Transmuting 36 biangles into 42 by a geometric method
Star-figures in circles, parallelograms a, b, and why the change cannot be done arithmetically
The verso continues the quadrature studies of the facing sheet. Leonardo shows how to keep the same total value while turning 36 biangles into 42, or the reverse, by reshaping a parallelogram (figures a, b) so its width and length hold seven stars instead of six. He insists the operation must be done geometrically, because arithmetically the sixth of 36 gives 6 whole units and the equivalence fails. Diagrams of squares and circles in rectangles, and stars valued at 72 and 84 'portions', accompany the argument.
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Reshaping the parallelogram a, b to hold seven stars
To transmute 36 biangles into 42 of equal value, Leonardo builds a parallelogram containing the six stars of 36 biangles, then narrows and lengthens it until its dimensions can hold seven stars, i.e. 42 biangles. So the 42 come to be worth the original 36, as the parallelograms a b show.
Why it must be done geometrically, not arithmetically
He warns that arithmetic makes the task impossible, since the sixth of 36 is 36/6, that is 6 whole units, which added to 36 would give 42 whole stars rather than 42 diminished ones worth 36. Hence one must remove the seventh of the value of 42 and operate geometrically.
Six curvilinear stars worth 72 least portions
The six curvilinear stars are worth 72 least portions, so one of their greatest portions equals twelve of the least. The seven stars give seven least portions for each of their greatest, and these seven portions are worth the six of the six stars.