compared together such as are equal; that is, such as may be placed, the one upon the other, so as to coincide entirely. And we remark that this coincidence is an indispensable condition of equal figures. Preparatory to a more extended comparison of triangles and other rectilinear figures; and a discussion of some other properties of individual polygons, it is necessary to consider the doctrine of proportion. Of Ratios and Proportions. 73. Before entering upon an examination of the laws of proportion, it will be proper to explain the meaning of certain signs which are found very convenient in expressing mathematical truths. When we wish to indicate the addition of two quantities expressed by A and B, we place between them this sign +, which is equivalent to the Latin word plus, signifying more; thus, A + B; which is usually read, A plus B, and means A added to B. It expresses, of course, the sum of the magnitudes A and B. When we would indicate that B is subtracted from A, we use the sign which is equivalent to the Latin minus, i. e. less; thus, AB, usually read, A minus B, signifies the remainder after subtracting the magnitude B from the magnitude A, and expresses the difference of the two magnitudes. To express the multiplication of two magnitudes, we use this sign X; thus, A X B; which is read, A multiplied by B; and signifies the product of these magnitudes. A period (.) placed between two magnitudes, also denotes their multiplication; thus, A B significs A multiplied by B. To denote the division of one magnitude by another, A we write them in the form of a fraction; thus, which B' Ᏼ is read A divided by B, and expresses the quotient arising from the division of A by B. Instead of the expression A × A, we write, for the sake of brevity, A2; which signifies the product of A multiplied by itself, and is read, A second power. If wc would indicate the second power of the sum of two magnitudes, we should write (A + B)2; the second power of the product, (AX B)2 *. To indicate the third power of a called exponents. To indicate any root of a magnitude, we write the letter indicating the magnitude with a fractional exponent expressing the root, as A, which we read, the second root of A. So also the third root of A –– B, is with the sign B). between them 1 To show that two magnitudes are equal, we write them between them; as A = as A = B; this is read, The expression 5+ 49, is read 5 plus When we wish to state that A is greater write them thus A > B. If we would say that A is less than B, we write them A <B. The expressions 2A, 3A, &c. indicate double, triple, &c. the magnitude represented by A. We now proceed to the consideration of ratios. 74. To find a common measure of two lines, and hence their ratio, we employ a method similar to that in arithmetic for finding the greatest common divisor of two numbers. Let AB and CD (fig. 43) be two straight lines of which a common measure is required. Apply the less CD to AB the greater, as many times as CD is contained in AB; suppose this to be three times with a remainder EB; so that we shall have AB equal to three times CD added to EB, or AB= 3CD + EB. Now let us apply the remainder EB to CD; as it will be contained four times with the remainder FD, we shall have CD 4EB + FD. Apply now the second remainder FD to EB; it will be contained once, with the remainder GB; this gives EB FD + GB. Then apply GB to FD; it will be contained twice, exactly, which gives FD 2GB. By retracing the steps of the operation, we shall perceive that EB = 3GB ; that CD=14 GB; and that AB 45 GB; from which it is evident, that-- The last remainder GB is a common measure of the straight lines AB and CD. As this measure 2 2 * Sometimes we find them written A + B, A × B, &c. Fig. 43. Fig. 43. GB is contained forty-five times in the first, and fourteen times in the second, we say that the line AB is to the line CD in the ratio of 45 to 14. This ratio we write in a fraction, 45. The ratio of two lines, therefore, is the number of times the first contains the second; as the ratio of two numbers is the quotient arising from the division of the first by the second. 14 75. It will be easy to apply this process to any other example. We should continue the comparison of the successive remainders, till the last remainder is contained an exact number of times in the next preceding. We shall soon arrive at this result; or at least so near that the error will be too small to be recognised by the senses; in this case, we say, we have the approximate ratio of the two lines. This explanation, notwithstanding the imperfection of the mechanical process of comparing lines, is sufficient to give us a clear idea of what is meant by the ratio of one line to another. 76. Remark. It is not at all necessary, in expressing a ratio, that the greater magnitude should stand as the numerator of the fraction. In the example above, we said, that the ratio of AB to CD, is 45; but the ratio of CD to AB is 14. We say that 3; that is, 6 con 45 tains 2 three times, or the ratio of 6 to 2, is equal to 3 ; we may say also, 2 ; that is, the ratio of 2 to 6 is equal to one third. 77. When four magnitudes are such that the ratio of the first to the second, is equal to the ratio of the third to the fourth, these four magnitudes are said to be in proportion. Proportion is equality of ratios. "A PROPORTION," mathematically speaking, is a formula, expressing the equality of two ratios. Let A represent a magnitude which contains another magnitude a, as many times as the magnitude B contains b; these four terms will form a geometrical proportion. This proportion we A B* write thus: We read this proportion,--The a b * A proportion has generally been written in this form; A:a:: B: b; and read, A is to a, as B is to b. The terms A, and B, are called antecedents; and a, b, are called consequents. The terms A and are called the extreme terms; a and B are called the mean terms. A ratio of A large to a small, equals the ratio of B large to b small; or, A divided by a equals B divided by b. That is, A contains a as many times as B contains b. 78. Suppose that A contains a three times; then B must contain b three times. Then the ratio of A to a is fore, on account of the common ratio, we have a b A B If we compare this equation with the one above, we shall see that the terms of each ratio are the same, but the numerators have taken the places of the denominators, and vice versa. We therefore have-I. In any proportion, the terms of each ratio may be inverted, and the expression will still be a proportion. This change we call inversion. 79. If, in the original proportion, A B b, we multiply a --II. In any proportion the ratio of the numerators, equals that of the denominators. This change we call alternation. 80. If A contain a three times, A+ a will contain a four times; so also, if B contain b three times, B + b will contain b four times; we shall therefore have the fundamental principle of such a proportion is-The product of the two means is always equal to the product of the two extremes. The proportion, therefore, will give A x b = a × B. a X B. If we divide both sides of this equation, by a, and also by b, we shall have A B the ex pression in the text. It is believed that the principles of a proportion are more readily apprehended from this fractional method of writing them. proportion A+a_В + b a b ; that is,-III. In a propor tion, the sum of the terms of the first ratio, contains its denominator as many times as the sum of the terms of the second ratio contains its denominator. α 81. How many times soever A may contain a, A will contain a one time less; and how many times soever B may contain b, B-b will contain b one time less. This gives A a a B — b ; that is-IV. The differ ence of the terms of the first ratio, contains the denominator as many times as the difference of the terms of the second ratio contains its denominator. 82. From the two last articles, we have by alternation the terms of the first ratio of a proportion contains the sum of the terms of the second ratio as many times as the first numerator contains the second, or as the first denominator contains the second denominator. Also-VI. The difference of the terms of the first ratio, contains the difference of the terms of the second ratio, as many times as the first numerator contains the second, or as the first denominator contains the second, and, consequently--VII. The sum of the terms of the first ratio contains the sum of the terms of the second as many times as the difference of the terms of the first contains the difference of the term. of the second. 83. If we apply the propositions of the preceding art A - B A B a - b a b A+ B a + b That is-VIII. In any proportion, the sum of the numerators contains the sum of the denominators as many times as one numerator contains its denominator. Also-IX. The difference of the numerators contains the difference of the denominators as many times as one numerator contains its denominator. And, there· |