<s id="A18-1.15.01">[15] Let us now prove how, with the aid of an instrument, to find for a given plane figure a similar one that is in a given ratio to it.</s>
<s id="A18-1.15.02">Let us make two round discs (ac, ab), that are cogged regularly, around the same center (a), that are attached to it and are moving around the same axle in the same plane that the figure, for which we want to construct a similar one, lies in.Let the ratio of the discs be that known ratio.</s>
<s id="A18-1.15.03">Let there be a ruler (pr, lo) at each of the two discs, with cogs towards that direction (a) and their cogs are to mesh with the cogs of the discs.</s>
<s id="A18-1.15.04">Let these rulers run in the groove of another ruler (ahk), that can be moved on the axle by means of a round hole.</s>
<s id="A18-1.15.05">Let there be markings (m, n) for the line of the similar figures on the edges of the cogged rulers and these markings are to run on a straight line (amn) that goes through the center of the discs.</s>
<s id="A18-1.15.06">In order, however, to have both of them always move so that the motion takes place on a straight line that goes through the center, and so that the three points always do the same and always remain on the same straight line, we have to put the markings on the cogged rulers at the same distance from the center of the discs, as the shortest distance of the center of both discs from the edges of the rulers.</s>
<s id="A18-1.15.07">Then we shift those so that they meet the plane that we want to draw the similar figures on.</s>
<s id="A18-1.15.08">If one now moves one of the markings so that it comes to rest on the perimeter of that figure and the other one so far from it that the distance between the first one and the center of the discs relates to the distance between this and the other marking like the diameters of the cogged discs to one another (let, however, the ruler that has the groove be a little bent, so the marking that lies on the line mentioned by us runs on this line), then the other marking describes the figure that is similar to the first one and describes it in the given ratio, because the cogged discs are in this ratio.</s>
