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38:
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1707:. 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
1657:
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.
1865:
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
2027:
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
1847:
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
2024:, 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.
920:, 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
181:
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
2413:
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
168:
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
1862:), 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.
3389:
2373:
173:, 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.
255:. 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
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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
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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
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946:. 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.
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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
787:, 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
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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
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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
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807:, each segment has equal length on opposite sides of the median, so balance follows by symmetry. This argument can be easily made rigorous by
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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
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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.
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which can be easily rectified using the mechanical method. Adding to each triangular section a section of a triangular pyramid with area
2132:
2028:
Archimedes did inscribe and circumscribe shapes to prove rigorous bounds for the volume, although the details have not been preserved.
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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
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where they intersect the lever, then they exert the same torque on the lever as does the whole weight of the triangle resting at
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splitting the base area up, and then each cone makes a contribution according to its base area, just the same as in the sphere.
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Both problems have a slicing which produces an easy integral for the mechanical method. For the circular prism, cut up the
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from the fulcrum on the other side. This means that the cone and the sphere together, if all their material were moved to
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1988:
1121: = 1:3. Therefore, it suffices to show that if the whole weight of the interior of the triangle rests at
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would balance the whole triangle. This means that if the original uncut parabola is hung by a hook from the point
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1858:
996:, which he used to find the center of mass of a hemisphere, and in other work, the center of mass of a parabola.
708:(so that the whole mass of the parabola is attached to that point), it will balance the triangle sitting between
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147:. The palimpsest includes Archimedes' account of the "mechanical method", so called because it relies on the
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969:, although higher powers become complicated without algebra. Archimedes only went as far as the integral of
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The center of mass of a triangle can be easily found by the following method, also due to
Archimedes. If a
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as its fulcrum. As
Archimedes had previously shown, the center of mass of the triangle is at the point
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1761:. Subtracting the volume of the cone from the volume of the cylinder gives the volume of the sphere:
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is that
Archimedes finds two shapes defined by sections of cylinders, whose volume does not involve
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The mass of this cross section, for purposes of balancing on a lever, is proportional to the area:
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503:, where an object's torque equals its weight times its distance to the fulcrum. For each value of
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The method of
Archimedes recently discovered by Heiberg; a supplement to the Works of Archimedes
1956:
886:. Solving these equations, we see that the intersection of these two medians is above the point
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1234:
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by using little rectangles instead of infinitesimal lines, and this is what
Archimedes does in
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is parallel to the axis of symmetry of the parabola. Further suppose that the line segment
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states that two objects on opposite sides of the fulcrum will balance if each has the same
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from the fulcrum; so it would balance the corresponding slice of the parabola, of height
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The two shapes he considers are the intersection of two cylinders at right angles (the
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1840:. By scaling the dimensions linearly Archimedes easily extended the volume result to
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2780:"Archimedes' Method for Computing Areas and Volumes - Cylinders, Cones, and Spheres"
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varies from 0 to 1. The triangle is the region in the same plane between the
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Consider an infinitely small cross-section of the triangle given by the segment
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1650:= 1, would balance a cylinder of base radius 1 and length 2 on the other side.
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1200:, then the lever is in equilibrium. In other words, it suffices to show that
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is parallel to the axis of symmetry of the parabola. Call the intersection of
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1437:-axis, to form a cone. The cross section of this cone is a circle of radius
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Slice the parabola and triangle into vertical slices, one for each value of
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1986:, or "four times its largest circle". Archimedes proves this rigorously in
1216:. But that is a routine consequence of the equation of the parabola.
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in the figure to the right. Pick two points on the parabola and call them
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Proposition 14: preliminary evidence from the
Archimedes palimpsest, I",
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is drawn from any one of the vertices of a triangle to the opposite edge
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117:
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argues that, besides the volume of the bicylinder, Archimedes knew its
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1217:
500:
2368:{\displaystyle \displaystyle \int _{-1}^{1}{1 \over 2}(1-x^{2})\,dx}
66:. Statements consisting only of original research should be removed.
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are to be weighed together, the combined cross-sectional area is:
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Archimedes then considered rotating the triangular region between
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Since each pair of slices balances, moving the whole parabola to
1184:. Thus, we wish to show that if the weight of the cross-section
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is exactly three times the area bounded by the parabola and the
3105:
2846:
31:
2842:
2732:; Saito, Ken; Tchernetska, Natalie (2001), "A new reading of
2804:(2002), "The surface area of the bicylinder and Archimedes'
1827:{\displaystyle V_{S}=4\pi -{8 \over 3}\pi ={4 \over 3}\pi .}
1018:
2626:
And this is the same integral as for the previous example.
2619:{\displaystyle \displaystyle \int _{-1}^{1}4(1-y^{2})\,dy.}
1848:
sphere, cone, and cylinder be engraved upon his tombstone.
1125:, and the whole weight of the section of the parabola at
2765:
Mathematical thought from ancient to modern times, vol 1
2122:{\displaystyle x^{2}+y^{2}<1,\quad y^{2}+z^{2}<1,}
639:, at a distance of 1 on the other side of the fulcrum.
55:
1661:
The volume of the cylinder is the cross section area,
1400:{\displaystyle \pi \rho _{S}(x)^{2}=2\pi x-\pi x^{2}.}
646:
Balanced triangle and parabolic spandrel by the Method
242:{\displaystyle \int _{0}^{1}x^{2}\,dx={\frac {1}{3}},}
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and the circular prism, which is the region obeying:
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1953:. Therefore, the surface area of the sphere must be
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2213:is the interior of a right triangle of side length
2190:{\displaystyle x^{2}+y^{2}<1,\quad 0<z<y.}
1262:between 0 and 2 is given by the following formula:
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2506:, which defines a region which is a square in the
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135:, and contains the first attested explicit use of
105:Περὶ μηχανικῶν θεωρημάτων πρὸς Ἐρατοσθένη ἔφοδος
1319:{\displaystyle \rho _{S}(x)={\sqrt {x(2-x)}}.}
127:takes the form of a letter from Archimedes to
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1912:, which must equal the volume of the sphere:
113:, is one of the major surviving works of the
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1558:{\displaystyle \pi \rho _{C}^{2}=\pi x^{2}.}
2767:. Oxford University Press. pp. 110–12.
159:, which were demonstrated by Archimedes in
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82:Learn how and when to remove this message
1996:Curvilinear shapes with rational volumes
1873:. The volume of the cone with base area
1730:, while the height is 2, so the area is
1567:So if slices of the cone and the sphere
1196:of the section of the parabola rests at
523:, the slice of the triangle at position
139:(indivisibles are geometric versions of
2677:
2000:One of the remarkable things about the
1869:Let the surface of the sphere be
1509:and the area of this cross section is
18:Archimedes' use of infinitesimals
3236:Infinitesimal strain theory (physics)
3032:List of things named after Archimedes
2201:-axis into slices. The region in the
1172:. If the weight of all such segments
7:
2638:Other propositions in the palimpsest
2292:{\displaystyle {1 \over 2}(1-x^{2})}
1192:and the weight of the cross-section
854:, while a second median is the line
469:-axis is a lever, with a fulcrum at
1000:First proposition in the palimpsest
27:Mathematical treatise by Archimedes
2547:{\displaystyle 2{\sqrt {1-y^{2}}}}
25:
3338:Transcendental law of homogeneity
3231:Constructive nonstandard analysis
3175:The Method of Mechanical Theorems
3162:Criticism of nonstandard analysis
2952:The Method of Mechanical Theorems
2243:{\displaystyle {\sqrt {1-x^{2}}}}
1684:times the height, which is 2, or
251:which is an elementary result in
96:The Method of Mechanical Theorems
3189:
3086:
3085:
2969:
2499:{\displaystyle z^{2}<1-y^{2}}
2453:{\displaystyle x^{2}<1-y^{2}}
1101:. We will think of the segment
1042:. The first proposition states:
36:
3221:Synthetic differential geometry
2554:, so that the total volume is:
2299:, so that the total volume is:
2168:
2086:
1129:, the lever is in equilibrium.
543:has a mass equal to its height
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2354:
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2267:
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1500:{\displaystyle \rho _{C}(x)=x}
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1308:
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1279:
1093:is equal to the distance from
610:, if the latter were moved to
274:
262:
1:
3390:Analyse des Infiniment Petits
3226:Smooth infinitesimal analysis
131:, the chief librarian at the
2903:On the Equilibrium of Planes
2778:Gabriela R. Sanchis (2016).
1946:{\displaystyle 4\pi r^{3}/3}
1610:{\displaystyle M(x)=2\pi x.}
814:On the Equilibrium of Planes
162:On the Equilibrium of Planes
2991:Archimedes's cattle problem
2035:), which is the region of (
1077:. Construct a line segment
62:the claims made and adding
3452:
2979:Discoveries and inventions
2917:On the Sphere and Cylinder
2910:Quadrature of the Parabola
2717:Cambridge University Press
2634:, which is also rational.
1989:On the Sphere and Cylinder
1979:{\displaystyle 4\pi r^{2}}
1085:, where the distance from
3354:Gottfried Wilhelm Leibniz
3187:
3081:
2967:
1859:Measurement of the Circle
1457:{\displaystyle \rho _{C}}
1251:{\displaystyle \rho _{S}}
1046:The area of the triangle
1026:Suppose the line segment
426:varies from 0 to 1.
104:
2931:On Conoids and Spheroids
1852:Surface area of a sphere
1164:and the intersection of
2889:Measurement of a Circle
2403:{\displaystyle x^{2}/2}
1754:{\displaystyle 8\pi /3}
1034:lies on a line that is
563:, and is at a distance
333:{\displaystyle y=x^{2}}
107:), also referred to as
3283:Standard part function
2825:10.1006/hmat.2002.2349
2661:Method of indivisibles
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3369:Augustin-Louis Cauchy
3181:Cavalieri's principle
3027:Archimedes Palimpsest
2996:Archimedes' principle
2790:on February 23, 2017.
2763:Morris Kline (1972).
2656:Archimedes Palimpsest
2621:
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2514:plane of side length
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2019:
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1723:{\displaystyle 4\pi }
1702:
1700:{\displaystyle 4\pi }
1679:
1677:{\displaystyle 2\pi }
1637:
1635:{\displaystyle 2\pi }
1612:
1560:
1502:
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1402:
1321:
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1113:on the "lever" where
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989:{\displaystyle x^{3}}
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915:
913:{\displaystyle x=2/3}
881:
879:{\displaystyle y=1-x}
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847:{\displaystyle y=x/2}
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280:{\displaystyle (x,y)}
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145:Archimedes Palimpsest
133:Library of Alexandria
3211:Nonstandard calculus
3206:Nonstandard analysis
3022:Archimedes' heat ray
2811:Historia Mathematica
2666:Method of exhaustion
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2464:
2418:
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2017:{\displaystyle \pi }
2008:
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1905:{\displaystyle Sr/3}
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701:{\displaystyle x=-1}
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672:{\displaystyle x=-1}
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632:{\displaystyle x=-1}
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307:-axis and the curve
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188:
3431:Works by Archimedes
3421:History of calculus
3395:Elementary Calculus
3276:Individual concepts
3216:Internal set theory
3073:Eutocius of Ascalon
3063:Apollonius of Perga
2713:Thomas Little Heath
2579:
2324:
1535:
1413: = 0 and
1176:rest at the points
1073:be the midpoint of
1038:to the parabola at
939:{\displaystyle x=0}
753:{\displaystyle x=1}
727:{\displaystyle x=0}
488:{\displaystyle x=0}
449:. Imagine that the
399:{\displaystyle y=x}
380:-axis and the line
287:plane between the
205:
3436:Rediscovered works
3288:Transfer principle
3152:Leibniz's notation
3068:Hero of Alexandria
3006:Claw of Archimedes
3001:Archimedes's screw
2938:On Floating Bodies
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1838:volume of a sphere
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1248:
1223:Volume of a sphere
1136:, where the point
1105:as a "lever" with
1024:
986:
959:
936:
910:
876:
844:
797:
777:
750:
724:
698:
669:
648:
629:
600:
573:
553:
533:
513:
485:
459:
439:
416:
396:
370:
350:
330:
297:
277:
239:
191:
177:Area of a parabola
47:possibly contains
3408:
3407:
3323:Law of continuity
3313:Levi-Civita field
3298:Increment theorem
3257:Hyperreal numbers
3099:
3098:
3058:Eudoxus of Cnidus
3037:Pseudo-Archimedes
2986:Archimedean solid
2896:The Sand Reckoner
2542:
2333:
2265:
2238:
1816:
1800:
1433:plane around the
1311:
1160:and the parabola
962:{\displaystyle x}
800:{\displaystyle E}
780:{\displaystyle E}
576:{\displaystyle x}
556:{\displaystyle x}
536:{\displaystyle x}
516:{\displaystyle x}
462:{\displaystyle x}
442:{\displaystyle x}
419:{\displaystyle x}
373:{\displaystyle x}
353:{\displaystyle x}
300:{\displaystyle x}
253:integral calculus
234:
149:center of weights
92:
91:
84:
49:original research
16:(Redirected from
3443:
3364:Pierre de Fermat
3359:Abraham Robinson
3199:Related branches
3193:
3126:
3119:
3112:
3103:
3089:
3088:
2973:
2867:
2860:
2853:
2844:
2837:
2836:
2827:
2798:
2792:
2791:
2786:. Archived from
2775:
2769:
2768:
2759:
2753:
2752:
2726:
2720:
2719:
2711:, translated by
2699:
2625:
2623:
2622:
2617:
2601:
2600:
2578:
2573:
2553:
2551:
2550:
2545:
2543:
2541:
2540:
2525:
2505:
2503:
2502:
2497:
2495:
2494:
2476:
2475:
2459:
2457:
2456:
2451:
2449:
2448:
2430:
2429:
2409:
2407:
2406:
2401:
2396:
2391:
2390:
2374:
2372:
2371:
2366:
2353:
2352:
2334:
2326:
2323:
2318:
2298:
2296:
2295:
2290:
2285:
2284:
2266:
2258:
2249:
2247:
2246:
2241:
2239:
2237:
2236:
2221:
2196:
2194:
2193:
2188:
2158:
2157:
2145:
2144:
2128:
2126:
2125:
2120:
2109:
2108:
2096:
2095:
2076:
2075:
2063:
2062:
2023:
2021:
2020:
2015:
1985:
1983:
1982:
1977:
1975:
1974:
1952:
1950:
1949:
1944:
1939:
1934:
1933:
1911:
1909:
1908:
1903:
1898:
1833:
1831:
1830:
1825:
1817:
1809:
1801:
1793:
1779:
1778:
1760:
1758:
1757:
1752:
1747:
1729:
1727:
1726:
1721:
1706:
1704:
1703:
1698:
1683:
1681:
1680:
1675:
1641:
1639:
1638:
1633:
1616:
1614:
1613:
1608:
1564:
1562:
1561:
1556:
1551:
1550:
1534:
1529:
1506:
1504:
1503:
1498:
1481:
1480:
1463:
1461:
1460:
1455:
1453:
1452:
1425:= 2 on the
1406:
1404:
1403:
1398:
1393:
1392:
1365:
1364:
1349:
1348:
1325:
1323:
1322:
1317:
1312:
1292:
1278:
1277:
1257:
1255:
1254:
1249:
1247:
1246:
995:
993:
992:
987:
985:
984:
968:
966:
965:
960:
945:
943:
942:
937:
919:
917:
916:
911:
906:
885:
883:
882:
877:
853:
851:
850:
845:
840:
806:
804:
803:
798:
786:
784:
783:
778:
759:
757:
756:
751:
733:
731:
730:
725:
707:
705:
704:
699:
678:
676:
675:
670:
638:
636:
635:
630:
609:
607:
606:
601:
599:
598:
582:
580:
579:
574:
562:
560:
559:
554:
542:
540:
539:
534:
522:
520:
519:
514:
497:law of the lever
494:
492:
491:
486:
468:
466:
465:
460:
448:
446:
445:
440:
425:
423:
422:
417:
405:
403:
402:
397:
379:
377:
376:
371:
359:
357:
356:
351:
339:
337:
336:
331:
329:
328:
306:
304:
303:
298:
286:
284:
283:
278:
248:
246:
245:
240:
235:
227:
215:
214:
204:
199:
157:law of the lever
106:
87:
80:
76:
73:
67:
64:inline citations
40:
39:
32:
21:
3451:
3450:
3446:
3445:
3444:
3442:
3441:
3440:
3411:
3410:
3409:
3404:
3400:Cours d'Analyse
3378:
3342:
3333:Microcontinuity
3318:Hyperfinite set
3271:
3267:Surreal numbers
3240:
3194:
3185:
3157:Integral symbol
3135:
3130:
3100:
3095:
3077:
3041:
3010:
2974:
2965:
2876:
2871:
2841:
2840:
2800:
2799:
2795:
2777:
2776:
2772:
2762:
2760:
2756:
2728:
2727:
2723:
2701:
2700:
2679:
2674:
2652:
2640:
2592:
2556:
2555:
2532:
2516:
2515:
2486:
2467:
2462:
2461:
2440:
2421:
2416:
2415:
2382:
2377:
2376:
2344:
2301:
2300:
2276:
2252:
2251:
2228:
2215:
2214:
2149:
2136:
2131:
2130:
2100:
2087:
2067:
2054:
2049:
2048:
2006:
2005:
1998:
1966:
1955:
1954:
1925:
1914:
1913:
1883:
1882:
1854:
1770:
1765:
1764:
1732:
1731:
1709:
1708:
1686:
1685:
1663:
1662:
1621:
1620:
1575:
1574:
1542:
1513:
1512:
1472:
1467:
1466:
1444:
1439:
1438:
1384:
1356:
1340:
1332:
1331:
1269:
1264:
1263:
1238:
1233:
1232:
1225:
1002:
976:
971:
970:
951:
950:
922:
921:
888:
887:
856:
855:
822:
821:
789:
788:
769:
768:
736:
735:
710:
709:
681:
680:
652:
651:
612:
611:
590:
585:
584:
565:
564:
545:
544:
525:
524:
505:
504:
471:
470:
451:
450:
431:
430:
408:
407:
382:
381:
362:
361:
342:
341:
320:
309:
308:
289:
288:
257:
256:
206:
186:
185:
179:
88:
77:
71:
68:
53:
41:
37:
28:
23:
22:
15:
12:
11:
5:
3449:
3447:
3439:
3438:
3433:
3428:
3423:
3413:
3412:
3406:
3405:
3403:
3402:
3397:
3392:
3386:
3384:
3380:
3379:
3377:
3376:
3374:Leonhard Euler
3371:
3366:
3361:
3356:
3350:
3348:
3347:Mathematicians
3344:
3343:
3341:
3340:
3335:
3330:
3325:
3320:
3315:
3310:
3305:
3300:
3295:
3290:
3285:
3279:
3277:
3273:
3272:
3270:
3269:
3264:
3259:
3254:
3248:
3246:
3245:Formalizations
3242:
3241:
3239:
3238:
3233:
3228:
3223:
3218:
3213:
3208:
3202:
3200:
3196:
3195:
3188:
3186:
3184:
3183:
3178:
3171:
3164:
3159:
3154:
3149:
3143:
3141:
3137:
3136:
3133:Infinitesimals
3131:
3129:
3128:
3121:
3114:
3106:
3097:
3096:
3094:
3093:
3082:
3079:
3078:
3076:
3075:
3070:
3065:
3060:
3055:
3049:
3047:
3046:Related people
3043:
3042:
3040:
3039:
3034:
3029:
3024:
3018:
3016:
3012:
3011:
3009:
3008:
3003:
2998:
2993:
2988:
2982:
2980:
2976:
2975:
2968:
2966:
2964:
2963:
2959:Book of Lemmas
2955:
2948:
2941:
2934:
2927:
2920:
2913:
2906:
2899:
2892:
2884:
2882:
2878:
2877:
2872:
2870:
2869:
2862:
2855:
2847:
2839:
2838:
2818:(2): 199–203,
2802:Hogendijk, Jan
2793:
2770:
2754:
2721:
2676:
2675:
2673:
2670:
2669:
2668:
2663:
2658:
2651:
2648:
2639:
2636:
2614:
2611:
2608:
2604:
2599:
2595:
2591:
2588:
2585:
2582:
2577:
2572:
2569:
2565:
2539:
2535:
2531:
2528:
2523:
2493:
2489:
2485:
2482:
2479:
2474:
2470:
2447:
2443:
2439:
2436:
2433:
2428:
2424:
2399:
2395:
2389:
2385:
2363:
2360:
2356:
2351:
2347:
2343:
2340:
2337:
2332:
2329:
2322:
2317:
2314:
2310:
2288:
2283:
2279:
2275:
2272:
2269:
2264:
2261:
2250:whose area is
2235:
2231:
2227:
2224:
2186:
2183:
2180:
2177:
2174:
2171:
2167:
2164:
2161:
2156:
2152:
2148:
2143:
2139:
2118:
2115:
2112:
2107:
2103:
2099:
2094:
2090:
2085:
2082:
2079:
2074:
2070:
2066:
2061:
2057:
2013:
1997:
1994:
1973:
1969:
1965:
1962:
1942:
1938:
1932:
1928:
1924:
1921:
1901:
1897:
1893:
1890:
1853:
1850:
1823:
1820:
1815:
1812:
1807:
1804:
1799:
1796:
1791:
1788:
1785:
1782:
1777:
1773:
1750:
1746:
1742:
1739:
1719:
1716:
1696:
1693:
1673:
1670:
1642:at a distance
1631:
1628:
1606:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1554:
1549:
1545:
1541:
1538:
1533:
1528:
1524:
1520:
1496:
1493:
1490:
1487:
1484:
1479:
1475:
1451:
1447:
1396:
1391:
1387:
1383:
1380:
1377:
1374:
1371:
1368:
1363:
1359:
1355:
1352:
1347:
1343:
1339:
1315:
1310:
1307:
1304:
1301:
1298:
1295:
1290:
1287:
1284:
1281:
1276:
1272:
1245:
1241:
1224:
1221:
1168:and the lever
1067:
1066:
1059:
1058:
1001:
998:
983:
979:
958:
935:
932:
929:
909:
905:
901:
898:
895:
875:
872:
869:
866:
863:
843:
839:
835:
832:
829:
796:
776:
749:
746:
743:
723:
720:
717:
697:
694:
691:
688:
668:
665:
662:
659:
628:
625:
622:
619:
597:
593:
572:
552:
532:
512:
484:
481:
478:
458:
438:
415:
395:
392:
389:
369:
349:
327:
323:
319:
316:
296:
276:
273:
270:
267:
264:
238:
233:
230:
225:
222:
219:
213:
209:
203:
198:
194:
178:
175:
141:infinitesimals
90:
89:
44:
42:
35:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3448:
3437:
3434:
3432:
3429:
3427:
3424:
3422:
3419:
3418:
3416:
3401:
3398:
3396:
3393:
3391:
3388:
3387:
3385:
3381:
3375:
3372:
3370:
3367:
3365:
3362:
3360:
3357:
3355:
3352:
3351:
3349:
3345:
3339:
3336:
3334:
3331:
3329:
3326:
3324:
3321:
3319:
3316:
3314:
3311:
3309:
3306:
3304:
3301:
3299:
3296:
3294:
3291:
3289:
3286:
3284:
3281:
3280:
3278:
3274:
3268:
3265:
3263:
3260:
3258:
3255:
3253:
3252:Differentials
3250:
3249:
3247:
3243:
3237:
3234:
3232:
3229:
3227:
3224:
3222:
3219:
3217:
3214:
3212:
3209:
3207:
3204:
3203:
3201:
3197:
3192:
3182:
3179:
3177:
3176:
3172:
3170:
3169:
3165:
3163:
3160:
3158:
3155:
3153:
3150:
3148:
3145:
3144:
3142:
3138:
3134:
3127:
3122:
3120:
3115:
3113:
3108:
3107:
3104:
3092:
3084:
3083:
3080:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3054:
3051:
3050:
3048:
3044:
3038:
3035:
3033:
3030:
3028:
3025:
3023:
3020:
3019:
3017:
3015:Miscellaneous
3013:
3007:
3004:
3002:
2999:
2997:
2994:
2992:
2989:
2987:
2984:
2983:
2981:
2977:
2972:
2961:
2960:
2956:
2954:
2953:
2949:
2947:
2946:
2942:
2940:
2939:
2935:
2933:
2932:
2928:
2926:
2925:
2921:
2919:
2918:
2914:
2912:
2911:
2907:
2905:
2904:
2900:
2898:
2897:
2893:
2891:
2890:
2886:
2885:
2883:
2881:Written works
2879:
2875:
2868:
2863:
2861:
2856:
2854:
2849:
2848:
2845:
2835:
2831:
2826:
2821:
2817:
2813:
2812:
2807:
2803:
2797:
2794:
2789:
2785:
2781:
2774:
2771:
2766:
2758:
2755:
2751:
2747:
2743:
2739:
2735:
2731:
2725:
2722:
2718:
2714:
2710:
2709:
2704:
2698:
2696:
2694:
2692:
2690:
2688:
2686:
2684:
2682:
2678:
2671:
2667:
2664:
2662:
2659:
2657:
2654:
2653:
2649:
2647:
2645:
2637:
2635:
2633:
2629:
2628:Jan Hogendijk
2612:
2609:
2606:
2597:
2593:
2589:
2586:
2580:
2575:
2570:
2567:
2563:
2537:
2533:
2529:
2526:
2521:
2513:
2509:
2491:
2487:
2483:
2480:
2477:
2472:
2468:
2445:
2441:
2437:
2434:
2431:
2426:
2422:
2411:
2397:
2393:
2387:
2383:
2361:
2358:
2349:
2345:
2341:
2338:
2330:
2327:
2320:
2315:
2312:
2308:
2281:
2277:
2273:
2270:
2262:
2259:
2233:
2229:
2225:
2222:
2212:
2209:plane at any
2208:
2204:
2200:
2184:
2181:
2178:
2175:
2172:
2169:
2165:
2162:
2159:
2154:
2150:
2146:
2141:
2137:
2116:
2113:
2110:
2105:
2101:
2097:
2092:
2088:
2083:
2080:
2077:
2072:
2068:
2064:
2059:
2055:
2046:
2042:
2038:
2034:
2029:
2025:
2011:
2003:
1995:
1993:
1991:
1990:
1971:
1967:
1963:
1960:
1940:
1936:
1930:
1926:
1922:
1919:
1899:
1895:
1891:
1888:
1880:
1876:
1872:
1867:
1863:
1861:
1860:
1851:
1849:
1845:
1843:
1839:
1834:
1821:
1818:
1813:
1810:
1805:
1802:
1797:
1794:
1789:
1786:
1783:
1780:
1775:
1771:
1762:
1748:
1744:
1740:
1737:
1717:
1714:
1694:
1691:
1671:
1668:
1659:
1656:
1651:
1649:
1645:
1629:
1626:
1617:
1604:
1601:
1598:
1595:
1592:
1586:
1580:
1572:
1570:
1565:
1552:
1547:
1543:
1539:
1536:
1531:
1526:
1522:
1518:
1510:
1507:
1494:
1491:
1485:
1477:
1473:
1464:
1449:
1445:
1436:
1432:
1428:
1424:
1420:
1417: =
1416:
1412:
1407:
1394:
1389:
1385:
1381:
1378:
1375:
1372:
1369:
1366:
1361:
1353:
1345:
1341:
1337:
1329:
1326:
1313:
1305:
1302:
1299:
1293:
1288:
1282:
1274:
1270:
1261:
1243:
1239:
1230:
1222:
1220:
1219:
1215:
1211:
1208: =
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1167:
1163:
1159:
1155:
1151:
1147:
1143:
1139:
1135:
1130:
1128:
1124:
1120:
1116:
1112:
1108:
1104:
1100:
1096:
1092:
1088:
1084:
1080:
1076:
1072:
1064:
1061:
1060:
1056:
1053:
1049:
1045:
1044:
1043:
1041:
1037:
1033:
1029:
1021:
1017:
1015:
1011:
1007:
1004:Consider the
999:
997:
981:
977:
956:
947:
933:
930:
927:
907:
903:
899:
896:
893:
873:
870:
867:
864:
861:
841:
837:
833:
830:
827:
818:
816:
815:
810:
794:
774:
766:
761:
747:
744:
741:
721:
718:
715:
695:
692:
689:
686:
666:
663:
660:
657:
644:
640:
626:
623:
620:
617:
595:
591:
570:
550:
530:
510:
502:
498:
482:
479:
476:
456:
436:
427:
413:
393:
390:
387:
367:
347:
325:
321:
317:
314:
294:
271:
268:
265:
254:
249:
236:
231:
228:
223:
220:
217:
211:
207:
201:
196:
192:
183:
176:
174:
172:
166:
164:
163:
158:
154:
150:
146:
142:
138:
134:
130:
126:
122:
119:
116:
115:ancient Greek
112:
111:
102:
98:
97:
86:
83:
75:
65:
61:
57:
51:
50:
45:This article
43:
34:
33:
30:
19:
3308:Internal set
3293:Hyperinteger
3262:Dual numbers
3174:
3173:
3166:
2962:(apocryphal)
2957:
2951:
2950:
2943:
2936:
2929:
2922:
2915:
2908:
2901:
2894:
2887:
2815:
2809:
2805:
2796:
2788:the original
2783:
2773:
2764:
2757:
2741:
2737:
2733:
2730:Netz, Reviel
2724:
2707:
2641:
2632:surface area
2511:
2507:
2412:
2210:
2206:
2202:
2198:
2044:
2040:
2036:
2030:
2026:
2001:
1999:
1987:
1878:
1874:
1870:
1868:
1864:
1857:
1855:
1846:
1835:
1763:
1660:
1654:
1652:
1647:
1643:
1618:
1573:
1568:
1566:
1511:
1508:
1465:
1434:
1430:
1426:
1422:
1418:
1414:
1410:
1408:
1330:
1327:
1259:
1228:
1226:
1213:
1209:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1144:, the point
1141:
1137:
1133:
1131:
1126:
1122:
1118:
1114:
1110:
1106:
1102:
1098:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1068:
1062:
1054:
1047:
1039:
1031:
1027:
1025:
1013:
1009:
1003:
948:
819:
812:
762:
649:
428:
250:
184:
180:
167:
160:
151:of figures (
137:indivisibles
129:Eratosthenes
124:
109:
108:
95:
94:
93:
78:
69:
46:
29:
3168:The Analyst
2945:Ostomachion
2784:Convergence
2047:) obeying:
1877:and height
1052:secant line
765:median line
3426:Archimedes
3415:Categories
3147:Adequality
2924:On Spirals
2874:Archimedes
2703:Archimedes
2672:References
2644:hemisphere
2033:bicylinder
809:exhaustion
406:, also as
171:exhaustion
155:) and the
125:The Method
121:Archimedes
110:The Method
56:improve it
3383:Textbooks
3328:Overspill
2590:−
2568:−
2564:∫
2530:−
2484:−
2438:−
2342:−
2313:−
2309:∫
2274:−
2226:−
2012:π
1964:π
1923:π
1842:spheroids
1819:π
1803:π
1790:−
1787:π
1741:π
1718:π
1695:π
1672:π
1630:π
1599:π
1540:π
1523:ρ
1519:π
1474:ρ
1446:ρ
1382:π
1379:−
1373:π
1342:ρ
1338:π
1303:−
1271:ρ
1240:ρ
1188:rests at
871:−
693:−
664:−
624:−
193:∫
72:June 2024
60:verifying
3091:Category
2744:: 9–29,
2705:(1912),
2650:See also
1148:lies on
1140:lies on
1081:through
1006:parabola
153:centroid
118:polymath
3140:History
2834:1896975
2750:1837052
2738:Sciamvs
2043:,
2039:,
1258:at any
1212: :
1204: :
1117: :
1036:tangent
54:Please
3053:Euclid
2832:
2806:Method
2761:E.g.,
2748:
2734:Method
2460:while
2002:Method
1218:Q.E.D.
1152:, and
501:torque
495:. The
3303:Monad
1063:Proof
101:Greek
2478:<
2432:<
2179:<
2173:<
2160:<
2111:<
2078:<
1569:both
1421:and
1069:Let
1012:and
734:and
2820:doi
2808:",
1881:is
1653:As
1097:to
1089:to
1048:ABC
340:as
58:by
3417::
2830:MR
2828:,
2816:29
2814:,
2782:.
2746:MR
2740:,
2715:,
2680:^
1992:.
1844:.
1214:JD
1210:EH
1206:GD
1202:EF
1194:EF
1186:HE
1174:HE
1166:HE
1158:HE
1154:HE
1150:AB
1142:BC
1134:HE
1119:DB
1115:DI
1103:JB
1079:JB
1075:AC
1055:AB
1032:BC
1028:AC
1016:.
817:.
760:.
165:.
123:.
103::
3125:e
3118:t
3111:v
2866:e
2859:t
2852:v
2822::
2742:2
2613:.
2610:y
2607:d
2603:)
2598:2
2594:y
2587:1
2584:(
2581:4
2576:1
2571:1
2538:2
2534:y
2527:1
2522:2
2512:z
2510:-
2508:x
2492:2
2488:y
2481:1
2473:2
2469:z
2446:2
2442:y
2435:1
2427:2
2423:x
2398:2
2394:/
2388:2
2384:x
2362:x
2359:d
2355:)
2350:2
2346:x
2339:1
2336:(
2331:2
2328:1
2321:1
2316:1
2287:)
2282:2
2278:x
2271:1
2268:(
2263:2
2260:1
2234:2
2230:x
2223:1
2211:x
2207:z
2205:-
2203:y
2199:x
2185:.
2182:y
2176:z
2170:0
2166:,
2163:1
2155:2
2151:y
2147:+
2142:2
2138:x
2117:,
2114:1
2106:2
2102:z
2098:+
2093:2
2089:y
2084:,
2081:1
2073:2
2069:y
2065:+
2060:2
2056:x
2045:z
2041:y
2037:x
1972:2
1968:r
1961:4
1941:3
1937:/
1931:3
1927:r
1920:4
1900:3
1896:/
1892:r
1889:S
1879:r
1875:S
1871:S
1822:.
1814:3
1811:4
1806:=
1798:3
1795:8
1784:4
1781:=
1776:S
1772:V
1749:3
1745:/
1738:8
1715:4
1692:4
1669:2
1655:x
1648:x
1644:x
1627:2
1605:.
1602:x
1596:2
1593:=
1590:)
1587:x
1584:(
1581:M
1553:.
1548:2
1544:x
1537:=
1532:2
1527:C
1495:x
1492:=
1489:)
1486:x
1483:(
1478:C
1450:C
1435:x
1431:y
1429:-
1427:x
1423:x
1419:x
1415:y
1411:y
1395:.
1390:2
1386:x
1376:x
1370:2
1367:=
1362:2
1358:)
1354:x
1351:(
1346:S
1314:.
1309:)
1306:x
1300:2
1297:(
1294:x
1289:=
1286:)
1283:x
1280:(
1275:S
1260:x
1244:S
1229:x
1198:J
1190:G
1182:I
1178:G
1170:G
1162:F
1146:E
1138:H
1127:J
1123:I
1111:I
1107:D
1099:D
1095:B
1091:D
1087:J
1083:D
1071:D
1065::
1057:.
1040:B
1014:B
1010:A
982:3
978:x
957:x
934:0
931:=
928:x
908:3
904:/
900:2
897:=
894:x
874:x
868:1
865:=
862:y
842:2
838:/
834:x
831:=
828:y
795:E
775:E
748:1
745:=
742:x
722:0
719:=
716:x
696:1
690:=
687:x
667:1
661:=
658:x
627:1
621:=
618:x
596:2
592:x
571:x
551:x
531:x
511:x
483:0
480:=
477:x
457:x
437:x
414:x
394:x
391:=
388:y
368:x
348:x
326:2
322:x
318:=
315:y
295:x
275:)
272:y
269:,
266:x
263:(
237:,
232:3
229:1
224:=
221:x
218:d
212:2
208:x
202:1
197:0
99:(
85:)
79:(
74:)
70:(
52:.
20:)
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