in geometrical progression. (Alg. 436.) And their logarithms L, L+1, L+21, L+31, &c. are in arithmetical progression. (Alg. 423.)* 18. Hyperbolic logarithms. Although the system of logarithms which has now been explained, is far more convenient than any other, for the common purposes of calculation, and is the only one in general use, at the present time; yet it is not that which was first proposed, by the celebrated inventor of logarithms Lord Napier. For particular reasons, he made the radix of his system 2.718281828459, instead of 10. This produces a change throughout the whole series, except the logarithm of 1, which, in every system, is 0. Napier's logarithms are also called hyperbolic logarithms, from certain relations which they have to the spaces between the asymptotes and the curve of an hyperbola; although these relations are not, in fact, peculiar to Napier's system. These logarithms have some particular uses, to which the common tables are not adapted. THE LOGARITHMIC CURVE. 19. The relations of logarithms, and their corresponding numbers, may be represented by the abscissas and ordinates of a curve. Let the line AC (Fig. 1.) be taken for unity. Let AF be divided into portions, each equal to AC, by the points 1, 2, 3, &c. Let the line a represent the radix of a given system of logarithms, suppose it to be 1.3; and let a2, a3, &c. correspond, in length, with the different powers of a. Then the distances from A to 1, 2, 3, &c. will represent the logarithms of a, a, a3, &c. (Art. 2.) The line CH is called the logarithmic curve, because its abscissas are proportioned to the logarithms of numbers represented by its ordinates. (Alg. 527.) 20. As the abscissas are the distances from AC, on the line * See note C. It may not be correct to ascribe to Napier the original discovery of the principle of logarithms. But he was the first who constructed logarithmic tables, and adapted them to the methods of computation to which they are now so extensively applied. These tables were published in 1614. For a particular History of logarithms, see the Introduction to Hutton's Mathematical Tables. AF, it is evident, that the abscissa of the point C is 0, which is the logarithm of 1=AC. (Art. 2.) The distance from A to 1 is the logarithm of the ordinate a, which is the radix of the system. For Briggs' logarithms, this ought to be ten times AC. The distance from A to 2 is the logarithm of the ordinate a2; from A to 3 is the logarithm of a3, &c. 21. The logarithms of numbers less than a unit are negative. (Art. 9.) These may be represented by portions of the line AN, on the opposite side of AC. (Alg. 507.) The ordinates a1, a-2, a-, &c. are less than AC, which is taken for unity; and the abscissas, which are the distances from A to -1,-2, -3, &c. are negative. 22. If the curve be continued ever so far, it will never meet the axis AN. For, as the ordinates are in geometrical progression decreasing, each is a certain portion of the preceding one. They will be diminished more and more, the farther they are carried, but can never be reduced absolutely to nothing. The axis AN is, therefore, an asymptote of the curve. (Alg. 545.) As the ordinate decreases, the abscissa increases; so that, when one becomes infinitely small, the other becomes infinitely great. This corresponds with what has been stated, (Art. 15.) that the logarithm of 0 is infinite and negative. 23. To find the equation of this curve, Let a the radix of the system, x=any one of the abscissas, Then, by the nature of the curve, (Art. 19.) the ordinate to any point, is that power of a whose exponent is equal to the abscissa of the same point; that is (Alg. 528.) y=a** *For other properties of the logarithmic curve, see Fluxions. C SECTION II. DIRECTIONS FOR TAKING LOGARITHMS AND THEIR NUMBERS FROM THE TABLES.* ART. 24. THE purpose which logarithms are intended to answer, is to enable us to perform arithmetical operations with greater expedition, than by the common methods. Before any one can avail himself of this advantage, he must become so familiar with the tables, that he can readily find the logarithm of any number; and, on the other hand, the number to which any logarithm belongs. In the common tables, the indices to the logarithms of the first 100 numbers, are inserted. But, for all other numbers, the decimal part only of the logarithm is given; while the index is left to be supplied, according to the principles in arts. 8 and 11. 25. To find the logarithm of any number between 1 and 100; Look for the proposed number, on the left; and against it, in the next column, will be the logarithm, with its index. Thus The log. of 18 is 1.25527. The log. of 73 is 1.86332. 26. To find the logarithm of any number between 100 and 1000; or of any number consisting of not more than three significant figures, with ciphers annexed. In the smaller tables, the three first figures of each number, are generally placed in the left hand column; and the fourth figure is placed at the head of the other columns. Any number, therefore, between 100 and 1000, may be found on the left hand; and directly opposite, in the next column, is the decimal part of its logarithm. To this the index must be prefixed, according to the rule in art. 8. *The best English Tables are Hutton's in 8vo, and Taylor's in 4to. In these, the logarithms are carried to seven places of decimals, and proportional parts are placed in the margin. The smaller tables are numerous; and, when accurately printed, are sufficient for common calculations. The log. of 458 is 2.66087, The log. of 935 is 2.97081, of 386 2.58659. of 796 2.90091, If there are ciphers annexed to the significant figures, the logarithm may be found in a similar manner. For, by art. 14, the decimal part of the logarithm of any number is the same, as that of the number multiplied into 10, 100, &c. All the difference will be in the index; and this may be supplied by the same general rule. The log. of 4580 is 3.66097, The log. of 326000 is 5.51322, of 79600 of 8010000 4.90091, 6.90363. 27. To find the logarithm of any number consisting of FOUR figures, either with, or without, ciphers annexed. Look for the three first figures, on the left hand, and for the fourth figure, at the head of one of the columns. The logarithm will be found, opposite the three first figures, and in the column which, at the head, is marked with the fourth figure.* The log. of 6234 is 3.79477, The log. of 783400 is 5.89398, of 5231 3.71859, of 6281000 6.79803. 28. To find the logarithm of a number containing MORE than FOUR significant figures. By turning to the tables, it will be seen, that if the differences between several numbers be small, in comparison with the numbers themselves; the differences of the logarithms will be nearly proportioned to the differences of the numbers. Thus Now 43 is nearly half of 87, one third of 130, one fourth of 173, &c. Upon this principle, we may find the logarithm of a number which is between two other numbers whose logarithms are given by the tables. Thus the logarithm of 21716 is not to be found, in those tables which give the numbers to four places of figures only. *In Taylor's, Hutton's and other tables, four figures are placed in the left hand column, and the fifth at the top of the page. But by the table, the log. of 21720 is 4.33686 and the log. of 21710 is 4.33666 The difference of the two numbers is 10; and that of the logarithms 20. Also, the difference between 21710, and the proposed number 21716 is 6. If, then, a difference of 10 in the numbers make a difference of 20 in the logarithms: A difference of 6 in the numbers, will make a difference of 12 in the logarithms. That is, 10:20 :: 6:12. If, therefore, 12 be added to 4.33666, the log. of 21710; The sum will be We have, then, this RULE. 4.33678, the log. of 21716. To find the logarithm of a number consisting of more than four figures; Take out the logarithm of two numbers, one greater, and the other less, than the number proposed: Find the differ ́ence of the two numbers, and the difference of their logarithms: Take also the difference between the least of the two numbers, and the proposed number. Then say, As the difference of the two numbers, To the difference of their logarithms; So is the difference between the least of the two numbers, and the proposed number, To the proportional part to be added to the least of the two logarithms. It will generally be expedient to make the four first figures, in the least of the two numbers, the same as in the proposed number, substituting ciphers, for the remaining figures; and to make the greater number the same as the less, with the addition of a unit to the last significant figure. Thus For 36843, take 36840, and 36850, For 792674, For 6537825, 792600, 6537000, 792700, The first term of the proportion will then be 10, or 100, or 1000, &c. |