Geometry: transforming cubes, pyramids and solids
Equal stacks of sixteen cubes, transmuting pyramids onto square bases, and reducing a pyramid to height
This is a densely worked sheet of solid geometry. One demonstration sets two stacks each of sixteen equal cubes and argues that as one grows wider it grows lower, and the other as it grows taller grows narrower, so the two remain proportionate. Further notes and figures treat transmuting a pyramid of long base into a body on a square base, warn that an irrational or 'lunar' base makes the problem most difficult, and reduce a pyramid a b c to a required height S b. Diagrams of squares, inscribed semicircles, grids and paired pyramids fill the page, with mirror-writing between them.
On this page
Two equal stacks of sixteen cubes
a b and b c d e are equal, each composed of sixteen equal cubes, with S common to both. The wider stack is correspondingly lower, and the taller stack correspondingly narrower, so that both remain proportionate.
Transmuting a pyramid onto a square base
To transmute a pyramid of long base into a body of square base, proceed in the stated way; if a cylinder had an irrational long base, so too would its pyramid's base. Square that base to pose the question; once posed it proves most difficult, still more if the base is made a squarable lunar figure.
Reducing a pyramid to a given height
Let the pyramid a b c be reduced to the height S b; the pyramid d e f, set in width and height upon the base n m c o, is then the pyramid that is sought.