Constructing the Square Root of 10
Pythagoras and Euclid's Elements II for extracting roots geometrically
Leonardo shows how to obtain the square root of 10 geometrically: find two numbers whose squares add to 10, namely 3 and 1, set them as two lines at right angles, and the hypotenuse of the resulting Pythagorean right triangle is the root of 10. He notes that the figure which yields the root of any fractional number exactly is the last proposition of Book II of Euclid's Elements. Two diagrams accompany the text: a Pythagorean square-on-triangle figure labelled with 3, 10, 9, 1, and a semicircle construction for the Euclidean proposition.
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Root of 10 as the hypotenuse of a 3-and-1 right triangle
To find the root of 10, take two numbers whose squares sum to 10, that is 3 and 1 (9 + 1 = 10). Set them as two lines joined at a right angle; the hypotenuse of this right triangle 'of Pythagoras' is then the root of 10.
Pythagorean figure (3, 10, 9, 1)
A figure of the theorem of Pythagoras is drawn with the values 3, 10, 9, 1, 1. Leonardo adds that the construction giving the root of any fractional number exactly is the last proposition of the second book of Euclid's geometric Elements.
Euclidean semicircle construction (o, S, n, r, p, m, q)
A second figure, labelled o, S, n, r, p, m, q, illustrates the cited Euclidean proposition: the semicircle construction that yields a mean proportional, and hence the square root.