cousins of the brachistochrone: the 100 yard dash in mathematics

An NRO Pioneer (2003) digs into the mountainous number of curves to help resolve the issue of whether or not a segment of the Cycloid is uniquely the only solution of the Brachistochrone problem. He decomposes each of nearly 300 curve segments which are personified as cousins of the Brachistochrone and statistically resolves the results as first best, second best, third best, etc., showing that the top-ranked cousins are, to a certain degree, essentially tied with the champion Cycloid. These 300 curve segments represent the optimal pieces of thousands of various functional forms. Along the way, he uncovers some of the essential features of mathematics that have created stumbling blocks as well as revelatory insight used to advance the mathematical principles evolving from the 300-year-old-renaissance of the 17th Century. He demonstrates, through a hypothetical 100-yard-dash, that the first-place candidate (a segment of the curve) is seven microseconds behind the Cycloid in the dash and just 27 microseconds when the race is at a mile.