Continuous and Discrete Quantity Compared
A common measure entering lines and numbers; the third lesson of Book X
Under the heading 'the third lesson of the tenth' Leonardo takes up the comparison of continuous quantity (a length in braccia) with discrete quantity (number). He argues that both can be measured by one common measure: a one-braccio measure enters four times into a line of four braccia and three times into three braccia, exactly as one unit enters four times into four and three times into three. The accompanying diagram shows horizontal segments marked 4, 3 and 1, and three segments labeled a, b and c divided into corresponding parts.
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A unit and a braccio measured into four and three
A continuous quantity can share a common measure with a discrete one. A measure of one braccio enters four times into a line of four braccia and three times into a measure of three braccia, just as one unit enters four times into four numbers and three times into three. The proportion of four braccia to four numbers is thus the same as that of one number to one braccio.
Segments labeled 4, 3, 1 and a, b, c
Two horizontal segments are marked 4 and 3 with a 1 noted between them. Below, three further segments are labeled a (divided into four parts), b (a single part) and c (divided into three parts), illustrating how a continuous line is divided into commensurable parts.