5. To find the sum of the infinite geometrical progression (38.) Supposing A and B to stand for any two given numbers or quantities of the same sort, and that n denotes any given positive integer, we shall show how to reduce the comto the single ratio of two numbers or quan pound ratio (4) (金) tities of the same sort. quently, since B and A are given, C is found; and the terms A A B B C A B X = X which reduces (1) to B B C D' (金) == we get A B C = X X B C the value of P, since A, B and n are given; and we shall A have (4)" =+ as required. n Hence, we see that if we have the equation (4)"= (C)", so that the equation which involves two compound ratios is reduced to an equation which involves the simple ratios of A to P, and of C to Q. A C (39.) We shall now reduce the compound ratio X to B D the single ratio of two numbers or quantities of the same kind; supposing, of course, that A, B, C, D are given numbers, or quantities such that A and B are of the same kind, and that C and D are of the same kind with each other. C B Thus, put = D E' D and by (31) we have E=Bx which C' A C B D gives the value of E; consequently, we shall have X A B A = X - = the required single ratio of two numbers or quantities of the same sort. In a similar way we can reduce any compound ratio which is compounded of the ratios of given numbers or quantities to the single ratio of two given numbers or quanA C E G tities. Hence, if we have the equation X = X B D F H' we see how to reduce the equation to the form A E in which L and K are found as before; so that the equation is reduced to the equality of the ratios of A to L, and of E to K. And we may proceed in a similar way for any number of ratios that may be compounded in the two members of the equation. (40.) If we have a proportion such that either antecedent and its consequent are numbers, then shall the product of the extremes equal that of the means, and vice versa. For let m and n be two numbers, and M and N two numbers or quantities of the same sort, such that we have the proportion mn:: M: N, or its equivalent the equation that M contains N, we shall have M N x these equals by the number n, we shall have Mxn=N×m, as required. Reciprocally, if we have M x n = Nx m, such that m M and n are numbers, we shall have = (2). m n N' M For since m = n x and MN × the equation is n N' Nn X and omitting Nu, n which is common to both sides of the equation, we must or (which is the same) m M M m have = N n lent the proportion m: n:: M: N, as required. Hence, if the first and second terms of a proportion are numbers (or are regarded as such), it follows that the product of the means divided by the first term gives the fourth term, which is in accordance with the common method of working the statement of a question in the Rule of Three. Remarks.-1. It results from (31), that a fourth proportional to three numbers or quantities can always be found by multiplying the third by the ratio of the second to the first. 2. Similarly, it follows from (32), that a third proportional to two numbers or quantities equals the second multiplied by its ratio to the first. (41.) If we have the proportion A: B:: C: D, or the A с equation = ; then supposing n to represent any positive B D or negative real number, we shall clearly (from Ax. II.) get find; ; and thence, Hence, if (^)" is given, we can easily find (金) if C is known, D can be found. Thus, if n = 3, and 27 is put for (4), we shall have ()=27 = 3o, whose cube root gives = 3, and, of course, D = D с 3 If A, B, C, D are numbers (or are regarded as such), then n the equation (^)" = (C)" A" can be written in the form Cn Dn B which gives the proportion A" : B" :: C" : D"; consequently, since A" × D" = B" × C", we have D" = B" × C" cording to the common method of proceeding. It may be added that if A, B, C, D are not numbers (nor are regarded as such), the proportion A": B" :: C" : D" can Again, if we have the proportions A: B:: C: D, E: F :: GH, K: L:: M: N, etc., or their equivalents, the A CE GK M equations, etc.; then, by com pounding these equal ratios, we shall (from Ax. II.) have A E K X X x etc. = B FL C G M N Hence, if A, B, C, D, etc., are numbers (or are regarded as such), it is clear that the preceding equation can be writAEK etc. CGM etc., which gives the proBFL etc. DHN etc. ten in the form = portion AEK etc.: BFL etc. :: CGM etc. : DHN etc., which can be immediately obtained from the given proportions by taking the product of their corresponding terms. But if A, B, C, D, etc., are neither numbers (nor considered as such), then the preceding proportion can not exist, because (42.) AE etc. CG etc. = BF etc. DH etc. ABRIDGED PROPORTIONS OR VARIATIONS. 1. If, in the proportion A: B:: C: D, A and B are variable, while C and D are invariable, then it is clear that the ratio of A to B will be constant or invariable. Hence, we say that A varies as B directly, which we express (according to custom) by writing A & B directly. 2. Since the proportion A: B:: C: D gives A =Bx C if B and C are invariable, while A and D are variable, then it is clear that, if A increases, D must decrease, and vice versa. Hence, we may say that as A increases, D must decrease, and reciprocally, or that A varies inversely as D, which we may express by AD inversely or reciprocally. 3. Since A =Bx it follows that A varies as the pro C (43.) HARMONICAL PROPORTIONS AND PROGRESSIONS. 1. Three numbers or quantities (of the same kind) are said to be in harmonical proportion or progression when the first is to the third as the difference of the first and second is to the difference of the second and third. And the third term is called a third harmonical proportional to the first and second; also, the second term is called a harmonical mean between the first and third terms, noticing that the three terms are sometimes called harmonicals. Thus, if A, B, C are the harmonicals, we must have either A: C::AB: B-C, or A:C: B-A: C-B; and since the terms are of the same kind, these proportions by (13) may be written in the forms A: A-B::C: B-C, and A: B -A::C:C- B. Hence, if by (19) and (20) we apply composition to the first of these proportions, and division to the second, they will be |