The Elements

...that square is equal to the square of c. d., which was to be proved. Here ends the first book.

Here begins the second book.

A geometric diagram illustrating the definitions of Book II. It shows a large rectangle ACDB divided into four smaller rectangles by two interior lines, EF (horizontal) and GH (vertical), which intersect at point K on the diagonal AD. The vertices and intersection points are labeled: 'a', 'g', 'b' across the top; 'e', 'k', 'f' through the middle; and 'c', 'h', 'd' across the bottom. A diagonal line runs from 'a' to 'd'.

Every rectangular parallelogram is said to be contained under the two lines containing the right angle.
¶ A parallelogram is a surface of parallel sides.
¶ A rectangular parallelogram is one having all right angles. It is produced from one of its two containing sides, from one of its angles into the other; and therefore it is said to be contained under them. In modern terms, the area of a rectangle is the product of its length and width.

Of every parallelogrammic space, those parallelograms which the diameter cuts through the middle are said to stand around the same diameter. Of those parallelograms which stand around the same diameter, any one of them together with the two complements original: "supplementis" — refers to the areas in a divided parallelogram that do not sit on the diagonal is called a gnomon. GnomonAn L-shaped figure representing the area remaining when a smaller similar parallelogram is removed from a larger one.
¶ Which parallelograms are said to stand around the diameter, and which are the complements, was explained above in the demonstration of Proposition 43 of the first book. ¶ For let there be a parallelogram a. b. c. d. whose diameter is a. d. Let two lines e. f. and g. h. be drawn parallel to the opposite sides of the said parallelogram, intersecting each other upon the diameter a. d. at point k. The parallelogram itself will be divided into four parallelograms, and each of the two parallelograms which are a. g. e. k. and k. f. h. d., which the diameter cuts through the middle, is said to stand around the diameter. The remaining two, which the diameter does not cut, are called complements. These two complements together with either of the said parallelograms standing around the diameter compose a certain figure which is called a gnomon. It lacks one remaining parallelogram standing around the diameter to complete the parallelogram; if that were added upon the diameter of the total composite, it would stand there, and it would be similar to the whole. Hence, although a parallelogram grows when a gnomon is added, it is not altered in shape; just as Aristotle said in the Categories. original: "predicamentis" — Aristotle's work The Categories, where he uses the gnomon to illustrate things that change in quantity but not in quality/form.

Proposition .j.

A geometric diagram for Proposition 1. A large rectangle AFGB is divided into three smaller vertical rectangles by lines DH and EK. The bottom vertices are labeled 'a', 'd', 'e', 'b' and the top vertices are 'f', 'h', 'k', 'g'. To the far left, a standalone vertical line 'c' is depicted, representing the constant multiplier.

If there be two lines, one of which is divided into any number of parts, the rectangle produced by the multiplication of one by the other will be equal to the rectangular parallelograms produced by the multiplication of the undivided line into each part of the divided line individually. This is the geometric expression of the distributive law: a(b+c+d) = ab + ac + ad.
¶ To lead a line into another line is to erect two lines perpendicularly from the ends of one, equal to the other, and to complete a rectangular surface of parallel sides which, by definition, is said to be contained under those two lines. ¶ There are two lines a. b. and c., one of which, namely a. b., is divided into any number of parts, which are a. d., d. e., and e. b. I say that the rectangle which is made from the multiplication of c. into the whole of a. b. is equal to those rectangular parallelograms joined together which are made from c. into a. d., and from c. into d. e., and from c. into e. b. ¶ Upon points a. and b., I shall erect lines a. f. and b. g. perpendicular to the line a. b., each of which shall be equal to line c., and I will complete the rectangular surface a. f. b. g. by drawing line f. g., which by definition is produced from c. into a. b. and is said to be contained under them. I shall also draw from points d. and e. the lines d. h. and e. k. parallel to the sides a. f. and b. g. Each of them will be equal to c. by Proposition 34 of the first book, and each of them is equal to a. f. Therefore, by definition, the rectangle a. d. f. h. is produced from c. into a. d. and is said to be contained under them; and the rectangle d. h. e. k. from c. into d. e.; and the rectangle e. k. b. g. from c. into e. b. And since these rectangles joined together are equal to the total rectangle a. f. b. g., it is clear that the proposition is true.