PROPOSITION X. 304. The products of the corresponding terms of two or more proportions are in proportion. 305. Like powers, or like roots, of the terms of a proportion are in proportion. 306. Equimultiples of two quantities are the products ob tained by multiplying each of them by the same number. Thus, ma and mb are equimultiples of a and b. PROPOSITION XII. 307. Equimultiples of two quantities are in the same ratio as the quantities themselves. Let a and b be any two quantities. 308. SCHOLIUM. In the treatment of proportion it is assumed that fractions may be found which will represent the ratios. It is evident that the ratio of two quantities may be represented by a fraction when the two quantities compared can be expressed in integers in terms of a common unit. But when there is no unit in terms of which both quantities can be expressed in integers, it is possible to find a fraction that will represent the ratio to any required degree of accuracy. (See §§ 251-256.) Hence, in speaking of the product of two quantities, as for instance, the product of two lines, we mean simply the product of the numbers which represent them when referred to a com mon unit. An interpretation of this kind must be given to the product of any two quantities throughout the Geometry. PROPORTIONAL LINES. PROPOSITION I. THEOREM. 309. If a line is drawn through two sides of a triangle parallel to the third side, it divides those sides proportionally. In the triangle ABC let EF be drawn parallel to BC. CASE I. When AE and EB (Fig. 1) are commensurable. Find a common measure of AE and EB, as BM. Suppose BM to be contained in BE three times, At the several points of division on BE and AE draw straight lines to BC. These lines will divide AC into seven equal parts, of which FC will contain three, and AF will contain four, § 187 (if parallels intercept equal parts on any transversal, they intercept equal parts on every transversal). CASE II. When AE and EB (Fig. 2) are incommensurable. Divide AE into any number of equal parts, and apply one of these parts as a unit of measure to EB as many times as it will be contained in EB. Since AE and EB are incommensurable, a certain number of these parts will extend from E to a point K, leaving a remainder KB less than the unit of measure. ratios EK FH AF Suppose the unit of measure indefinitely diminished, the and continue equal; and approach indefi 310. COR. 1. One side of a triangle is to either part cut off by a straight line parallel to the base as the other side is to the corresponding part. or For EB: AE= FC: AF, by the theorem. = .. EBAE: AE FC+ AF: AF, AB: AE AC: AF = § 300 311. COR. 2. If two lines are cut by any number of parallels, the corresponding intercepts are proportional. Let the lines be AB and CD. Draw AN to CD, cutting the Ils at L, M, and N. Then AL CG, LM-GK, MN= KD. §187 = By the theorem, A C FL G AH: AM AF: AL= FH: LM = HB: MN. If the two lines AB and CD were parallel, the correspond ing intercepts would be equal, and the above proportion be true. PROPOSITION II. THEOREM. 312. If a straight line divide two sides of a triangle proportionally, it is parallel to the third side. In the triangle ABC let EF be drawn so that (one side of a ▲ is to either part cut off by a line || to the base, as the other side is to the corresponding part). .. EF, which coincides with EH, is to BC. Hyp. Ax. 1 Cons. Q. E. D. |