BOOK III. EQUIVALENCE AND AREA SECTION I. DEFINITIONS AND THEIR DISCUSSION; FORMULAS 225. Equivalence. Two closed figures have been defined as being equivalent when they contain the same amount of surface (§ 34). The question at once arises as to what ways of proving figures equivalent are known. The distinction between the different kinds of equality, as shown in §§ 32-37, and the equality axioms (§ 38), give the foundation for this class; the two following methods show how these definitions and axioms can be used. (1) Congruent figures are equivalent, and equivalent figures added, subtracted, and multiplied or divided by the same number, give equivalent results (although the results need not be congruent). (2) The whole equals the sum of all its parts, and the other equality axioms can be applied to the equation obtained. : The equivalence class must then depend on these two methods the equivalence of two figures usually on the first, equivalence equations usually on the second. When the first is used, the congruent figures are often added or subtracted in different positions, thus forming new figures which are equivalent, but not congruent. 226. Addition of Polygons. The addition of two polygons is accomplished by placing the polygons entirely outside of each other, as regards their surface, but with some portion of the perimeter (either a point, or some one or more sects) in common, and then taking the whole figure thus formed. The common part of the boundary line is considered as omitted, unless it is a point. It is evident that this is simply the ordinary understanding of a sum, the only difference being that the polygons have to be placed in such a position that their sum can be shown as a single figure. 227. Subtraction of Polygons. The subtraction of two polygons is accomplished by placing the smaller polygon entirely inside the larger, and then omitting the part occupied by the smaller; the remaining part of the larger polygon is the difference. Here again the usual idea of subtraction is employed, the figures being placed so that the difference shows as a new figure. 228. Addition and Subtraction of Parallelograms. From the definitions of addition and subtraction of polygons, it can be seen that two parallelograms which have a side and an angle of one equal to a side and an angle of the other can be added or subtracted so as to give a new parallelogram having the same side, and the same angle, as the result. This applies to rectangles having one equal side, and the fact can be put in formal statement as follows: Two rectangles having an equal side can be added so as to form a rectangle having the equal side for one of its sides, the sum of the two other length sides of the given rectangles for its other side. Similarly for the difference. 229. Rectangles and Squares. A rectangle having the sides X and Y is spoken of as the X, Y. This is permissible, since all rectangles having those sides are con gruent. Similarly, a square on the side x is spoken of as the X. Remembering that all rectangles which have two sides equal are congruent, it is easily seen that all that it is necessary to do in order to multiply a rectangle, is to multiply one side, for that makes a new rectangle, composed of as many congruent rectangles as the number by which the side was multiplied. This is most easily seen by keeping in mind that multiplication is here strictly a kind of addition, and that the line is multiplied by continuing it until it contains the required number of equal parts. In exactly the same way, a rectangle can be divided into any number of equal parts by dividing one side into that number of equal parts, and so forming a new rectangle, having one side the required part of the given side. NOTE. Dividing both sides of a rectangle, or multiplying both sides, performs the operation on the rectangle twice. For example, if each side of a rectangle is doubled, the rectangle is made four times as large. 230. Formulas. By the use of the axiom, "the whole equals the sum of all its parts," it is possible to obtain many equations between rectangles and squares, the following of which are the most important: *(1) The square on the sum of two sects is equivalent to the sum of their squares plus twice their rectangle. Take two given sects, draw the square on their sum, and see if it contains the required parts. *(2) The square on twice a sect is equivalent to four times the square on the sect. *(3) The square on the difference of two sects is equivalent to the sum of their squares, less twice their rectangle. Try to add the squares of the sects, and subtract the two rectangles in such a manner that a square will be left. *(4) The difference of the squares on two sects equals a rectangle, having one side equal to the sum of the sects, the other side equal to the difference of the sects. Subtract the smaller square, then make one rectangle out of the remainder. These geometric formulas, which deal entirely with the surfaces of the figures, correspond very closely to certain algebraic formulas; the only differences between the two are that where "square on a sect" appears in the Geometry, "square of a number" appears in Algebra, and where "rectangle of two sects" appears in Geometry, "product of two numbers" appears in Algebra. This correspondence follows throughout; wherever an algebraic formula is entirely of the second degree, the corresponding geometric statement concerning squares and rectangles is also true. The reason for this correspondence will be evident from the third theorem and its corollaries. 333. Find the formula for the square on the sum of three sects. Try to show what general formula can be obtained. 334. What does the square on three times a sect equal? What general formula is there? SECTION II. THEOREMS 231. Theorem I. Parallelograms (or triangles) on the same base, or on equal bases, and between the same parallels, are equivalent. Do not add, for this method does not work for all positions of the parallelograms. 232. COR. 1. Parallelograms (or triangles) on the same base, or on equal bases, and having equal altitudes, are equivalent. Place their bases on the same line, then show what? 335. Triangles having two sides equal, and the included angles supplemental, are equivalent. 336. Any two medians of a triangle form, with the side from whose ends they are drawn, and the halves of the sides to which they are drawn, two equivalent triangles. 337. Any median divides the triangle into two equivalent triangles. 338. The three medians of a triangle divide it into six equivalent triangles. NOTE. In the following theorems, it is the surface of the figure which is being considered. 233. Theorem II. Two rectangles having equal altitudes are proportional to their bases. (Commensurable case only.) 234. COR. 1. Two rectangles having equal bases are proportional to their altitudes. 235. Theorem III. The ratio between two rectangles equals the product of the ratio of their bases by the ratio |