ON THE PRINCIPLES OF FORTIFICATION.

THE RULES ARE:

1. The Angle of the Center is found by dividing the circle—that is, 360 degrees—by the number of sides, which is four. Since the figure is a Perfect SquareA four-sided polygon where all sides and angles are equal., it is understood that the four sides will be equal, and thus the Chordsoriginal: "Subtenses." These are the straight lines connecting the points on the circle's edge. B C, C E, E D, and D B are equal. But chords of the same length cut equal parts of their circle, and consequently, arc B C will be the fourth part of the circle. Now, angle B A C will have as many degrees as arc B C; since such an arc is a quadrant [a quarter-circle], it is evident that the angle will be a right angle.


Degrees of the circle	360	( 90 B A C Angle of the Center.
Sides	4

2. The Angle of the figure is found by subtracting the Angle of the Center from two right angles, or one hundred eighty degrees. Because the three angles of every triangle sum to two right angles, by subtracting angle B A C from the sum of the three aforesaid, the two remaining angles A B C and B C A will be produced. But the two triangles B A C and C A E are equal: thus A C E will be equal to A B C, and the two angles B C A and A C E together will be equal to the two A B C and B C A. It is necessary that the sum of B C A and A C E (that is, the entire angle B C E) be the remainder. In short, the figure being a Perfect Square, their angles will be right angles.


180	two right angles.
90	the Angle of the center, subtracted.
---	---
90	Angle of the figure.

3. It follows that triangle B A C is a right-angled triangle, but since the sides B A and C A are Radiioriginal: "semidiametres." of the same circle, the triangle will be IsoscelesA triangle with at least two equal sides., meaning it will have two sides of the same length. The angles A B C and B C A will be of the same size, and since they together equal the size of one right angle, each one individually will be half of it—namely, 45 degrees. One will therefore find A B or A C by the SineA trigonometric function used here to calculate the length of the sides based on the known angles., in this manner:


The Sine of angle A B C 45° will be	70711.
Which, multiplied by B C	48000③.
Gives the product	3394128000.
Which, divided by the Radiusoriginal: "Raid." In 17th-century fortification, this refers to the standard radius of the circle used for calculations.	100000.
A C. Results in A C	33941③.

4. The Diagonal is the Diameter of the Circle, and consequently two Radii added together; this is why one simply doubles A C, and you will have B E.


	A C is found to be	33941③.
B E.	The double will be B E	67882③.

SIXTH PROPOSITION.

To complete the Design of a Redoubt from the present calculation,

or from the following table.

FIGURE NO. VI.

From the scale, one shall take the Radius of the RedoubtA small, enclosed defensive work, often square, used to protect soldiers outside a main fortress., and with such a span of the compasses, describe a circle. In the circumference of this circle, the sides—taken from the same scale—shall be marked four times, and the nearest points shall be joined with lines.

SEVENTH PROPOSITION.

To make the Design of a Redoubt in the field.

FIGURE NO. VII.

There are two ways to satisfy this requirement: the first by means of the center, the second without the center.
Starting with the center, which is chosen at A, one shall make the four right angles B A C, C A D, E A D, and E A B; but the legs of these angles shall be made the length of the Radius, such as E A, B A, C A, and D A. The nearest points, marked with stakes [bastons], shall be joined as shown in the figure, and the black lines will be excavated.
Without the center, one must first make one side, such as A B; from each end, a perpendicular line shall be made, which will be quite easy by forming two right angles. A C and B D shall be made the same length as A B, and C D is joined, completing the Design as described above.