Transmutations of a slab: simple and composite changes
A systematic list of how a board changes one dimension when another is fixed
Leonardo sets out the vocabulary of solid transformation, distinguishing the surface (length and width, no depth) from the slab or 'tavola' (length, width and uniform depth), and declares that operations on a surface carry over to the slab. He then enumerates, in a long ordered catalogue, the 'simple transmutations' (numbered first through sixth) and the 'composite transmutations', each asking how a board changes in one dimension when it is shortened, lengthened, thickened, thinned, widened or narrowed to a given limit while another dimension is held fixed. A left-margin key lists the operations and dimensions, with partial trial phrasings. The sheet is torn and faded but the tabulated structure is clear.
On this page
Surface and slab defined
The surface is a plane figure with width and length, uniformly without depth. The slab is a plane body that likewise has width and length and is uniformly of equal depth; therefore all the workings done on the surface are understood to be done on the slab. This distinction frames the whole page of dimensional problems.
Simple transmutations, numbered first to sixth
A list of single-variable changes: shorten to a given distance without changing width, then how much does it thicken; thicken to a given thickness without changing width, then how much does it shorten; widen to a given width without changing thickness, then how much does it shorten, and so on through the sixth case.
Composite transmutations
Paired changes to two dimensions at once: shorten and narrow to given limits, how much does it thicken; lengthen and widen to given limits, how much does it thin; widen and thicken to given limits, how much does it shorten. Each line fixes two dimensions and asks for the response of the third.
Margin key of operations and dimensions
A left-margin glossary lists the six operations (to shorten, lengthen, thicken, thin, widen, narrow) and the three dimensions (length, thickness, width), followed by partial trial statements such as let it be shortened and not widened, or shortened and not thickened.