rea. lore &c. 4 2, and then 1=1, 2n +1 Froni the preceding articles hyperbolic logarithms may be calculated, as in the following examples. Example 1. Required the hyperbolic logarithm of 2. Put 1+1 2n + 2 2n + 112 2n+1)-1 9 In order to proceed by the series in article 11, let 1 + x 4 3. x x=0.14285711286 1 and then s=5. Consequently, 1-X or Sum of the above terins - 0.14384103622 2 0.28768207244 ent1 3 The double of which is 0,57536114488, and answers to the first part of the expression in article 12. 1+1 9 Secondly, let and then 8+87= 9-9.r, and 1 which answers to the second part of the expression in article 12. Consequently the hyperbolic logarithm of the number 2 is 0.57536414488 +0.11778308566 =0.69314718054. 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 4 =1.38629436108 of 8=2.07944154162 &c. From the above the logarithm of 3 may easily be obtained. 4 For 4 =4x = 3; and therefore as the logarithm of 3 3 was determined above, and also the logarithm of 4, From the logarithm of 4, viz. - - 1.38629436108, 4 Subtract the logarithm of viz. 0.28768207244, 4 And the logarithm of 3 is 1.09861228864. 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 2 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 19. 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 i, Let b be the prime number, whose logarithm is sought, and a and c even numbers whose logarithms are known, or easily obtuined 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-1x6+1 62 ac+1 =b2-1, and act1=b?; and therefore This is a general expression for the fraction which it will be ac ac proper to put ac 1+1, that the series,expressing the hyper1 1 + x bolic logarithm may converge quickly. For as 1 ac +1 ac tacx =dc+1- ac x, and therefore 2 acx 1 + x = 1, and x = 2act1 Example 2. Required the hyperbolic logarithm of 5. 1 1 Here a = 4,c= 6, and x = Consequently, 2ać +1 49 = 0.0204081632 23 = 0.0000028332 5 = 0.0000000007 Sum of the above terms 0.020410997 25 I++ - 0.0408219942 24 25 But X8*3 =25, and the addition of logarithms án 24 swers to the multiplication of the natural numbers to which they belong. Consequently, 25 0.0408219942 2.0794415129 1.0986122990 And the sum is the log. of 25 - 3.2188758234 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 1 ac #1 It& Here a=6, C=8, and = and Zac +1 97 dc 49 Consequently, 48 The sum is the log. of 49 3.89182029798 49 For 48 x 6 x 8 = 49. Consequently the half of this, viz. 1.94591014899, is the hyperbolic logarithm of 7; for 7 *7 = 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 it, 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 2.5649993575381 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 3 5 6 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, ita,', itas, itals, i talt, ita)', &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 tberefore be represented with different sets or kinds of logarithms in the following manner: 1, ita, it al, 1+al, it al, 1+als, ital, &c. 1, ital, itala, 1+231 17a", itals, italo, &c. 1, TFalíi tam, tam itam, it alm, itam, &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+a14 be equal to 7, then 1+as, is equal to 7, as is also 1+2im. If italo be equal to 10, then 17 al is equal to 10, as is also ital", &c. If therefore 1, 21, 31, &c. be byperbolic logarithms, calculated by the methods already explained, the logarithms expressed by 3 m' m' m be derived from them; for the hyperbolic logarithm of any given number is to the logarithm in the last-inentioned set, of the same number, in a given the las 4 6 ratio. Thus 41: also 61: 4lm 6 olm 4. 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 6 OL. 2 &c. inay 1 ::1: : 1: m |