Leonardo relished the challenge of geometric puzzles. Sheet 24v of Manuscript E is one of many devoted to the ancient Archimedean conundrum of “squaring the circle”, that is how to construct a square equal in area to a given circle.

For this method, Leonardo used a sphere instead of a circle. He claims that a calculation of the area one eighth of the sphere will provide the solution, and so he proceeds to divide it into eight equal parts. He then takes one part and divides it into equal sections, noting how the curvature of the two sides becomes straightened, rather mysteriously “by motion over a plane location”. The next diagram shows the section transformed into a quarter circle divided into equal radiant segments. Finally, these are re-arranged to form a rectangle, but not before their curved edges are “made straight by motion”.

Leonardo never did solve the problem. In fact it was impossible. The effort that he expended on it now seems rather poignant. However, his understanding of geometry gained by this type of study lead to a deeper understanding of organic structures such as the valves of the heart.

In Leonardo's words
The night of St. Andrew’s day, I came to the end of the squaring of the circle and it was the end of the night and of the paper on which I was writing. It was concluded at the end of the hour.