‹‹‹ Vitr. 9.0.4 | Table of Contents | Vitr. 9.0.6 ›››
5Since, therefore, we cannot find them by the aid of numbers, in the square of ten feet a diagonal is to be drawn from angle to angle, so that the square may thereby be divided into two equal triangles of fifty feet area each. On this diagonal another square being described, it will be found, that whereas in the first square there were two triangles, each containing fifty feet, so in the larger square formed on the diagonal there will be four triangles of equal size and number of feet to those in the larger square. In this way Plato shewed and demonstrated the method of doubling the square, as the figure appended explains.
5Therefore, since this is inexplicable by arithmetic, let a diagonal line be drawn from angle to angle of that square of ten feet in length and width, dividing it into two triangles of equal size, each fifty feet in area. Taking this diagonal line as the length, describe another square. Thus we shall have in the larger square four triangles of the same size and the same number of feet as the two of fifty feet each which were formed by the diagonal line in the smaller square. In this way Plato demonstrated the doubling by means of lines, as the figure appended at the bottom of the page will show.