Eddington in 1923, first identified four dimensionless numbers, derived from combinations of the basic physical constants, which are known as the “Eddington constants.” In formulating these dimensionless numbers, Eddington, a leading physicist of his time, claimed that they are characteristic of the structure and dynamics of the Universe at large, on the microscopical scale, and at the macroscopical scale. These four dimensionless ratios are labeled here, and elsewhere, as α (also called the fine structure constant), and β (the electron-proton mass ratio), and γ, (the ratio of electrical-to-gravitational force of the proton on the electron), and δ (a ratio involving the cosmological constant and other constants). Here, in this communication, is defined a 5th Eddington ratio, labeled as ε (a ratio of characteristic energies of diatomic and monatomic hydrogen). The uncanny fitting of these 5 fundamental ratios to simple formulas involving the mathematical constants e, the base of natural logarithms, π, the familiar circular constant, ϕ, the golden ratio and “2,” the only even prime number, is described to some degree along with a tabulation of characteristics of these 5 ratios.Instituto de Investigaciones Fisicoquímicas Teóricas y Aplicadas