Parallelepipeds and the doubling of the cube
Rows of little solid boxes, a surd-root problem, and a column of calculations.
The sheet is filled with dozens of small parallelepipeds (rectangular boxes) drawn in perspective, together with a few triangular perspective schemes at left. Alongside them Leonardo notes the word 'rectangle', a measurement of 500 braccia by a river, and a column of arithmetic. At the foot he poses a problem of solid geometry and algebra: to find the surd (irrational) root of a cube that is double or triple another cube, and to prove how the squaring of a parallelepiped's face relates to the squaring of the whole body and of its half. Further faint jottings appear on the sheet beyond the transcribed blocks.
On this page
A field of parallelepipeds and the squaring of the box and its half
Dozens of little rectangular solids are drawn in perspective across the sheet, one labelled simply 'rectangle'. Leonardo takes a parallelepiped and the same solid halved, marked n and m, and asks first to prove how the squaring of the face n contains the squaring of the whole body of that n. He then extends the method to show that the squaring of its half m is squarable with respect to its whole quantity, since as much as it narrows it rises, and both have a right angle.
Doubling the cube by a surd root
At the foot of the page Leonardo sets himself the classic problem of the duplication of the cube. He writes: give me the surd root of a cube that is double or triple another cube. The demand couples the geometric problem of doubling a solid with the search for its irrational (surd) root.
Column of calculations and a river measure
At the right margin a column of figures works through products such as 500 x 300 and totals of 150000, set out in the stacked manner of a manual reckoning. Nearby Leonardo notes a length of 500 braccia by the river. The numbers accompany the solid-figure studies rather than any account of money.