In the same manner it can be shown that a square yard contains 9 square feet; and so in general the square described on any line contains n2 times the square described on the nth part of the line. Thus, as a simple case, the square on a line is four times the square on its half. On account of this property the second power of a quantity is called its square; and, conversely, the square on a line AB is expressed symbolically by AB2. 4. If a rectangular parallelogram be such that two adjacent sides contain respectively m and n linear units, by dividing one side into m and the other into n equal parts, and drawing parallels to the sides, the whole area is evidently divided into mn square units. Hence the area of the parallelogram is found by multiplying its length by its breadth, and this explains why we say (see Def. iv.) a rectangle is contained by any two adjacent sides; for if we multiply the length of one by the length of the other we have the area. Thus, if AB, AD be two adjacent sides of a rectangle, the rectangle is expressed by AB.AD. Definitions. i. If a point C be taken in a line AB, the parts AC, CB are called segments, and C a point of division. ii. If C be taken in the line AB produced, AC, CB are still called the segments of the line AB; but C is called a point of external division. iii. A parallelogram whose angles are right angles is called a rectangle. iv. A rectangle is said to be contained by any two adjacent sides. Thus the rectangle ABCD is said to be contained by AB, AD, or by AB, BC, &c. v. The rectangle contained by two separate lines such as AB and CD is the parallelogram formed by erecting a perpendicular to AB, at A, equal to CD, and drawing parallels: the area of the rectangle will be AB.CD. vi. In any parallelogram the figure which is composed of either of the parallelograms about a diagonal and the two complements [see I., xliii.] is called a gnomon. Thus, if we take away either of the parallelograms AO, OC from the parallelogram AC, the remainder is called a gnomon. PROP. I.—Theorem. If there be two lines (A, BC), one of which is divided into any number of parts (BD, DE, EC), the rectangle contained by the two lines (A, BC), is equal to the sum of the rectangles contained by the undivided line (A) and the several parts of the divided line.