A D can be drawn from A, falling on the same side of BA with CD, and not cutting the circle, which shall not cut CD. This second tangent, then, must be either AD or some other line, AM, cutting CD in a point M differing from D. If now AM is a tangent, ABCM is a circumscribed quadrilateral, and by the preceding proposition we shall have But we also have, by the hypothesis of the present proposition, AB+ DC = AD + BC. = Taking the difference of these equalities, we have DM = AM— AD; that is, one side of a triangle is equal to the difference of the other two, which is absurd. Therefore, the hypothesis that the tangent drawn from A and cutting the line CD, cuts it in any other point than D, leads to an absurdity; therefore, that hypothesis must be false, and the tangent in question must cut CD in D, and consequently coincide with AD. Hence, a circle has been described which is tangent to the four sides of the quadrilateral; and the quadrilateral is circumscribed about the circle. 103. Scholium. The method of demonstration employed above is called the indirect method, or the reductio ad absurdum. At the outset of a demonstration, or at any stage of its progress, two or more hypotheses respecting the quantities under consideration may be admissible so far as has been proved up to that point. If, now, these hypotheses are such that one must be true, and only one can be true, then, when all except one are shown to be absurd, that one must stand as the truth. While admitting the validity of this method, geometers usually prefer the direct method whenever it is applicable. There are, however, propositions, such as the preceding, of which no direct proof is known, or at least no proof sufficiently simple to be admitted into elementary geometry. We have already employed the reductio ad absurdum in several cases without presenting the argument in full; see (I. 47), (I. 85), (27). 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 AX 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 Α 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 necessarily of the same kind as A and B); that is, if we have 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 : mB = nA' : nB'. MA: nB mA': nB'. |