LOGARITHMS, USE OP. 



WXHC 



n 



So the logarithm riuiml, a* far as seven figure Ublei will give it 



A litU. practice will enable the calculator to add together, without 

 writing them down, the two ooatiogenU derived from the table of 

 proportion*! put*. The compuUr murt be careful not to wiu the 

 chang* of th third figure of a logarithm, which umally take* pUoe in 

 Ik* middle of a hoe. though it can 00)7 be marked at the beginniiiK. 

 Thus. in to* specimen the logarithm of 45603 ha* the decimals 

 6MM34, but that of 45604 ha* SJH>029, not 6580039. Some, mark in 

 usually made to give warning that the change ha* taken place ; but 

 those whoa* attention U ao little alive to the circuinrtance a* to itand 

 much in need of this mark will often make the mi*take in ipite of it. 



*19- To take the number to a given logarithm out of (even-figure 

 table* proceed a* follow* : Find the nearest decimal* under the given 

 i****-*!*. and by subtraction find the umatiaiiud part. Take out the 

 five number-figure* belonging to the attained part, for the first five 

 significants of the number. Look in the table of proportional part* 

 tor Ik* nearest under the unattained part, and take the digit opposite 

 to it fur the sixth significant. Annex a cipher to the remainder of the 

 nnattaiiMiH part, and take the digit opposite to the result (or its 

 nearest) in the Uble of proportional part* for the seventh significant. 

 For hutanee, required the number to the logarithm 1*6590176 : 



(MM 



: I I7fl 



51 

 48 



Number required 45-60558. 



2<\ The following U the working of the four questions already 

 worked with four-figure and five-figure tables : 



intt 



at 



4-2503712 

 t<ttSS071 



: tan u 



IUU 0526170 



215 



6 193 



G 220 



Auswer 1128856000. 



17-923 

 9429*2 



12378703 

 19744650 



1833941 1-2633853 



1-9273 

 190-S682 



02849493 

 8 



2-2795944 



00006569181 

 4104439 



11)57457913 

 16132538 



21 . More complicated question* may be Jirertiy solved by logarithms, 

 that is, without requiring to seek the number to a logarithm until the 

 and of the process, so long as the multiplications, divisions, involutions, 

 and evolution* are not broken by addition* or subtraction*. Thus to 

 calculate *V {a4 tyc-i-d] we form for the logarithm of the answer 



1 r 



jj \ 2 log a + log 6 -f } log e log rf 



Bat if addition* or subtraction, intervene, the process must be broken 

 to produce the remilt; that b, the logarithmic process must be 

 nnended, and the result* exhibited in numbers for the performance 

 of the additions and *ubtraction* (unless indeed Gauss's logarithms be 

 employed). [LOGARITHM*, OATHS'*. J Thus to find V(o'4 + erf), it 

 would b* neceesary, not merely to form the logarithm* of oi and /, 

 or J log a + log 4 and log c + log rf, but to find the numbers to them : 

 the** number* being then added, and the logarithm of the sum being 

 taken, half the last logarithm i* the logarithm of the answ. r. 



There i* nothing in the trigonometrical logarithms more than i* 

 described in TRIUOXOHKTHK AI. TABLU. Many young (tudenU, from 

 Idem or never using the actual sinrs, cosine*, Ac., of angles, but only 

 their logarithms, are apt to confound the former with the latter. 



28. Some liuU additional correctness may be obtained by annexing 

 to the uw of the table*, a* given above, an attempt to add something 

 for one more figure of the number. Thus, suppose it required to find 

 the logarithm of 11-374928. In the four-figure table 113 has 0581 and 

 ... 7 adds 24. The next figure is 4, which, had it been the fourth 

 significant, would have added 15 ; being the fifth, let it add the tenth 

 of 15, or iu nearest integer, 2. Hence we add 28 instead of 26 to 



0531, and get 0559 instead of 0557, a* a nearer approach to the 

 logarithm. Again, in the five-figure table, 1137 give* 05576, and the 

 tabular difference i* 38. The 4 give* 15, or 4-tenth* of 38, and the 9 

 ought to give 9-hundredth* of 38, or S. Consequently 15 + 3, or 18, i* 

 added, giving 05594. Had 49 been read as 50, in the usual way, 

 19 would have been added. In the seven-figure table* 11874 

 0559132 ; 9 add* 345, the 2 should add the tenth of 77, and the 8 

 should add the hundredth of 306 : hence 3 more should be added than 

 arises from the figure* 92. This plan may be followed \\lnn ii;.- 

 greatest accuracy is desirable, especially in the four-figure table. It in 

 most useful when the tabular difference* are large, that U, at the 

 beginning of the table. 



-I It may sometimes save a mistake if the computer remember 

 that the significant* of the number generally read leu than those of 



the decimals in the logarithm. Thus 200 has 301 for the 



decimal figures of the logarithm. It is not so however at the 

 beginning of the table, and the figures of the logarithm overtake those 

 of the number between 13712 and 13713. 



25. The rule of the characteristics, when well learnt, furnishes one 

 of the easiest modes of assigning the place of a decimal point in a 

 common division. For example, in dividing '0017643 by 018H6, the 

 characteristics being a and 2, we shall have to carry 1 to Z, because 

 significants of the divisor, 18, are greater than thoro of the dividend, 17. 

 Take 1 then from 8, which leaves 2, the characteristic of the quotient, 

 which has therefore -0 before its significants. The rule is, to find the 

 characteristic of the quotient, from that of the dividend subtract that 

 of the divisor, carrying 1 if the first significants of the divisor be 

 greater than those of the dividend. 



26. In using the trigonometrical table*, or any other, careful 

 attention must be paid to the right choice of the method of inter- 

 polation. If the argument and tabular result [TABLE], or the numbers 

 with which we enter the table and those which wu take from it, 

 increase together or diminish together, then the alterations introduced 

 by the table of proportional parts consist in adding to either for every 

 addition to the other, and subtracting for every subtraction. But if 

 the tabular result diminish as the argument increases, then every 

 addition to the one is accompanied by a subtraction from the other, 



rend. In the tables of logarithmic cosines and cotangents, 

 the young computer frequently fails into error from neglect of the 

 last-mentioned direction. Thua the logarithm of the cosine of 84 9' 

 in a five-figure table is 9-00828, and the tabular difference is 124, 

 (limiiii.Hhing. Hence the log cosine of 84 9''5 is not 9-00828 + '00062, 

 but 9-00828 -00062. And if we ask for the angle whose log cosine 

 is 9 00840, we find the next underneath this in the table to be 9-00828, 

 and 22 for the uuattained part. Divide 220 by 124, and 2 is the 

 nearest integer. But the angle required is not 84 9'+0'-2, but 

 849'-0'-2, or 84 8' 8. 



27. The equation a* = b, in which .c is unknown, is solved by 

 taking the logarithm of both sides, which gives x log a = log b, or 

 z=log i-rlog a. The process may perplex a beginner when the log- 

 arithms have negative integers. We therefore subjoin on example. 

 Let it be required to solve the equations 



00163 and 182 = -6. 



(The reader is supposed to know the complete interpretation of 

 algebraical exponent*.} We have then 



x= log (-001 63) -r log (y) =3-21219 -j-1-75696 

 y = log -6 T- log 183 = r-77815-i- 2-26007. 



Now throw the three logarithms which have negative integers into the 

 more usual algebraical forma. 



21219-8 , 

 Or -278781 , 



75696 - 1 

 - -24304 



77815-1 

 - -22185 



And make the divisions, which give for the quotients 11-471, and 

 - -098161; or 



'4 \II-47I 



j) =-00168 



(182)"""= -6 



LOGIC. Since the ' Penny Cyclopjcdia ' was completed, the study 

 of logic in this country has undergone three important changes. First, 

 HMO i - paid to the subject : secondly, innovations 



have been listened to in a spirit which seems to admit that Kant's 

 dictum aliout the perfection of the Aristotelian loflc may possibly be 

 false : thirdly, a disposition has arincn to distinguish logic from meta- 

 physics and psychology, without losing sight of the psychological and 

 metaphysical dUcussion which is necessary to a sound view of the 

 meanit: and first principles of the science. In such a state 



of transition the best thing to be done is to allow the main logical 

 articles of the ' 1 Vnny ( 'ycloptedia' to stand unaltered, as representatives 

 of a starting joint, and to devote the present article chiefly to some 

 account of recent developments and current dispute.-. W. therefore 

 refer to ORUAXON, SYLLOGISM, &c., on preparatory to the present article. 



