Triangle of Curved Sides: Counting Similar Triangles
How many small triangles a d e fit the triangle a b c: multiply the linear division by itself (8 x 8 = 64).
A pen-and-red-chalk diagram at the upper right shows a triangle a b c with one curved (falcate) side and an interior division marked in red letters. In a long pen note Leonardo gives a rule for counting how many small similar triangles a d e fit inside the larger triangle a b c: multiply the line a c by itself. When d e is parallel to b c and the side a e goes eight times into a c, the count is eight times eight, that is sixty-four, and he calls this a general rule that holds also for the falcate (lune-shaped) figure of equal sides. A brief red-chalk note states an equality of the parts a-b and c-b.
On this page
Similar triangles multiply as the square of the side-ratio
To learn how many triangles a d e fit inside triangle a b c, Leonardo multiplies the line a c by itself, since a e is an aliquot part of a c. With d e parallel (equidistant) to b c and a e going eight times into a c, the number of small triangles equals eight times eight. He states this as a general rule.
8 x 8 = 64 as a general counting rule
The side a e divides a c into eight equal parts, so the number of contained triangles is eight multiplied by eight, giving sixty-four. Leonardo presents this squaring of the linear division as a general rule. He adds that the same is meant of the falcate (lune-shaped) figure whose sides are equal in length and in curvature.