SECTION VII. THE COMPOUNDING OF RATIOS. DUPLICATE RATIO. Props. 35-37. Art. 124. The following proposition, No. 35, is necessary for, and proposition 56 below is also very useful in, the theory of the Compounding of Ratios. Art. 125. PROPOSITION XXXV*. (Euc. V. 22.) ENUNCIATION 1. If A, B, C are three magnitudes of the same kind; if T, U, V are three magnitudes of the same kind; If A, B, C are three magnitudes of the same kind; if T, U, V are three magnitudes of the same kind; As in Prop. 22 it is convenient to use the second form of the conditions for the sameness of two scales in Prop. 8. Take any integerr in the first column, and any integer s in the second column of the scales to be proved the same. It is necessary to show that (1) If rA <sC, then rT< 8V. then since rA, sC, and B are magnitudes of the same kind, by Prop. 7 integers n, t exist such that the scale of A, B shows the fact exhibited in Fig. 97. .. the scale of T, U shows the fact exhibited in Fig. 98, the scale of B, C shows the fact exhibited in Fig. 99. .. the scale of U, V shows the fact exhibited in Fig. 100, ... tU <ns V... (I). (II). From (I) and (II) nrT<ns V, .. rT<sV, .. if rA <sC, then rT<sV......... (III). In like manner * if rA SC, then rT> sV From (III), (IV), (V), (VI) it follows by Prop. 8 (ii) that [A, C]=[T, V]. Art. 126. EXAMPLE 54. (IV), .(V), (VI). Two circles whose centres are C and C'intercept equal chords AB and A'B' on a straight line cutting both circles. Art. 127. THE COMPOUNDING OF RATIOS. The development of this process is made in four stages. STAGE 1. When it is necessary to determine the relative magnitude of two magnitudes, A and C, of the same kind, it is often convenient not to make the comparison directly, but indirectly by taking another magnitude B of the same kind as A and C'; and then comparing A with B, and afterwards B with C. From this point of view the relative magnitude of A and C is considered to be determined by the relative magnitude of A and B and the relative magnitude of B and C. STAGE 2. Euclid expresses the general idea stated in the first stage by saying that the ratio of A to C is compounded of the ratio of A to B and the ratio of B to C. * The proof of (IV) is obtained from that of (III) by reversing all the signs of inequality, and making the corresponding changes in the figures. To obtain (V), since rT<s V, observe that integers n, t exist by Prop. 7 such that and Or else it is obvious that the conditions given may be re-written [T, U]~ [A, B], Comparing these with the original form, it appears that they can be deduced from the original form by interchanging A and T, B and U, C and V. Hence it is permissible to make these changes in (III) and (IV), and the result is to give (V) and (VI). + See the 23rd proposition of Euclid's Sixth Book, where the meaning is more easily understood than in the 5th definition of that book. STAGE 3. Def. 21. THE PROCESS OF COMPOUNDING RATIOS. Let the ratios to be compounded be P: Q and T: U. Take any arbitrary magnitude A, and then find B so that : Then the ratio compounded of P Q and T: U is the ratio compounded of A B and B C, and is therefore AC by the statement in the second stage. This process* contains an arbitrary element, viz. A. STAGE 4. In order to justify the process described in the preceding stage, it is necessary to show that the presence of the arbitrary element in the third stage has no influence on the value of the resulting ratio. Suppose that instead of A, the magnitude A' had been selected, and that B' and C' had then been found so that Then the resulting ratio would be that compounded of A': B' and B': C', and would therefore be A': C'. In order that this may agree with the previous result, it is necessary to show that From (VI) and (VII) by Prop. 35, the proportion (V) follows. Hence the process in the third stage always leads to the same value of the resulting ratio, whatever be the value of the arbitrary element. This is the justification of the process described in the third stage. * It should be noted that this process assumes the existence of B and C, when A has been chosen arbitrarily, the proof of which depends on the Fundamental Proposition in the Theory of Scales. Art. 128. ARITHMETICAL APPLICATION OF THE PROCESS FOR COMPOUNDING RATIOS. To compound the ratio rs with the ratio u v where r, s, u, v are positive integers. by the definition of the ratio compounded of other ratios in Art. 127, Stage 2. Hence this arithmetical theorem corresponds to the theorem that rs compounded with u: v= ru : sv. Art. 129. Def. 22. DUPLICATE RATIO. If a ratio be compounded with itself the resulting ratio is called the duplicate ratio of the original ratio. Thus if A: B be compounded with A: B, the resulting ratio is called the duplicate ratio of A: B. |