Fortification or Military Architecture

GEOMETRY

A rectilinear angle is that which is formed and bounded by two straight lines; see figure 4.
A curvilinear angle is that which is formed by two curved lines, as in figure 5.
A mixed angle is that which is bounded by one straight line and one curved line. 6.
A right angle is formed when a line falling upon another makes equal angles on both sides, as can be seen in Figure 7; when angle A is equal to angle B, then each angle is called a right angle.
And the falling line is called a perpendicular, or orthogonal from the Greek orthogonios, meaning "straight-angled.".
An obtuse angle is that which is larger or more open than a right angle, like angle F. D. E. in Figure 7.
An acute angle is one which is smaller or more narrow than a right angle, like angle E. D. G. in Figure 7.
Parallel or equidistant straight lines are those which, when extended, never meet one another at any point on either side, as in Figure 9.

On Surfaces.

A surface or area original: superfice ou aire is that which has only length and width, and its boundaries are lines, as seen in the figures from 10 to 32.
A plane surface is that which lies evenly between its lines, and all angles drawn upon it are called plane angles; it is the opposite of curved surfaces.
Parallel surfaces or planes are those which are equidistant and which, if continued, never meet, as in Figure 32.

On Rectilinear Surfaces.

A rectilinear triangle is a figure or surface enclosed by three straight lines, and which has three angles, as in Figures 10, 11, 12, 13, and 16.
Triangles are named by the differences in their angles: namely, "right-angled" original: rectangle for one that has a right angle, like Figure 11.
Obtuse-angled or "amblygone" original: ambligone, from the Greek for "blunt-angled." for one that has an obtuse angle, like Figure 12.
Acute-angled or "oxygone" original: Oxigone, from the Greek for "sharp-angled." for one that has all its angles acute, like Figure 13.
And by the differences in their sides, they are named as follows: "equilateral," which has three equal sides, like Figure 10.
"Isosceles," which has only two equal sides, like Figure 16.
"Scalene," which has three unequal sides, Figure 13.
A square is a surface with four equal sides and four right angles, like Figures 15 and 26.
An oblong rectangle, or "long-square," is one that has four right angles and equal opposite sides, but not all sides are equal to each other, like Figure 14.
A rhombus or lozenge is that which has equal sides and equal opposite angles, but not all angles are equal.
A rhomboid is that which has only its opposite sides and opposite angles equal. These four types are also called parallelograms because their sides are parallel. 18.
The diagonal of these last four is the straight line drawn from one angle to the opposite one, which divides and cuts the figure into two triangles equal to one another, as in Figures 14, 15, and 20.
A gnomon is the residue or remainder of a parallelogram from which another parallelogram has been subtracted, having its angles on the diagonal of the first parallelogram, com- This catchword "com-" likely introduces the next page's explanation of how a gnomon is "composed" or "comprised."