PROPOSITION XXVIII.-PROBLEM. B B' D D' 80. On a given straight line, to construct a polygon similar to a given polygon. Let it be required to construct upon A'B' a polygon similar to ABCDEF. Divide ABCDEF into triangles by diagonals drawn from A. Make the angles B'A'C' and A'B'C' equal to BAC and ABC respectively; then, the triangle A'B'C' will be similar to ABC 25). In the same manner construct the triangle A'D'C' similar to ADC, A'E'D' similar to AED, and A'E'F' similar to AEF. Then, A'B'C'D'E'F' is the required polygon (38).. PROPOSITION XXIX.-PROBLEM. B В B! 81. To construct a polygon similar to a given polygon, the ratio of similitude of the two polygons being given. Let ABCDE be the given polygon, and let the given ratio of similitude be M:N. Take any point o, either within or without the given polygon, and draw straight lines from 0 through each of the vertices of the polygon. Upon any one of these lines, as 04, take OA' a fourth proportional to M, N, and OA, that is, so that с DI MI NH In the angle A OB draw A'B' parallel to AB; then, in the angle BOC, B'C' parallel to BC, and so on. The polygon A'B'C"D'E' will be similar to ABCDE; for the two polygons will be composed of the same number of triangles, additive or subtractive, similarly placed; and their ratio of similitude will evidently be the given ratio M:N. (40). 82. Scholium. The point 0 in the preceding construction is called the centre of similitude of the two polygons. 11 * BOOK IV. COMPARISON AND MEASUREMENT OF THE SURFACES OF RECTILINEAR FIGURES. 1. DEFINITION. The area of a surface is its numerical measure, referred to some other surface as the unit; in other words, it is the ratio of the surface to the unit of surface (II. 43). The unit of surface is called the superficial unit. The most convenient superficial unit is the square whose side is the linear unit. 2. Definition. Equivalent figures are those whose areas are equal. PROPOSITION 1.—THEOREM. D 3. Two rectangles having equal altitudes are to each other as their bases. Let ABCD, AEFD, be two rectangles having equal altitudes, AB and AE their bases; then, ABCD AB E B D F A Suppose the bases to have a common measure which is contained, for example, 7 times in AB, and 4 times in AE; so that if AB is divided into 7 equal parts, AE will contain 4 of these parts; then, we have If, now, at the several points of division of the bases, we erect perpendiculars to them, the rectangle ABCD will be divided into 7 equal rectangles (I. 120), of which AEFD will contain 4; consequently, we have ABCD 7 AEFD 4 and therefore ABCD AB The demonstration is extended to the case in which the bases are incommensurable, by the process already exemplified in (II. 51) and (III. 15). 4. Corollary. Since AD may be called the base, and AB and AE the altitudes, it follows that two rectangles having equal bases are to each other as their altitudes. Note. In these propositions, by "rectangle" is to be understood “surface of the rectangle.” PROPOSITION II.—THEOREM. 5. Any two rectangles are to each other as the products of their bases by their altitudes. Let R and R' be two rectangles, k and k' their bases, h and h' their altitudes; then, R R' k For, let S be a third rectangle having the same base k as the rectangle R, and the same altitude h' as the rectangle R'; then we have, by (4) and (3), R h s k hi' R' k' and multiplying these ratios, we find (III. 14), 6. Scholium. It must be remembered that by the product of two lines, is to be understood the product of the numbers which represent them when they are measured by the linear unit (III. 8). PROPOSITION III.—THEOREM. 7. The area of a rectangle is equal to the product of its base and altitude. Let R be any rectangle, k its base and h its altitude numerically expressed in terms of the linear unit; and let Q be @ the square whose side is the linear unit; then, by the preteding theorem, hl R k R k Xh k X h. Q 1 X 1 R But since Q is the unit of surface, the numerical measure, or Q area, of the rectangle R (1); therefore, Area of R=k X h. 8. Scholium I. When the base and altitude are exactly divisible by the linear unit, this proposition is rendered evident by dividing the rectangle into squares each equal to the superficial unit. Thus, if the base contains 7 linear units and the altitude 5, the rectangle can obviously be divided into 35 squares each equal to the superficial unit; that is, its area = 5 x 7. The proposition, as above demonstrated, is, however, more general, and includes also the cases in which either the base, or the altitude, or both, are incommensurable with the unit of length. 9. Scholium II. The area of a square being the product of two equal sides, is the second power of a side. Hence it is, that in arithmetic and algebra, the expression “square of a number” has been adopted to signify “second power of a number.” We may also here observe that many writers employ the expression “rectangle of two lines” in the sense of “product of two lines,” because the rectangle constructed upon two lines is measured by the product of the numerical measures of the lines. |