337 



LOGARITHMS, USE OF. 



LOGARITHMS, USE OF. 



333 



SEVEN-FIGURE TABLE. 



12. To find the logarithms of numbers and the numbers to loga- 

 rithms, from the four-figure table, proceed as follows : From inspection 

 of the number, take the proper characteristic, and then note the first 

 four iignijicant figures. In the row which begins with the Jimt tiro, 

 find the figures which are in the column headed by the third ; and 

 add to them the figures out of the side table which are in the column 

 headed by the fourth. For instance, required, as well as it can be 

 given from a four-figure table, the logarithm of 4275898'! 16. The 

 characteristic is 6 : and the first four significants are 4276 (reading 

 58 rather as 60 than as 50). In the row 42, and under 7, we see 6304, 

 and in the side table opposite to 6 is 6, and 6304 + 6 is 6310. Hence 

 6-6310 is the logarithm of 4275898-116, so far as the four-figure table 

 will give it. Similarly the logarithm of -4 (which must be read '400) 

 is 1-6021, that of 430 is 2'6335, that of -04179 is 2-6210. 



To find the number to a logarithm, in the fuur-Jiyure table, use the 

 decimals of the logarithm with the antilogarithmic table in the same 

 manner as the four significants of the number were used in the other 

 table to find the logarithm ; and then settle the place of the decimal 

 point by means of the integer of the logarithm. The four-figure table 

 goes into so small a space that it is worth while to print an inverse 

 table of anti!ti</'t >-itlnnx, or of numbers to logarithms ; of which table 

 we have also given a specimen above. Thus the number to the loga- 

 rithm 5'6234 being required, we neglect 5, and, going into the anti- 

 table with "6234, opposite to -62 and under 3 we find 419S, and under 

 4 in the side table we find 4. Hence 4198 + 4 or 4202 are the first 

 four significants of the number required ; and its characteristic is 5 : 

 whence 420200 is the number to the logarithm, as near as the four- 

 figure table will give it. Similarly the number to '6111 is 4'084, that 

 to 15-6208 ia -004177, that to 9-6000 is 3981000000. 



13. We give an instance of the application of each of the rules in 

 2. Let it be required to find, as nearly as four-figure tables will do 

 it, the product of 17798 and 63426 ; the quotient of 17'293 divided by 

 942942; the eighth power of 1-9273, and the eleventh root of 

 00005569181. The processes are as follows : 



Log of 17798 (say 17800) 4-2504 1 . 

 Log of 63426 (say 63430) 4'8023J A a 



Log of product 9-0527 

 Answer 1129000000 from the table. 

 True Answer 1128855948 



Error 144052, about -0001 of the whole. 



Again, log 17'293 (say 17'29) 

 log -942942 (9429) 



Log of quotient 

 Answer 18'33. 



Subtract. 



1-2632 



Log 1-9273 (say 1-927) 0-2849 



Multiply by 8 



Log of (1-927) 8 2-2792 

 Answer 190-2. 



In raising powers, the errors are generally larger than in other pro- 

 cesses, seeing that the necessary error of the logarithm is multiplied as 

 many times as the logarithm itself. 



Log -00005569181 (say -00005569) 5-7458 



Divide by 11 V6133, log ' 

 Answer '4105. 



-00005569 



,14. Before beginning, to use five-figure tables, it is advisable to 

 practise the formation of the tenths of numbers not exceeding 50, in 

 the head. For instance, which is the nearest integer to 7-teuths of 37. 

 The process at length would of course be 



37 



7 



10)259 



ARTS ASD SCI. DIV. VOt. V. 



2.?9 Nearest integer 26. 



It should be done thus : Having multiplied 7 by 7, and got 49, reject 

 the units, and carry 5 as the nearest number of tens. Then add 5 to 

 21 obtained from the three. When 5 units are thrown away, consider 

 the ten next above as the nearest. Thus 9-tenths of 45 should be 

 considered as 41, not 40, and 5-tenths of 17 as 9, not 8. Similarly, 

 8-tenths of 21 is 17 ; 3-tenths of 19 is 6. But 2-tenths of 32 is 6; 

 7-tenths of 42 is 29 ; 9-tenths of 28 is 25. 



15. In the specimen of the five-figure table will be seen four figures 

 of number, followed by five figures of logarithms, and an additional 

 column marked D, which contains nothing but the differences between 

 the successive sets of five figures in the logarithm. This column D is 

 referred to under the name of the column of tabular differences. To 

 take the logarithm of a number, take the characteristic as before, and 

 five significants of the number. Find out the first four significants in 

 the table, and to their five figures of logarithm add as many tenths of 

 the tabular difference as there are units in the fifth significant. Thus 

 to find the logarithm of -011217, we find 1121 in the specimen, 

 opposite to which is 04961, with 38 for a tabular difference. Now 

 7-tenths of 38 is 27, and 61 and 27 is 88 ; so -04988 is the decimal 

 part of the logarithm. Similarly if the significants of the number be 

 11223, the decimals of the logarithm are -05011 ; also 11201 gives 

 04926; 11209 gives '04957; 58332 gives '76591; 58333 also gives 

 76591 ; 58334 gives '76592 ; and so on. 



16. There is no antilogarithmic table to a five-figure table ; and the 

 way of finding the number to a logarithm is as follows : Seek out 

 among the logarithms the decimals next under the decimals of the 

 given logarithm, and take the four figures of number belonging to 

 them for the first four significants of the number. Find by how much 

 the decimals just used fall short of the given decimals, and call this 

 difference the unattained part. Annex a cipher to the unattained part, 

 and divide by the tabular difference ; the digit which most nearly 

 expresses the quotient is the fifth significant of the number. For 

 instance, what is the number to the logarithm 3'05016. Looking into 

 the table, we find that the next under '05016 is 04999, opposite to 

 1122. The unattaiued part is 17; the tabular difference 39 ; and 170 

 contains 39 4 times more nearly than 5 times ; so that 4 is the fifth 

 figure. The five significants of the number required are then 11224 ; 

 and, looking at the integer part of the logarithm, the characteristic is 

 seen to be 3 ; whence -0011224 is the number to the logarithm, as 

 correctly as five-figure tables will give it. 



17. The four questions worked above with four-figure logarithms 

 are thus worked with five-figure logarithms : 



17798 

 63426 



4-25037 

 4-80227 



1128900000 9-05264 

 05231 



38)330 

 1-9273 0-28495 



190-37 2-27960 

 27944 



23)160 



17-293 1-23787 

 94294 "1-97449 



18-339 1-26338 

 26316 



24)220 



000055692, 11)^74580 



41044 1-61325 



61321 



10)40 



18. The seven-figure tables have five figures of number, with seven 

 decimals of logarithm ; and the sixth and seventh significants of the 

 number are to be provided for by means of the tabular differences. 

 But as these tabular differences run to three and four places of figures, 

 their tenths are written down in small separate tables. [PROPORTIONAL 

 PARTS.] To take out a logarithm, take out the seven decimals 

 belonging to the first five significants of the number, and add from the 



are 4551737, look in the table for 45517, and we have 



45517 

 Tab. Pro. P. 



gives 



C581736 



29 



7 



0581772 



