From rectangle to square and back, after Euclid II
Reducing oblong a b g h to the square e f a d, then stretching a square into a rectangle (props. 5-6)
This page presents two paired propositions. The fifth, "From a rectangle to a square," reduces the oblong a b g h to the equilateral square e f a d by placing its front at a i, taking the midpoint c of the line i b, and drawing the semicircle i e b, citing the last proposition of the second book of Euclid. The sixth, "From a square to a rectangle," stretches the square a b c d into a long rectangle from c to m and asks how much its breadth diminishes, with a marginal statement and a main column that begins the solution. Three semicircle constructions are drawn down the right margin.
On this page
Reducing rectangle a b g h to the square e f a d
The oblong a b g h is to become an equilateral square e f a d. Its front a g is placed at a i, the midpoint c of the whole line i b is taken, and the semicircle i e b is drawn; the side a g is extended to the circle, so that a e is a side of the square sought, as proved in the last proposition of the second book of Euclid.
Stretching square a b c d into a rectangle from c to m
The square a b c d is to be stretched into the form of a long rectangle from c to m, and the diminution of its breadth is asked. The main column begins the solution before breaking off, and a marginal note restates the problem: a square stretched in length by a given space, with its breadth to be established with certainty.