From the preceding articles hyperbolic logarithms may be calculated, as in the following examples. Example 1. Required the hyperbolic logarithm of 2. Put 2n+2 4 2, and then n=1, 2n+1 and In order to proceed by the series in article 11, let The double of which is 0,57536414488, and answers to the first part of the expression in article 12. which answers to the second part of the expression in article 12. Consequently the hyperbolic logarithm of the number 2 is 0.57536414488+0.117783085660.69314718054. The hyperbolic logarithm of 2 being thus found, that of 4, 8, 16, and all the other powers of 2, may be obtained by multiplying the logarithm of 2, by 2, 3, 4, &c. respectively, as is evident from the properties of logarithms stated in article 6. Thus by multiplication, the hyperbolic logarithm of 41.38629436108 of 82.07944154162 &c. From the above the logarithm of 3 may easily be obtained. 4 4 4 For 4+4x=3; and therefore as the logarithm of 3 was determined above, and also the logarithm of 4, From the logarithm of 4, viz. - - 1.38629436108, Subtract the logarithm of 4 g' viz. 0.28768207244, 1.09861228864. And the logarithm of 3 is Having found the logarithms of 2 and 3, we can find, by addition only, the logarithms of all the powers of 2 and 3, and also the logarithms of all the numbers which can be produced by multiplication from 2 and 3. Thus, To the logarithm of 3, viz. - 1.09961228864 And the sum is the logarithm of 6 1.79175946918. To this last found add the logarithm of 2, and the sum 2.48490664972 is the logarithm of 12. The hyperbolic logarithms of other prime numbers may be more readily calculated by attending to the following article. 13. Let a, b, c, be three numbers in arithmetical progression, whose common difference is 1. Let b be the prime number, whose logarithm is sought, and a and c even numbers whose logarithms are known, or easily obtained from others already computed. Then, a being the least of the three, and the common difference being 1, a =b− 1, and c=b+1. Consequently axc-b-1xb+1 b2 ac+1. = b2-1, and ac +1b2; and therefore is a general expression for the fraction which it will be = ac ac This 142 proper to put = 1 that the series.expressing the hyper 1+x bolic logarithm may converge quickly. For as 1 Ꮖ 25 -0.0408219942 But x8x3=25, and the addition of logarithms an 24 swers to the multiplication of the natural numbers to which they belong. Consequently, And the sum is the log. of 25 - 3.2188758254 The half of this, viz. 1.6094379127, is the hyperbolic loga rithm of 5; for 5 x 5 = 25. Example 3. Required the hyperbolic logarithm of 7. 1+æ 1 Here a=6, c=8, and ✯= 1. ac + 1 = 2ac+1 97 49 48 For x6x8=49. Consequently the half of this, viz. 1.94591014899, is the hyperbolic logarithm of 7; for 7 x7=49. If the reader perfectly understand the investigations and examples already given, he will find no difficulty in calculating the hyperbolic logarithms of higher prime numbers. It will only be necessary for him, in order to guard against any embarrassment, to compute them as they advance in succession above those already mentioned. Thus, after what has been done, it would be proper, first of all, to calculate the hyperbolic logarithm of 11, then that of 13, &c. Proceeding according to the method already explained, it will be found that The byperbolic logarithm of 11 is 2.397895273016 of 13 is 2.564999357538 of 17 is 2.833213344878 of 19 is 2.944438979941.5 Logarithms were invented by Lord Neper, Baron of Merchiston, in Scotland. In the year 1614, he published at Edinburgh a small quarto, containing tables of them, of the hyperbolic kind, and an account of their construction and use. The discovery afforded the highest pleasure to mathematicians, as they were fully sensible of the very great utility of logarithms; but it was soon suggested by Mr. Briggs, afterwards Savilian Professor of Geometry in Oxford, that another kind of logarithms would be more convenient, for general purposes, than the hyperbolic. That one set of logarithms may be obtained from another will readily appear from the following article. 14. It appears from articles 1, 3, and 7, that if all the logarithms of the geometrical progression 1, 1+a,', 1+a2, 1+a3, 1+a+, 1+a, &c. be multiplied or divided by any given number, the products and also the quotients will likewise be logarithms, for their addition or subtraction will answer to the multiplication or division of the terms in the geometrical progression to which they belong. The same terms in the geometrical progression may therefore be represented with different sets or kinds of logarithms in the following manner: 2 1, 1+a1, 1+ a2, i+a3, 1+a^, i+al3, 1+a, &c. 1, 1+al2, 1+al22, 1+a3⁄41⁄2 1+a“, 1+a3, 1+a, &c. 1, 1+am, i+am, 1+am, 1+am, 1+am, i+am, &c. In these expressions land m denote any numbers, whole or fractional; and the positive value of the term in the geometrical progression, under the same number in the index, is understood to be the same in each of the three series. Thus if 1+a+ be equal to 7, then 1+a, is equal to 7, as is also 1+am. If 1+a° be equal to 10, then 6 1+a is equal to 10, as is also 1+a", &c. If therefore 1, 2, 31, &c. be hyperbolic logarithms, calculated by the methods already explained, the logarithms expressed by 2 3 &c. may be derived from them; for the hy m' m' m' perbolic logarithm of any given number is to the logarithm in the last-mentioned set, of the same number, in a given 15. Mr. Briggs's suggestion, above alluded to, was that 1 should be put for the logarithm of 10, and consequently 2 for the logarithm of 100, 3 for the logarithm of 1000, &c. This proposed alteration appears to have met with the full approbation of Lord Neper; and Mr. Briggs afterwards, with incredible labour and perseverance, calculated extensive tables of logarithms of this new kind, which are |