Mar BOOK III. PROPORTIONAL LINES. SIMILAR FIGURES. THEORY OF PROPORTION. 1. DEFINITION. One quantity is said to be proportional to another when the ratio of any two values, A and B, of the first, is equal to the ratio of the two corresponding values, A' and B', of the second; so that the four values form the proportion This definition presupposes two quantities, each of which can have various values, so related to each other that each value of one corresponds to a value of the other. An example occurs in the case of an angle at the centre of a circle and its intercepted arc. The angle may vary, and with it also the arc; but to each value of the angle there corresponds a certain value of the arc. It has been proved (II. 51) that the ratio of any two values of the angle is equal to the ratio of the two corresponding values of the arc; and in accordance with the definition just given, this proposition would be briefly expressed as follows: "The angle at the centre of a circle is proportional to its intercepted arc." 2. Definition. One quantity is said to be reciprocally proportional to another when the ratio of two values, A and B, of the first, is equal to the reciprocal of the ratio of the two corresponding values, A' and B', of the second, so that the four values form the proportion A: BB': A', For example, if the product p of two numbers, x and y, is given, so that we have xy=p, then, x and y may each have an indefinite number of values, but as x increases y diminishes. If, now, A and B are two values of x, while A' and B' are the two corresponding values of y, we must have ▲ × A' = p, BX B' = p, whence, by dividing one of these equations by the other, that is, two numbers whose product is constant are reciprocally propor tional. 3. Let the quantities in each of the couplets of the proportion A = A' or A: BA': B', [1] be measured by a unit of their own kind, and thus expressed by numbers (II. 42) ; let a and b denote the numerical measures of A and B, a' and b' those of A' and B'; then (II. 43), and the proportion [1] may be replaced by the numerical proportion, 4. Conversely, if the numerical measures a, b, a', b', of four quantities A, B, A', B', are in proportion, these quantities themselves are in proportion, provided that A and B are quantities of the same kind, and A' and B' are quantities of the same kind (though not neces sarily of the same kind as A and B); that is, if we have a: b = a'b', we may, under these conditions, infer the proportion A: BA': B'. 5. Let us now consider the numerical proportion and multiplying both members of this equality by bb', we obtain whence the theorem: the product of the extremes of a (numerical) proportion is equal to the product of the means. Corollary. If the means are equal, as in the proportion a: b = b: c. we have b2 = ac, whence b = Vac; that is, a mean proportional between two numbers is equal to the square root of their product. 6. Conversely, if the product of two numbers is equal to the product of two others, either two may be made the extremes, and the other two the means, of a proportion. For, if we have given. Corollary. The terms of a proportion may be written in any order which will make the products of the extremes equal to the product of the means. Thus, any one of the following proportions may be inferred from the given equality ab' = a'b: == Also, any one of these proportions may be inferred from any other. 7. Definitions. When we have given the proportion a: ba': b', the second proportion is said to be deduced by alternation. When we infer the proportion this proportion is said to be deduced by inversion. 8. It is important to observe, that when we speak of the products of the extremes and means of a proportion, it is implied that at least two of the terms are numbers. If, for example, the terms of the proportion A: BA': B', are all lines, no meaning can be directly attached to the products A × B', B × A', since in a product the multiplier at least must be a number. But if we have a proportion such as A: B =m: N, in which m and n are numbers, while A and B are any two quantities of the same kind, then we may infer the equality nA mB. Nevertheless, we shall for the sake of brevity often speak of the product of two lines, meaning thereby the product of the numbers which represent those lines when they are measured by a common unit. 9. If A and B are any two quantities of the same kind, and m any number whole or fractional, we have, identically, MA A mB B that is, equimultiples of two quantities are in the same ratio as the quantities themselves. Similarly, if we have the proportion A: BA': B', and if m and n are any two numbers, we can infer the proportions mA: mBnA' : nB'. mA : nB mA': nB'. 10. Composition and division. Suppose we have given the propor tion A A' [1] in which A and B are any quantities of the same kind, and A' and B' quantities of the same kind. Let unity be added to both mem results which are briefly expressed by the theorem, if four quantities are in proportion, they are in proportion by composition; the term composition being employed to express the addition of antecedent and consequent in each ratio. If we had subtracted unity from both members of [1], we should have found results which are briefly expressed by the theorem, if four quantities are in proportion, they are in proportion by division; where the term. division is employed to express the subtraction of consequent from antecedent in each ratio, this subtraction being conceived to divide. or to separate, the antecedent into parts. The quotient of [2] divided by [3] is that is, if four quantities are in proportion, they are in proportion by composition and division. |