b a – d c – f [e]
Divide me a square into an odd number of other squares.
From a given square I will make whatever number, even or odd as required, of other squares, all equal among themselves and of value equal to their whole.
Let then the given square be a b c d, of which nine equal squares are to be made. And to do this I will add beneath it another square equal to it, and this I will divide into 9 equal and similar parallels, and each of these parallels I will square by the rule given in the next-to-last proposition of the first book of the Elements, set out in the second figure here in the margin.
And if in the second you put 9 squares similar to the square m (which is the parallel-square), you will make a large parallelogram containing within it the 9 smaller parallels, and in sum the whole square c d e f, which larger parallelogram you will square in the way you squared the smaller ones, and you will have remade the greatest square, etc. And if in the third you wish to make a greater number of squares from a smaller number, make two similar and equal parallelograms, joined together by their width, and divide the upper parallelogram into as many squares as it can hold, and the lower parallelogram divide into as many parts as you wish. As if the upper parallelogram were divided into 7 squares and the lower into thirteen parallels, you will make 13 squares from these 13 parallels, which will be worth exactly the said 7 squares above. And the fourth is of similar nature to the third.
But if you wished to increase the greater number of squares in figure, it is necessary that you divide the lower parallelogram into a smaller number of parts than the upper. And these such divisions will remain in the form of parallels, which by the second you will reduce into squares, as is shown.

* Odd numbers have no square roots, and for this a square will not be resolved into an even number of other smaller squares. It is proved. Three times three makes nine.

3 times 3, 9.

Three times 4 makes 12.

Such proportion is from circle to circle as is from square to square made by the multiplication of its diameter by itself. And such proportion the similar parts have among themselves as their similar wholes have among themselves.

c

b a – c d – f S e

This parallelogram is too long. Remove one of the 4 greater partitions S f, because they must be 15 particles, and they are 19.

The squaring of the fifteen parallels, set above in the margin in the square c d e f, transmuting the 15 parallels into 15 squares — even though these squares contain exactly the 15 parallels — they will not compose the square above with four right angles, being resolved into 15 squares, because it is an odd number and would stand thus (Figure). Whence it is a good method to lay out these 15 squares in a straight line, one after another, in this way. And doing so, you will have the square a b c d equal to the 15 squares e f, and it will be well made with the help of the last proposition of the first book of the geometric Elements.

a b

a b are squarable by removing the six portions from a, or the 12 lesser ones the half, or the 24 lesser ones the three quarters: it is all one and the same thing and performs one and the same office.