BİRİNCİ FEN · BİRİNCİ BÂB · DÖRDÜNCÜ FASIL · Birinci Madde
Article One
It summarizes the definitions of point, line, surface, and body, as well as the parts and characteristics of lines and surfaces.
O Noble One! Know that geometers (masters of measurement) have said:
If a thing, being among the forms (ārāz), is perceptible to the senses and does not admit of division in any respect—that is, if it has no dimension—then it is called a “point.” In reality, it is something that occupies space but has no parts. This thing is the beginning and end of a line.
That which is perceptible to the senses and admits of division in one direction is called a "line" (khaṭṭ), having length but neither surface nor depth; it terminates with a point. That which is perceptible to the senses and can be divided in two directions—that is, possessing two dimensions (length and width) but no depth—is called a “surface” (sāṭiḥ). This thing has both breadth and length, and ends with a line. That which is among the forms and possesses three dimensions—meaning it is divisible into length, breadth, and depth—is called a "body" (jism). This is the body referred to in the mathematical discipline.
A line is divided into two kinds: straight and curved (munḥani). A straight line has all its points aligned along the distance it extends (in the same direction). It is a joining of points that continues in the same direction, neither rising nor falling, such that nothing obstructs the view of its middle or end. A curved line, contrary to the straight line, has bends and flexures in its parts as it extends; what lies upon it obscures the view of its middle and end from one extremity. Straight lines are either parallel or not parallel. In parallel straight lines, the distance between two or more lines is equal throughout, and the perpendicular distance between them is the same on all sides. If these lines were extended in the same direction to infinity, they would never intersect. In non-parallel straight lines, the situation is precisely the opposite.
As for surfaces; a surface is either plane or not. In every part of a plane surface, all its components are aligned along the distance it extends until the end of that space—that is, all the segments of lines assumed to exist upon that surface, in terms of length and breadth, are mutually parallel and aligned. The situation is the reverse in non-plane surfaces. Some non-plane surfaces are called spherical surfaces, such as the convex surface of a solid sphere, or the convex and concave surfaces of a hollow sphere. Their halves are named semi-spherical convex and semi-spherical concave respectively. The condition of parallel and non-parallel surfaces can be understood by comparison with the condition of parallel and non-parallel lines.