Enlarging and Shrinking a Square Board
Adding or removing a set amount while keeping a board square (Euclid I.5-6)
Three linked problems transform a square board while keeping it square. First (square a): add a third to a square board yet keep it square at its original thickness, extending the board with Euclid I.6 to three times the width of square a, adding a part equal to one of those thirds, then squaring it again. The middle figure shows the elongated rectangle divided into parts a, b, c, reduced back to the square b by Euclid I.5. The last problem (square b) is the converse: remove a given quantity from a square board and keep it square, extending it lengthwise and returning the elongated tetragon to a square.
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Add a third to a square, keeping it square (square a)
Add to a square board a third of its quantity and keep it square at its first thickness. With the sixth proposition of Book I, extend board a to a length three times the width of square a, then add a part equal to one of those thirds, laying the board out lengthwise.
Rectangle divided into parts a, b, c
The elongated board is carried by Euclid I.6 to the length a b, divided into 3 equal parts; the part b c is then added at the end, of the same width as the other three. With the fifth proposition of Book I the elongated tetragon a c is returned to square form, giving the square b.
Converse: remove a quantity, keeping it square (square b)
Remove a given quantity from a square board and keep it square. This is the converse of the above: extend the square lengthwise with the sixth of Book I, take away the given quantity, and with the fifth of Book I return the elongated tetragon to its square.