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Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord. [3] Archimedes gives two proofs of the main theorem: one using abstract mechanics and the other one by pure geometry. In the first proof, Archimedes considers a lever in equilibrium under the action of gravity ...
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. [1]: 8 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
The area bounded by the intersection of a line and a parabola is 4/3 that of the triangle having the same base and height (the quadrature of the parabola); The area of an ellipse is proportional to a rectangle having sides equal to its major and minor axes;
In Quadrature of the Parabola, Archimedes proved that the area enclosed by a parabola and a straight line is 4 / 3 times the area of a corresponding inscribed triangle as shown in the figure at right. He expressed the solution to the problem as an infinite geometric series with the common ratio 1 / 4 :
The book begins with a simpler proof of the law of the lever in Proposition 1, making reference to results found in Quadrature of the Parabola. Archimedes proves the next seven propositions by combining the concept of centre of gravity and the properties of the parabola with the results previously found in On the Equilibrium of Planes I ...
Archimedes used the method of exhaustion to calculate the area under a parabola in his work Quadrature of the Parabola. Laying the foundations for integral calculus and foreshadowing the concept of the limit, ancient Greek mathematician Eudoxus of Cnidus ( c. 390–337 BC ) developed the method of exhaustion to prove the formulas for cone and ...
The area of the surface of a sphere is equal to four times the area of the circle formed by a great circle of this sphere. The area of a segment of a parabola determined by a straight line cutting it is 4/3 the area of a triangle inscribed in this segment. For the proofs of these results, Archimedes used the method of exhaustion attributed to ...
The area of the surface of a sphere is equal to quadruple the area of a great circle of this sphere. The area of a segment of the parabola cut from it by a straight line is 4/3 the area of the triangle inscribed in this segment. For the proof of the results Archimedes used the Method of exhaustion of Eudoxus.