: consequently the third angles at A and D are equal (cor. 1, th. 17): also AC, CD, and CE, CB, are like or corresponding sides, being opposite to equal angles : therefore AC. CB : CD. CE (th. 62). But CD. CE = CD2 + CD. DE (th. 30); therefore AC. CB = CD2 + CD, DE, CD2 + AD , DB, since CD. DE = AD. DB (th. 61), THEOREM LXV. The rectangle of the two diagonals of any quadrilateral inscribed in a circle, is equal to the sum of the two rectangles of the opposite sides. D E A B : LET ABCD be any quadrilateral inscribed in a circle, and AC, BD, its two diagonals: then AC. BD = AB. DC + AD. BC. For, let CE be drawn, making the angle BCE equal to the angle DCA. Then the two triangles ACD, BCE, are equiangular; for the angles A and B are equal, standing on the same arc DC; and the angles DCA, BCE, are equal by construction; consequently, the third angles ADC, BEC, are also equal ; also AC, BC, and AD, BE, are like or corresponding sides, being opposite to the equal angles : therefore the rectangle AC.BE is equal to the rectangle AD.BC (th. 62). Again, the two triangles ABC, DEC, are equiangular : for the angles BAC, BDC, are equal, standing on the same arc BC; and the angle DCE is equal to the angle BCA, by adding the common angle ACE to the two equal angles DCA, BCE; therefore the third angles E and ABC are also equal : but AC, DC, and AB, DE, are the like sides : therefore AC.DE = AB.DC (th. 62). Hence, by equal additions, AC.BE + AC.DE : AD.BC + AB.DC. But AC.BE + AC.DE = AC.BD (th. 30): therefore AC.BD = AD.BC + AB.DC (ax. 1). Cor. Hence, if ABD be an equilateral triangle, and C any point in the arc BCD of the circumscribing circle, we have AC = BC + DC. For AC.BD = AD.BC + AB.DC ; and dividing by BD = AB = AD, there results AC = BC + DC. RATIOS AND PROPORTIONS. DEFINITIONS. 76. Ratio is the relation subsisting between two magnitudes of the same kind, in respect of quantity. Of the two magnitudes compared, that which is taken as the standard of comparison is called the antecedent term of the ratio, or simply, the antecedent ; and that which is compared with it, the consequent. The leading idea of ratio is, the a с number of times that the antecedent is contained in the consequent; and hence the doctrine of ratio becomes, essentially, a branch of arithmetic *. The manner of writing a ratio is a : 6, where a is the antecedent and b the consequent. The reading it is, a is to b; and the expression of the fundamental 6 idea is written as a fraction, the numerator and denominator of which are a the consequent and antecedent respectively. 77. Proportion is the equality of two ratios, expressed as fractions. Thus, if b d the magnitudes a, b, c, d, are said to be proportionals, or to be in proportion. In geometrical investigations it is, however, more usual to write them a:b:: 0 :d, the verbal interpretation of which is either a is to b as c is to d, or a has the same ratio to b that c has to d. 78. When there is any number of magnitudes of the same kind, the ratio of the first to the last of them is said to be compounded of the ratios of the first to the second, the second to the third, the third to the fourth, and so on to the last. This is expressed by the term compound ratio. 79. When all these ralios, viz. that of the first term to the second, the second to the third, and so on, are all equal, the terms are said to form a geometrical progression, and are said to be continued proportionals. 80. When the ratios are equal, and there are only three terms, (or two ratios,) the third is called a third proportional to the first and second ; and the first is said to have to the third the duplicate ratio of that which the first has to the second. The middle term is called a mean proportional between the first and third. In like manner, when there are three equal ratios, the first term is said to have to the fourth, the triplicate ratio of the first to the second, and so on, however many equal ratios there may be. There are other technical terms employed to signify certain modifications * The method of treating the doctrine of ratio by the Greek geometers was precisely similar in all its essential characters to their method of treating theoretical arithmetic. The modern method of discussing the properties of numbers has superseded the Greek one; but in treating the doctrine of ratio, the original mode is still adhered to by the great majority of geometrical ters, on account of its supposed superiority of logical conclusiveness. The great beauty of that method of investigating the properties of ratios, no one doubts; but its superior conclusiveness may be very fairly questioned, and its great complexity renders it a serious obstacle to the progress of geometrical study. The great logical difficulty that has been felt in treating ratio directly and formally as a branch of arithmetic, has arisen from the possible incommensurability of the two terms of the ratio. Now if it were essential that the specific ratio itself should be assigned between the two terms, there would be some force in this objection; but as in all our investigations, and in all the uses we make of the doctrine in theoretical researches, resolve themselves into investigations respecting the equality or inequality of two or more ratios, as the result of given conditions, it is obviously sufficient that we should be able to determine the essential equality or inequality of those several ratios, without discussing the actual values of the fractional expressions themselves. Such ratios themselves may be unknown, indeterminable, or irrational; and yet their equality or inequality may be determined as completely by arithmetical considerations, as by the method of the Greeks. In fact, all the reasonings in which ratio is employed are conducted altogether independently of the actual value of the fraction, and which may, therefore, with perfectly logical accuracy, be denoted by any symbol whatever, as m, or n, f(r), or any other. under which magnitudes originally proportional will still continue so. These cases being enunciated and proved in the following series, the several terms or phrases by which they are designated, are annexed to the propositions themselves. 81. A line is said to be divided in extreme and mean ratio, when the whole line is to the greater segment as the greater segment is to the less ; or conversely to be extended in extreme and mean ratio, when the extended part is to the original line as the original line is to the whole line composed of the original one and the extended part. 82. The altitude of a triangle or a parallelogram is the perpendicular distance (or simply the distance, def. 50) of the vertex of the triangle, or the opposite side of the parallelogram from the base. 83. Two pairs of magnitudes are said to be reciprocally proportional, when the first of the first pair is to the first of the second pair, as the latter of the second pair is to the latter of the first pair. Thus, if a, b, c, d, taken in order , were the two pairs, they are reciprocally proportional when a : c::d:b. 84. A line is said to be divided in harmonical ratio, (or simply divided harmonically,) when it is divided and extended in the same ratio. 85. A transversal is any straight line or circle which is drawn to cut a system of straight lines. 86. When a straight line is divided harmonically, and lines are drawn from the points of division to any fifth point, the four lines so drawn are called an harinonical fasceau. A convenient mode of writing this is as follows: Let A, B, C, D, be the four points of the harmonical line, and E the point of the fasceau ; then E{ABCD} denotes lines drawn as in the definition. THEOREM LXVI. Equimultiples of two magnitudes have the same ratios as the magnitudes themselves. Let a, b, be the two magnitudes, and ma, mb, their equimultiples. Then b. mb a: 6 :: ma : mb. For the ratios and are equal, whatever be the value of the multiple m, whether integer, fractional, or irrational. Cor. Hence any equisubmultiples of two magnitudes have the same ratio as the magnitudes themselves. a ma THEQREM LXVII. the same ratio as the consequents. b d Let a : 6::c:d; then a : c::6:d. Then since (def. 77) gives d 영 and this fulfils the definition of proportional terms, or a : c::b:d. THEOREM LXVIII. If four quantities taken in order be proportionals, then will the first consequent be to the first antecedent as the second consequent is to the second antecedent. [This is called proportion by inversion.] Let a : 6::C: d, then also we shall have b: a :: d: c. 6 d For since b or again, by the definition, b:a::d: c. d' a с we have a THEOREM LXIX. с a a If four magnitudes be proportional, then the sum or difference of the first and second will be to the first or second as the difference of the third and fourth is to the third or fourth. [This is termed proportion by composition or division, according as the sums or differences are used). Let a : 6::c:d; then we shall have to prove that a +b:a::c+d:c, and that a + b:b::c+d:d. 6 d 6 d Now, since and hence also 1 + -=l+ 17., and + c' b ď b a + 6 a+b c+d 1 = ä + 1. Whence also, and that is, again, b d after inversion, a + b:a::c+d: c, and a +b:6::c+d:d. Cor. 1. Hence also, a + b b::c + dic d. For by the preceding b -b -d ; atď a + b:a-b::+d:c- d. Cor. 2. Also a : c:: a+bic+ d, and c:d:: a+c:b+ d. = a we have a с c+d a с If, of four proportional magnitudes there be taken any equimultiples whatever of the two antecedents, and any whatever of the two consequents, these multiples will be proportionals. b d nb Ler a : 6 :: :d, then also ma : nb :: mc : nd. For and hence nd : or, which is the same thing, ma : nb :: mc : nd. a ma mc THEOREM LXXI. If there be four proportional magnitudes, and the two consequents be either aug mented or diminished by magnitudes which have the same ratio as the two antecedents, the sums or differences form with the two antecedents a set of proportionals. Let a : 6::c:d, and e : f::a:c; then will.a : 6+e:: c:d.£f. с a , hence boe a e d For, from the two given proportions we have and :: d:f, and b +e:df::b:d :: a: c. Cor. The variation of this theorem is obvious. viz.: a te:c+f::6:d. THEOREM LXXII. If any number of magnitudes be proportional, then any one of the antecedents is to its consequent as all the antecedents taken together are to all the consequents taken together. LET a : 6::c:d::e:f::g:h Then a : 6:: a t ctetg :5+ d + f + h 6 d b+d_f_b+ d +f_h b + d + f + h For and so on to a + c a + cte 9 a tote + ģ any extent. Hence the conclusion follows. a с e THEOREM LXXIII. If a whole magnitude be to a whole as a part taken from the first is to a part taken from the other : then the remainder will be to the remainder as the whole to the whole. If there be several pairs of ratios which are equal each to each, then the ratio com pounded of all the first ratios will be equal to the ratio compounded of all the others. LET a : 6 :: 0 :d Then will the ratio compounded of a : b, a, :b,, aj : b, :: 0 :d, dz : ba, ....am: bm, be equal to that compounded а of c : d, c, : di, c, : d,, ... Cm : dm. dim For Whence it follows that c'a, са, C2 b b, b, bom dd, d, a'a, ag Om C.C, C2 Cor. 1. If there be magnitudes common to the numerator and denominator of either multiplied fraction, they may be cancelled, on the principle of the common da a measure. Cor. 2. If the magnitudes be numerically expressed, we shall have, as at p. 166, adida .... am : bb,b, bm :: CCC2 · Cm : dd,dz...d mo 2... |