THEOREM IX. 23. If any number of quantities are proportional, any antecedent is to its consequent as the sum of all the antecedents is to the Adding (A), (B), (C) a (b+d+f)=b(a+c+e) Hence, by (14) a:b=a+c+e:b+d+f THEOREM X. 24. If there are two sets of quantities in proportion, their pro ducts, or quotients, term by term, will be in proportion. BOOK II. RELATIONS OF POLYGONS. DEFINITIONS. 1. The Area of a polygon is the measure of its surface. It is expressed in units, which represent the number of times the polygon contains the square unit that is taken as a standard. 2. Equivalent Polygons are those which have the same area. 3. The Altitude of a triangle is the perpendicular distance from the opposite vertex to the base ; as B D. B 4. The Altitude of a parallelogram is the perpendicular distance from the opposite side to the base; as IK. 5. The Altitude of a trapezoid is the perpendicular distance between its parallel sides; as PR. THEOREM I. gon ABCDEF equal to the polygon GHIK L M. For if the polygon ABCDEF is applied to the polygon GHIKLM so that A B shall be on GH with the point A on G, B will fall on H, as AB and GH are equal; and as the angle B is equal to the angle H, BC will take the direction HI; and as BC is equal to HI, the point C will fall on I; and so also the points D, E, F will fall on the points K, L, M; and the polygon ABCDEF will coincide with the polygon GHIKLM, and therefore be equal to it. THEOREM II. 7. The area of a rectangle is equal to the product of its base and altitude. Let ABCD be a rectangle; its area = AD × AB. BOPQRC G V F M S L AHIJK D Suppose AB and AD to be divided into any number of equal parts, A E, EF, AH, HI, &c., and through the E points of division, lines EL, FM, HO, IP, &c. be drawn parallel to the sides of the rectangle; then the rectangle will be divided into squares; these squares will be equal to each other (6). If one of the equal parts, AE, represents the linear unit, then one of the squares, A ESH, represents the square unit; and there will be as many square units in the rectangle AELD as there are linear units in AD; and as many square units in the rectangle ABCD as there are square units in A ELD multiplied by the number representing the number of linear units in AB; that is, the area of the rectangle is equal to the product of its base and altitude, that is = A D XAB. 8. Scholium. If AD and A B have no common measure, the linear unit may be taken as small as we please, that is, so small that the remainders will be infinitesimal, and can be neglected. 9. Corollary. The area of a square is the square of one of its sides. THEOREM III. 10. The area of a parallelogram is equal to the product of its base and altitude. Let DF be the altitude of the parallelogram ABCD; then the area of ABCD=A D X D F. EB A 1 FC D At A draw the perpendicular A E meeting CB produced in E; AEFD is a rectangle equivalent to the parallelogram ABCD. For the two triangles AEB and DFC, having the sides A E, AB equal respectively to the sides DF, DC (I. 64), and the included angle E AB equal to the included angle FDC (I. 12), are equal. Adding DFC to the common part ABFD gives the parallelogram ABCD; and adding its equal AEB to the common part ABFD, gives the rectangle A EFD; therefore the parallelogram A B C D is equivalent to the rectangle AEFD; but the area of the rectangle = A D X DF (7); therefore the area of the parallelogram = A D X D F. THEOREM IV. 11. The area of a triangle is equal to half the product of its base and altitude. Let BD be the altitude of the triangle ABC; then the area of ABC=ACX BD. B E Draw CE parallel to AB, and BE parallel to A C, forming the parallelogram ABEC. The triangle A B C is one half the parallelogram ABEC (I. 63); the area of the parallelogram = AC X BD (10); therefore the area of the triangle = 1 ACX BD. AD C 12. Cor. 1. Triangles are to each other as the products of their bases and altitudes. For if A and a represent the altitudes of two triangles T and t, and B and 6 their bases, their areas are A × Banda × b; therefore or (Pn. 21) T:t=AXB:aXb 13. Cor. 2. Triangles having equal bases are as their altitudes; those having equal altitudes as their bases. For in the proportion above, if B = b, or A = a, the equals can be cancelled from the second ratio (Pn. 21). THEOREM V. 14. The area of a trapezoid is equal to half the product of its altitude and the sum of its parallel sides. Let EF be the altitude of the trapezoid ABCD; then the area of ABCD = EFX (BC + AD). Draw the diagonal BD; it will divide the trapezoid into two triangles, ABD, BCD, having the same altitude EF as the trapezoid. By (11) the area of and the area of B EC A F D BCD=EFX BC 2 ABD=EF X AD Therefore the area of the trapezoid = 1 E F X (B C + AD). 15. Corollary. As (I. 66) the line joining the middle points of the sides AB and CD of the trapezoid = 1 (BC + AD), |