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123:, were aware of the problem and made many futile attempts at solving what they saw as an obstinate but soluble problem. Greeks made references on the subject. However, according to Eutocius, Archytas was the first to solve the problem of doubling the cube (the so-called Delian problem) with an ingenious geometric construction. The nonexistence of a compass-and-straightedge solution was finally proven by
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to find a solution for their internal political problems at the time, which had intensified relationships among the citizens. The oracle responded that they must double the size of the altar to Apollo, which was a regular cube. The answer seemed strange to the
Delians, and they consulted
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Descartes theory of geometric solution of equations uses a parabola to introduce cubic equations, in this way it is possible to set up an equation whose solution is a cube root of two. Note that the parabola itself is not constructible except by three dimensional methods.
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1425:, Book VII, states that "if any whole city should hold these things honourable and take a united lead and supervise, they would obey, and solution sought constantly and earnestly would become clear."
490:-coordinates of the points in the order that they were defined until we reach the original pair of points (0,0) and (1,0). As every field extension has degree 2 or 1, and as the field extension over
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that are additions, subtractions, multiplications, and divisions involving the coordinates of the previously defined points (and rational numbers). Restated in more abstract terminology, the new
1478:
Menn, S. (2015). "How
Archytas doubled the cube". In Holmes, B.; Fischer, K.-D. (eds.). The Frontiers of Ancient Science: Essays in honor of Heinrich von Staden. pp. 407–436 – via Google books.
1487:
Masià, R. (2016). "A new reading of
Archytas' doubling of the cube and its implications". Archive for History of Exact Sciences. 70 (2): 175–204. doi:10.1007/s00407-015-0165-9. ISSN 1432-0657.
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on one previously defined point and passing through another, and to create lines passing through two previously defined points. Any newly defined point either arises as the result of the
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Place the marked ruler so it goes through A and one end, G, of the marked length falls on ray CF and the other end of the marked length, H, falls on ray CE. Thus GH is the given length.
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935:, who was able to interpret the oracle as the mathematical problem of doubling the volume of a given cube, thus explaining the oracle as the advice of Apollo for the citizens of
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solved the problem in the 4th century BC using geometric construction in three dimensions, determining a certain point as the intersection of three surfaces of revolution.
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1298:(a musical interval caused by doubling the frequency of a tone), and a natural analogue of a cube is dividing the octave into three parts, each the same
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Guilbeau, Lucye (1930). "The
History of the Solution of the Cubic Equation". Mathematics News Letter. 5 (4): 8–12. doi:10.2307/3027812. JSTOR 3027812
1649:"The Algebra of Geometric Impossibility: Descartes and Montucla on the Impossibility of the Duplication of the Cube and the Trisection of the Angle"
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proportionals between a line segment and another with twice the length. In modern notation, this means that given segments of lengths
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of two such circles, as the intersection of a circle and a line, or as the intersection of two lines. An exercise of elementary
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with the unit line segment - so equivalently we may consider the task of constructing a line segment from (0,0) to (
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to occupy themselves with the study of geometry and mathematics in order to calm down their passions.
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Mathologer video: "2000 years unsolved: Why is doubling cubes and squaring circles impossible?"
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of the coordinates of the original pair of points is clearly of degree 1, it follows from the
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generated by a constructible point must be a power of 2. The field extension generated by
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False claims of doubling the cube with compass and straightedge abound in mathematical
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1822:. J. J. O'Connor and E. F. Robertson in the MacTutor History of Mathematics archive.
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using a marked ruler for a length which is the cube root of 2 times another length.
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A significant development in finding a solution to the problem was the discovery by
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Doubling the cube, proximity construction as animation (side = 1.259921049894873)
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over the field generated by the coordinates of previous points, of no greater
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is therefore of degree 3. But this is not a power of 2, so by the above:
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Respectively, the tools of a compass and straightedge allow us to create
1512:. Walter de Gruyter. pp. 84, quoting Plutarch and Theon of Smyrna.
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Construct an equilateral triangle ABC with the given length as side.
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curves. Other more complicated methods of doubling the cube involve
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The problem owes its name to a story concerning the citizens of
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So, given a coordinate of any constructed point, we may proceed
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A unit cube (side = 1) and a cube with twice the volume (side =
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Mark a ruler with the given length; this will eventually be GH.
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Menaechmus' original solution involves the intersection of two
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1069:{\displaystyle {\frac {a}{r}}={\frac {r}{s}}={\frac {s}{2a}}.}
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is double that of the first. As with the related problems of
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Phillips, R. C. (October 1905), "The equal tempered scale",
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The Origin of the
History of Science in Classical Antiquity
93:
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1607:, Dover Books on Mathematics, Courier Dover Publications,
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of 2. The impossibility of doubling the cube is therefore
190:. This is because a cube of side length 1 has a volume of
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Solutions via means other than compass and straightedge
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generated by the previous coordinates. Therefore, the
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1826:To Double a Cube – The Solution of Archytas
910:in order to learn how to defeat a plague sent by
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1779:A pretty accurate solution to the Delian problem
1685:Textual Studies in Ancient and Medieval Geometry
475:corresponding to each new coordinate is 2 or 1.
292:We begin with the unit line segment defined by
1534:"Plutarch, De E apud Delphos, section 6 386.4"
540:of any coordinate of a constructed point is a
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1681:(1989). "Pappus' texts on cube duplication".
1724:100 Great Problems of Elementary Mathematics
518:that the degree of the field extension over
366:, 0), which entails constructing the point (
1604:The Ancient Tradition of Geometric Problems
1374:{\displaystyle 2^{4/12}=2^{1/3}={\sqrt{2}}}
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1207:Origami may also be used to construct the
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1449:Kern, Willis F.; Bland, James R. (1934).
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1397:The Delian problem shows up in Plato's
1246:Extend the line DC forming the line CF.
1243:Extend the line BC forming the line CE.
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3363:Compass and straightedge constructions
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3124:Latin translations of the 12th century
1749:Musical Opinion and Music Trade Review
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1586:Die Kurzdialoge der Appendix Platonica
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1381:, the side length of the Delian cube.
27:Ancient geometric construction problem
2854:Straightedge and compass construction
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1506:Zhmudʹ, Leonid I︠A︡kovlevich (2006).
1240:Extend AB an equal amount again to D.
983:that it is equivalent to finding two
2819:Incircle and excircles of a triangle
1776:Frédéric Beatrix, Peter Katzlinger:
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1119:{\displaystyle r=a\cdot {\sqrt{2}}.}
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1647:Lützen, Jesper (24 January 2010).
1453:. New York: John Wiley & Sons.
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1253:Then AG is the given length times
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1784:Parabola Volume 59 (2023) Issue 1
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1209:cube root of two by folding paper
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1842:Delian Problem Solved. Or Is It?
1665:10.1111/j.1600-0498.2009.00160.x
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1689:. Boston: Birkhäuser. pp.
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134:requires the construction of a
130:In algebraic terms, doubling a
3156:A History of Greek Mathematics
2669:The Quadrature of the Parabola
1831:A History of Greek Mathematics
1634:A History of Greek Mathematics
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1788:University of New South Wales
1451:Solid Mensuration With Proofs
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437:of degree at most 2 over the
2937:Intersecting secants theorem
946:, Plato gave the problem to
742:{\displaystyle \mathbb {Q} }
720:{\displaystyle \mathbb {Q} }
629:{\displaystyle \mathbb {Z} }
582:{\displaystyle \mathbb {Z} }
533:{\displaystyle \mathbb {Q} }
506:{\displaystyle \mathbb {Q} }
458:{\displaystyle \mathbb {R} }
7:
2932:Intersecting chords theorem
2799:Doctrine of proportionality
1967:Quadratic irrational number
1953:Pisot–Vijayaraghavan number
1808:Encyclopedia of Mathematics
1699:10.1007/978-1-4612-3690-0_5
1181:for not providing a proper
918:, however, the citizens of
893:the cube cannot be doubled.
284:, however, is of degree 3.
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2628:On the Sphere and Cylinder
2581:On the Sizes and Distances
1275:{\displaystyle {\sqrt{2}}}
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771:{\displaystyle {\sqrt{2}}}
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3330:Ancient Greece portal
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3134:Philosophy of mathematics
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2104:Ancient Greek mathematics
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1894:
1803:"Duplication of the cube"
1722:Dörrie, Heinrich (1965).
1559:, De genio Socratis 579.B
1079:In turn, this means that
705:is also irreducible over
3373:Euclidean plane geometry
3368:Cubic irrational numbers
3001:Aristarchus's inequality
2574:On Conoids and Spheroids
1143:straightedge and compass
1133:proved in 1837 that the
247:. This implies that the
110:compass and straightedge
63:= 1.2599210498948732...
3109:Ancient Greek astronomy
2922:Inscribed angle theorem
2912:Greek geometric algebra
2567:Measurement of a Circle
296:(0,0) and (1,0) in the
3343:Mathematics portal
3129:Non-Euclidean geometry
3084:Mouseion of Alexandria
2957:Tangent-secant theorem
2907:Geometric mean theorem
2892:Exterior angle theorem
2887:Angle bisector theorem
2591:On Sizes and Distances
2063:Mathematics portal
1726:. Dover. p. 171.
1570:Quaestiones convivales
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288:Proof of impossibility
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202:to the statement that
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3031:Pappus's area theorem
2967:Theorem of the gnomon
2844:Quadratrix of Hippias
2767:Circles of Apollonius
2715:Problem of Apollonius
2693:Constructible numbers
2517:Archimedes Palimpsest
1679:Knorr, Wilbur Richard
1599:Knorr, Wilbur Richard
1538:www.perseus.tufts.edu
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2985:Apollonius's theorem
2824:Method of exhaustion
2794:Diophantine equation
2784:Circumscribed circle
2601:On the Moving Sphere
1909:Constructible number
1584:Carl Werner Müller,
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1215:Using a marked ruler
1179:Pappus of Alexandria
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3378:History of geometry
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3139:Neusis construction
3059:Spiral of Theodorus
2952:Pythagorean theorem
2897:Euclidean algorithm
2839:Lune of Hippocrates
2708:Squaring the circle
2464:Theon of Alexandria
2139:Aristaeus the Elder
2035:Supersilver ratio (
2026:Supergolden ratio (
1228:neusis construction
974:Eutocius of Ascalon
435:minimal polynomials
102:squaring the circle
88:problem. Given the
3383:Unsolvable puzzles
3026:Menelaus's theorem
3016:Irrational numbers
2829:Parallel postulate
2804:Euclidean geometry
2772:Apollonian circles
2314:Isidore of Miletus
1929:Eisenstein integer
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2777:Apollonian gasket
2703:Doubling the cube
2676:The Sand Reckoner
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2044:Twelfth root of 2
1924:Doubling the cube
1914:Conway's constant
1899:Algebraic integer
1888:Algebraic numbers
1820:Doubling the cube
1797:Wikimedia Commons
1519:978-3-11-017966-8
1385:Explanatory notes
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2016:Square root of 6
2011:Square root of 5
2006:Square root of 3
2001:Square root of 2
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1005:
999:
992:
889:
887:
886:
881:
879:
877:
869:
857:
855:
854:
849:
847:
845:
837:
825:
823:
822:
817:
815:
804:
802:
794:
789:
777:
775:
774:
769:
767:
765:
757:
748:
746:
745:
740:
738:
726:
724:
723:
718:
716:
704:
689:
678:
674:
663:
657:
642:
636:
635:
633:
632:
627:
625:
608:
594:would involve a
588:
586:
585:
580:
578:
561:
539:
537:
536:
531:
529:
512:
510:
509:
504:
502:
489:
485:
466:
464:
462:
461:
456:
454:
432:
428:
420:
416:
394:
392:
391:
386:
384:
382:
374:
365:
363:
362:
357:
355:
353:
345:
328:
326:
325:
320:
318:
316:
308:
283:
281:
280:
275:
273:
271:
263:
230:
228:
227:
222:
220:
218:
210:
193:
188:cube root of two
185:
184:
182:
181:
176:
174:
172:
164:
150:
143:
84:, is an ancient
72:
62:
60:
59:
54:
52:
50:
42:
21:
3398:
3397:
3393:
3392:
3391:
3389:
3388:
3387:
3353:
3352:
3351:
3346:
3335:
3324:
3322:
3311:
3277:Arabian/Islamic
3265:
3254:numeral systems
3143:
3093:
3063:
3011:Heron's formula
2989:
2971:
2863:
2859:Triangle center
2849:Regular polygon
2726:and definitions
2725:
2719:
2681:
2661:
2651:
2613:
2603:
2593:
2583:
2559:
2549:
2532:
2498:
2469:Theon of Smyrna
2114:
2106:
2101:
2071:
2066:
2055:
2048:
2036:
2027:
1995:
1992:
1988:
1972:Rational number
1959:
1958:Plastic ratio (
1940:
1904:Chebyshev nodes
1890:
1885:
1828:. Excerpt from
1801:
1773:
1768:
1767:
1745:
1741:
1734:
1720:
1716:
1709:
1676:
1672:
1645:
1641:
1630:
1626:
1619:
1596:
1592:
1583:
1579:
1567:
1563:
1555:
1551:
1542:
1540:
1532:
1531:
1527:
1520:
1504:
1500:
1495:
1491:
1486:
1482:
1477:
1473:
1462:
1458:
1447:
1440:
1435:
1430:
1429:
1418:
1414:
1407:
1396:
1392:
1387:
1364:
1359:
1346:
1342:
1338:
1325:
1321:
1317:
1315:
1312:
1311:
1288:
1286:In music theory
1265:
1260:
1258:
1255:
1254:
1217:
1151:
1106:
1101:
1087:
1084:
1083:
1053:
1048:
1035:
1022:
1020:
1017:
1016:
1007:
1001:
994:
988:
914:. According to
900:
873:
868:
866:
863:
862:
841:
836:
834:
831:
830:
811:
798:
793:
785:
783:
780:
779:
761:
756:
754:
751:
750:
734:
732:
729:
728:
712:
710:
707:
706:
695:
680:
676:
669:
659:
648:
638:
621:
619:
616:
615:
610:
598:
574:
572:
569:
568:
548:
525:
523:
520:
519:
498:
496:
493:
492:
487:
483:
473:field extension
450:
448:
445:
444:
442:
430:
426:
418:
414:
378:
373:
371:
368:
367:
349:
344:
342:
339:
338:
312:
307:
305:
302:
301:
290:
267:
262:
260:
257:
256:
253:field extension
214:
209:
207:
204:
203:
191:
168:
163:
161:
158:
157:
152:
145:
139:
64:
46:
41:
39:
36:
35:
28:
23:
22:
18:Double the cube
15:
12:
11:
5:
3396:
3386:
3385:
3380:
3375:
3370:
3365:
3348:
3347:
3320:
3317:
3316:
3313:
3312:
3310:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3273:
3271:
3270:Other cultures
3267:
3266:
3264:
3263:
3262:
3261:
3251:
3250:
3249:
3239:
3238:
3237:
3227:
3226:
3225:
3215:
3214:
3213:
3203:
3202:
3201:
3191:
3190:
3189:
3179:
3178:
3177:
3167:
3166:
3165:
3151:
3149:
3145:
3144:
3142:
3141:
3136:
3131:
3126:
3121:
3119:Greek numerals
3116:
3114:Attic numerals
3111:
3105:
3099:
3095:
3094:
3092:
3091:
3086:
3081:
3075:
3073:
3069:
3068:
3065:
3064:
3062:
3061:
3056:
3051:
3046:
3041:
3033:
3028:
3023:
3018:
3013:
3008:
3003:
2997:
2995:
2991:
2990:
2988:
2987:
2981:
2979:
2973:
2972:
2970:
2969:
2964:
2959:
2954:
2949:
2944:
2942:Law of cosines
2939:
2934:
2929:
2924:
2919:
2914:
2909:
2904:
2899:
2894:
2889:
2883:
2881:
2869:
2865:
2864:
2862:
2861:
2856:
2851:
2846:
2841:
2836:
2834:Platonic solid
2831:
2826:
2821:
2816:
2814:Greek numerals
2811:
2806:
2801:
2796:
2791:
2786:
2781:
2780:
2779:
2774:
2764:
2759:
2758:
2757:
2747:
2746:
2745:
2740:
2729:
2727:
2721:
2720:
2718:
2717:
2712:
2711:
2710:
2705:
2700:
2689:
2687:
2683:
2682:
2680:
2679:
2672:
2665:
2655:
2645:
2642:Planisphaerium
2638:
2631:
2624:
2617:
2607:
2597:
2587:
2577:
2570:
2563:
2553:
2543:
2536:
2526:
2519:
2514:
2506:
2504:
2500:
2499:
2497:
2496:
2491:
2486:
2481:
2476:
2471:
2466:
2461:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2416:
2411:
2406:
2401:
2396:
2391:
2386:
2381:
2376:
2371:
2366:
2361:
2356:
2351:
2346:
2341:
2336:
2331:
2326:
2321:
2316:
2311:
2306:
2301:
2296:
2291:
2286:
2281:
2276:
2271:
2266:
2261:
2256:
2251:
2246:
2241:
2236:
2231:
2226:
2221:
2216:
2211:
2206:
2201:
2196:
2191:
2186:
2181:
2176:
2171:
2166:
2161:
2156:
2151:
2146:
2141:
2136:
2131:
2126:
2120:
2118:
2112:Mathematicians
2108:
2107:
2100:
2099:
2092:
2085:
2077:
2068:
2067:
2053:
2050:
2049:
2047:
2046:
2041:
2032:
2023:
2018:
2013:
2008:
2003:
1998:
1991:
1987:Silver ratio (
1984:
1979:
1974:
1969:
1964:
1955:
1950:
1945:
1939:Golden ratio (
1936:
1931:
1926:
1921:
1916:
1911:
1906:
1901:
1895:
1892:
1891:
1884:
1883:
1876:
1869:
1861:
1855:
1854:
1849:
1839:
1823:
1817:
1799:
1790:
1772:
1771:External links
1769:
1766:
1765:
1755:(337): 41–42,
1739:
1732:
1714:
1707:
1670:
1639:
1624:
1617:
1590:
1577:
1561:
1549:
1525:
1518:
1498:
1489:
1480:
1471:
1464:Stewart, Ian.
1456:
1437:
1436:
1434:
1431:
1428:
1427:
1412:
1389:
1388:
1386:
1383:
1367:
1363:
1358:
1353:
1349:
1345:
1341:
1337:
1332:
1328:
1324:
1320:
1287:
1284:
1268:
1264:
1251:
1250:
1247:
1244:
1241:
1238:
1235:
1216:
1213:
1150:
1147:
1131:Pierre Wantzel
1127:
1126:
1115:
1109:
1105:
1100:
1097:
1094:
1091:
1077:
1076:
1065:
1059:
1056:
1052:
1047:
1042:
1039:
1034:
1029:
1026:
985:geometric mean
922:consulted the
899:
896:
895:
894:
891:
876:
872:
859:
844:
840:
814:
810:
807:
801:
797:
792:
788:
764:
760:
737:
715:
624:
577:
528:
501:
453:
381:
377:
352:
348:
315:
311:
289:
286:
270:
266:
217:
213:
171:
167:
125:Pierre Wantzel
82:Delian problem
49:
45:
26:
9:
6:
4:
3:
2:
3395:
3384:
3381:
3379:
3376:
3374:
3371:
3369:
3366:
3364:
3361:
3360:
3358:
3345:
3344:
3339:
3332:
3331:
3318:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3274:
3272:
3268:
3260:
3257:
3256:
3255:
3252:
3248:
3245:
3244:
3243:
3240:
3236:
3233:
3232:
3231:
3228:
3224:
3221:
3220:
3219:
3216:
3212:
3209:
3208:
3207:
3204:
3200:
3197:
3196:
3195:
3192:
3188:
3185:
3184:
3183:
3180:
3176:
3173:
3172:
3171:
3168:
3164:
3160:
3159:
3158:
3157:
3153:
3152:
3150:
3146:
3140:
3137:
3135:
3132:
3130:
3127:
3125:
3122:
3120:
3117:
3115:
3112:
3110:
3107:
3106:
3103:
3100:
3096:
3090:
3087:
3085:
3082:
3080:
3077:
3076:
3074:
3070:
3060:
3057:
3055:
3052:
3050:
3047:
3045:
3042:
3040:
3039:
3034:
3032:
3029:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3007:
3004:
3002:
2999:
2998:
2996:
2992:
2986:
2983:
2982:
2980:
2978:
2974:
2968:
2965:
2963:
2960:
2958:
2955:
2953:
2950:
2948:
2947:Pons asinorum
2945:
2943:
2940:
2938:
2935:
2933:
2930:
2928:
2925:
2923:
2920:
2918:
2917:Hinge theorem
2915:
2913:
2910:
2908:
2905:
2903:
2900:
2898:
2895:
2893:
2890:
2888:
2885:
2884:
2882:
2880:
2879:
2873:
2870:
2866:
2860:
2857:
2855:
2852:
2850:
2847:
2845:
2842:
2840:
2837:
2835:
2832:
2830:
2827:
2825:
2822:
2820:
2817:
2815:
2812:
2810:
2807:
2805:
2802:
2800:
2797:
2795:
2792:
2790:
2787:
2785:
2782:
2778:
2775:
2773:
2770:
2769:
2768:
2765:
2763:
2760:
2756:
2753:
2752:
2751:
2748:
2744:
2741:
2739:
2736:
2735:
2734:
2731:
2730:
2728:
2722:
2716:
2713:
2709:
2706:
2704:
2701:
2699:
2696:
2695:
2694:
2691:
2690:
2688:
2684:
2678:
2677:
2673:
2671:
2670:
2666:
2664:
2660:
2656:
2654:
2650:
2646:
2644:
2643:
2639:
2637:
2636:
2632:
2630:
2629:
2625:
2623:
2622:
2618:
2616:
2612:
2608:
2606:
2602:
2598:
2596:
2592:
2588:
2586:
2584:(Aristarchus)
2582:
2578:
2576:
2575:
2571:
2569:
2568:
2564:
2562:
2558:
2554:
2552:
2548:
2544:
2542:
2541:
2537:
2535:
2531:
2527:
2525:
2524:
2520:
2518:
2515:
2513:
2512:
2508:
2507:
2505:
2501:
2495:
2492:
2490:
2489:Zeno of Sidon
2487:
2485:
2482:
2480:
2477:
2475:
2472:
2470:
2467:
2465:
2462:
2460:
2457:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2415:
2412:
2410:
2407:
2405:
2402:
2400:
2397:
2395:
2392:
2390:
2387:
2385:
2382:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2280:
2277:
2275:
2272:
2270:
2267:
2265:
2262:
2260:
2257:
2255:
2252:
2250:
2247:
2245:
2242:
2240:
2237:
2235:
2232:
2230:
2227:
2225:
2222:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2195:
2192:
2190:
2187:
2185:
2182:
2180:
2177:
2175:
2172:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2125:
2122:
2121:
2119:
2117:
2113:
2109:
2105:
2098:
2093:
2091:
2086:
2084:
2079:
2078:
2075:
2065:
2064:
2059:
2051:
2045:
2042:
2040:
2033:
2031:
2024:
2022:
2019:
2017:
2014:
2012:
2009:
2007:
2004:
2002:
1999:
1997:
1985:
1983:
1980:
1978:
1977:Root of unity
1975:
1973:
1970:
1968:
1965:
1963:
1956:
1954:
1951:
1949:
1948:Perron number
1946:
1944:
1937:
1935:
1932:
1930:
1927:
1925:
1922:
1920:
1917:
1915:
1912:
1910:
1907:
1905:
1902:
1900:
1897:
1896:
1893:
1889:
1882:
1877:
1875:
1870:
1868:
1863:
1862:
1859:
1853:
1850:
1847:
1843:
1840:
1837:
1833:
1832:
1827:
1824:
1821:
1818:
1814:
1810:
1809:
1804:
1800:
1798:
1794:
1791:
1789:
1785:
1781:
1780:
1775:
1774:
1762:
1758:
1754:
1750:
1743:
1735:
1729:
1725:
1718:
1710:
1708:9780817633875
1704:
1700:
1696:
1692:
1687:
1686:
1680:
1674:
1666:
1662:
1658:
1654:
1650:
1643:
1636:
1635:
1628:
1620:
1618:9780486675329
1614:
1610:
1606:
1605:
1600:
1594:
1587:
1581:
1574:
1571:
1565:
1558:
1553:
1539:
1535:
1529:
1521:
1515:
1511:
1510:
1502:
1493:
1484:
1475:
1468:. p. 75.
1467:
1466:Galois Theory
1460:
1452:
1445:
1443:
1438:
1424:
1423:
1416:
1408: 380 BC
1402:
1401:
1394:
1390:
1382:
1365:
1361:
1356:
1351:
1347:
1343:
1339:
1335:
1330:
1326:
1322:
1318:
1309:
1305:
1301:
1297:
1293:
1283:
1266:
1262:
1248:
1245:
1242:
1239:
1236:
1233:
1232:
1231:
1229:
1221:
1212:
1210:
1205:
1203:
1199:
1194:
1190:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1146:
1144:
1140:
1139:constructible
1136:
1132:
1113:
1107:
1103:
1098:
1095:
1092:
1089:
1082:
1081:
1080:
1063:
1057:
1054:
1050:
1045:
1040:
1037:
1032:
1027:
1024:
1015:
1014:
1013:
1010:
1004:
998:
991:
986:
982:
977:
975:
971:
967:
966:
961:
960:pure geometry
957:
953:
949:
945:
942:According to
940:
938:
934:
929:
925:
921:
917:
913:
909:
905:
892:
874:
870:
860:
842:
838:
829:
828:
827:
808:
799:
795:
762:
758:
702:
698:
693:
692:Gauss's Lemma
687:
683:
672:
667:
662:
655:
651:
646:
641:
613:
606:
602:
597:
596:linear factor
593:
592:factorisation
589:
565:
559:
555:
551:
545:
543:
517:
513:
481:
476:
474:
470:
465:
440:
436:
424:
412:
408:
404:
401:
396:
379:
375:
350:
346:
336:
332:
313:
309:
299:
295:
285:
268:
264:
254:
250:
246:
242:
238:
234:
215:
211:
201:
197:
189:
169:
165:
155:
148:
142:
137:
133:
128:
126:
122:
118:
113:
111:
107:
103:
99:
95:
91:
87:
83:
79:
71:
67:
47:
43:
32:
19:
3334:
3321:
3163:Thomas Heath
3154:
3037:
3021:Law of sines
2877:
2809:Golden ratio
2702:
2674:
2667:
2658:
2652:(Theodosius)
2648:
2640:
2633:
2626:
2619:
2610:
2600:
2594:(Hipparchus)
2590:
2580:
2572:
2565:
2556:
2546:
2538:
2533:(Apollonius)
2529:
2521:
2509:
2484:Zeno of Elea
2244:Eratosthenes
2234:Dionysodorus
2054:
1982:Salem number
1923:
1846:cut-the-knot
1829:
1806:
1777:
1752:
1748:
1742:
1733:0486-61348-8
1723:
1717:
1684:
1673:
1656:
1652:
1642:
1632:
1627:
1603:
1593:
1585:
1580:
1569:
1564:
1552:
1541:. Retrieved
1537:
1528:
1508:
1501:
1492:
1483:
1474:
1465:
1459:
1450:
1420:
1415:
1398:
1393:
1292:music theory
1289:
1252:
1225:
1206:
1200:literature (
1195:
1191:
1152:
1137:of 2 is not
1128:
1078:
1008:
1002:
996:
989:
978:
970:Eratosthenes
963:
941:
901:
700:
696:
685:
681:
670:
668:); that is,
660:
653:
649:
639:
611:
604:
600:
557:
553:
549:
546:
477:
423:coefficients
407:intersection
397:
291:
187:
153:
146:
140:
136:line segment
129:
114:
81:
77:
76:
3230:mathematics
3038:Arithmetica
2635:Ostomachion
2604:(Autolycus)
2523:Arithmetica
2299:Hippocrates
2229:Dinostratus
2214:Dicaearchus
2144:Aristarchus
1659:(1): 4–37.
1631:T.L. Heath
1304:major third
658:; but also
564:irreducible
480:inductively
237:polynomials
3357:Categories
3282:Babylonian
3182:arithmetic
3148:History of
2977:Apollonius
2662:(Menelaus)
2621:On Spirals
2540:Catoptrics
2479:Xenocrates
2474:Thymaridas
2459:Theodosius
2444:Theaetetus
2424:Simplicius
2414:Pythagoras
2399:Posidonius
2384:Philonides
2344:Nicomachus
2339:Metrodorus
2329:Menaechmus
2284:Hipparchus
2274:Heliodorus
2224:Diophantus
2209:Democritus
2189:Chrysippus
2159:Archimedes
2154:Apollonius
2124:Anaxagoras
2116:(timeline)
1543:2024-09-17
1433:References
1175:Pandrosion
1171:Philo line
956:Menaechmus
673:= 1, 2, −1
643:must be a
542:power of 2
516:tower rule
200:equivalent
138:of length
2743:Inscribed
2503:Treatises
2494:Zenodorus
2454:Theodorus
2429:Sosigenes
2374:Philolaus
2359:Oenopides
2354:Nicoteles
2349:Nicomedes
2309:Hypsicles
2204:Ctesibius
2194:Cleomedes
2179:Callippus
2164:Autolycus
2149:Aristotle
2129:Anthemius
1813:EMS Press
1653:Centaurus
1410:) VII.530
1169:, or the
1135:cube root
1099:⋅
637:, and so
609:for some
245:quadratic
231:is not a
196:cube root
132:unit cube
127:in 1837.
117:Egyptians
86:geometric
3307:Japanese
3292:Egyptian
3235:timeline
3223:timeline
3211:timeline
3206:geometry
3199:timeline
3194:calculus
3187:timeline
3175:timeline
2878:Elements
2724:Concepts
2686:Problems
2659:Spherics
2649:Spherics
2614:(Euclid)
2560:(Euclid)
2557:Elements
2550:(Euclid)
2511:Almagest
2419:Serenus
2394:Porphyry
2334:Menelaus
2289:Hippasus
2264:Eutocius
2239:Domninus
2134:Archytas
1757:ProQuest
1637:, Vol. 1
1601:(1986),
1575:, 718ef)
1568:(Plut.,
1557:Plutarch
1422:Republic
1419:Plato's
1400:Republic
1300:interval
1187:Archytas
1012:so that
965:Sisyphus
952:Archytas
944:Plutarch
916:Plutarch
439:subfield
335:parallel
144:, where
3287:Chinese
3242:numbers
3170:algebra
3098:Related
3072:Centers
2868:Results
2738:Central
2409:Ptolemy
2404:Proclus
2369:Perseus
2324:Marinus
2304:Hypatia
2294:Hippias
2269:Geminus
2259:Eudoxus
2249:Eudemus
2219:Diocles
1815:, 2001
1761:7191936
1573:VIII.ii
948:Eudoxus
898:History
560:− 2 = 0
471:of the
403:centred
400:circles
251:of the
243:than a
121:Indians
70:A002580
68::
3302:Indian
3079:Cyrene
2611:Optics
2530:Conics
2449:Theano
2439:Thales
2434:Sporus
2379:Philon
2364:Pappus
2254:Euclid
2184:Carpus
2174:Bryson
1782:. In:
1759:
1730:
1705:
1615:
1516:
1296:octave
1165:, the
1161:, the
1159:neusis
928:Delphi
924:oracle
912:Apollo
908:Delphi
590:– any
486:- and
469:degree
429:- and
417:- and
395:, 0).
331:origin
294:points
249:degree
241:degree
186:, the
98:volume
3297:Incan
3218:logic
2994:Other
2762:Chord
2755:Axiom
2733:Angle
2389:Plato
2279:Heron
2199:Conon
1691:63–76
1198:crank
1155:conic
937:Delos
933:Plato
920:Delos
904:Delos
690:. By
566:over
547:Now,
298:plane
192:1 = 1
92:of a
3259:list
2547:Data
2319:Leon
2169:Bion
1728:ISBN
1703:ISBN
1613:ISBN
1609:p. 4
1514:ISBN
1129:But
1006:and
993:and
954:and
950:and
749:for
645:root
556:) =
115:The
104:and
94:cube
90:edge
66:OEIS
3161:by
2875:In
1844:at
1834:by
1695:doi
1661:doi
1306:in
1290:In
1204:).
972:by
926:at
675:or
647:of
441:of
149:= 2
3359::
1811:,
1805:,
1753:29
1751:,
1701:.
1693:.
1657:52
1655:.
1651:.
1611:,
1536:.
1441:^
1405:c.
1331:12
1282:.
1211:.
1185:.
1173:.
1145:.
694:,
677:−2
614:∈
603:−
544:.
333:,
156:=
119:,
73:).
2096:e
2089:t
2082:v
2039:)
2037:ς
2030:)
2028:ψ
1996:)
1993:S
1989:δ
1962:)
1960:ρ
1943:)
1941:φ
1880:e
1873:t
1866:v
1848:.
1838:.
1795:—
1736:.
1711:.
1697::
1667:.
1663::
1622:.
1546:.
1522:.
1403:(
1366:3
1362:2
1357:=
1352:3
1348:/
1344:1
1340:2
1336:=
1327:/
1323:4
1319:2
1267:3
1263:2
1114:.
1108:3
1104:2
1096:a
1093:=
1090:r
1064:.
1058:a
1055:2
1051:s
1046:=
1041:s
1038:r
1033:=
1028:r
1025:a
1009:s
1003:r
997:a
995:2
990:a
875:3
871:2
843:3
839:2
813:Q
809::
806:)
800:3
796:2
791:(
787:Q
763:3
759:2
736:Q
714:Q
703:)
701:x
699:(
697:p
688:)
686:x
684:(
682:p
671:k
661:k
656:)
654:x
652:(
650:p
640:k
623:Z
612:k
607:)
605:k
601:x
599:(
576:Z
558:x
554:x
552:(
550:p
527:Q
500:Q
488:y
484:x
452:R
431:y
427:x
419:y
415:x
380:3
376:2
351:3
347:2
314:3
310:2
269:3
265:2
216:3
212:2
170:3
166:2
154:x
147:x
141:x
48:3
44:2
20:)
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