LIBRO

DIVERSI COMPARTIMENTI DI CIRCOLI VARIOUS DIVISIONS OF CIRCLES.

A geometric diagram shows a circle with a horizontal diameter labeled CB passing through the center A. Points H and I are marked at the top and bottom of the circumference, respectively.

The circle H I, or the circular line that perfectly forms the roundness of spherical bodies about its center A, will always be divided in half by drawing the line C B, which passes through its center, such that the space of the surface A H will be equal to A I.

The circular form is the most perfect of all others, since it is formed with a single line around its center, where it necessarily comes to be drawn all around with equal proportion, without moving further or closer in one part than in the other. Therefore, all divisions made from the aforementioned center to its circumference will be equal among themselves. This form serves as a foundation and a guiding figure in all operations for forming those bodies or surfaces that may be needed, and especially for drawing the angles that will be shown in the following Fortresses. First, for the easiest, we propose to draw a perfect square from the present circle AB, from which one will be able to have the rule for dividing all other forms of various angles; that is, let the said circumference be divided into four equal parts AC and BC, and by drawing the lines from one point to the other, the aforementioned square will have been formed.

A diagram shows a circle with an inscribed square rotated to a diamond position. The vertices on the circumference are labeled: C (top), B (left), A (right), and D (bottom).

And wishing to form figures of five, six, or more angles, one will always divide the circumference into that number of parts or angles that one desires, then drawing the lines in the manner described. This is called dividing by practice.

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There are also other ways and other rules for making such divisions by theory, which we will show, not because it is necessary to know them, but they will be noted as Geometric pleasantries, given that the true and just division is that which is done with the opening of the compass by practice. First, to describe the pentagon, which is a figure of five angles, one will form the circle of the size that its diameter is to be, which we assume to be the CD, a perpendicular diameter, and the EF, horizontal, so that it comes to divide its circumference into four equal parts. Then, dividing one of the two parts of the half-diameter in half, which will be at F, where one will place one point of the compass, extending the other up to the middle circumference at D, and drawing the semicircle E D, one will note the point E, from which one will draw a straight line to D, which will be E D, which will be one of the five parts of the pentagon that we had to show.

A complex geometric construction for a pentagon within a circle features a vertical diameter CD and a horizontal diameter FE with center H. A point F is marked as the midpoint of the radius HE. An arc is drawn centered at F from D down to the circumference, where point E is marked. A chord connects D and E.

The division of six angles will be the easiest, since of all circles described with the compass, its circumference contains six times its opening, from which it comes about that the compass is called a sixth.

The seventh angle that follows will be found if, with the same opening of the compass, one marks the line B C on its circumference. Then noting the half of that, which will be at D, where one will draw the perpendicular line to its center, which will be AD, and replicated seven times in the circumference, it will form the proposed figure of the seven angles.

A construction diagram for a heptagon within a circle shows the center A. An isosceles triangle is inscribed with its apex at A and its base as a horizontal chord BC at the top. D marks the midpoint of the chord BC, with a vertical line connecting A and D.