Equalizing opposed triangles by swapping their bases
Rule for triangles h i o and i p K set between converging lines and a c
Two blocks of text flank a diagram in which triangles are inscribed between two straight lines that converge to a point at the right. Leonardo argues that triangles meant to balance opposite quantities are not equal merely because their bases are equal, since they may differ in height, and gives a rule that widens the base of the lower triangle in proportion to how much lower it stands. The two lines e g and h k are required to be parallel and at right angles to a c. The triangles h i o and i p K are the pair to be equalized by exchanging their bases in proportion, the crossing of the lines showing the swap.
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Why equal bases do not make equal triangles
Triangles that equalize opposed quantities are not equal even when their bases are equal, because they need not be of equal height. By the rule shown, the triangle K L e is made as much wider in base than the triangle i K b as it stands lower than it, and the same is done for the other opposed pair.
Exchanging the bases of triangles h i o and i p K
The triangle h i o and the triangle i p K are the pair to be equalized by exchanging their bases in proportion, so that whichever stands lower is given the lower base. The intersection of the lines in the figure marks where the two bases are exchanged.