The Fortifications

P R I M O .

It will also be possible by a general rule to divide any circle into as many parts as we wish. That is, if we wish to form five angles, let the circle first be divided into four equal parts A, B, C, D, and then let the circumference of one of these four parts, which we will call B C, be divided into five, of which parts one must take four, which will reach E, and draw the straight line E B, which will be one of the five parts of the entire circumference. To conclude, if you want six, seven, or eight angles, or however many you wish to make, one will always divide one of those four parts of the circle into the number of angles with which you want to divide them; and then taking only four of those parts, one will draw the diametrical line, which will always make one of the aforementioned parts, with which the whole circle will be divided, as is also seen in the part ED, by the DF, which serves for the figure of six angles.

A circular diagram shows a circle divided into four quadrants by axes A-B and C-D. Construction lines and points E and F on the circumference illustrate the method of dividing the circle's perimeter into equal segments for polygons like pentagons or hexagons.

And if it were proposed to take from a given circumference or square another equally proportioned by half less—that is, from the circle A, B, C, D, over the center R, to take another that contains half its surface—let the diametrical lines A, B, C, D be drawn, and the sides of the square from one point to the other, as is seen, then describing the second circle inside that square E G, H F, which will make half of the first, which was to be shown; and if inside this second circumference the four lines are drawn, the second square will be formed, which will also be half of the first, and similarly the third circle, which will be the fourth part of the first.

A geometric diagram shows a large circle (A, B, C, D) with a square inscribed within it. Inside that square, a second circle is inscribed (passing through points H, E, G, F), with point R at the common center. This illustrates halving the area of circular and square forms.

From the circular body, as has been said, the right angle is drawn, which is truly the soul of all operations done in the use of Geometric measuring instruments, as well as in Architecture, in building public and private edifices, with that greatest beauty and comfort that is desired. Therefore, all forms (even those of unequal sides, where one can form the said right angle on one side) can be proportioned and measured by means of it.

For example, this will be seen by the three different angles. The triangle of equal sides A, B, C is reduced to an oblong square by drawing the perpendicular line C H over the base A B, so that at H, a right angle is formed; and by drawing with the same angle the part A D, and D C, equal to the part of the base H B, the proposed oblong square A D, C H will be formed, because the base D C is equal to the H B, and the triangle G is equal to the triangle F.

A diagram shows an equilateral triangle ABC with a vertical altitude CH dropped to the base. It demonstrates how to transform the triangle into a rectangle ADCH of equivalent area, labeling internal areas G and F to show their geometric correspondence.

Next is the second triangle of unequal sides, A, B, C, which, having divided the height C over the base A B into two equal parts at F, and transporting the part, or height, of F C to the ends of said base, namely B D and A E, at right angles, by drawing the line D E, one will come to form the oblong square A B, D E, equal to the said triangle A, B, C, because the triangle N is equal to O, and H is equal to I.

A diagram illustrates the conversion of a scalene triangle ABC into a rectangle ABDE of equal area. Construction lines show the bisection of the triangle's height at point F and the realignment of resulting triangular sections (N, O, H, I) to form the rectangle.