centered at point G, for example an hexagon ABCDEF, and draw another one similar to it, concentric but smaller, that we call HIKLMN. We prolong the side AB of the major hexagon towards S, and in the same way the corresponding side HI of the minor hexagon in the same direction, so that the HT line is parallel to AS; and let another line GV pass through the center. Once this is done, let's imagine that the larger polygon rotates on the AS line, taking with it the minor polygon. 69 It is evident that when point B, at the end of the side AB, remains at the beginning of the rotation, point A rises and point C falls by describing the arc CQ until the BC side coincides with the line BQ, equal to BC. But during this rotation the point I of the smaller polygon rises above the IT line, and returns to the IT line when point C has reached position Q. Point I, having described the arc IO above the HT line, will reach the position O at the same time in which the IK side has taken the OP position; but in the meantime the G center has left the GV line and not will come back to it until the GC arc is completed. After this, the major polygon was brought to stand with its side BC on the BQ line, and the IK side of the minor polygon has been made to coincide with the OP line, having passed the entire IO part without touching it; also the G center will have reached position C after having crossed its entire course over the parallel GV line. And finally the whole figure will assume a position similar to the first; so if we continue the rotation and get to the second step, the side of the major polygon CD will coincide with the QX portion and the KL side of the minor polygon, having first skipped the arc PY, falling on YZ, while the center, always above the GV line, will return to R after missing CR. After a full rotation, the larger polygon will have traced above the AS line, without interruptions, six lines equal to its perimeter; the minor polygon will draw in the same way six lines equal to its perimeter, but separated by the interposition of five arcs, which represent the parts of HT not touched by the polygon. The center G does not reach the GV line, except in six points. It is evident that the space covered by the minor polygon is almost the same as the path taken by the major, i.e., the HT line approximates the AS line, from which it differs only in length of a chord of one of these arcs,