus
2646:Eutocius
2621:Domninus
2516:Archytas
2376:(2009).
2354:(2006).
2294:(1991).
1927:(1985).
1879:11 April
1842:See also
180:text on
176:) is an
3669:Chinese
3624:numbers
3552:algebra
3480:Related
3454:Centers
3250:Results
3120:Central
2791:Ptolemy
2786:Proclus
2751:Perseus
2706:Marinus
2686:Hypatia
2676:Hippias
2651:Geminus
2641:Eudoxus
2631:Eudemus
2601:Diocles
2013:Algebra
1233:δύναμις
1023:epsilon
1019:digamma
966:Epsilon
847:Symbol
299:Mashhad
218:Summary
83:scholar
3684:Indian
3461:Cyrene
2993:Optics
2912:Conics
2831:Theano
2821:Thales
2816:Sporus
2761:Philon
2746:Pappus
2636:Euclid
2566:Carpus
2556:Bryson
2422:
2403:
2384:
2362:
2340:
2321:
2302:
2109:: 150.
1934:6 July
1907:
1095:up to
1025:ε and
1009:modern
973:
935:
897:
855:
353:was a
150:Author
85:
78:
71:
64:
56:
3679:Incan
3600:logic
3376:Other
3144:Chord
3137:Axiom
3115:Angle
2771:Plato
2661:Heron
2581:Conon
2167:exact
1962:(PDF)
1274:κύβος
886:Alpha
168:Greek
138:Greek
136:from
134:Latin
90:JSTOR
76:books
3641:list
2929:Data
2701:Leon
2551:Bion
2420:ISBN
2401:ISBN
2382:ISBN
2360:ISBN
2338:ISBN
2319:ISBN
2300:ISBN
2173:and
1970:2015
1936:2014
1905:ISBN
1881:2013
1045:ἴσος
1029:ζ.)
1027:zeta
1004:Iota
1002:10 (
928:Beta
62:news
3543:by
3257:In
2445:doi
2078:doi
2009:239
1988:234
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Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
