fore—X. The sum of the numerators contains the sum of the denominators as many times as the difference of the numerators, contains the difference of the denominators. 84. If we have a series of equal ratios, a = 7 =} D E o g - ... A + B j - d . &c.; by applying law VIII, we obtain a + b TT B . C ë , o . A -- B 5 * “Or as # = z we may write instead of this a + b T C e o A + B+C co’ and applying the same law to this, we have a + b + c = D. and consequently A + B + C + D + P_ E 5 d a + b + c + d-He © wherefore—XI. In any series of equal ratios, the sum of the numerators contains the sum of the denominators as many times as one numerator contains its denominator. . A 85. If we have the two proportions a = ! and g o we can multiply them together ; that is, the first ratio of the first proportion by the first ratio of the second, and the second ratio of the first, by the second ratio of the second ; and as they are equal quantities, multiplied by equal quantities, they will give equal products. But to multiply two fractions together, we multiply their numerators for a new numerator, and their denominators for a A × C. B. × D a × c T b X d’ This process is called multiplying the proportions in order ; the result is a compound proportion, and each of the ratios is a compound ratio. We say then—XII. Proportions multiplied in order, will give a proportion. 86. It is evident from the last article, that a proportion may be multiplied by itself, term by term, and give a new denominator. This will give proportion ; thus, the proportion a = 7, multiplied by A × A B × B & X & too #2; but Ax A is written A* (73), B. × B is B?, &c. This proportion 3. B2 itself, term by term, gives may therefore take the form a? = 5: . If we multiply th & A* B", b he origi . A e proportion g = f; , by the original proportion a = |B . . | gri A * B3 5’ it will give as T 53 As we may take the powers, so also we may take the roots (of the same degree) of the several terms of a proportion and form another proporI T t I e A 2 B2 A3 B3 tion ; thus, F = —s and — = −. We therefore a 3 b2 a 3 O3 have this general law—XIII. The same powers or the same roots, of the terms of a proportion, are also in pro portion. A 2 . 87. Toemark. The ratio is is called the duplicate ra e o A 3 e * * go tio of A to a ; the ratio -a is called the triplicate ratio l, . A2 . . & te of A to a. The ratio or is called a sub-duplicate ratio; a? I & A3 e * * * e and to, a sub-triplicate ratio. a 3 We now return to the examination of geometrica; figures. Of Similar Figures. 88. In constructing the triangles &c. in the preceding operations, we have considered ourselves as taking the absolute magnitudes of the sides given in the problems; but in most cases in practical geometry this is inconvenient and undesirable. The methods used in surveying fields and townships, and constructing maps, afford a good illustration of this part of the subject. The lines by which the field &c. is bounded, are measured, and also the several angles; and from the notes of the survey a perfect representation of the outline of the field, township, &c. is constructed upon paper. This is called a plan of the field, township, &c. Suppose, for instance, that we have a triangular field whose sides are eight, ten, and twelve chains, respectively ; the method of construction would be similar to that of the triangle with three given sides (38). But instead of taking the absolute lengths of these known sides, we take lines embracing Fig. 44. AC g so . AB times as AC contains a c, that is a 5- or In the same manner we may show that AB contains a b as many AB BC times as BC contains b c ; that is, o, o ż, , , These two C AB BC proportions may be written together, thus, I = W. = AC –––. We see therefore, that—In similar triangles, the O! C corresponding or homologous sides are proportional. 91. If, in making a survey of the field, we had measured one of the angles and the two sides containing that angle, the construction of the plans would have been analogous to the solution of the problem in Art. 40; and plans upon two different scales will accurately represent the field, if we construct, in each, an angle equal to the measured angle of the field, and take for the lines which represent the two sides in the larger plan, as many divisions of the larger scale as there are chains or rods in these lines respectively, and for the corresponding sides in the smaller plan the same number of divisions respectively, of the smaller scale. The corresponding sides will have the ratio of the two scales, and will therefore be in proportion. We see then, that—Two triangles will be similar, when an angle of the one is equal to an angle of the other, and the sides containing the equal angles are proportional. If we draw from the vertices C, c, of these triangles, perpendiculars to the opposite sides, as they measure the distance of that corner of the field from the opposite side, they must have the same ratio to each other as the corresponding sides of the two plans ; and so of any other similar lines in the two plans respectively ; whence we derive the proposition—In similar triangles, the homologous angles are equal, and all homologous lines are proportional. Remark. Either of these conditions, however, is sus. ficient to determine their similarity; for by altering the angles, we alter the ratio of the sides and vice versä. 92. If the field were a rectangle, a parallelogram, or any other rectilinear figure, it might also be represented by plans upon different scales; and these plans will be similar to each other, and similar to the field, when, in each, the angles are respectively equal to the angles of A + B + C + D + E_A a + b + c + d -H e T a ' that is—In similar polygons, the perimeters are proportional to their homologous sides. 95. In the base of the triangle ABC (fig. 45), take A c Fig. 43. of any convenient magnitude, and draw c b parallel to " C B; we shall have the triangle A c b, equiangular with ACB (24), and therefore similar to it (91). Hence—If a triangle is cut by a straight line parallel to one of the sides, the small triangle cut off is similar to the whole. These similar triangles give the proportion AC AB AC —— A c AB —— A b * = x : ; therefore (sl), -w- = ***, * : , C c B. b to ge .. e that is, A c T Ab ' and the line c b divides the two sides AB and AC, proportionally. Thus we have the general |