<s id="A18-1.18.01">[18] In order to construct similar solid figures, we proceed in the following way.We take two plane boards of wood that can be moved around a common line, so that the line remains one and the same line in any motion.We achieve this when the centers of the hinges, around which the boards move, fall on this common line.</s>
<s id="A18-1.18.02">Let the size of the boards fit the size of the largest of the similar figures.</s>
<s id="A18-1.18.03">The manufacture and use of the tool we shall now explain.Let us take two frames of iron that resemble the letter called upsilon and let the parts of each of them spread out be similar to one another.</s>
<s id="A18-1.18.04">Let us now bend their ends so that the bend has a point and the bending of both of them may result in the figure of a triangle.</s>
<s id="A18-1.18.05">Let further the known ratio of the one of the similar figures to the other one be equal to the tripled (i.e. cubic) ratio of the proportional sides of the two triangles and let us assume this now for the lines <ab>, <ag> and <ad>, while the lines that were bent are <ge>, <bz> and <dh>; let the other frame consist of the lines <qk>, <ql> and <qm>, and the bent lines be <kn>, <lc> and <mo>; let the two similar triangles be <hez> and <noc>.</s>
<s id="A18-1.18.06">Let us now draw above the line (cb), that is common to the movable boards, on one (ab) of the boards, a figure (<hez>) congruent to the iron frame and let us further draw through one of the sides of the triangle a line (<oc>) that is parallel to the baseline (<ez>) of the triangle and which cuts off another triangle (<noc>), that is equal (congruent) to the iron triangle which resembles the letter upsilon.</s>
<s id="A18-1.18.07">On each of the upsilon-frames let a tin rod (S<a> and s<q>), the end of which is very pointed, be attached so that when it is bent and then released, it is firm, i.e. does not tremble, like the tin rods that are used for the human pictures(?).</s>
<s id="A18-1.18.08">Let the form of this letter called upsilon (after bending) be similar to the tool called Galeagra.Let the motion of the mentioned boards against one another be such that, when the motion stops, they stand firm and cannot be shaken, like the "crabs".</s>
<s id="A18-1.18.09">This is the manufacture of the tool; we want to explain its use directly.</s>
<s id="A18-1.18.10">If we now want to make a solid figure similar to another one, which is in a known ratio to it, we bring the surface of the solid figure close to the upsilon-frame, so that the markings on all sides touch the plane and we also bring the other upsilon-frame close to the body to be constructed.</s>
<s id="A18-1.18.11">If we now want to make it larger than the existing body, we bring the larger body to the larger triangle, the other one to the second.</s>
<s id="A18-1.18.12">Let us assume we want to make a similar body from stone or wood or any other matter and bring the markings to each body.</s>
<s id="A18-1.18.13">Let us locate the assumed markings in similar positions on the bodies and we construct the remaining parts on the basis of this procedure.</s>
<s id="A18-1.18.14">In order to make our explanation clearer, let us assume that we want to attach an eye to the picture of a human or the picture of something else.</s>
<s id="A18-1.18.15">Let us therefore put the markings of the upsilons on the existing, I mean the given [object], for which we want to make a similar figure, and let us bend the tip (S) of the tin rod that is on the upsilon until the tip touches the eye concerned; then we take the upsilon and put it on the triangle (<hez>), which is drawn on the board (ab); then we lower or raise the other board (cd), on which nothing is drawn, until in its rising or sinking it meets the tip of the rod.</s>
<s id="A18-1.18.16">Then we remove the upsilon and draw, starting from the point (m) that the tin rod has made on the board (cd), two lines (m<h>, m<z>) towards the end points of the side of the triangle that are lying on the line common to both boards, and make sure that the boards do not move against one another, draw through the other point (<c>), which is located on the line common to both boards, a line (n<c>) parallel to m<z> (text: to the largest lines that are near the lines parallel to the baseline), until it intersects the other drawn line (<h>m).</s>
<s id="A18-1.18.17">Then we take the other upsilon, put the pointed tips of the teeth that were bent on the triangle (<noc>) that is on the board (ab) and is equal (congruent) to the triangle formed by the ends of those parts (<kn>, <mo>, <lc>), bend the tin rod until it meets the point (n) that was determined by the parallel line (n<c>) on the other board (cd), remove the upsilon and put it on the given points of the body not used yet.</s>
<s id="A18-1.18.18">The point where the end of the rod meets on the body is the point determined on the picture for the place of the eye, which has a similar position as the one on which we bent the first rod.</s>
<s id="A18-1.18.19">In the same manner, we bend the rod towards the other parts of the picture and mark the similarly situated points on the stone; then we construct the plane according to the assumed points, which are the points that make the figure similar to the one given first and that has a ratio to it as the one mentioned.</s>
<s id="A18-1.18.20">Now concerning the mentioned parallel line, it is lightly drawn on the other board, when we draw on the board any parallel to the common line (?).</s>
<s id="A18-1.18.21">That the figures gained in this manner are similar becomes evident from the fact that they originate from similar, similarly situated pyramids whose bases are the triangles (<hez>, <noc>) defined by the upsilons on the bodies, and whose tips are the points (m, n) marked by the ends of the rods on each of the bodies.</s>
<s id="A18-1.18.22">That they are in a known ratio is clear, because the ratio of the pyramids that the bodies were made from is the tripled (i.e. cubic) ratio of the proportional sides, for the sides of the similar triangles (<hez>, <noc>) were so assumed.</s>
<s id="A18-1.18.23">Thus the bodies are in this known ratio with one another.</s>
