The Four Kinds of Parallel Space
Leonardo's 'first book': definitions of parallels between straight and curved lines
Opening a 'first book' with the 'definition of 4 kinds of parallels', this folio distinguishes four sorts of parallel space, illustrated by four small figures in the right margin. The first lies between two straight equidistant lines, the second between two lines of equal curvature, the third between lines of differing curvature made around a circle's centre, and the fourth is a single curved line at equal distance around a point. Leonardo asserts all such lines are of one nature, since by motion the straight becomes curved and the curved straight, and closes by citing the Elements that triangles on equal bases between parallels are equal.
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Four kinds of parallel space
The parallel space is of four kinds: between two straight equidistant lines; between two equidistant lines of equal curvature; between two equidistant lines of differing curvature (as parallels made around a circle's centre); and a single curved line at equal distance around a point, that is, the circumference around its centre. Leonardo holds all these lines to be of equal nature, since by motion the straight line becomes curved and the curved straight, and invokes the Elements that all triangles on equal bases between straight parallels are equal.