the logarithm of 1 is 1; that of 01 is 2; that of .001 is-3; &c. The log. of 1 being 0 in every system. Whence it follows, that the logarithm of any number between 1 and 10, must be 0 and some fractional parts; and that of a number between 10 and 100, will be 1 and some fractional parts; and so on, for any other number whatever. And since the integral part of a logarithm, usually called the Index, or Characteristic, is always thus readily found, it is commonly omitted in the tables; being left to be supplied by the operator himself, as occasion requires. Another Definition of Logarithms is, that the logarithm of any number is the index of that power of some other number, which is equal to the given number. So, if there be Nrn, then n is the log. of N; where n may be either positive or negative, or nothing, and the root r any number whatever, according to the different systems of logarithms. When n is 0, then N is 1, whatever the value of r is; which shows that the log. of 1 is always 0, in every system of logarithms. When n is 1, then N is = r; so that the radix r is always that number whose log is 1, in every system. When the radix r is = 2.71828 1828459, &c. the indices n are the hyperbolic or Napier's log. of the numbers N; so that n is always the hyp. log. of the number N or (2.718 &c.)n. = But when the radix r is 10, then the index n becomes the common or Brigg's log. of the number N: so that the common log. of any number 10 n or N, is n the index of that power of 10 which is equal to the said number. Thus 100, being the second power of 10, will have 2 for its logarithm; and 1000, being the third power of 10, will have 3 for its logarithm: hence also, if 50 be 101.69897, then is 1.69897 the common log. of 50. And, in general, the following decuple series of terms, 10–1, viz. 10, 103, 102, 101, 10o, 10-2, 10-3, 10–4, or 10000, 1000, 100, 10, 1, 1, .01, .001, 0001, have 4, 3, 2, 1, 0,-1, -2, --3, —4, for their logarithms, respectively. And from this scale of numbers and logarithms, the same properties easily follow, as above mentioned. PROBLEM PROBLEM. To compute the Logarithm to any of the Natural Numbers 1, 2, 3, 4, 5, &c. RULE 1. TAKE the geometric series, 1, 10, 100, 1000, 10000, &c. and apply to it the arithmetic series, 0, 1, 2, 3, 4, &c. as logarithms.Find a geometric mean between 1 and 10, or between 10 and 100, or any other two adjacent terms of the series, between which the number proposed lies.-In like manner, between the mean, thus found, and the nearest extreme, find another geometrical mean; and so on, till you arrive within the proposed limit of the number whose logarithm is sought.-Find also as many arithmetical means, in the same order as you found the geometrical ones, and these will be the logarithms answering to the said geometrical means. EXAMPLE. Let it be required to find the logarithm of 9. Here the proposed number lies between 1 and 10. First, then, the log of 10 is 1, and the log of 1 is 0; theref. 1+0÷ 2 ==·5 is the arithmetical mean. and ✔ 10X1= ✓103.1622777 the geom. mean. hence the log. of 3.1622777 is ⚫5. Secondly, the log. of 10 is 1, and the log. of 3.1622777 is ·5; theref. 15275 is the arithmetical mean, and 10X3-1622777 5.6234132 is the geom. mean; hence the log. of 5.6234132 is ⚫75. Thirdly, the log. of 10 is 1, and the log. of 5.6234132 is ·75; theref. 1752 875 is the arithmetical mean, and 10 X 5.6235132 = 7·4989422 the geom. mean : bence the log. 7.4989422 is 875. Fourthly, the log. of 10 is 1, and the log. of 7-4989422 is ·875; theref. 18752.9375 is the arithmetical mean, and 10 X 7-4989422 8.6596431 the geom. mean ; hence the log. of 8.6596431 is .9375. The reader who wishes to inform himself more particularly concerning the history, nature, and construction of Logarithms, may consult the Introduction to my mathematica! Tables, lately published, where he will find his curiosity amply gratified. Fifthly, Fifthly, the log. of 10 is 1 and the log. of 8-6596431 is 9875 theref. 1+9375 ÷ 2 = ·96875 is the arithmetical mean, and 10 × 8·6596431 = = 9.3057204 the geom. mean ;· hence the log. of 9 3057204 is ⚫96875. ; Sixthly, the log. of 8-6596431 is 9375, and the log. of 9-3057204 is ⚫96875; theref. 9375 + ·96875 ÷ 2 = ·953125 is the arith. mean, and/8-6596431 X 9.30572048-9768713 the geome tric mean; hence the log. of 8-9768713 is 953125. And proceeding in this manner, after 25 extractions, it will be found that the logarithm of 8.9999998 is 9542425; which may be taken for the logarithm of 9, as it differs so little from it, that it is sufficiently exact for all practical purposes. And in this manner were the logarithms of almost all the prime numbers at first computed. RULE II.* LET b be the number whose logarithm is required to be found; and a the number next less than b, so that ba = 1, the logarithm of a being known; and lets denote the sum of the two numbers a+b. Then 1. Divide the constant decimal 8685889638 &c. by s, and reserve the quotient: divide the reserved quotient by the square of s, and reserve this quotient: divide this last quotient also by the square of s, and again reserve the quotient and thus proceed, continually dividing the last quotient by the square of s, as long as division can be made. 2. Then write these quotients orderly under one another, the first uppermost, and divide them respectively by the odd numbers, 1, 3, 5, 7, 9, &c. as long as division can be made; that is, divide the first reserved quotient by 1, the second by 3, the third by 5, the fourth by 7, and so on. 3. Add all these last quotients together. and the sum will be the logarithm of ba; therefore to this logarithm add also the given logarithm of the said next less number a, so will the last sum be the logarithm of the number 6 proposed. For the demonstration of this rule, see my Mathematical Tables, p. 109, &c. That n denotes the constant given decimal 8685889638, &c. EXAMPLES. Ex. 1. Let it be required to find the log. of number 2. Here the given number 6 is 2, and the next less number a is 1, whose log. is 0; also the sum 2+1=3=s, and its square $29. Then the operation will be as follows: 3) 868588964 1) 289529654 •289529654 9 289529654 3 32169962 10723321 Ex. 2. To compute the logarithm of the number 3. Here b=3, the next less number a=2, and the sum a+b 1=5=s, whose square s2 is 25, to divide by which, always multiply by 04. Then the operation is as follows: 5 ⚫868588964 1) 173717793 (-173717793 3) 6948712 2316237 25) 6948712 5 25) 277948 7 25) 11118 9 25) 445 11) Then, because the sum of the logarithms of numbers, gives the logarithm of their product; and the difference of the logarithms, gives the logarithm of the quotient of the VOL. I. numbers; 22 numbers; from the above two logarithms, and the logarithm of 10, which is 1, we may raise a great many logarithms, as in the following examples: Because 2 x 2 = 4, therefore Because 3a 9, therefore sum is log. 4 602059991 gives log. 9 954242509 And thus, computing, by this general rule, the logarithms to the other prime numbers, 7, 11, 13, 17, 19, 23, &c. and then using composition and division, we may easily find as many logarithms as we please, or may speedily examine any logarithm in the table*. * There are, besides these, many other ingenious methods, which later writers have discovered for finding the logarithms of numbers, in a much easier way than by the original inventor; but, as they cannot be understood without a knowledge of some of the higher branches of the mathematics, it is thought proper to omit them, and to refer the reader to those works which are written expressly on the subject. It would likewise much exceed the limits of this compendium, to point out all the particular artifices that are made use of for constructing an entire table of these numbers; but any information of this kind, which the learner may wish to ob tain, may be found in my Tables, before mentioned, Description |