Transforming circular segments with fractional ratios
Making wholes and halves from segments, using fractions like 3/4 and 7/8.
This page treats the enlargement, halving and doubling of circular segments ('portions') as though they were fractional and whole numbers. Rules ask, for instance, that from one whole and 3/4 a single whole be made, and that from 3 wholes and 7/8 a third of another be produced (2). Lettered figures - b a d c, m o p n, a b c d e - carry converse operations, such as making a whole from a half portion or a half from a whole (4, 6, 8, 12). Leonardo notes that the method parallels the ordinary arithmetic of fractions and whole numbers (2).
On this page
Segments handled like fractional numbers
From one whole and 3/4 a single whole is to be made, and one whole is to be divided into one and 3/4; from 3 wholes and 7/8 a third of another is to be produced. Leonardo says this rule supplies for all fractions and wholes, as is done in the arithmetic of fractional and whole numbers.
Making two similar portions from three (a b c to d e)
Using the first rule taken in the converse manner, it is asked that from the 3 portions a b c the 2 be made, that is d and e. These are to be similar in figure to each of the first ones.
From half a portion to a whole, and back
One rule makes a whole from a half portion, equal to the entire half (a as the proposed half, b the thing sought). Its converse makes a half from a whole portion, equal to the whole (figures a, b, c, d).
Operation on the lettered segments b a d c
Of the equal parts, a is double and common; b remains equal to d. Taking c from d leaves d equal to b, and with a equal to d, c remains squared, equal to b squared.