It is shown that the new precise formulation of the Large Number Hypothesis (LNH), relating by means of the large number N = 5.73x10^60 the modern cosmological parameters (age, size, mass, average density, and minimum temperature of the universe) with the corresponding Planck units, allows to determine the time course of these cosmological parameters during the expansion. It was found that the dimensions and mass of the universe increase linearly with time, from Planck time to the present day, starting from Planck values and increasing 5.73x10^60 times to now. The amazing result was found that for each discrete time step (beat) with a unit Planck time, the size of the universe increases by one Planck length and its mass increases by one Planck mass. It is shown that the average density of the universe decreases proportionally to the square of time, and starting from the Planck density ~ 10^96 kg m^(-3) decreases N^2 = 3.28x10^121 times to 9.46x10^(-27) kg m^(-3) in the current epoch. The minimum measurable temperature, which is equal to the Hawking temperature for the universe decreases linearly with time 5.73x10^60 times, and starting from the Planck temperature ~ 10^32 K, it falls to 1.75x10^(-29) K at the present time. It is shown that the found time course of cosmological parameters and the Planck values of the size, mass, average density, and temperature of the universe at the initial moment of the expansion (Planck time) follow from the requirement to preserve the Euclidean geometry of space throughout the time of the cosmological expansion. Therefore, the suggested cosmological model based on the new formulation of LNH is free of singularity because the size and density of the universe remain finite/Planckian in the initial moments of its emergence. Besides, this model conserves the flatness and homogeneity of the universe during cosmological expansion and does not need an inflationary epoch in the early universe.