Summing a Progression and Finding Roots Geometrically
Adding 1 to 8, and constructing the root of 9 with a semicircle and 3-4-5 triangle
This transmitted-light plate joins two calculating methods. Under the heading 'On progression' Leonardo sums the numbers from 1 to 8: square 8 to get 64, take away half to leave 32, then add half the multiplier (8) to reach 36. He then gives a geometric way to find any square root, worked for the root of 9: lay 9 on a line, add 1, draw a semicircle on the whole, and the perpendicular at the join marks the root a b. A 3-4-5 right triangle proves the result. Lettered figures, including a gridded triangle and a semicircle, accompany the text.
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Summing the progression from 1 to 8
To add the numbers 1 through 8: multiply 8 by itself to get 64, remove half to leave 32, then add half the multiplier, which was 8, giving 36. The same rule serves for the sum of any such progression.
The root of 9 by semicircle construction
Set 9 on line b e and add 1 to make the whole line d e; take its midpoint c as compass centre and draw the semicircle d f e. The perpendicular raised at b meets the circle at a, and the segment a b is the root of 9.
Proof by the 3-4-5 right triangle
The right triangle a b c has sides 3, 4 and 5. Since the square of the hypotenuse (5, making 25) less the square of side b c (4, making 16) leaves 9, side a b equals its root, namely 3.