Chapter 2

# Corollaries

by Johannes Kepler

The harmonic divisions of a single string are seven in number, not more.

II.

The expansion of the numbers which are characteristic of divisions occurs in the following manner. To begin with, the whole is expressed in the form of a fraction, that is to say with unity above as numerator, and unity below for denominator. Then each number separately is put as a numerator, and the sum of the two is put as denominator in each case.

Hence from any given fraction two branches arise, until from the sum occurs the number which indicates an unconstructible figure.

I. Causes of this

I found these seven divisions of the string first sevenfold with hearing as guide, in other words the same number found number as there are harmonies not greater than successively.

a single diapason. After that I dug out the causes both of the individual divisions and of the num­ ber of the total, not without toil, from the deep­ est fountains of geometry. Let the diligent reader 13 read what I wrote about these divisions 22 years 2Same ago in The Secret of the Universe, Chapter XII,’’

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