FIGURE OF THREE EQUILATERAL TRIANGLES

This figure is included in this science to provide a doctrine for measuring lines contained by other lines, as in the smaller triangle whose 3 lines are only worth a half of the three lines of the bigger, general triangle, and likewise with b.c.d.

You can know this by experience if you make one straight line with a compass out of the three lines of the major triangle and then make another straight line out of the three lines of a, and you will see that the line of the greater triangle is worth twice the line of the smaller one, and the same will happen if you draw 3 triangles in a. like those in the major triangle, and so on successively until the smallest triangle that can possibly be made.

From this, a doctrine can be derived for multiplying triangles that contain more than the contained ones, and the same considerations apply to squares drawn within each other. We are done with the first part where we showed how a circle can be quadrangulated and triangulated, and how a square is triangulated, and we made 39 figures to provide a doctrine enabling one to obtain an art and a method for measuring and imagining mathematical quantities by using quantities which are sensed, and for measuring one figure with another. The doctrine we gave with these 39 figures can serve to study other, peregrine figures.