Dividing circular portions into 3 and 5 similar parts
Converse rules and a numeric example (6 divided by 3 = 2) for equal segments.
Continuing the segment geometry, Leonardo divides equal 'portions' into three and five parts that each keep the shape of their whole, illustrated with lettered figures a b c and a b c d e (3, 5, 10, 11). A numeric example - a and b each 3, making 6, divided by 3 to give 2 - anchors the method (1). Several propositions state the converse operations: making 2 similar portions from one, or 2 equal portions from 3, using the same rules taken in reverse (13, 15, 17, 18). He closes that the demand set above has been satisfied (12).
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Numeric example: 6 divided by 3 gives 2
Leonardo supposes a is 3 and b is 3, which makes six; dividing 6 by 3 gives 2. The arithmetic underpins the geometric division of the portions into equal parts.
Portion a b c divided into three equal parts
The portion a b c equals the 2 above and, with the seventh proposition, is divided into 3 equal parts, of which c is one, equal to a and also to b. So what was sought is done, c being 2/3 of b, that is of a.
Dividing d (double c) into five similar parts
Since d is double c, it is to be made into 5 equal parts, all similar in figure to one of those proposed. Finding a whole triple the whole of c and removing the part proportional to c yields e f, the quantity sought.
Converse: from one portion to two in double ratio
This is the converse of the first proposition, satisfied by the same rules taken in reverse. From a given portion 2 portions are to be made, similar to the first but standing in double proportion one to the other.