Search results
Results from the WOW.Com Content Network
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 lever and its properties were already well known before the time of Archimedes, and he was not the first to provide an analysis of the principle involved. [5] The earlier Mechanical Problems, once attributed to Aristotle but most likely written by one of his successors, contains a loose proof of the law of the lever without employing the concept of centre of gravity.
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 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 texts under the forged pictures, as well as previously unreadable texts, were revealed by analyzing images produced by ultraviolet, infrared, visible and raking light, and X-ray. All images and transcriptions are now freely available on the web at the Archimedes Digital Palimpsest under the Creative Commons License CC BY. [10] [11] [12]
A page from Archimedes' On Conoids and Spheroids. On Conoids and Spheroids (Ancient Greek: Περὶ κωνοειδέων καὶ σφαιροειδέων) is a surviving work by the Greek mathematician and engineer Archimedes (c. 287 BC – c. 212 BC).