Asian Research Journal of Mathematics

The proportiones perfectus law is introduced. Let  . By definition, in the spectrum 1≤ y ≤ x ,   is a proportione perfectus  With   so defined, for an arbitrary positive integer h₁ it is shown that there exists an integer sequence H_n satisfying the quasi-geometric relation  such that the arithmetic relation  holds  The golden mean, designated  becomes the most basic and fundamental of proportiones perfectus. New concepts to the study of the golden section are presented: chirality, number genetics and law of polarity, special numerical harmony, and chemical geometry. A geometrical basis for the fine-structure constant in the golden section is established. Our stating of over forty theorems in this reading serves no other purpose than that of expanding the theory of the golden section while equipping the interested reader with instruments for further research and development of this science of number.