The text discusses the application of a previous demonstration regarding measuring proportions using a pencil holder as a tool. It explains how an observer can compare the height of a column's base to its total height through visual alignment and incremental measurement. The explanation details that the base height is one-third of the total height.
Application of the Previous Demonstration.
The operation we have just described is both too important and simple enough that one should not refrain from presenting its application to one of the most straightforward cases.
Imagine an observer A (Fig. 3), looking at column BC and trying to discern the proportions of this column. If they want to compare the height of DE, the base of the column, to the total height BC, they will place their pencil holder perpendicularly and on point, so that the top of this pencil holder touches the line that transfers point E to their eye. The top E will be, for the observer, at the height of point E on the column. Then, moving the index finger up or down until it meets the line that brings point D to their eye, the observer will find point ab marking point D on their pencil holder, which represents the height ED.
With this height identified, it is about comparing it to the entire column to know how many times this initial height is contained within the total height. This is achieved by raising the hand with the pencil holder, but always along a perfectly vertical line, starting from the point at which the top of the pencil holder was previously located. You find that this height ab is contained two more times, from b to c. Therefore, the height of the base DE is one-third of the height of column BC, with the height bc represented to eye A as the height of the column BC.