Calculus is a heavy hammer, and can be used to solve many problems. Sometimes, a much more elegant solution to a problem can be found using basic geometric techniques.
One such example is Kepler's observations of the nature of planetary motion. Although Kepler's observations were accurate, it was not apparent why the planets should behave the way they did. Newton proved using calculus and his laws of motion that the orbits of planets are elliptical and that they sweep out equal areas in fixed time periods. However, he also showed this using nothing but the laws of motion and plane geometry, presumably so as to convince an audience that might be skeptical of calculus.
Richard Feynman rederived Newton's method and presented it a lecture that is now part of a collection called Feynman's Lost Lectures. Nigel Higson and Rachel Hall at Penn State included this as part of coursework they developed for an honors calculus class.
The technique itself can be found here. It is a beautiful use of plane geometry, one among many such examples of the power of elementary geometric reasoning (let us not forget that the first theorems were geometric proofs)
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