  "Good luck normalizing this relationship and integrating the various functions to provide an alternate development of the Brachistochrone's solution (+1=6)."

Brachistochrone is Greek for "minimal time" and, in physics, is used to specify the function corresponding to the trajectory taken by a bead falling down a frictionless wire on a vertical plane influenced only by gravity from point A to point B, when B is not beneath A. This is not a straight line but a segment of a cycloid curve, the curve traced on a circle as it rolls along a straight line without slipping. This curve is one of a family of curves known as rolling curves. The cycloid curve's equations satisfy Euler's requirements for a minimal-time descent, but other proofs are low in number compared to, say, the Pythagorean theorem of the right triangle, with over 400 proofs.

The author traces this failure to what he terms the "death of Indefinite Integral Calculus" when no analytically determined solutions to indefinite integrals were found. By excluding infinity by introducing various kinds of parameters, the author has measured some 284 potential "cousin" types of curves to find if others might match or overtake the speed of the cycloid. Finding the roots of polynomials through the Newton-Raphson numerical method called a "quadratic convergence" that bypasses the need for analytic solutions, the author divides these curves into various "segments." Each curve is named, given a short history, and then followed by three graphs that provide the measured curve, its segmentation, and then its normalization.

Stoner provides useful equations and performance for these curves, and many entries are followed by an insightful discussion of their special properties and how close they match the cycloid. The author's language is technical, yes, but also conversational. The absence of mathematical proofs makes the narrative flow and the difficult concepts easier to picture. The inclusion of the importance of the Renaissance and research methodologies help enrich this dense and informative book.

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