333 



LOGARITHMS, GAUSS'S. 



LOGARITHMS, USE OF. 



334 



From which the following rules may easily be obtained : a number 

 which has m figures before the unit's place lies between 10 ! and 

 10", and its logarithm therefore lies between m 1 and m, or it is 

 ro' 1 + a fraction less than unity. Also, if a fraction be less than 

 unity, and if its first significant figure lie in the nth decimal place, 

 this fraction lies between 10-" and 10 ("-H ; so that its logarithm 

 is + a fraction less than unity. Now the convenience of Briggs's 

 system lies in this, that the fraction less than unity, which is a 'part of 

 every logarithm, does not depend on the position of the decimal point, 

 but entirely upon the significant figures : the reason being, that an 

 alteration of the position of the decimal point being a multiplication 

 or division by some whole power of 10, alters the logarithm by the 

 addition or subtraction of a whole number. This question is discussed 

 in every treatise on the mode of using logarithms. 



Let a be the base of a system of logarithms, and let log x signify 

 simply the Naperian or natural logarithm of .c ; then by the theorem 

 already proved 



log (hue a) = = g 



The factor 1 -f- log a, which converts Naperian logarithms into those 

 whose base is a, is called the modulo* of the system whose base is a. 

 In Briggs's system this modulus is -4342945 nearly, and the logarithms 

 of this system being called common or tabular logarithms, we have 



common log x = -4342945 x Nap. log x 



43 

 = go x Nap. log x, very nearly. 



Nap. log x = 2-3025851 x com. log j-, 

 _ A 00 " 1 



LOGARITHMS, GAUSS'S. These are tables suggested by the 

 celebrated Gauss for supplying an inconvenience connected with the 

 use of logarithms described in 21 of the article LOGARITHMS, USE OF. 

 Though they have been suggested more than fifty years, they are only 

 beginning to receive the attention which they merit. 



If log a and log 6 be given, and log (a + b) or log (a ft) be wanted, 

 the ordinary tables can only be applied by finding a and b from their 

 logarithms, adding or subtracting them as found, and then finding the 

 logarithm of the sum or difference. This requires three uses of the 

 tables, and one process of addition or subtraction. Gauss's table 

 substitutes one use of a table, and two processes of addition or sub- 

 traction. When the above necessity occurs only now and then, it 

 may be hardly worth while to have recourse to such a table; but 

 in any series of calculations in which the determination of log (a + 6) 

 from log a and leg 6 is a frequent constituent part, the table is a very 

 great relief. 



The construction of the table is as follows : There are three 

 columns styled A, B, c ; the first, A, containing the arguments. [TABLE.] 

 Also B and c are furuished with differences and tables of proportional 

 parts, in the same manner as the seven-figure logarithms. In column 

 A are entered successive decimal fractions, in a manner depending 

 upon the extent of the table. In the one we shall mention, A contains 

 0000, -0001, -0002, &c.,up to 2'0000; then 2-001, 2-002, &c., up to 

 3-000; then 3'01, 3'02, &c., up to 4'00; then 4'1, 4'2, &c., up to 5'0, 

 followed by 6 and 7. And whatever an A may be the logarithm of, 



say jf, then its B is the logarithm of 1 + , and its c is the logarithm of 



1 + >-. Thus, opposite to 2 00 in A, which is the logarithm of 100, the 

 B is log I'Ol, the c is log 101 ; accordingly iu one part of the table 

 we see 



A B c 



2-00 | -0043214 | 2-0043214 



Again, in another part of the table we see 

 B 



| -1031058 | 



4845 



c 

 6076058 



And it will be found that '4845 being the logarithm of K (not 

 mentioned; it is 3-041406), -1031058 is the logarithm of 1 + 1 -=- N, 

 and -6076058 of 1+x. 



The manner of using this table is as follows : Let x and y be two 

 numbers, of which x is the greater, and let log x and log y be given. 

 Then 



1. To find log (x + y). Let A = log x log y 

 log (x +.y) = log x + B = log y + c 



2. To find log (x y). Let B = log x log y 



log (x y) = log y A = log x o 

 Or thus : Let log x log ;/ = c 



log (x y) = log y + A = log x, B. 



It seems as if this were two tables, each of which might serve all purposes. 

 And it is true that log (x + y) can always be found from either table. 

 But the B table begins from -3010300 or log 2, and deacendt, while the 

 c table begins from the same, and attends. Consequently, '23, for 



instance, cannot be found about the c's, nor '32 among the B.'s. So 

 that in finding log (x y), log x log y must not be made B if it be 

 greater than log 2, nor c if it be less. 



For a list of tables published in aid of the above method, see 

 TABLES. 



LOGARITHMS, HYPERBOLIC. As tables of hyperbolic loga- 

 rithms are not very frequently met with, the following table is given 

 to facilitate the finding of the hyperbolic logarithm by means of the 

 common one. It is in fact a table of the hyperbolic logarithms of the 

 p.. \vers of 10 : thus opposite to 12 we see 27-68102112, which is the 

 hyperbolic logarithm of 10 l= , or a million of millions. 



To find the hyperbolic logarithm of a number, multiply the common 

 logarithm of that number by 2-30258509, by means of the table. Set 

 down the number opposite to the integer of the logarithm, then that 

 opposite to the first pair of decimal figures, leaving out the two last 

 figures, then that opposite to the second pair, leaving out the four last 

 figures, and so on. Add the results together. But if the characteristic 

 of the common logarithm be negative, subtract the united results of 

 the pairs of decimal places from the result of the characteristic, and 

 make the answer negative. For example, required the hyperbolic 

 logarithms of 327 and of "00142. The common logarithm of 327 is 

 2-5145478. 



51 



45 



47 



80 



Hyp. log. 327 



4-60517019 



1-17431839 



1036163 



10822 



184 



5-78996027 



The answer can be depended on to about a unit in the seventh 

 decimal place, and 5 7899602 is the answer to seven places. 

 Again, the common logarithm of t '-00142 is "-1522883 



LOGARITHMS, USE OF. \Ve have thought it advisable to add 

 to this work directions on the method of using logarithms, independent 

 of all considerations connected with the meaning and theory of these 

 numbers. Thus a person who has a table of logarithms, and finds 

 its preliminary explanations not sufficiently clear or complete, may 

 possibly receive help from this article, which is not written to accom- 

 pany any system of tables in particular. There are many reasons 

 against our inserting the table of logarithms itself in a large work of 

 reference, as was frequently done a century ago : we are pretty certain 

 that it would not be used. 



1. The object of logarithms is the performance of the second and 



