Cousins of the Brachistochrone

It is often said one of the most common and universal languages that not only our civilization, but any civilization we may ever encounter in the future from other worlds, would be that of mathematics. Math enters our daily lives constantly, from deciding how much to tip a restaurant server to putting a down payment on a mortgage - and so much more. Not everyone is able to speak fluently in this mathematical language, but for those enthusiasts who can, being able to find the beauty in math is an essential part of their interests. As Audrey MacLean once said, “Math is the great equalizer. If you can do the numbers, the boys have to respect you.”

In author James W. Stoner, PhD’s Cousins of the Brachistochrone: The 100 Yard Dash in Mathematics, an NRO Pioneer digs into the large volume of curves to study whether or not the Cycloid is truly the only solution to the Brachistochrone problem in mathematics. Using a hypothetical 100-yard dash to illustrate the problem, the author runs through the 300 curve segments that are known as the cousins of the Brachistochrone and shows how the ranked “cousins” are in essence tied to the Cycloid itself. The author also delves into the history behind the mathematics of this problem, from the 17th century and beyond, and explores not only the sound reasoning of the problem’s solutions but the almost artistic quality of math as a whole.

The author crafted a truly unique and educational read. Finding the right balance that spoke to both casual math enthusiasts and studious math scholars, the author utilized a healthy blend off history and mathematical equations/solutions. The descriptive way the author writes, as well as the passion that can be felt for the mathematics of it all, made this book flow smoothly. What really impressed me however was the author ability to showcase the beauty and artistry that exists within mathematics. While not a math expert, it was inspiring to see the author’s passion show through the exploration of this problem and the intricate details of the solutions.

This book is the perfect read for those who enjoy non-fiction, especially those who enjoy mathematics and math enthusiasts as a whole. I did find myself intrigued by the author’s history of the creation of this problem and the detailed graphs and imagery used to illustrate the solutions to the problem.

Creative, knowledgeable, and engaging, author James W. Stoner, PhD’s Cousins of the Brachistochrone: The 100 Yard Dash in Mathematics is a must-read book on mathematics. The passion and imagery used to bring out the mathematical solutions to this problem and the history behind the math problem and its founder were truly fascinating to read about.

- Pacific Book Review

"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.

- The US Review of Books