gression are produced by division, but their corresponding logarithms are produced by subtraction. 3. The same observations apply to logarithms when they are fractions. Thus if a denote any number, 3 an an an 1, 1, a", a", a", a", &c. constitute a series of numbers 0, 1, 2, 2, 1, &c. are the logarithms; and it is evident that n ก n the assertions in the last article hold true, both with respect to the numbers in geometrical progression and their corresponding logarithms. As a and n may be taken at pleasure, it follows that numbers in very different geometrical progressions may have the same logarithms; and that the same series of numbers in geometrical progression may have different series of logarithms corresponding to them. 4. If a be an indefinitely small decimal fraction, and successive powers of 1+a be raised, then the excess of any power of i+a above that immediately preceding it will be indefinitely small. Thus let a=00000000001, and then 1+a210000000000200000000001; and T+a3= 1'000000000030000000000300000000001; and proceeding by actual multiplication to obtain higher powers of 1-00000000001, it will be found that the difference between two successive powers is very small. If instead of supposing, as above, that a=00000000001, we suppose it only one millionth part of this value, then the successive powers of 1+a will differ from one another by much smaller decimal fractions T 5. If therefore a be indefinitely small and successive powers of 1+a be raised, a series of numbers in geometrical progression will be produced, of which the common numbers 2, 3, 4, 5, &c. will become terms. For on every multiplication by 1+a, an indefinitely small addition is made to the power multiplied, and by this indefinitely small addition, the next higher power is produced. Some power of 1+a will therefore be equal to the number 2, or so nearly equal to it that they may be considered as equal. Continuing the advancement of the powers of 1+a, the numbers 3, &c. for the same reasons, will fall into the series. 6. The sum of the logarithms of any two numbers is equal 5, to the logarithm of the product of the same two numbers. Thus if 1+ a raised to the nth power be equal to the number N, and if 14a raised to the mth power be equal to the number M, then, by the preceding articles, n is the loga. rithm of 1+an or of its equal N, and for the same reason m is the logarithm of M. Hence it follows that n+m= the logarithm of Nx M, for NxM=1+a" xl+am = 1+a+m by the nature of indices. If the logarithm of N be subtracted from the logarithm of M, the difference is equal to the logarithm of the quotient which arises from the division of M by N. For M 1+am N n 1+a =1+amn, by the nature of indices.-The addition of logarithms, therefore, answers to the multiplication of the natural numbers to which they belong; and the subtraction of logarithms answers to the division by the natural numbers to which they belong. 7. If the logarithms of a series of natural numbers be all multiplied by the same number, the several products will have the last-mentioned properties of logarithms. Thus if the indices of all the powers of 1+a be multiplied by l, then using the notation stated in the last article, the logarithm of N is nl, and the logarithm of M is ml, and the logarithm of N x M is nl+ml; for N×Mi+ax1+ ami =1+am, by the nature of indices. Also ml-nlM M 1+aml the logarithm of =1+am. Hence the products arising from the multiplication of l into the indices of the powers of 1+a, are termed logarithms, as are also all numbers which have the properties stated at the end of article 6. It is on account of these properties that logarithms are so very useful in calculations of the highest importance. 8. If the indices of the powers of 1+a, be multiplied by a, the products are called the hyperbolic logarithms of the numbers equal to the powers of 14a. Thus if the number N be equal to 1+al", then na is the hyperbolic logarithm of N; and if the number M be equal to 1+ām, then ma is the hyperbolic logarithm of M. Hyperbolic logarithms are not those in common use, but they can be calculated with less labour than any other kind, and common logarithms are obtained from them. 9. If successive powers of a very small fraction be raised, they will successively be less and less in value. This truth appears most evident by putting the value in the form of a 10. Let it be required to determine the hyperbolic logarithm L, of any number N. Using the same notation as in the preceding articles, 1+al"N, and by extracting the nth root of each side of the equation, 1+ a=N", Put m 1 m and 1+N, and then N1+a" (by the binomial +&c. =1+a. Now as a is indefinitely small, the power of 14a, which is equal to the number N, must be indefinitely high; or, which is the same thing, n must be indefinitely great. Consequently m must be indefinitely small, and therefore may be rejected from the expressions m-1, m-2, m-3, &c. Hence 1 being taken from each side of mx2 mx3 mx* the above equation, we have a=mx• + 3 4 + &c. Each side of this equation being divided by m mxs 5 hyperbolic logarithm of N, by article 8. This series, however, if x be a whole number, does not converge. 1 Let M be a whole number, and M, and then x is less than 1. For, multiplying both sides of the equation by 1 1-x, we have M-Mar-1, and therefore 1-=r. Now 1 1 1 let M ==1). Then we have 1+a== =x=1+x)2. 1 === (by putting r (by putting r=—3) = 1 − rx + r × 2 7-1 T-2 2 3 "-2 xx2-rx2-2x2-x+&c. But for the same reasons as above, r must be indefinitely small, and therefore may be rejected from the factors r-1, r-2, r-3; &c. Consequently, taking 1 from each side of the above equarx2 rx3· rx2 rx3 2 3 4 5 -&c. But-r= tion, a――rx-. and therefore, dividing the left hand side of the equation x2 x3 x4 by and the other by-r we have ap=+=+=+ + 2 ++ &c. the hyperbolic logarithm of M. 5 11. As, by the last article, the hyperbolic logarithm of N 207 + ++&c. the hyperbolic logarithm of N x M, 6 is equal to the sum of these two series, that is, 3 equal to 2r+ + + +&c. This series converges faster than either of the preceding, and its value may garithms answers to the multiplication of the numbers to which they belong, the logarithm of the square of any number, is the logarithm of the number multiplied by 2. Hence From the preceding articles hyperbolic logarithms may be calculated, as in the following examples. Example 1. Required the hyperbolic logarithm of 2. Put 9 8 In order to proceed by the series in article 11, let 1 -X 2n+1 3 The double of which is 0.57535414488, and answers to the first part of the expression in article 12. 1+x 9 Secondly, let and then 8+8x=9-9x, and 8 17 Consequently, |