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I constantly meet people who are doubtful, generally without due reason, about their potential capacity [as mathematicians]. The first test is whether you got anything out of geometry. To have disliked or failed to get on with other [mathematical] subjects need mean nothing; much drill and drudgery is unavoidable before they can get started, and bad teaching can make them unintelligible even to a born mathematician.
A Mathematician's Miscellany, Methuen and Co., 1953.
Loci: Convergence
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Archimedes' Method for Computing Areas and Volumes
Exercise on Proposition 6 of The Method
Archimedes' Proposition 6, illustrated below, concerns a cone inscribed in a hemisphere. If you click on Proposition 6, you will see a dynamic version of the same sketch.
a) Describe the balancing relationship that is illustrated in the sketch.
b) Use this to determine the location of the center of gravity of the hemisphere. You will need to use the fact that the center of gravity of a cone lies on the cone's axis, three times as far from the vertex as from the base.
For the solution to this exercise, click here .
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