Abstract: Research into ancient physical structures, some having been known as the seven wonders of the ancient world, inspired new developments in the early history of mathematics. At the other end of this spectrum of inquiry the research is concerned with the minimum of observations from physical data as exemplified by Eddington’s Principle. Current discussions of the interplay between physics and mathematics revive some of this early history of mathematics and offer insight into the fine-structure constant. Arthur Eddington’s work leads to a new calculation of the inverse fine-structure constant giving the same approximate value as ancient geometry combined with the golden ratio structure of the hydrogen atom. The hyperbolic function suggested by Alfred Landé leads to another result, involving the Laplace limit of Kepler’s equation, with the same approximate value and related to the aforementioned results. The accuracy of these results are consistent with the standard reference. Relationships between the four fundamental coupling constants are also found.
Inverse fine-structure constant is a root of: x4 – 136x3 – 136x2 – 818x + 1 = 0. This gives a value for x = 137.035 999 168 ….
CODATA (2014) Inverse alpha = 137.035 999 160 (33). The other root of the equation is approximately 1/818 and 818 = (136+1/3)6 = (4×136)+(2×137). OSF Preprints Physical_Mathematics_and_the_Fine-Structure_Constant