Leonardo da Vinci's statement refers to the geometric relationship between two squares and a circle, where both squares are tangent to the circle at their four respective vertices. When one square is double the other, it implies that the side length of the larger square is twice the side length of the smaller square.
To understand why the statement is true, let's consider a circle with a radius r, a smaller square with side length a, and a larger square with side length 2a. We'll examine the geometric relationships between these three shapes.
The smaller square is inscribed in the circle, meaning that the circle passes through all four vertices of the square. Thus, the diameter of the circle is equal to the diagonal of the smaller square. Using the Pythagorean theorem for the smaller square, we can derive the diameter (d) of the circle:
a^2 + a^2 = (d/2)^2
2a^2 = (d/2)^2
d = 2 * √(2a^2) = 2a√2
Now, let's consider the larger square. In this case, the circle is inscribed within the larger square, meaning that the diameter of the circle is equal to the side length of the larger square. Thus:
d = 2a
By equating the two expressions for d, we can confirm that the side length of the larger square is indeed double the side length of the smaller square:
2a√2 = 2a
2a = a√2
Therefore, when each of two squares touches the same circle at four points, one square is indeed double the other, as Leonardo da Vinci's statement suggests.