BİRİNCİ FEN · BİRİNCİ BÂB · DÖRDÜNCÜ FASIL · İkinci Madde
**Article Two**
This article summarizes the parts of triangles, the varieties of quadrilaterals, the shapes of polygons, and types of angles, as well as the center, circumference, arc, chord, and sine of a circle.
Know, esteemed one, that geometricians have stated: Any surface surrounded by one or more lines is called a plane. If a surface is surrounded by three lines, it is called a triangle (müselles), which comes in three types. A triangle whose three sides are equal is an equilateral triangle; one with two equal sides is an isosceles triangle; and one with three unequal sides is a scalene triangle. If a plane is surrounded by four lines, it is called a quadrilateral (zü’l-erba’a). A surface surrounded by five lines is called a pentagon (zü’l-hamse). In this manner, the varieties of polygons are extended to that extent, and a plane with ten sides is called a decagon (zü’l-aşere). If their sides are equal, they become a regular quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon. The triangles and quadrilaterals described here are also classified according to their angles. If a triangle has a right angle, it is called a right triangle (kãimü’z-zâviye); if it has an obtuse angle, it is called an obtuse angled triangle (münfericü’z-zevâya). A triangle that does not have obtuse or right angles is called an acute angled triangle (hâddu’z-zevâya).
Similarly, a quadrilateral with four equal sides and four right angles is called a square (murabba‘). A quadrilateral whose angles are right but whose sides are unequal is called a rectangle (mustatîl). Conversely, a quadrilateral with four equal sides but non-right angles is called a rhombus (şekl-i muayyen). A quadrilateral whose opposite sides and angles are equal, but whose sides are not all equal and whose angles are not all right, is called a parallelogram (şibh-i muayyen). Any quadrilateral other than those described here is called a trapezoid (münharif). Shapes with more than four sides are called polygons (kesîrü’l-evzân).
**Angle**
An angle is a plane surrounded by two lines that meet at a single point, such that these two lines do not join infinitely. An angle has two types: The first is a projected angle (zâviye-i musattaha), which is the space between two lines originating from a point and continuing without joining; it is convex. The second is a spatial angle (zâviye-i mücesseme), which is an angle formed on a solid due to the enclosure of one or more planes. For example, the apex angle of a cone and the angles at the corners of a house (between intersecting surfaces).
Plane angles are also three types: One is a right angle. When a perpendicular line is drawn upon any given line, each of the angles formed on either side of this perpendicular is a right angle. The line drawn upwards from the given line to the perpendicular is called a perpendicular (amûd). The second is an acute angle (zâviye-i hâdde), which is smaller than a right angle. The third is an obtuse angle (zâviye-i münferice), which is larger than a right angle. It is not necessary for the lines defining these last two types of angles to be straight.
**Figure**
A figure is an area surrounded by one or more boundaries, and these boundaries are not infinite.
**Circle**
In a plane shape, a curved line surrounds a flat surface, and if a point within that surface is assumed, all the lines drawn from that point to the circumference are equal; this surrounding surface is called a circle. The curved line enclosing it is called the circumference of the circle (muhit-i daire or hatt-ı müstedîr). The assumed point is called the center of the circle (merkez-i daire). Each of the lines drawn is called a radius (nısf-ı kutr), and the line passing through the center and reaching from one side to the other of the circumference, which is the sum of two radii, is called the diameter (kutr-ı daire). This diameter divides the circle into two equal parts and encloses the entire diameter and half of the circumference. The straight line that bisects the circle but does not pass through the center is called a chord (veter). This line divides the circle into two unequal parts, one larger and one smaller. Each of these two parts is surrounded by a chord and a portion of the circumference. Each of these two portions is called a “circular segment.” Each part of the circumference is called an arc (kavs). Half the length of a chord is called the plane sine "ceyb-i müstevî." A line drawn from half the chord to the midpoint of the arc is called a sehm and ceyb-i ma’kûs. Absolute sine is the radius; do not forget this.