To explain the manner, let it be required to resolve in whole numbers the indeterminate equation ax+by=c, in which a, b, c are whole numbers; and a, b prime to each other. Suppose that the ratio a 1, is reduced to a continued fraction, and that all the convergents are calculated, the last must be the ratio itself. Subb a a tracting from the last convergent but one (which may be denoted by ៩), the numerator of the difference is ab'-ba'; and by Article 197, This solution being known, all the others are given by the formula (Art. 96), x=+b'c—bt, y=Fa'c+at, t being any whole number. b The upper or lower sign is to be taken according as the convergent is of the even or the odd rank. Example. Let the equation 261x-82y=117 be given to find the integer values of x, y. Multiplying by c=117, 261 × 11 × 117-82 × 35 ×117=117. Consequently the proposed equation is satisfied by making If 1287 is divided by 82, and 4095 by 261, 1287=82 × 15+57, 4095=261 × 15+180. Whence, as t is any whole number whatever, the general values can be more simply expressed by x=57+82t, y=180+261t. Examples. It is required to reduce the following expressions to continued fractions, and to form the convergents. Ans. The quotients are 3, 7, 15, 1, 3 22 333 355 103993 104348 208341 And the convergents, 17106' 113' 33102' 33215' 66317 &c. 206. Let +0.1.2. 3 4 5. 6. 7. 8. 9 1 2 1:39:27: 81: 243: 729 : 2187: 6561: 19683 : be two progressions, the first arithmetical and beginning with 0, the second geometrical and beginning with 1. If any terms of the second progression, 27 and 243 for example, are multiplied together, and the corresponding terms, 3 and 5, of the first progression are added together, the product, 6561, of the terms of the second, and the sum, 8, of the terms of the first, are corresponding terms of the two progressions; and, generally, the number which expresses the product of any terms of the geometrical progression and the number which expresses the sum of the corresponding terms of the arithmetical progression are corresponding terms of the two series. Since the product of any terms of the second series is obtained by forming the sum of the corresponding terms of the first series, and taking, in the second series, the number which is the term corresponding to that sum, it follows that the multiplication of numbers contained in the second series is reducible to the addition of numbers contained in the first series. 207. Numbers through every change of magnitude may be considered as the terms of a geometrical progression; for in the series 1:1+a: (1+a)2 : (1+a)3 : (1+a)+ : . . if a is supposed a very small quantity, the terms increase by very small increments; and since a may be diminished indefinitely, the terms may be considered in this case to vary in a continuous manner. A progression in which continuity exists cannot indeed be written; but the understanding is capable of conceiving such a progression, and this is sufficient. All numbers greater than 1 may therefore be considered as comprised in the geometrical progression. If again an arithmetical progression, +0.6.23.33.43... is taken, the terms of which begin from 0 and increase by very small differences, and the terms of this series are considered in connexion with the terms of the geometrical progression, the former are called the logarithms of the latter. Whence, the logarithms of numbers are the terms of an arithmetical progression commencing with 0, which correspond to other numbers considered as forming the terms of a geometrical progression which commences with 1. No logarithms appear to be assigned by this definition to numbers less than 1. In order, however, that the geometrical progression may include such numbers, it is sufficient to conceive the progression prolonged below 1. This is accomplished by dividing 1 by the successive powers of the ratio 1+a. If the geometrical progression is decreasing, as, the corresponding terms of the arithmetical progression, +....... 23.8.0.. are obtained by subtracting the ratio 3 from each term to form the succeeding term. When from 3 (which is the term corresponding to 1+a) 3 is subtracted, the remainder, which is the term corresponding to 1, is 0. The next term of the geometrical progression is or (1+a)-1, and the corresponding term of the arithmetical progression is 0-3 or -ß. 1 1+a The corresponding terms of the decreasing progressions are, ÷0. -ẞ -23 -38.. #1: 1 1+a' (1+a)2 ' (1+a)3 Negative terms, therefore, being employed, the arithmetical progression is made to descend below zero; and these negative terms are the logarithms of numbers less than 1. If the descending terms are written on the left, the two series may be exhibited thus, (1+a)3: (1+a)2: (1+a) :1:1+a:(1+a)2: (1+a)3... 4 and the first gives the logarithms of the corresponding terms of the second. It is an essential condition in the definition of logarithms that the terms 1 and 0 must always correspond to each other; in other words that the logarithm of 1 must be 0. The ascending part of both the series 3, 4, augments to infinity; but in 4 the descending part tends indefinitely towards zero, while in 3 it augments to negative infinity. Therefore log. = c; and log. 0=— ∞c. It appears from a comparison of the series 3, 4, that if any number, n, is situated at a certain distance from 1 in the ascending part of 4, the number 1 ought to occupy the same rank in the descending part. Therefore this last number has the same logarithm, but taken negatively; that is, 208. The fundamental property of logarithms is this. The logarithm of a product is equal to the sum of the logarithms of the factors of that product. To give a general demonstration of this theorem, 1st. Let a, b be two numbers in the ascending part of progression 4; for example, let a=(1+a)2; b=(1+a)5. Then ab=(1+a)° × (1+a)3=(1+a)5+2.' Now, in progression 3, the logarithms of a, b, are 23, 53; therefore, log. a+log. b=23+5,3=(5+2),3. But (5+2)ẞ or 7,3 is the logarithm of (1+a)5+2 or (1+a)2; therefore, log. ab=log. a+log.b. 2d. Let a be in the descending part, b in the ascending part of series 4, and b more remote than a from the term 1. But log. a=-23; log. b=53; therefore log. a+log. b=(5—2),3. Whence, as (5-2)3 is evidently the logarithm of (1+a)5-2, it is concluded again that log. ab=log. a+log. b. 3d. Let a be in the descending part, b in the ascending part of series 4 b being nearer than a to the term 1. 1 (1+a) 2 1 then ab= *(1+a) š× (1+a)2=(1+a)=(1+a)3==• But log. a=-53; log. b=23; therefore log. a+log, b=−(5—2)3, which is the log. of Whence log. ab=log. a+log. b. 4th. Let a, b, be both in the descending part of series 4. In this case log. a=-23; log. b=-5,3. Consequently log. a+log. b= −(2+5)/3, which is the log. of the product ab; therefore log. ab=log. a+ log. b. The equality log. ab=log. a+log. b being true in all cases if b is changed into bc. log. (axbe)=log. a+log. bc. But log. (axbc)=log. abc, and log, be=log. b+log. c; a 209. Let q= and therefore bq=a. b' Then (Art. 208) log. b+log. q=log. a; and transposing, log. q=log. a-log. b. .... Whence the log. of a quotient is equal to the log. of the dividend less the log. of the divisor. 210. If a product is composed of n factors, each equal to a, αχαχαχ ... to n factors a"; ..log. (axaxax to n factors) =log. a". .... .... But log. (axaxax to n factors) =log. a+log. a+log. a.... repetitions =n log. a. Therefore log. a”=n log, a. To extend this formula to fractional exponents, let 'r=a .'.x"=a”; log. x"=log. a"; n log. x=m log. a; . to n Let x=a = ... xa"=1; and log. (ra")=log. 1=0. But log. (xa")=log. x+log. a"=log. x+n log.a. Therefore log. x+n log. a=0, and log. x=-n log. a. Whence the logarithm of any power whatever of a number is equal to the product of the logarithm of the number by the exponent of the power. 211. Let ra, and therefore r"=a. Then, by Article 210, n log. r=log. a. Dividing by n, log. r= n Whence log. r=log. (Va)= Wherefore the logarithm of the root of a number is obtained by dividing the logarithm of the number by the index of the root. 212. In the geometrical series 1:1+a:(1+a)2 : (1+a)3, &c., a may be assumed so small that the difference of any two consecutive terms shall be less than any assigned number. Then, as 1+a is greater than 1, and the series may be indefinitely prolonged, it is evident that each term is greater than the preceding, and that a term may be found which shall exceed any given number. Since the series can be prolonged until a term greater than 2 is found, then either one of the terms must be equal to 2, or 2 must fall between two consecutive terms. The difference of these terms being less than any assigned number, and the difference between 2 and either term being less than the difference of the two terms, either term may be considered equal to 2. In the same manner it can be made to appear that other terms of the series are either equal exactly to 3, 4, 5 . . . or differ from them by less than any assigned numbers. Consequently certain terms of the series are equal respectively to the numbers 2, 3, 4, 5, . . . 10, . . . 100,... 1000, &c. ... An arithmetical series +0.6.26.33.43.... whose ratio ẞ is a very small quantity, being connected with the preceding geometrical series by the conditions that the terms 0 and 1 shall correspond, and that the exponent of any terms of the geometrical and the coefficient of the corresponding term of the arithmetical series shall be the same number, the terms of the arithmetical series corresponding to those terms of the geometrical series which are considered equal to the numbers 2, 3, 4, . . . are the logarithms of the numbers 2, 3, 4, . .... 213. Another condition may be imposed on the series, viz., that between the simultaneous increments of 1 and 0, the first terms of the geometrical and arithmetical series, an arbitrary ratio must subsist; as, for example, that the one increment shall be equal to M times the other. Then, if the increment of the term 1 of the geometrical series is denoted by w, that of the term O of the arithmetical series is Mw, and the progressions become, According as different particular values are given to M, different systems of logarithms are formed. The ratio M is termed the modulus of the system of logarithms. +0.0.2w.3w 3. By means of the hypothesis M=1, the multiplications by M are avoided. In the system published by Napier, the inventor of logarithms, M=1. To change the series 3 into 2, it is only necessary to multiply the terms of 3 by M. Wherefore, to change the logarithms of Napier into logarithms of any other system, it is sufficient to multiply Napier's logarithms by the modulus of that other system. |