The Fortifications (1609) - Page 22

LIBRO

For the third and last T S C, with the right angle S, C T being the diagonal of the square, one will divide S C into two parts, namely at D, and transporting D C into T V at a right angle over the base S T, by drawing D V, one will come to form the oblong square D V, T S, equal to the triangle T S C, because the two triangles A R are likewise equal.

A geometric diagram shows a right-angled triangle labeled T-S-C and a rectangle labeled D-V-T-S. The interiors of the shapes are labeled with A and R, indicating equal areas.

It remains for us finally to show how ovals are formed in four ways, that is, with triangles, quadrangles, and circles. First, let two triangles of equal sides be formed over the base A B, namely A B D and A B C, making C the center to draw the part of the circle E G, and similarly D to draw the F H, and B A for the ends H E and G F.

A geometric diagram of an oval (ovato) is constructed using two equilateral triangles sharing a common base A-B. Vertices D and C act as centers for circular arcs. The points on the perimeter are labeled E, G, F, H.

Next are two squares R S to form the proposed oval, that is, for the part of the circumference P Q, center L will be made, and similarly for the N O M, and for the ends P O and Q N, R S.

A geometric diagram of an oval inscribed within a larger rectangle divided into two squares labeled R and S. Centers L and M are used to define the arcs. Vertices of the inner figure are labeled N, O, P, Q.

And then the third with three circles for the circumference F I, the center will be C, and for the H L D, the ends L I, F H are made by the same circles.

A geometric diagram of an oval formed by three intersecting circles. Labels indicate various points including C, D, F, H, I, and L.

One will be able to form the proposed oval simply with a double rope, without any of the said observations of circles or angles. That is, divide on a plane the length one wishes to give, for example D C, into eight parts, of which one will take six, which will make A B, leaving at each of its ends one, namely A C, B D, where one will stick the two pegs or nails A B. To these, one will wrap a thin rope or twine, well tied at its ends, so that it is doubled, as long as the space A D, which is of seven parts. Then, taking another peg or nail, and placing its point in the end between these two ropes, one will go holding them thus taut, describing the line D E, C D, making the said rope always slide thus doubled between the two pegs A B, with which one will come to form, as one pleases, the form of the oval circle.

A geometric diagram illustrates the gardener's method for drawing an ellipse or oval using a loop of string. Pegs are placed at points A and B on a central axis D-C. A point E traces the curve as the string is kept taut.