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, D E B A B C a &c.; by applying law VIII, we obtain b C or as we may write instead of this ; and applying the same law to this, we have D d' b C A+ B a + b A+ B a+b A+B+C a + b + c and consequently 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. 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 AXC B×D new denominator. a x c b x ď This will give 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 A B proportion; thus, the proportion is written A2 (73), BX B is B2, &c. This proportion 1 B2 also we may take the roots (of the same degree) of the several terms of a proportion and form another propor } I We therefore have this general law-XIII. The same powers or the same roots, of the terms of a proportion, are also in proportion. 87. Remark. The ratio a2 is called the duplicate ra We now return to the examination of geometrical 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 10 lengths of these known sides, we take lines embracing eight, ten, and twelve divisions upon a scale of equal parts, of any magnitude, of an inch, for instance, which will give us a plan of a convenient size, as in the figure ABC (fig. 44). This is a complete representation Fig. 44. of the field surveyed; so also it would be, if the sides of the field had been eight, ten, and twelve rods, or yards, instead of chains. We should also have respresented the field as perfectly, if instead of the scale of tenths of an inch, we had used a scale divided into twentieths of an inch, as in the figure a b c. If then all these different triangles are faithfully represented by each other, they must have a common character; they must be similar. Let us see in what this similarity consists. 89. 1st. The angles of the one are respectively equal to the angles of the other. It is evident that this is necessary; for if these artificial triangles have angles different from those of the field, they cannot represent its form, which is essential to a map or plan. If we had measured one side and the two adjacent angles, the side represented by AB, and the angles A and B; in drawing these two plans of the field, we should have made the angles A and a, each equal to that angle of the field which they represent; and the angles B and b, each equal to the other measured angle. And the two sides ac, bc, having the same inclinations to a b, as AC, BC have to AB, must have the same inclination to each other, as AC and BC have; that is, the angle c must be equal to the angle C; and consequently the three angles of the one must be respectively equal to the three angles of the other. This then is an essential property of similar triangles, their angles are respectively equal. 90. 2dly. Recurring to the construction of the two triangles, ABC, abc; the divisions of the scale with which the larger was constructed, are just double the divisions of the scale with which the other was constructed; so that each side of the triangle ABC, is just double the corresponding side of the triangle abc. If the divisions in the two scales had borne any other ratio to each other, the corresponding sides of the two triangles would have borne the same ratio. But the ratio of AB to a b being equal to the ratio of the two scales; and also the ratio of AC to a c being equal to the ratio of the two scales; it follows that AB contains ab as many AB AC times as AC contains a c, that is In the same ab ас AB BC These two AB BC b c manner we may show that AB contains ab as many times as BC contains be; that is, ab proportions may be written together, thus, ab AC ac We see therefore, that-In similar triangles, the 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 suf ficient to determine their similarity; for by altering the angles, we alter the ratio of the sides and vice versa. 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 the field, and their dimensions also proportional to the corresponding dimensions of the field. Hence we have the general truth-Two similar rectilinear figures have the angles of the one equal to the angles of the other, each to each, and all homologous lines proportional. 93. Remark. If diagonals were drawn through corresponding vertices in the two polygons; being homologous dimensions, they would be proportional; and being proportional, the triangles into which these diagonals divided the two figures would be similar respectively. We say therefore, that-Similar polygons are composed of the same number of similar triangles, similarly placed. 4. If the several sides of a polygon were represented by A, B, C, D, E, and the corresponding sides of a similar A B polygon, by a, b, c, d, e; we should have (92) α ; from which we obtain (84), A+B+C+D+E a+b+c+d+e A 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. 45. of any convenient magnitude, and draw cb parallel to CB; 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 A c AB AC therefore (81), A c B b that is, A c A b Ас АВ AB-- Ab ; and the line c b divides the two sides AB and AC, proportionally. Thus we have the general rule,--A straight line drawn through any triangle parallel to one of the sides, divides the other two sides proportionally. The converse of these two propositions may be proved from Art. 91. 96. If another parallel, b' c', were drawn between b c and BC, it would give the proportion A c' C c' and B b' |