The paper solves the problem of mathematical inference of large numbers, which was formulated in 1985 by P. C. W. Davies [1]. The law of scaling of large numbers is derived. The law of scaling gives a new method of obtaining large numbers from dimensionless constants. It complements the known method based on relations of dimensional physical quantities. The law of scaling of large numbers shows that large numbers of scale 10^39, 10^40, 10^61, 10^122 are only part of the complete family of large numbers. The large numbers are supplemented by new large numbers of scales 10^140, 10^160, 10^180, which are naturally derived from the fundamental parameters of the observable Universe. New coincidences of relations of dimensional quantities on scales 10^140, 10^160, 10^180 are found. It is shown that large numbers of different scales are functionally related to each other. The primary large number D20 =(αDo)^(1/2) = 1.74349...x 10^20, from which large numbers of other scales are formed according to a uniform law, is chosen on the scale of 10^20. The primary large number D20 = 1.74349...x 10^20 consists of two dimensionless constants: the fine structure constant alpha and the Weyl number Do = 4.16561...x 10^42. The coincidences of the relations of the dimensional quantities with large numbers on scales 10^160 and 10^180 allowed us to derive simple and beautiful formulas for calculating the Hubble constant H and the cosmological constant Ʌ. An equation is derived which shows that the constants H and Ʌ are related. The origin of H and Ʌ from the fundamental physical constants of the electron is proved. The law of scaling of large numbers makes it possible to calculate analytically the parameters of the observable Universe with high accuracy.A new equation is derived, which unites the 5 most important parameters of the observable Universe: MuRuGɅ^2 = H^2.