TTSBS OP THB TABLES.] 



MATHEMATICS. ALGEBRA, 



521 



99999 can be constructed ; which is practically the same 

 as from 1 to 100000. 



It is to be observed, that with such a table, by means 

 of a very simple subsidiary calculation, we can obtain 

 the logarithm-of any number from 1 to 10000000. 



Thus, the tables give us log. 73894, log. 73895, i.e., 

 log. 7389400, log. 7389500, for these only differ from 

 those in the characteristic. 



The subsidiary calculation referred to enables us to fill 

 up any one of the logarithms of 7389401, <kc., up to 

 7389499. 



21. To explain the construction and use of the Table of 

 Proportional Parts. 



Suppose N to be a number such as that above referred 

 to, > 1000000 ; and suppose its log. to be given in the 

 table ; then log. (N + 100) is also given in the tables. 

 From these data we want to find log. (N + t) where t 

 lies between and 100. 



Now log. (N + 1) - log. N (l + ~ 



---- <tc. 



4)'- 



Now I < 100 ,/ > 1000000 



-T;< -00000001 

 (10000)' 



-.ilu.ii can be omitted, since we only take in the first 

 seven places of logarithms. 



/. log. (N 4- 1) = log. N + * 



/.log. (N + 100) = log. N + . 100. 



Now, log. (N -f 100) log. N _ A (suppose) is given 

 by the tables, and we see that 



which is true for every value of I from 1 to 99. 



If a and 6 are the digits of f, so that 8 = 10 .a + 6. 

 This formula can be written 



We have already seen that the difference between two 

 consecutive logarithms is the same for several logarithms 

 together ; accordingly, a small subsidiary table, giving 



A AX2 AX3 AX9. ,,,,, 



^;- ,,. t JQ ~~j7r~> is calculated for each 



different value of A, and is printed, as A occurs, in the 

 margin of the table. For instance, A corresponding to 

 log. 28568, is "0000152 ; or, as it is written, 152 ; it 

 being understood that the last figure, 2, falls under the 

 seventh decimal of the logarithm, In this case 152. 

 the subsidiary table is the accompanying. It is 

 called a Table of Proportional Parts. Since a 



A a 

 and 6 are digits, this table gives at once -TTT and 



A 6 1 A 6 



-JQ-, and therefore j^ -JQ-. Hence, by means of 



this table, we can determine log. (N + f) from 

 log. N by addition only. 



Thus, log. 2856800 = 6-4558798. . 



15 



30 



46 



61 



76 



91 



106 



122 



137 



Now, ~- = -0000046, and ~ = -00000106 



= "0000011, as we only take in seven places of decimals. 

 Hence, log. 2856837 = 6-4558798 + -0000046 + -0000011 

 = 6-4558855. 



We have now given a full explanation of the principles 

 on which Logarithmic Tables are calculated, so far as 

 that explanation is possible in a purely elementary 

 treatise, and have exemplified those principles in the 

 case of the ordinary tables which (practically) give the 

 logarithms of numbers from 1 up to 10,000,000, to seven 

 places of decimals. The student may ask, what would 

 be done if a case occur in which we have numbers ex- 

 ceeding 10,000,000 } The answer to the question is the 

 following : If the number were, for instance, 97536982, 

 and if the calculation demanded BO much accuracy that 

 we could not consider this as equal to 97536980, then a 

 more refined set of tables would be necessary ; in point 

 of fact, however, for all ordinary calculations, the degree 

 of accuracy which the common tables allow of is 

 sufficient. 



We now proceed to explain the practical method of 

 using the tables : 



THE USE OF A. TABLE OF LOGARITHMS. 



For the purpose of explanation, the following is printed from p. 78 of Hiilsse's edition of Vega's Logarithms. 



(1). To find the Logarithm, when the number it given in 

 the Tablet. 



(a). To find the mantissa. 



Suppose we want to find the logarithm of 46017. The 

 number N. 46000 at the top of the page will direct us to 

 the page on which we shall find 46017 ; the number 4001, 

 in the column marked N, will give us the line in which 

 we shall find what we want. Pass your eye along the 

 line, 4601, until it comes to column 7, there we find 

 9183. You will observe that there is 662 written in 

 column O ; this is to be written before every one of the 

 numbers under the other columns, and is only written 



VOL. i. 



once to render the tables more compact. Write this in 

 front of 9183, which we before found, and we obtain 

 6629183. This is the mantissa of the logarithm of 

 46017 ; it is, therefore, a decimal, and must be written 

 6629183. 



So again, to find mantissa of logarithm of 46035, look 

 down col. N for 4603, look along the line 4603 till you 

 come to col. 5, when you find 0881 ; before this prefix 

 663, and '6630881 is the mantissa of logarithm of 46035. 



Again, to find the mantissa of a logarithm of 46028, 

 look for 4602 in col. N ; along this line, in col. 8, you find 

 *0221 ; the asterisk in front of this 0221 shows that we 



3 x 



