(5.) If for A in (1) and (12) we put 10, they will become what is called the Briggs or Common Logarithm of B. = = Thus, since 10° 1, 101= 10, 10° 100, 103 = 1000, etc., 0, 1, 2, 3, etc., are the common logarithms of 1, 10, 100, 1000, 1 etc., and because 10-1 = = 0.1, 10-2 = 10 1 100 =0.01, 10-30.001, etc., -1, -2, -3, etc., are the common logarithms of 0.1, 0.01, 0.001, etc. If we represent any positive whole number (including 0) by n, it is clear, from what has been done, that the common logarithm of any number which lies between 10" and 10"+1, must lie between n and n + 1; consequently, the logarithm evidently equals n + an interminable decimal; noticing that the integer n is called the characteristic of the logarithm. Similarly any number which lies between 10--1 and 10-" has for its logarithm a number which lies between − (n + 1) and n; consequently, the logarithm equals (n + 1) + an interminable decimal; noticing that the characteristic (n + 1) is negative. Thus 2, which lies between 10° and 101, has 0.3010300 for its (approximate) logarithm, its characteristic being 0; 89, which lies between 101 and 102, has 1.9493900 for its logarithm; 625.43, which lies between 102 and 103, has 2.7961787 for its logarithm; and 0.843, which lies between 10-1 and 10o, has 10.92582761.9258276 for its logarithm, 0.08934, which lies between 10-2 and 10-1, has - 2.9510459 for its logarithm, and so on; noticing, that when the characteristic is negative, the sign is sometimes written over instead of before it; thus, for 2.9510459, and so on. 2.9510459, we may write Hence, it follows that the characteristic of the logarithm of any number greater than unity is one less than the number of places of integers in the number; and the characteristic of the logarithm of any decimal fraction is negative, and equal to unity added to the number of naughts which imme diately follow the decimal point; noticing, that the characteristic is sometimes called the index of the logarithm. Again, if N stands for any whole number, then we shall N have l 10" = IN — 710′′ = 7N-n; consequently, the deci N mal part of the logarithm of is the same as that of N 10n (since n is supposed to be an integer), while the characteristic of the logarithm of N is diminished by the number of units in n. Hence, all numbers which consist of the same figures placed in the same order (whether they are wholly integral or partly or wholly decimal), are such that the decimal parts of their logarithms are the same. (6.) For the method of finding the logarithms of numbers from the tables, we shall refer the reader to the directions which generally precede the tables. We will now proceed to illustrate what has been done by the following EXAMPLES. = 1. To find the value of x in 10% 2, in which x = the common logarithm of 2. I From the equation we get 10 = 2, and since 8 = 23, we + 2 1/1 1/1 4 1/1 2 `1 (1)2 + 1 (1)^ + 1 (1)' + etc.) + (1 + 1 (1)* + = 0.015625, ()* = etc. = 0.25 -0.03125 +0.00520833333. — 0.0009765625 + etc. 0.22314355134, which is correct to ten decimal places; logarithm of 2, correct to ten decimal places. Because 1'2 0.6931471805, it is manifest that '10 = 2.3025850929, and of course m = = 0.4342944819, = 1 7'10 the modulus of common logarithms; it being the number by which any hyperbolic logarithm being multiplied, the result will be the corresponding common logarithm, and vice versa. 3 Because − − 3/3 — 7(1 − 1) = 7′′ = 0.4054651081, and l′2 + l' =1'3, we get l'3 = 1.0986122886, and thence 13 = 0.4771212547 = the common logarithm of 3; also, l'4= 2/21.3862943611, and 74 = 2/2 = 0.6020599913, 7'5 = 71072 1.6094379124, 75 710 72 = 1 − 72 = = 0.6989700043, 16 = 1'3 +12=1.7917594692, 76 = 13+ 12 = 0.7781512503, l'8 = 1'2° 37'22.0794415416, 78=372 0.9030899869, 1'91′32 = 2.1972245773, 19 = 213 = 0.9542425094; similarly, since 12 = 4 × 3, 15 = 5 × 3, 16 =24, 189 × 2, 2010 x 2, etc., it will be easy to find the hyperbolic and common logarithms of 12, 15, 16, etc., from the preceding logarithms. = Remarks.-1st. It is clear, from what has been done, that the calculation of the common logarithm of 2 enables us to find the hyperbolic and common logarithms of an unlimited number of other numbers. 2d. It is also clear that if we calculate the logarithms of the prime numbers, we can thence get the logarithms of all the composite numbers resulting from the products of the primes or their powers, by adding their logarithms. = Thus, since 14=12+ 17 and 12272 + 711, etc., it appears that we get the logarithm of 14 by adding the logs. of 2 and 7, etc. 3d. The method of calculating the logarithms of the primes 7, 11, 13, 17, etc., may be given as follows: thus, since 7=8—1 = 8(1 − 1), 11= 12(1-1), etc, if we take their common logarithms, we shall get 77 = 18 + ml' m2(1—1), It may be well to notice that there are other methods, which are often more expeditious than the preceding. Thus, since 7* = 2401 = 2400(1 + =2400(1+1), we have 477 = 12400 + = 1 2400 1 1 + 0.000000037699 + 0.00000000001 - etc. 3.38039216002; consequently, 17 = 3.38039216002 =0.8450980400 (= the common logarithm 4 of 7), correct to ten decimal places. Similarly, the logarithms of 11, 13, 17, 19, etc., can easily be found from 2/11 4th. We will terminate these remarks by showing how to find the number corresponding to any given common logarithm, without using the tables. Thus, to find the number whose common logarithm is 3.44496555; since the characteristic is 3, it follows that 1000 is a factor of the sought number; consequently, subtracting 3 from the logarithm, we reduce it to 0.44496555, from which we have to find the remaining factors. Since 0.44496555 × 2 = 0.88993110, we get 0.44496555= 10.11006890 0.05503445; consequently, since the 1 = 2 1 2 number corresponding to the logarithm is 10, we have 10 for another factor of the sought number; and subtract ing 1 2 from the preceding logarithm, it is reduced to -0.05503445, which, being negative, it results if N stands for the number whose logarithm is 0.05503445, that 1000 10 N will be the sought number. Dividing 0.05503445 by the modulus 0.43429448, we get the quotient 0.1267215 for the hyperbolic logarithm of N. To find N from its hyperbolic logarithm, we shall take the exponential theorem (8), given at p. 523, which, by putting a1 and N for A, and using 0.1267215 for l'A, gives N = 1 +0.1267215 + 0.0080291 +0.0003391 + 0.0000107 + 0.0000002 + etc. = 1.1351006. Hence, since 10=3.162277, 10001/10 becomes 3162.277 1.1351006 = 2785.9002 for the number corresponding to the given logarithm, which is correct to four decimals. For another example, we shall find the number having 4.7892346 for its common logarithm. Here 10000 10 has 4.5 for its logarithm; and by subtracting 4.5 from the given logarithm, it is reduced to 0.2892346; consequently, if we find the number having 0.2892346 for its logarithm, and multiply it by 10000 TO, the product will be the sought number. Because 10 has 0.25 for its logarithm, if from 0.2892346 we subtract 0.25, the given |