Thread: Proof of Volume of a Sphere View Single Post
 Recognitions: Homework Help Science Advisor Oh boy! Here is a quote from the introduction to the work of archimedes where it states explicitly, that archimedes found a volume of a certain section of a cylinder, by reducing it to the problem of the area of a parabola. "Proposition XI is the interesting case of a segment of a right cylinder cut off by a plane through the center of the lower base and tangent to the upper one. He shows this to equal one-sixth of the square prism that circumscribes the cylinder. This is well known to us through the formula $v = 2r^2h/3$, the volume of the prism being $4r^2h$, and requires a knowledge of the center of gravity of the cylindric section in question. Archimedes is, so far as we know, the first to state this result, and he obtains it by his usual method of the skilful balancing of sections. There are several lacunae in the demonstration, but enough of it remains to show the ingenuity of the general plan. The culminating interest from the mathematical standpoint lies in proposition XIII, where Archimedes reduces the whole question to that of the quadrature of the parabola." By the way, the famous work of Galileo in the 1600's of discovering that a moving projectile travels in the path of a parabola, and that the distances traveled by a falloing object, in succeeding units of time, stand to one another as the squares of the positibe integers, are also mathematical consequences of the work of archimedes. this causes one to wonder why they were thought to be new in galileo's time, and why a genius like galileo did not realize they were corollaries of archimedes work. of course the connection of the mathematics with the physics is in itself a significant discovery, but galileo seems to re-derive all the mathematics by geometry. this puzzles me.