CASE II. If the sim. rectilineal. figures, &c. E. 1 Hyp. 2 Conc. C. 1 12, VI. 2 18, VI. D. 1 C. 1. 2 C. 2. 3 C. 2. 5 Hyp. 6 9, V. 7 C. 2. Let fig. KAB: fig. LCD = fig. MF: fig. NH; Make AB CD = EF: PR, and on PR descr. fig. SR sim. and similarly situated to MF or NH. AB: CD EF: PR; = and on AB, CD are sim. and sim. desc. figs. KAB, LCD; and they are sim. and sim. situated, 8 C. 1. D. 7. And ... AB: CD = EF: PR, and PR = 9 7, V. 10 Rec. GH; If therefore four st. lines be proportionals, &c. Q.E. D. COR. As a particular case, if four st. lines A,B,C,D, be proportionals their squares A2, B2, C2, D2, shall also be proportionals ; and conversely; i. e., if A: BC: D; then Ao : B2 = C2 : D2; and if A2 : B2 = C2: D2; then A : B = C: D. SCH. Thus, this proposition is equivalent to the theorem, that, "if two ratios be equal, their duplicates and subduplicates will also be equal," or in other words their powers and roots. USE & APP. The principle contained in this Theorem is often employed in Arithmetic and Algebra; for if a 3: 4 c6 d8; then a29: 62 16 c2 36: d264. If four quantities or numbers are in proportion, their like powers or roots are also proportionals. LEMMA. THEOR. If rectilineal figures be equal and similar their homologous sides are equal. 2 H. 2.16, V. ·.· EF : ED = BC: BA; .. altern. EF: BC=ED: BA; 3 D. 1.20, VI but EF>BC.. ED>BA; .. fig. DF>fig. AC; now fig, DF also = fig. AC;-which is impossible; 4 H. 1. 5 Conc. .. EF not BC; i. e., EF = BC. Q. E. D. PROP. 23. THEOR. Equiangular parallelograms have to one another the ratio which is compounded of the ratio of their sides. CON. 14, I. 31, I. 12, VI. DEM. Def. A. V. of Compound Ratio. When there are any no. of Ms of the same kind, the 1st is said to have to the last of them the R compounded of the R. which the 1st has to the 2nd, and of the R which the 2nd has to the 3rd, and of the R. which the 3rd has to the 4th, and so on unto the last magnitude. 1, VI. 11, V. 22, V. ex æquali. E. 1 Hyp. Let AC be eq. ang. to CF, having 2 Conc. BCD = / ECG; then AC: CF = R compounded of the ratio of the sides, BC: CG & DC : CE. C. 1 Pon. 14,I. Place BC & CG in a st. line 2 Def. A.V. but K: M is compounded of K: L & L : M; .. K: M is a R compounded of the R of the sides. And BC: CG =AC: 3 4 1, VI, C. 3. but BC: GC = K : L; .. K : L = AC: CH, but DC: CE = L: M; .. L: M = OCH : And K: L = 10 Remk. 11] 12 Rec. But K: M is compounded of the R of the sides; .. the sides; i. e. of BC: CG and DC: CE. .. Equiangular parallelograms have &c.. Q. E. D. N, B. This 23rd proposition would follow as a Corollary from the Theorem," that any two rectangles are to one another in the ratio which is compounded of the ratios of the sides." Let AC & CF be two eq. ang. parallelograms; Complete the CH. D. 1 Def. A,V.| AC: CF has the R compounded of 2 but AC: and CH: 3 Conc. AC: CH = BC: CG, CF = DC: CE; CF is a ratio compounded of Rs which are the same with the Rs of the sides. Arith. Hyp. Let BC = 8, CG = 5, DC = 4; and CE = 7. thus 8 8.75 is the ratio compounded of both; or 32: 35. COR. 1. If the terms of two analogies are lines, the rectangles under their corresponding terms are proportional. COR. 2. Hence, also, Rectangles whose bases are proportional and also their altitudes, are themselves proportionals. USE & APP. To describe a rhombus equal to a given rectilineal figure R, and having an angle equal to a given angle L. Parallelograms about the diameter of any parallelogram, are similar to the whole and to one another. the int. and opp. DEM. 29, I. If a st. line fall upon two || st. lines, it makes the alternate Ls to one another; and the ext. upon the same side; and likewise the two int. s upon the same side = 2 rt. /s. 4. VI. 34, I. The opp. sides and s of's are eq. to one another, and the diam. bisects them. 7. V. Def. 1, VI. 21, VI. Rectl. figures sim. to the same, are similar to each other. |