COR. III. Because all squares are similar figures, the ratio of any two squares to one another is the same with the duplicate ratio of the sides; and hence also, any two similar rectilineal figures are to one another as the squares of the homologous sides. COR. IV. In similar figures, their perimeters are to one another as the ratio of the homologous sides. :: AB,BC &c. : FG,GH &c. = AB : FG =BC: GH; COR. V. The homologous diagonals being sides of similar triangles are also homologous sides; and therefore, perimeters of similar figures are as their homologous diagonals. COR. VI. A Circle and its inscribed Polygon of an infinite number of sides do not differ from each other in any degree however small, but that a smaller difference might be assigned; and practically, they may be considered of identical values. As therefore it has been predicated of similar polygons, 20, VI, that "they are to each other as the squares of their corresponding sides," whether diagonals or sides;—so it may be predicated of all circles, that being similar figures, they are to each other as the squares of their respective diameters, radii, and circumferences. COR. VII. If on the three sides, AB, AC, CB, of art. Ld triangle ACB, similar figures be described as semicircles, A e C g B, Ad C and CfB; the figure on the hypotenuse, AB, will be equal to the sum of the similar figures on the two other sides, AC, CB. D. 1 Cor. 6, 20, VI. 23 47, I. Semicircles, or any sim. fig., on the three sides, are as the squares on those three sides; & AB2 = AC2 + CB2, .. Semicircle on AB = Semic. on AC + Semic. on CB. g HIPPOCRATES, of Chios, a Pythagorean philosopher, who lived about 460 B.C., is said to have made this application of the universal principle, which under different forms appears in 47, I., and 31, VI., that "if three similar figures be described upon the sides of a rt. angled triangle, the contents of that which is described upon the hypotenuse will be equal to the sum of the contents of the figures described upon the sides." Hence, was deduced, COR. VIII. That the Lunes Ae Cd, Cg Bf, formed by describing semicircles, as Ae Cg B, AdC & CfB, on the sides of a rt. d triangle, are equal in area to the right angled triangle ACB. D. 1 Cor. 7. 20, Semic. on AB = semic. on AC+ semic. on CB; VI. 2 Sum. 3 4 Sum. 5 6 Ax. 4, VI. from semic. on CB take away the shaded segments A e C & CgB; The remainder is the rt. d ▲ ACB; From semicircles on AB, CB take away the same segments; The remainders are the two lunes AeCd, BgCf; .. the Lunes AeCd + BgCf = ▲ ACB. N. B. This is the first known instance in which a curvilinear space was reduced to an equivalent rectilinear space. SCH. 1. Further elucidation of Prop. 20, will be found in LARDNER'S Euclid, pp. 210-214, and in pp. 127, 134 of this work. 2. In circles, which are all similar, when the segments are similar, the radii and the chords become homologous sides. USE & APP. I. By this proposition, a rt. lined figure may be increased or diminished in any ratio Thus, to make a pentagon five times the size of the pent. on DC, (in last figure but one.); Take a side AE= 2; and between AE and 5 times AE, i. e., 10, find the mean proportional;—it is √2 × 10 20 4.472136; A sim. fig. on a side of 4.472136 will be five times the pent. on AE. II. When the homologous sides are known the Proportion of one figure to another will be obtained by finding a third proportional to any two corresponding sides. III. The increase or diminution of Circles is effected in the same way. Thus one circle is on a diam. of 1,-another is to be constructed 4 times larger; required the diam. of the second circle. The mean propl. to 1 & 4 is /1X4=√/4= 2. .. the diam. is 2 for the required circle. Or, Given the diameters 1 & 2, required the magnitude of the second circle when compared with the first. = = 4,-the second circle is four times larger than the IV. In two similar figures, if. of the areas and corresponding sides, any The principle employed is, three be given, the fourth may readily be found. that the areas of similar figures are to one another as the squares of their homologous sides, and vice versâ. For, (fig. 20, VI.) ▲ ABE: ▲ FGL = AB2 : FG2; A BCE: A and ▲ CDE: A GHL BC2: GH2; HIL = CD2 : HI2. Hence, As ABE + BCE + CDE: As FGL + GHL + HIL =AB2: FG2. i.e. fig. ABCDE : fig. FGHIL = AB2 : FG2; Thus FG2 FGHILX AB2 ABCDEX FG2 AB2 = ; Ex. 1. Two similar hexagons are respectively of the Areas of 2500 sq. yards and 3000 sq. yards; a side of the first is 5 lineal yards; required the length of the corresponding side of the other hexagon. Here, the unknown side x = 3600 X 5X5 36 6 lineal yards. 2500 Ex. 2. One of the sides of the base of a pyramid measures 50 yards, and the area of the base 7500 sq. yards;-in a similar pyramid, what is the area of the base when the corresponding side is 60 yards ? PROP. 21.-THEOR. Rectilineal figures which are similar to the same rectilineal figure, are also similar to one another. DEM. Def 1, VI. Ax. 1, I. 11, V. E. 11 Hyp. 2 Conc. Let figures A & B be each sim. to fig. C. D. 1 H.Def.1,VI. A is sim. to C, . A is eq. ang. with C, & their homol. sides are proportionals. 2 H. Def. 1, VIB is sim. to C, .. B is eq. ang. with C, 3 Ax. 1, I. 11, V. and their homol. sides. propls. .. A & B are each eq. ang. with C, and their 4 Def. 1, VI... rectl. fig. A is sim. to rectl. fig. B. SCH. This proposition follows evidently from Def. 1, VI., of similar rectilineal figures; it is equivalent to an Axiom, and in this respect agrees with Prop. 30, I. St. Lines parallel to the same st. line are parallel to each other. 11, VI. Ratios that are the same to the same ratio are the same to one another; variations of the General Principle, -Things equal to the same thing are equal to one another. X PROP. 22.-THEOR. If four st. lines be proportionals, the similar rectilineal figures similarly described upon them shall also be proportionals; and conversely, if the similar rectilineal figures similarly described upon four st. lines be proportionals, those st. lines shall also be propor DEM. 11, V. 22, V. Cor. 2, 20, VI. 9, V. Magnitudes which have the same R to the same M are eq. to one another; and those to which the same M have the same R are eq. to one another. 7, V. Eq. Ms have the same R to the same M; and the same has the same R. to eq. Ms. Let four st. lines be proportionals, &c. CASE I. E. 1 Hyp. 1. Let AB: CD = EF: GH; K on AB & CD let sim. rect. figs. KAB & LCD, be similarly described; A & on EF & GH sim. rect. figs. MF & NH, be also similarly described; fig. MF: fig. NH; 4 11, V. | .. fig. KAB: fig LCD = fig. MF: fig. NH. X |