2960:
632:
3076:
27:
3180:
1696:. Archimedes could also find the volume of the cone using the mechanical method, since, in modern terms, the integral involved is exactly the same as the one for area of the parabola. The volume of the cone is 1/3 its base area times the height. The base of the cone is a circle of radius 2, with area
1646:
ranges from 0 to 2, the cylinder will have a center of gravity a distance 1 from the fulcrum, so all the weight of the cylinder can be considered to be at position 1. The condition of balance ensures that the volume of the cone plus the volume of the sphere is equal to the volume of the cylinder.
1854:
Archimedes states that the total volume of the sphere is equal to the volume of a cone whose base has the same surface area as the sphere and whose height is the radius. There are no details given for the argument, but the obvious reason is that the cone can be divided into infinitesimal cones by
2016:
Archimedes emphasizes this in the beginning of the treatise, and invites the reader to try to reproduce the results by some other method. Unlike the other examples, the volume of these shapes is not rigorously computed in any of his other works. From fragments in the palimpsest, it appears that
1836:
Archimedes argument is nearly identical to the argument above, but his cylinder had a bigger radius, so that the cone and the cylinder hung at a greater distance from the fulcrum. He considered this argument to be his greatest achievement, requesting that the accompanying figure of the balanced
2013:, despite the shapes having curvilinear boundaries. This is a central point of the investigation—certain curvilinear shapes could be rectified by ruler and compass, so that there are nontrivial rational relations between the volumes defined by the intersections of geometrical solids.
909:, so that the total effect of the triangle on the lever is as if the total mass of the triangle were pushing down on (or hanging from) this point. The total torque exerted by the triangle is its area, 1/2, times the distance 2/3 of its center of mass from the fulcrum at
170:
Archimedes' idea is to use the law of the lever to determine the areas of figures from the known center of mass of other figures. The simplest example in modern language is the area of the parabola. A modern approach would be to find this area by calculating the integral
2402:
For the intersection of two cylinders, the slicing is lost in the manuscript, but it can be reconstructed in an obvious way in parallel to the rest of the document: if the x-z plane is the slice direction, the equations for the cylinder give that
157:
Archimedes did not admit the method of indivisibles as part of rigorous mathematics, and therefore did not publish his method in the formal treatises that contain the results. In these treatises, he proves the same theorems by
1851:), the volume of the sphere could be thought of as divided into many cones with height equal to the radius and base on the surface. The cones all have the same height, so their volume is 1/3 the base area times the height.
2362:
162:, finding rigorous upper and lower bounds which both converge to the answer required. Nevertheless, the mechanical method was what he used to discover the relations for which he later gave rigorous proofs.
244:. Instead, the Archimedian method mechanically balances the parabola (the curved region being integrated above) with a certain triangle that is made of the same material. The parabola is the region in the
1821:
2613:
2768:
2116:
1845:
To find the surface area of the sphere, Archimedes argued that just as the area of the circle could be thought of as infinitely many infinitesimal right triangles going around the circumference (see
1394:
236:
1825:
The dependence of the volume of the sphere on the radius is obvious from scaling, although that also was not trivial to make rigorous back then. The method then gives the familiar formula for the
2184:
3378:
1313:
1552:
2286:
1009:
2541:
2237:
2493:
2447:
1494:
1940:
1604:
935:. This torque of 1/3 balances the parabola, which is at a distance 1 from the fulcrum. Hence, the area of the parabola must be 1/3 to give it the opposite torque.
1973:
1451:
1245:
2397:
1748:
327:
1717:
1694:
1671:
1629:
983:
907:
873:
841:
597:
274:
2011:
1899:
695:
666:
626:
933:
747:
721:
482:
393:
956:
794:
774:
570:
550:
530:
510:
456:
436:
413:
367:
347:
294:
938:
This type of method can be used to find the area of an arbitrary section of a parabola, and similar arguments can be used to find the integral of any power of
776:, the triangle will balance on the median, considered as a fulcrum. The reason is that if the triangle is divided into infinitesimal line segments parallel to
2776:
1216:
Again, to illuminate the mechanical method, it is convenient to use a little bit of coordinate geometry. If a sphere of radius 1 is placed with its center at
2291:
2631:
A series of propositions of geometry are proved in the palimpsest by similar arguments. One theorem is that the location of a center of mass of a
1755:
2546:
3383:
2039:
796:, each segment has equal length on opposite sides of the median, so balance follows by symmetry. This argument can be easily made rigorous by
1322:
176:
809:
So the center of mass of a triangle must be at the intersection point of the medians. For the triangle in question, one median is the line
2635:
is located 5/8 of the way from the pole to the center of the sphere. This problem is notable, because it is evaluating a cubic integral.
2364:
which can be easily rectified using the mechanical method. Adding to each triangular section a section of a triangular pyramid with area
2121:
2017:
Archimedes did inscribe and circumscribe shapes to prove rigorous bounds for the volume, although the details have not been preserved.
1608:
If the two slices are placed together at distance 1 from the fulcrum, their total weight would be exactly balanced by a circle of area
1169:
where they intersect the lever, then they exert the same torque on the lever as does the whole weight of the triangle resting at
1254:
3112:
1503:
1855:
splitting the base area up, and then each cone makes a contribution according to its base area, just the same as in the sphere.
3020:
2853:
2186:
Both problems have a slicing which produces an easy integral for the mechanical method. For the circular prism, cut up the
1635:
from the fulcrum on the other side. This means that the cone and the sphere together, if all their material were moved to
3326:
3219:
3150:
70:
1457:
52:
1110: = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at
1565:
668:
would balance the whole triangle. This means that if the original uncut parabola is hung by a hook from the point
3209:
985:, which he used to find the center of mass of a hemisphere, and in other work, the center of mass of a parabola.
697:(so that the whole mass of the parabola is attached to that point), it will balance the triangle sitting between
2979:
37:
136:. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the
3214:
958:, although higher powers become complicated without algebra. Archimedes only went as far as the integral of
3419:
3409:
3291:
3105:
2891:
802:
752:
The center of mass of a triangle can be easily found by the following method, also due to
Archimedes. If a
150:
44:
2242:
3424:
3224:
2506:
3240:
2905:
2898:
2705:
2205:
1977:
1098:
as its fulcrum. As
Archimedes had previously shown, the center of mass of the triangle is at the point
2452:
2406:
3342:
1847:
1750:. Subtracting the volume of the cone from the volume of the cylinder gives the volume of the sphere:
1993:
is that
Archimedes finds two shapes defined by sections of cylinders, whose volume does not involve
3169:
2984:
2919:
2846:
1317:
The mass of this cross section, for purposes of balancing on a lever, is proportional to the area:
1904:
492:, where an object's torque equals its weight times its distance to the fulcrum. For each value of
3098:
3010:
2877:
48:
2697:
The method of
Archimedes recently discovered by Heiberg; a supplement to the Works of Archimedes
1945:
875:. Solving these equations, we see that the intersection of these two medians is above the point
3271:
2649:
1429:
1223:
800:
by using little rectangles instead of infinitesimal lines, and this is what
Archimedes does in
125:
132:). The work was originally thought to be lost, but in 1906 was rediscovered in the celebrated
3357:
3140:
3015:
2989:
2644:
2367:
1722:
299:
133:
121:
3199:
3194:
2822:
2799:
2738:
2654:
1699:
1676:
1653:
1611:
1019:
is parallel to the axis of symmetry of the parabola. Further suppose that the line segment
961:
878:
846:
812:
797:
575:
488:
states that two objects on opposite sides of the fulcrum will balance if each has the same
247:
159:
1996:
1873:
671:
642:
602:
8:
3414:
3204:
3079:
3061:
3051:
2839:
2701:
912:
726:
700:
572:
from the fulcrum; so it would balance the corresponding slice of the parabola, of height
461:
372:
3388:
3276:
3056:
2994:
2926:
2020:
The two shapes he considers are the intersection of two cylinders at right angles (the
1826:
941:
779:
759:
555:
535:
515:
495:
441:
421:
398:
352:
332:
279:
3311:
3301:
3286:
3046:
3025:
2974:
2884:
1829:. By scaling the dimensions linearly Archimedes easily extended the volume result to
753:
241:
2769:"Archimedes' Method for Computing Areas and Volumes - Cylinders, Cones, and Spheres"
3352:
3347:
3245:
2808:
349:
varies from 0 to 1. The triangle is the region in the same plane between the
3321:
3306:
3145:
2959:
2818:
2734:
2632:
1121:
Consider an infinitely small cross-section of the triangle given by the segment
3362:
3255:
2947:
2695:
1639:= 1, would balance a cylinder of base radius 1 and length 2 on the other side.
631:
137:
103:
89:
1189:, then the lever is in equilibrium. In other words, it suffices to show that
1145:
is parallel to the axis of symmetry of the parabola. Call the intersection of
485:
145:
3403:
3121:
2790:
2616:
129:
1426:-axis, to form a cone. The cross section of this cone is a circle of radius
418:
Slice the parabola and triangle into vertical slices, one for each value of
3296:
3281:
2813:
2620:
1975:, or "four times its largest circle". Archimedes proves this rigorously in
1205:. But that is a routine consequence of the equation of the parabola.
117:
997:
in the figure to the right. Pick two points on the parabola and call them
3250:
3156:
2933:
2725:
Proposition 14: preliminary evidence from the
Archimedes palimpsest, I",
2718:
1040:
756:
is drawn from any one of the vertices of a triangle to the opposite edge
3135:
2912:
2862:
2691:
2619:
argues that, besides the volume of the bicylinder, Archimedes knew its
2021:
109:
3316:
1830:
2357:{\displaystyle \displaystyle \int _{-1}^{1}{1 \over 2}(1-x^{2})\,dx}
55:. Statements consisting only of original research should be removed.
994:
141:
106:
3179:
1560:
are to be weighed together, the combined cross-sectional area is:
1398:
Archimedes then considered rotating the triangular region between
3090:
1024:
639:
Since each pair of slices balances, moving the whole parabola to
1173:. Thus, we wish to show that if the weight of the cross-section
1039:
is exactly three times the area bounded by the parabola and the
3041:
1206:
1008:
489:
2831:
2721:; Saito, Ken; Tchernetska, Natalie (2001), "A new reading of
2793:(2002), "The surface area of the bicylinder and Archimedes'
1816:{\displaystyle V_{S}=4\pi -{8 \over 3}\pi ={4 \over 3}\pi .}
2615:
And this is the same integral as for the previous example.
2608:{\displaystyle \displaystyle \int _{-1}^{1}4(1-y^{2})\,dy.}
1984:
1837:
sphere, cone, and cylinder be engraved upon his tombstone.
1114:, and the whole weight of the section of the parabola at
2754:
Mathematical thought from ancient to modern times, vol 1
2111:{\displaystyle x^{2}+y^{2}<1,\quad y^{2}+z^{2}<1,}
2626:
628:, at a distance of 1 on the other side of the fulcrum.
1650:
The volume of the cylinder is the cross section area,
1389:{\displaystyle \pi \rho _{S}(x)^{2}=2\pi x-\pi x^{2}.}
988:
635:
Balanced triangle and parabolic spandrel by the Method
231:{\displaystyle \int _{0}^{1}x^{2}\,dx={\frac {1}{3}},}
2550:
2549:
2509:
2455:
2409:
2370:
2295:
2294:
2245:
2208:
2124:
2118:
and the circular prism, which is the region obeying:
2042:
1999:
1948:
1907:
1876:
1758:
1725:
1702:
1679:
1656:
1614:
1568:
1506:
1460:
1432:
1325:
1257:
1226:
964:
944:
915:
881:
849:
815:
782:
762:
729:
703:
674:
645:
605:
578:
558:
538:
518:
498:
464:
444:
424:
401:
375:
355:
335:
302:
282:
250:
179:
2717:
1942:. Therefore, the surface area of the sphere must be
1220: = 1, the vertical cross sectional radius
2766:
2202:is the interior of a right triangle of side length
2179:{\displaystyle x^{2}+y^{2}<1,\quad 0<z<y.}
1251:between 0 and 2 is given by the following formula:
2607:
2535:
2495:, which defines a region which is a square in the
2487:
2441:
2399:balances a prism whose cross section is constant.
2391:
2356:
2280:
2231:
2178:
2110:
2005:
1967:
1934:
1893:
1815:
1742:
1711:
1688:
1665:
1623:
1598:
1546:
1488:
1445:
1388:
1307:
1239:
977:
950:
927:
901:
867:
835:
788:
768:
741:
715:
689:
660:
620:
591:
564:
544:
524:
504:
476:
450:
430:
407:
387:
361:
341:
321:
288:
268:
230:
124:, and contains the first attested explicit use of
3401:
94:Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος
1308:{\displaystyle \rho _{S}(x)={\sqrt {x(2-x)}}.}
116:takes the form of a letter from Archimedes to
3106:
2847:
1901:, which must equal the volume of the sphere:
102:, is one of the major surviving works of the
2751:
2686:
2684:
2682:
2680:
2678:
2676:
2674:
2672:
2670:
1547:{\displaystyle \pi \rho _{C}^{2}=\pi x^{2}.}
2756:. Oxford University Press. pp. 110–12.
1840:
148:, which were demonstrated by Archimedes in
3113:
3099:
2854:
2840:
2690:
2812:
2789:
2667:
2594:
2346:
205:
71:Learn how and when to remove this message
2783:
1985:Curvilinear shapes with rational volumes
1862:. The volume of the cone with base area
1719:, while the height is 2, so the area is
1556:So if slices of the cone and the sphere
1185:of the section of the parabola rests at
630:
512:, the slice of the triangle at position
128:(indivisibles are geometric versions of
1989:One of the remarkable things about the
3402:
1858:Let the surface of the sphere be
1498:and the area of this cross section is
3225:Infinitesimal strain theory (physics)
3094:
3021:List of things named after Archimedes
2835:
2190:-axis into slices. The region in the
1211:
1161:. If the weight of all such segments
165:
2627:Other propositions in the palimpsest
2281:{\displaystyle {1 \over 2}(1-x^{2})}
1181:and the weight of the cross-section
843:, while a second median is the line
458:-axis is a lever, with a fulcrum at
20:
2711:
989:First proposition in the palimpsest
16:Mathematical treatise by Archimedes
13:
3120:
2536:{\displaystyle 2{\sqrt {1-y^{2}}}}
1007:
14:
3436:
3327:Transcendental law of homogeneity
3220:Constructive nonstandard analysis
3164:The Method of Mechanical Theorems
3151:Criticism of nonstandard analysis
2941:The Method of Mechanical Theorems
2232:{\displaystyle {\sqrt {1-x^{2}}}}
1673:times the height, which is 2, or
240:which is an elementary result in
85:The Method of Mechanical Theorems
3178:
3075:
3074:
2958:
2488:{\displaystyle z^{2}<1-y^{2}}
2442:{\displaystyle x^{2}<1-y^{2}}
1090:. We will think of the segment
1031:. The first proposition states:
25:
3210:Synthetic differential geometry
2543:, so that the total volume is:
2288:, so that the total volume is:
2157:
2075:
1118:, the lever is in equilibrium.
532:has a mass equal to its height
2861:
2760:
2744:
2591:
2572:
2343:
2324:
2275:
2256:
1578:
1572:
1489:{\displaystyle \rho _{C}(x)=x}
1477:
1471:
1346:
1339:
1297:
1285:
1274:
1268:
1082:is equal to the distance from
599:, if the latter were moved to
263:
251:
1:
3379:Analyse des Infiniment Petits
3215:Smooth infinitesimal analysis
2660:
120:, the chief librarian at the
2892:On the Equilibrium of Planes
2767:Gabriela R. Sanchis (2016).
1935:{\displaystyle 4\pi r^{3}/3}
1599:{\displaystyle M(x)=2\pi x.}
803:On the Equilibrium of Planes
151:On the Equilibrium of Planes
7:
2980:Archimedes's cattle problem
2638:
2024:), which is the region of (
1066:. Construct a line segment
51:the claims made and adding
10:
3441:
2968:Discoveries and inventions
2906:On the Sphere and Cylinder
2899:Quadrature of the Parabola
2706:Cambridge University Press
2623:, which is also rational.
1978:On the Sphere and Cylinder
1968:{\displaystyle 4\pi r^{2}}
1074:, where the distance from
3371:
3343:Gottfried Wilhelm Leibniz
3335:
3264:
3233:
3187:
3176:
3128:
3070:
3034:
3003:
2967:
2956:
2869:
1848:Measurement of the Circle
1446:{\displaystyle \rho _{C}}
1240:{\displaystyle \rho _{S}}
1035:The area of the triangle
1015:Suppose the line segment
415:varies from 0 to 1.
93:
2920:On Conoids and Spheroids
1841:Surface area of a sphere
1153:and the intersection of
2878:Measurement of a Circle
2392:{\displaystyle x^{2}/2}
1743:{\displaystyle 8\pi /3}
1023:lies on a line that is
552:, and is at a distance
322:{\displaystyle y=x^{2}}
96:), also referred to as
3272:Standard part function
2814:10.1006/hmat.2002.2349
2650:Method of indivisibles
2609:
2537:
2489:
2443:
2393:
2358:
2282:
2233:
2180:
2112:
2007:
1969:
1936:
1895:
1817:
1744:
1713:
1690:
1667:
1625:
1600:
1548:
1490:
1447:
1390:
1309:
1241:
1012:
979:
952:
929:
903:
869:
837:
790:
770:
743:
717:
691:
662:
636:
622:
593:
566:
546:
526:
506:
478:
452:
432:
409:
389:
363:
343:
323:
290:
270:
232:
3358:Augustin-Louis Cauchy
3170:Cavalieri's principle
3016:Archimedes Palimpsest
2985:Archimedes' principle
2779:on February 23, 2017.
2752:Morris Kline (1972).
2645:Archimedes Palimpsest
2610:
2538:
2503:plane of side length
2490:
2444:
2394:
2359:
2283:
2234:
2181:
2113:
2008:
1970:
1937:
1896:
1818:
1745:
1714:
1712:{\displaystyle 4\pi }
1691:
1689:{\displaystyle 4\pi }
1668:
1666:{\displaystyle 2\pi }
1626:
1624:{\displaystyle 2\pi }
1601:
1549:
1491:
1448:
1391:
1310:
1242:
1102:on the "lever" where
1011:
980:
978:{\displaystyle x^{3}}
953:
930:
904:
902:{\displaystyle x=2/3}
870:
868:{\displaystyle y=1-x}
838:
836:{\displaystyle y=x/2}
791:
771:
744:
718:
692:
663:
634:
623:
594:
592:{\displaystyle x^{2}}
567:
547:
527:
507:
479:
453:
433:
410:
390:
364:
344:
324:
291:
271:
269:{\displaystyle (x,y)}
233:
134:Archimedes Palimpsest
122:Library of Alexandria
3200:Nonstandard calculus
3195:Nonstandard analysis
3011:Archimedes' heat ray
2800:Historia Mathematica
2655:Method of exhaustion
2547:
2507:
2453:
2407:
2368:
2292:
2243:
2206:
2122:
2040:
2006:{\displaystyle \pi }
1997:
1946:
1905:
1894:{\displaystyle Sr/3}
1874:
1756:
1723:
1700:
1677:
1654:
1612:
1566:
1504:
1458:
1430:
1323:
1255:
1224:
962:
942:
913:
879:
847:
813:
780:
760:
727:
701:
690:{\displaystyle x=-1}
672:
661:{\displaystyle x=-1}
643:
621:{\displaystyle x=-1}
603:
576:
556:
536:
516:
496:
462:
442:
422:
399:
373:
353:
333:
300:
296:-axis and the curve
280:
248:
177:
3420:Works by Archimedes
3410:History of calculus
3384:Elementary Calculus
3265:Individual concepts
3205:Internal set theory
3062:Eutocius of Ascalon
3052:Apollonius of Perga
2702:Thomas Little Heath
2568:
2313:
1524:
1402: = 0 and
1165:rest at the points
1062:be the midpoint of
1027:to the parabola at
928:{\displaystyle x=0}
742:{\displaystyle x=1}
716:{\displaystyle x=0}
477:{\displaystyle x=0}
438:. Imagine that the
388:{\displaystyle y=x}
369:-axis and the line
276:plane between the
194:
3425:Rediscovered works
3277:Transfer principle
3141:Leibniz's notation
3057:Hero of Alexandria
2995:Claw of Archimedes
2990:Archimedes's screw
2927:On Floating Bodies
2605:
2604:
2551:
2533:
2485:
2439:
2389:
2354:
2353:
2296:
2278:
2229:
2176:
2108:
2003:
1965:
1932:
1891:
1827:volume of a sphere
1813:
1740:
1709:
1686:
1663:
1621:
1596:
1544:
1510:
1486:
1443:
1386:
1305:
1237:
1212:Volume of a sphere
1125:, where the point
1094:as a "lever" with
1013:
975:
948:
925:
899:
865:
833:
786:
766:
739:
713:
687:
658:
637:
618:
589:
562:
542:
522:
502:
474:
448:
428:
405:
385:
359:
339:
319:
286:
266:
228:
180:
166:Area of a parabola
36:possibly contains
3397:
3396:
3312:Law of continuity
3302:Levi-Civita field
3287:Increment theorem
3246:Hyperreal numbers
3088:
3087:
3047:Eudoxus of Cnidus
3026:Pseudo-Archimedes
2975:Archimedean solid
2885:The Sand Reckoner
2531:
2322:
2254:
2227:
1805:
1789:
1422:plane around the
1300:
1149:and the parabola
951:{\displaystyle x}
789:{\displaystyle E}
769:{\displaystyle E}
565:{\displaystyle x}
545:{\displaystyle x}
525:{\displaystyle x}
505:{\displaystyle x}
451:{\displaystyle x}
431:{\displaystyle x}
408:{\displaystyle x}
362:{\displaystyle x}
342:{\displaystyle x}
289:{\displaystyle x}
242:integral calculus
223:
138:center of weights
81:
80:
73:
38:original research
3432:
3353:Pierre de Fermat
3348:Abraham Robinson
3188:Related branches
3182:
3115:
3108:
3101:
3092:
3091:
3078:
3077:
2962:
2856:
2849:
2842:
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2826:
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2816:
2787:
2781:
2780:
2775:. Archived from
2764:
2758:
2757:
2748:
2742:
2741:
2715:
2709:
2708:
2700:, translated by
2688:
2614:
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2611:
2606:
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2562:
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2065:
2064:
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2012:
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1495:
1493:
1492:
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1470:
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1452:
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1449:
1444:
1442:
1441:
1414:= 2 on the
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1393:
1392:
1387:
1382:
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1314:
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568:
563:
551:
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543:
531:
529:
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523:
511:
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486:law of the lever
483:
481:
480:
475:
457:
455:
454:
449:
437:
435:
434:
429:
414:
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275:
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204:
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193:
188:
146:law of the lever
95:
76:
69:
65:
62:
56:
53:inline citations
29:
28:
21:
3440:
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3435:
3434:
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3431:
3430:
3429:
3400:
3399:
3398:
3393:
3389:Cours d'Analyse
3367:
3331:
3322:Microcontinuity
3307:Hyperfinite set
3260:
3256:Surreal numbers
3229:
3183:
3174:
3146:Integral symbol
3124:
3119:
3089:
3084:
3066:
3030:
2999:
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2209:
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1987:
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1514:
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1501:
1465:
1461:
1459:
1456:
1455:
1437:
1433:
1431:
1428:
1427:
1377:
1373:
1349:
1345:
1333:
1329:
1324:
1321:
1320:
1280:
1262:
1258:
1256:
1253:
1252:
1231:
1227:
1225:
1222:
1221:
1214:
991:
969:
965:
963:
960:
959:
943:
940:
939:
914:
911:
910:
891:
880:
877:
876:
848:
845:
844:
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781:
778:
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761:
758:
757:
728:
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724:
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699:
698:
673:
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669:
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583:
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517:
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249:
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215:
199:
195:
189:
184:
178:
175:
174:
168:
77:
66:
60:
57:
42:
30:
26:
17:
12:
11:
5:
3438:
3428:
3427:
3422:
3417:
3412:
3395:
3394:
3392:
3391:
3386:
3381:
3375:
3373:
3369:
3368:
3366:
3365:
3363:Leonhard Euler
3360:
3355:
3350:
3345:
3339:
3337:
3336:Mathematicians
3333:
3332:
3330:
3329:
3324:
3319:
3314:
3309:
3304:
3299:
3294:
3289:
3284:
3279:
3274:
3268:
3266:
3262:
3261:
3259:
3258:
3253:
3248:
3243:
3237:
3235:
3234:Formalizations
3231:
3230:
3228:
3227:
3222:
3217:
3212:
3207:
3202:
3197:
3191:
3189:
3185:
3184:
3177:
3175:
3173:
3172:
3167:
3160:
3153:
3148:
3143:
3138:
3132:
3130:
3126:
3125:
3122:Infinitesimals
3118:
3117:
3110:
3103:
3095:
3086:
3085:
3083:
3082:
3071:
3068:
3067:
3065:
3064:
3059:
3054:
3049:
3044:
3038:
3036:
3035:Related people
3032:
3031:
3029:
3028:
3023:
3018:
3013:
3007:
3005:
3001:
3000:
2998:
2997:
2992:
2987:
2982:
2977:
2971:
2969:
2965:
2964:
2957:
2955:
2953:
2952:
2948:Book of Lemmas
2944:
2937:
2930:
2923:
2916:
2909:
2902:
2895:
2888:
2881:
2873:
2871:
2867:
2866:
2859:
2858:
2851:
2844:
2836:
2828:
2827:
2807:(2): 199–203,
2791:Hogendijk, Jan
2782:
2759:
2743:
2710:
2665:
2664:
2662:
2659:
2658:
2657:
2652:
2647:
2640:
2637:
2628:
2625:
2603:
2600:
2597:
2593:
2588:
2584:
2580:
2577:
2574:
2571:
2566:
2561:
2558:
2554:
2528:
2524:
2520:
2517:
2512:
2482:
2478:
2474:
2471:
2468:
2463:
2459:
2436:
2432:
2428:
2425:
2422:
2417:
2413:
2388:
2384:
2378:
2374:
2352:
2349:
2345:
2340:
2336:
2332:
2329:
2326:
2321:
2318:
2311:
2306:
2303:
2299:
2277:
2272:
2268:
2264:
2261:
2258:
2253:
2250:
2239:whose area is
2224:
2220:
2216:
2213:
2175:
2172:
2169:
2166:
2163:
2160:
2156:
2153:
2150:
2145:
2141:
2137:
2132:
2128:
2107:
2104:
2101:
2096:
2092:
2088:
2083:
2079:
2074:
2071:
2068:
2063:
2059:
2055:
2050:
2046:
2002:
1986:
1983:
1962:
1958:
1954:
1951:
1931:
1927:
1921:
1917:
1913:
1910:
1890:
1886:
1882:
1879:
1842:
1839:
1812:
1809:
1804:
1801:
1796:
1793:
1788:
1785:
1780:
1777:
1774:
1771:
1766:
1762:
1739:
1735:
1731:
1728:
1708:
1705:
1685:
1682:
1662:
1659:
1631:at a distance
1620:
1617:
1595:
1592:
1589:
1586:
1583:
1580:
1577:
1574:
1571:
1543:
1538:
1534:
1530:
1527:
1522:
1517:
1513:
1509:
1485:
1482:
1479:
1476:
1473:
1468:
1464:
1440:
1436:
1385:
1380:
1376:
1372:
1369:
1366:
1363:
1360:
1357:
1352:
1348:
1344:
1341:
1336:
1332:
1328:
1304:
1299:
1296:
1293:
1290:
1287:
1284:
1279:
1276:
1273:
1270:
1265:
1261:
1234:
1230:
1213:
1210:
1157:and the lever
1056:
1055:
1048:
1047:
990:
987:
972:
968:
947:
924:
921:
918:
898:
894:
890:
887:
884:
864:
861:
858:
855:
852:
832:
828:
824:
821:
818:
785:
765:
738:
735:
732:
712:
709:
706:
686:
683:
680:
677:
657:
654:
651:
648:
617:
614:
611:
608:
586:
582:
561:
541:
521:
501:
473:
470:
467:
447:
427:
404:
384:
381:
378:
358:
338:
316:
312:
308:
305:
285:
265:
262:
259:
256:
253:
227:
222:
219:
214:
211:
208:
202:
198:
192:
187:
183:
167:
164:
130:infinitesimals
79:
78:
33:
31:
24:
15:
9:
6:
4:
3:
2:
3437:
3426:
3423:
3421:
3418:
3416:
3413:
3411:
3408:
3407:
3405:
3390:
3387:
3385:
3382:
3380:
3377:
3376:
3374:
3370:
3364:
3361:
3359:
3356:
3354:
3351:
3349:
3346:
3344:
3341:
3340:
3338:
3334:
3328:
3325:
3323:
3320:
3318:
3315:
3313:
3310:
3308:
3305:
3303:
3300:
3298:
3295:
3293:
3290:
3288:
3285:
3283:
3280:
3278:
3275:
3273:
3270:
3269:
3267:
3263:
3257:
3254:
3252:
3249:
3247:
3244:
3242:
3241:Differentials
3239:
3238:
3236:
3232:
3226:
3223:
3221:
3218:
3216:
3213:
3211:
3208:
3206:
3203:
3201:
3198:
3196:
3193:
3192:
3190:
3186:
3181:
3171:
3168:
3166:
3165:
3161:
3159:
3158:
3154:
3152:
3149:
3147:
3144:
3142:
3139:
3137:
3134:
3133:
3131:
3127:
3123:
3116:
3111:
3109:
3104:
3102:
3097:
3096:
3093:
3081:
3073:
3072:
3069:
3063:
3060:
3058:
3055:
3053:
3050:
3048:
3045:
3043:
3040:
3039:
3037:
3033:
3027:
3024:
3022:
3019:
3017:
3014:
3012:
3009:
3008:
3006:
3004:Miscellaneous
3002:
2996:
2993:
2991:
2988:
2986:
2983:
2981:
2978:
2976:
2973:
2972:
2970:
2966:
2961:
2950:
2949:
2945:
2943:
2942:
2938:
2936:
2935:
2931:
2929:
2928:
2924:
2922:
2921:
2917:
2915:
2914:
2910:
2908:
2907:
2903:
2901:
2900:
2896:
2894:
2893:
2889:
2887:
2886:
2882:
2880:
2879:
2875:
2874:
2872:
2870:Written works
2868:
2864:
2857:
2852:
2850:
2845:
2843:
2838:
2837:
2834:
2824:
2820:
2815:
2810:
2806:
2802:
2801:
2796:
2792:
2786:
2778:
2774:
2770:
2763:
2755:
2747:
2740:
2736:
2732:
2728:
2724:
2720:
2714:
2707:
2703:
2699:
2698:
2693:
2687:
2685:
2683:
2681:
2679:
2677:
2675:
2673:
2671:
2666:
2656:
2653:
2651:
2648:
2646:
2643:
2642:
2636:
2634:
2624:
2622:
2618:
2617:Jan Hogendijk
2601:
2598:
2595:
2586:
2582:
2578:
2575:
2569:
2564:
2559:
2556:
2552:
2526:
2522:
2518:
2515:
2510:
2502:
2498:
2480:
2476:
2472:
2469:
2466:
2461:
2457:
2434:
2430:
2426:
2423:
2420:
2415:
2411:
2400:
2386:
2382:
2376:
2372:
2350:
2347:
2338:
2334:
2330:
2327:
2319:
2316:
2309:
2304:
2301:
2297:
2270:
2266:
2262:
2259:
2251:
2248:
2222:
2218:
2214:
2211:
2201:
2198:plane at any
2197:
2193:
2189:
2173:
2170:
2167:
2164:
2161:
2158:
2154:
2151:
2148:
2143:
2139:
2135:
2130:
2126:
2105:
2102:
2099:
2094:
2090:
2086:
2081:
2077:
2072:
2069:
2066:
2061:
2057:
2053:
2048:
2044:
2035:
2031:
2027:
2023:
2018:
2014:
2000:
1992:
1982:
1980:
1979:
1960:
1956:
1952:
1949:
1929:
1925:
1919:
1915:
1911:
1908:
1888:
1884:
1880:
1877:
1869:
1865:
1861:
1856:
1852:
1850:
1849:
1838:
1834:
1832:
1828:
1823:
1810:
1807:
1802:
1799:
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1778:
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1760:
1751:
1737:
1733:
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1703:
1683:
1680:
1660:
1657:
1648:
1645:
1640:
1638:
1634:
1618:
1615:
1606:
1593:
1590:
1587:
1584:
1581:
1575:
1569:
1561:
1559:
1554:
1541:
1536:
1532:
1528:
1525:
1520:
1515:
1511:
1507:
1499:
1496:
1483:
1480:
1474:
1466:
1462:
1453:
1438:
1434:
1425:
1421:
1417:
1413:
1409:
1406: =
1405:
1401:
1396:
1383:
1378:
1374:
1370:
1367:
1364:
1361:
1358:
1355:
1350:
1342:
1334:
1330:
1326:
1318:
1315:
1302:
1294:
1291:
1288:
1282:
1277:
1271:
1263:
1259:
1250:
1232:
1228:
1219:
1209:
1208:
1204:
1200:
1197: =
1196:
1192:
1188:
1184:
1180:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1148:
1144:
1140:
1136:
1132:
1128:
1124:
1119:
1117:
1113:
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1073:
1069:
1065:
1061:
1053:
1050:
1049:
1045:
1042:
1038:
1034:
1033:
1032:
1030:
1026:
1022:
1018:
1010:
1006:
1004:
1000:
996:
993:Consider the
986:
970:
966:
945:
936:
922:
919:
916:
896:
892:
888:
885:
882:
862:
859:
856:
853:
850:
830:
826:
822:
819:
816:
807:
805:
804:
799:
783:
763:
755:
750:
736:
733:
730:
710:
707:
704:
684:
681:
678:
675:
655:
652:
649:
646:
633:
629:
615:
612:
609:
606:
584:
580:
559:
539:
519:
499:
491:
487:
471:
468:
465:
445:
425:
416:
402:
382:
379:
376:
356:
336:
314:
310:
306:
303:
283:
260:
257:
254:
243:
238:
225:
220:
217:
212:
209:
206:
200:
196:
190:
185:
181:
172:
163:
161:
155:
153:
152:
147:
143:
139:
135:
131:
127:
123:
119:
115:
111:
108:
105:
104:ancient Greek
101:
100:
91:
87:
86:
75:
72:
64:
54:
50:
46:
40:
39:
34:This article
32:
23:
22:
19:
3297:Internal set
3282:Hyperinteger
3251:Dual numbers
3163:
3162:
3155:
2951:(apocryphal)
2946:
2940:
2939:
2932:
2925:
2918:
2911:
2904:
2897:
2890:
2883:
2876:
2804:
2798:
2794:
2785:
2777:the original
2772:
2762:
2753:
2746:
2730:
2726:
2722:
2719:Netz, Reviel
2713:
2696:
2630:
2621:surface area
2500:
2496:
2401:
2199:
2195:
2191:
2187:
2033:
2029:
2025:
2019:
2015:
1990:
1988:
1976:
1867:
1863:
1859:
1857:
1853:
1846:
1844:
1835:
1824:
1752:
1649:
1643:
1641:
1636:
1632:
1607:
1562:
1557:
1555:
1500:
1497:
1454:
1423:
1419:
1415:
1411:
1407:
1403:
1399:
1397:
1319:
1316:
1248:
1217:
1215:
1202:
1198:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1146:
1142:
1138:
1134:
1133:, the point
1130:
1126:
1122:
1120:
1115:
1111:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1071:
1067:
1063:
1059:
1057:
1051:
1043:
1036:
1028:
1020:
1016:
1014:
1002:
998:
992:
937:
808:
801:
751:
638:
417:
239:
173:
169:
156:
149:
140:of figures (
126:indivisibles
118:Eratosthenes
113:
98:
97:
84:
83:
82:
67:
58:
35:
18:
3157:The Analyst
2934:Ostomachion
2773:Convergence
2036:) obeying:
1866:and height
1041:secant line
754:median line
3415:Archimedes
3404:Categories
3136:Adequality
2913:On Spirals
2863:Archimedes
2692:Archimedes
2661:References
2633:hemisphere
2022:bicylinder
798:exhaustion
395:, also as
160:exhaustion
144:) and the
114:The Method
110:Archimedes
99:The Method
45:improve it
3372:Textbooks
3317:Overspill
2579:−
2557:−
2553:∫
2519:−
2473:−
2427:−
2331:−
2302:−
2298:∫
2263:−
2215:−
2001:π
1953:π
1912:π
1831:spheroids
1808:π
1792:π
1779:−
1776:π
1730:π
1707:π
1684:π
1661:π
1619:π
1588:π
1529:π
1512:ρ
1508:π
1463:ρ
1435:ρ
1371:π
1368:−
1362:π
1331:ρ
1327:π
1292:−
1260:ρ
1229:ρ
1177:rests at
860:−
682:−
653:−
613:−
182:∫
61:June 2024
49:verifying
3080:Category
2733:: 9–29,
2694:(1912),
2639:See also
1137:lies on
1129:lies on
1070:through
995:parabola
142:centroid
107:polymath
3129:History
2823:1896975
2739:1837052
2727:Sciamvs
2032:,
2028:,
1247:at any
1201: :
1193: :
1106: :
1025:tangent
43:Please
3042:Euclid
2821:
2795:Method
2750:E.g.,
2737:
2723:Method
2449:while
1991:Method
1207:Q.E.D.
1141:, and
490:torque
484:. The
3292:Monad
1052:Proof
90:Greek
2467:<
2421:<
2168:<
2162:<
2149:<
2100:<
2067:<
1558:both
1410:and
1058:Let
1001:and
723:and
2809:doi
2797:",
1870:is
1642:As
1086:to
1078:to
1037:ABC
329:as
47:by
3406::
2819:MR
2817:,
2805:29
2803:,
2771:.
2735:MR
2729:,
2704:,
2669:^
1981:.
1833:.
1203:JD
1199:EH
1195:GD
1191:EF
1183:EF
1175:HE
1163:HE
1155:HE
1147:HE
1143:HE
1139:AB
1131:BC
1123:HE
1108:DB
1104:DI
1092:JB
1068:JB
1064:AC
1044:AB
1021:BC
1017:AC
1005:.
806:.
749:.
154:.
112:.
92::
3114:e
3107:t
3100:v
2855:e
2848:t
2841:v
2811::
2731:2
2602:.
2599:y
2596:d
2592:)
2587:2
2583:y
2576:1
2573:(
2570:4
2565:1
2560:1
2527:2
2523:y
2516:1
2511:2
2501:z
2499:-
2497:x
2481:2
2477:y
2470:1
2462:2
2458:z
2435:2
2431:y
2424:1
2416:2
2412:x
2387:2
2383:/
2377:2
2373:x
2351:x
2348:d
2344:)
2339:2
2335:x
2328:1
2325:(
2320:2
2317:1
2310:1
2305:1
2276:)
2271:2
2267:x
2260:1
2257:(
2252:2
2249:1
2223:2
2219:x
2212:1
2200:x
2196:z
2194:-
2192:y
2188:x
2174:.
2171:y
2165:z
2159:0
2155:,
2152:1
2144:2
2140:y
2136:+
2131:2
2127:x
2106:,
2103:1
2095:2
2091:z
2087:+
2082:2
2078:y
2073:,
2070:1
2062:2
2058:y
2054:+
2049:2
2045:x
2034:z
2030:y
2026:x
1961:2
1957:r
1950:4
1930:3
1926:/
1920:3
1916:r
1909:4
1889:3
1885:/
1881:r
1878:S
1868:r
1864:S
1860:S
1811:.
1803:3
1800:4
1795:=
1787:3
1784:8
1773:4
1770:=
1765:S
1761:V
1738:3
1734:/
1727:8
1704:4
1681:4
1658:2
1644:x
1637:x
1633:x
1616:2
1594:.
1591:x
1585:2
1582:=
1579:)
1576:x
1573:(
1570:M
1542:.
1537:2
1533:x
1526:=
1521:2
1516:C
1484:x
1481:=
1478:)
1475:x
1472:(
1467:C
1439:C
1424:x
1420:y
1418:-
1416:x
1412:x
1408:x
1404:y
1400:y
1384:.
1379:2
1375:x
1365:x
1359:2
1356:=
1351:2
1347:)
1343:x
1340:(
1335:S
1303:.
1298:)
1295:x
1289:2
1286:(
1283:x
1278:=
1275:)
1272:x
1269:(
1264:S
1249:x
1233:S
1218:x
1187:J
1179:G
1171:I
1167:G
1159:G
1151:F
1135:E
1127:H
1116:J
1112:I
1100:I
1096:D
1088:D
1084:B
1080:D
1076:J
1072:D
1060:D
1054::
1046:.
1029:B
1003:B
999:A
971:3
967:x
946:x
923:0
920:=
917:x
897:3
893:/
889:2
886:=
883:x
863:x
857:1
854:=
851:y
831:2
827:/
823:x
820:=
817:y
784:E
764:E
737:1
734:=
731:x
711:0
708:=
705:x
685:1
679:=
676:x
656:1
650:=
647:x
616:1
610:=
607:x
585:2
581:x
560:x
540:x
520:x
500:x
472:0
469:=
466:x
446:x
426:x
403:x
383:x
380:=
377:y
357:x
337:x
315:2
311:x
307:=
304:y
284:x
264:)
261:y
258:,
255:x
252:(
226:,
221:3
218:1
213:=
210:x
207:d
201:2
197:x
191:1
186:0
88:(
74:)
68:(
63:)
59:(
41:.
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