In general terms it may be said that the sixt book establishes; 1st, the proportion between the sides of similar triangles; and, 2nd, the proportion existing between the areas of similar rectilineal figures it also lays down the methods, either of finding magnitudes proportional to other magnitudes, or of describing figures similar to other figures, or equal to them. DEFINITIONS. I. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals thus the As ABC, DEF, are similar, if A= < D, ▲ B = ≤ E, ≤ C = ≤ F, and if AB : AC, = DE : DF, and AB BC= DE: EF. D "This definition, like some others to be found in the Elements, is excessive. To contain no more than is strictly necessary (or, indeed, than as yet has appeared to be probable), it should be modified as follows:- Two rectilineal figures are said to be similar when the first has all its sides but one proportional to the sides of the other, and the angles included by those sides equal to the angles included by the corresponding sides of the other." Geom. Pl., Sol. & Sph. p. 57. B CE T According to the definition, for one rectilineal figure to be similar to another, the conditions to be fulfilled are equal to twice the number of sides, or rather to the sum of the E D L H 10. Z A = F; 2°. ▲ B = /G; 3°. / C= / H; 4°. / D=/K; and 5°. E = L. Also 6°. EA: AB=LF: FG; 7°. AB :BCFG: GH; 8°. BC: CD =GH:HK; 9°. CD: DE= HK: KL; and 10° DE: EA=KL:LF. See HOSE's Euclid, p. 198. A /B FG II. "Reciprocal figures, viz., triangles and parallelograms, are such as have their sides about two of their angles proportionals in such a manner, that a side of the first figure is to a side of the other, as the remaining side of this other is to the remaining side of A the first;" thus, AB: CD=DE: B F; the analogy beginning in one figure and ending in the same. B FD E "Figures are reciprocal when the antecedents and the consequents of ratios are in each of the figures." EUCLID. Another way of putting the definition is:-" The sides of two figures, The sides are directly proportional, when in each figure the two sides compared are one an extreme and the other a mean: thus, if AB : BF = CD: EF, the proportion is direct. III. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment, as the greater segment is to the less; thus in the line AB and its parts, AB AC AC: CB. For EUCLID's definition, LARDNER substitutes," when the whole line is to one segment as that segment to the remaining one." A line thus divided is also said to be divided medially; and the ratio of its segments is named the medial ratio.-Prop. 30, VI. is the problem by which the segments are made, and is but another form of Prop. 11, II.; to divide a line so that the rectangle under the whole line and one segment shall equal the square of the other segment. IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base; thus AD is the altitude of the triangle A B C. Whichever side of a figure is assumed as the base, the same figure may vary with the change in position of its base. SUBSIDIARY DEFINITIONS. DEF. A. A straight line, O C, divided into three parts, is said to be harmonically divided, when the whole line CO is to one of its extreme segments O E, as the other extreme CD D E C. is to the middle part D E; i. e., OC : OE = CD: DE. Four st. lines are said to be harmonicals, when they pass through the same point, and divide any one st. line harmonically. DEF. B. "A figure is given in species, when its several angles and the ratios of the sides about them are given." DEF. C. "A figure is given in magnitude, when its area, or any figure equal to it in area, is given." DEF. D. "A parallelogram is said to be applied to a straight line, when it is described upon it as one of its sides; ex. gr. the parallelogram A C is applied to the straight line A B. IIJ A G DEF. E. "But a parallelogram A E, is said to be applied to a straight line A B, deficient by a parallelogram, when AD, the base of the parallelogram A E, is less than A B, the base of parallelogram A C; and because the AE is less than the A C, (described upon A B, with the same A, and between the same parallels AB, EC,) by the DC; therefore the DC is called the defect of A E. DEF. F. And a parallelogram A G is said to be applied to a straight line A B exceeding by a parallelogram, when A F, the base of AG is greater than the base A B of A C; and because the AG exceeds the AC, (described upon A B, with the same A, and between the same parallels A F, EG,) by the BG; therefore the BG is named the excess of ☐ A C. PROPOSITIONS. PROP. 1.-THEOR. Triangles and parallelograms of the same altitude are one to another as their bases. CON. Pst 1, I. Pst. 2, I. A st. L. may be drawn from one ⚫ to any other point. A terminated st. L. may be produced in a st. L. 3, I. From the gr. of two st. Ls. to cut off a pt. the less. Triangles upon equal bases and between the same parallels are equal to one another. 1, V. DEM. 38, I. Def. 5, V. The 1st of four Ms is said to have the same ratio to the 2nd which the 3rd has to the 4th, when any equims. whatsoever of the 1st and 3rd being taken, and any equims whatsoever of the 2nd and 4th; if the m of 1st be < = or that of the 2nd, the m of the 3rd is also <= or that of the 4th; 41, I. If a and a ▲ be upon the same base and between the same shall be double of the A. 15, V. Ms have the same ratio to one another which their equims. have. 11. V. Ratios that are the same to the same ratio, are the same to one another. 28, I. If a st. line falling upon two other st. lines makes the ext. = the int. and opp. / upon the same side of the line; or makes the s upon the same side together 2 rt. s; the two st. lines shall be parallel. 33, I. The st. lines which join the extrs. of two eq. and parallel st. lines 36, I. CASE I. Triangles of the same alt. are to one another as their bases. E. 1 Hyp. 2 Conc. Let As ABC & ADE have the same alt. AH; then BC: DE A ABC: AADE. K ONMB CHD E P Q L = Produce BE indef. to K & L; = BC; DE; Then BC = BM = MN NO, & KL | ST; ДАСВ; i. e. OC & A ACO are equims. of BC & ▲ ABC. So, DQ & A ADQ are equims. of DE & A ADE; and if OC >= or < DQ, ▲ ACO > = or <▲ ADQ. Now, CO & A ACO are equims. of CB & A ACB; |