b a - f c - d e - g

Creation of the profile. Position of the profile. Motion of the profile.

q; o - n; p - m

Of the base c d.

The profile of the base c d, generator of the pyramid c d e, is the one that becomes the guide of the instrument that produces this pyramid. Now, wishing to come to the creation of this profile, it must be taken as it is given by the rule c d e n f, which begins: let there be given several lines converging to a point at a given distance, as is set in the margin at n o, m p, lines converging to the point q. But to return to the subject we shall say that the ...

b a — o c - f S - d - e r h - m n

a b c d incident, reflected in c d e, and you can make the base of the pyramid, by the given rules, 4 braccia, and the length of the pyramid c d e 4 miles. And to resolve the object placed at e, make its support enter through the bottom of the fire vessel, that is through the hole n, and let its support be at n e, and so placed, such an object will be clothed all around by the power of all the reflected rays.

This takes the principles of its proof from the nature of the angle of incidence, which is always between two equal angles. And this alone is sufficient for the said proof, which is given in common perspective, et cetera.

b - m 7 p 32 n n - d y a - 5 v - S - t K - 4 - (⁸ - r f 8 - o g - x c h e - e

On oblique convergence in its recession.

Let the square a d e c be given, of which its diagonal a c is somewhat greater than the diameter of your circle, whence such a convergence of rays must reflect at whatever distance you please, which for the present will be the point x. Then let the straight diametral line e d be drawn in continuous straight extension outside this square, about 3 times the length of this diameter. And if you lengthen it more it will be still better, because the portion of the circumference of that circle, of which this straight line is made the semidiameter, will be so much the less curved the longer the semidiameter is in itself. Now let the line that passes through the diameter be b d e; then set the foot of the compass at b, and with the other point make the curved diameter of the square from corner a to corner c.

Besides this, let a parallel line be drawn somewhat distant above the front of the square, which shall be n m, from which the straight line n a descends orthogonally to the corner a of the square. This done, you will open the compass one thirtieth of this diameter of the square and set the moving foot of the compass upon the corner a of this square, and the fixed foot you will set upon the curve a at the point v, and you will make the semicircle a t. Then you will draw the parallel line 2 v above the point v, which line cuts the circumference of this semicircle at the point y. Now you must place the point of convergence of the reflected rays, which will be on the continuation of the lower side of this square at the point x. And having marked the line 2 v above the point v, draw the line v x, terminated at that x. Then take with the compass the space a y on the circumference a y t and set the foot of the compass where the convergence line x v cuts this circumference at S, which space shall be S t, and at this point it begins to deviate from the curvature of the diameter a c. Read above.

Then open the compass to the space t h and carry this space above to n 3, and mark the straight line 3 t, and at once draw the convergence line t x, and open the compass to the space t v, and make the semicircle v 5 4 around the point t. But this rule is made clearer in the lowest semicircle below, where, having opened the compass to the space g 8 and carried it above to n p, and drawn the perpendicular p g above the point g, and from that g drawn the convergence line g x, and then opened the compass to the space g f and made the curve f r o h around the point g, one takes with the compass the space f r and carries it upon the same curve to h, and from g h continues the curve from a g up to g h. Then the compass is opened to the space g h and another curve is made above the point h, as was done above g. But first the space g 9 is taken and carried above to n 7, and the perpendicular is drawn from 7 to h, and then, the said semicircle being made, the space f r is taken and carried to o h, and so continuing until the perpendicular line falls upon the converging line in a right-angled junction, though this is not necessary, because every part of the curved line a g has its converging lines at the point x, et cetera.