Proof that 84 portions less 12 equal 72
Bisangles reduced by a seventh, with lune and vault figures and a weight analogy
The upper half of the sheet is a long proof that 84 portions, reduced by 12, are worth 72, worked by turning 84 portions into 42 bisangles and 72 into 36, then adjusting each by a seventh of its size. Small figures of lunes and vault-like segments (falcate) illustrate how each of the 42 bisangles is diminished so that the least portions answer to the greatest. A further passage argues that the number of equal parts grows as the size of their whole diminishes, proved by a weight of 4 pounds losing 2, and closes with a column of calculations continued with the sheet turned upside down.
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84 less 12 is worth 72
Leonardo proves that 84 portions less 12 are equal in value to 72, since the two numbers differ by a seventh of their count. He reduces 84 portions to 42 bisangles and 72 to 36, then increases each of the 36 by a seventh, or diminishes each of the 42 by a seventh, so that unequal numbers hold equal total value.
Diminishing the 42 bisangles
For the lettered figure b a, Leonardo shows the manner of diminishing each of the 42 bisangles, the proportion between the sixth-portions of two circles of different size being as that from circle to circle. A note between the two equal drawings adds that all the least portions answer to the greatest portions.
More parts as the whole shrinks; a weight analogy
Leonardo states that the number of equal parts increases as much as the size of their whole decreases, and proves it with a weight of 4 pounds from which 2 are removed, leaving 2, the 4 halved being diminished by half. He adds that 72 twelfths of a portion make 12 whole portions.