Surfaces in triple ratio: uncovered areas and squaring
Overlapping figures, and the numbers 30, 10 and 28
The page treats surfaces standing in triple ratio. When a surface one-third of another partly overlaps it, the exposed part of the larger equals twice the whole smaller plus whatever of the smaller projects beyond, and a lettered figure shows a equal to twice c b plus b. Leonardo then considers two smaller surfaces jointly equal to a third, overlapping it on different sides, and how to square the part a by taking part b from the larger parts c d. A closing arithmetic example uses 30 and 10: removing 2 from each leaves 28 and 8, and 28 contains the smaller part 10 twice over plus the remaining 8. Diagrams of sectors and folded, kite-like figures fill the right margin.
On this page
Exposed area when a third-part surface overlaps a larger
If one surface is a third of another and partly superposed on it, the part of the larger left exposed equals twice the whole smaller, plus as much more as the smaller projects beyond the larger.
a equals twice c b plus b
The lettered figure encapsulates the rule: a is as much as two times c b and b alone joined together.
Squaring the part a from the larger parts c d
When two smaller surfaces jointly equal to a third overlap it on different sides, what of the larger lies outside their contact equals what of the two smaller lies outside the larger. To square the part a, take the part b from the larger parts c d, and the remainder of c d equals a.
Triple proportion and the example of 30 and 10
If from two things in triple proportion equal quantities are removed, the remainders are equal once the excess of the greater over the smaller is set aside. Taking 2 from 10 leaves 8 and 2 from 30 leaves 28, and 28 contains the smaller 10 twice plus the 8 left of the 10.