Whats special about 62 per cent?
The golden section is a proportion of about 62% to 38%. Its an irrational number. Irrational numbers have decimal expansions that neither terminate nor become periodic.
(Skip this paragraph if you like.) The golden section divides a whole into two parts. The smaller part is in the same proportion to the bigger part as the bigger is to the whole. For practical purposes, this is often shortened to 1:1.618, or about 62 per cent. With twelve decimal points, the bigger part is 61.804697156984 and the smaller 38.195302843016%.
Is the golden section useful in type design?
Not really. But lets try it anyway.
Here we apply it to the character height we have in mind, from ascender to descender, to decide the x-height.
We can then use it again, and give ourselves the ascender height and descender depth.
And for the third time well use it for the capital height, between the ascender height and the x-height. Are these proportions outstanding? Perhaps.
Here are four sets of proportions. The first is 1:1.4142135623731 (thats 1 against the square root of 2, which underlies European paper sizes). The second proportion is two-thirds. The third is the golden section. The fourth is 1:1.89, which is hardly a popular choice. Has one of them got timeless, universal qualities that the others lack?
Still, the golden section can be useful. It may not give you exactly what you want. But it helps by limiting your choices. Some people get more attached to the theory than is good for them. For the full tour, see DArcy Wentworth Thompsons monumental work, On Growth and Form. (Cambridge University Press 1942; first published 1917).