170 



• KNOWLEDGE 



[Dec. 23, 1881. 



In nearly all problems roqiiiring tlie use of IngnritlniiEi, however, 

 wc do not rp<inirc rxiict nocnnicy, but may be content with nji- 

 proximntion tu the thinl or fourth plnoo of (Iccinmls. 



I/Ct UR tnk<" n cnso not soipiititic, bnt pnictical. 



Rcqairod thi' nmoiint of £1,K28 nt oom]K)mid interest, five per 

 cent, per unniiin (pnynblo yearly), nt the end of ten years. 



Any sum nt the pven mte of interest, is iiicronsed in the ratio 



at the end of the first year : therefore, at the end of two years, 



(105\' 

 I ; and so on ; and at tlie end of 

 100/ 



ten years it is increased in the ratio ( — 1 Thus we have to find 



VlOO/ • 

 the value of the following expression : — 

 / 105 \'» 



Now log. 1828= 3 a; 10702 



Ten times log. 105 = 202 11Hi «0 



Sum = 23"472S0!i2 



Ten times log. 100 = 20- 



Log. (answer) = 3-17280a2 = log. 2i»70 



There is an error in tlie addition, log. 

 (answer) = 3'4738C92. True answer 

 somewhat greater than stated. 



Answer is £2,970. 15s. 5d. 



Here is another question relating to compound interest : — In 

 what time will a sum of money, say £100, be doubled at 5 per cent, 

 per annum, payable yearly, compound interest ? 



If X be the required number of years, we have 

 /105\ 



(ioo)(nro)=2oo 



or 105x = 2xl00i- 

 This is the same as saying that 



.r log. 105 = log. 2+1 log. 100 

 or (20211893) v = 0-3010300 + 2,v 

 _3010300 

 * ~ 211893 



We can again use logarithms to determine the value of this 

 fraction. 



We have log. 3010300 = C--1786098 



log. 211893 = 5-3251107 



1 



20 



15,420 



12 



5,040 



difference =l-152t'J31 = loR 



14-207 

 12 



Answ-er is 14 years 2i months, very nearly. 



For gi-eater exactness, multiply -207 liy 305, giving 75555, and 

 making answer, 14 years, 75 days, 13 hours, 19 minutes. 



The student of our subject is advised to go carefully through 

 each comjiutation. lie will note that in taking out the'logaritlim 

 of 211893, we jnit down for the first three figures not 325, but 320, 

 though the logarithm is found in a section which seems to have 

 325 for its leading digits. In this section we find 325 followed by 

 numbe£8 continually increasing up to 9875 ; then on the same line 

 comes 0080, which, of course, means that the logarithm has increased 

 from 3259875 to 3260080. The student must be careful on this 

 ]>oint, esjjccittlly in the earlier jmrt of the tables, where the changes 

 are more rapid. 



One other example illustrating an application of logarithms, in 

 ■which great time is saved. 



Suppose we want to find the cube-root of a number, say, 21,793, 

 correct to the third or fourth decimal place. By the ordinan- 

 arithmetical process this would be a long job, and we should have 

 oarefully to test the result to insure accuracy. But by logarithms 

 the process is very easy, thus : — 



Log. 21793 = 4-3383170 



Divide by 3 giving 1-.WG1057 = log 2793224. 

 Thus the cube-root of 21793 is 2793224. 



Take a more complex case, the solution of which by ordinary 

 arithmetical processes, with the same degree of accuracy, would 

 t.ike halt-a-day at the very least, even in the cuso of an arithme- 

 tician knowing how to take out the fifteenth and seventh roots of 

 numbers. 



Find the value of the expression : — 



( 1828)"<''^(0-17C3)^ 

 (715) MOOOol)* 



log. 1828-3 2G19762 

 2-lSth8 or 4-30th8 of this -04349302 



log. 0-1763 = 1-24C2523 

 1-third of this, or of-3 -^ 2 21C2523 = 17487508 



0-1836810 



log. 745=2-8721563 

 Ismnth of this _ -0-4103080 



log. 00051 = 3-7075702 

 1-fifthof this, or of-5 + 2-7075702= 1-5415140 



(A) 



1-9518220 (B) 

 A-B =0-2318590 



Answer = 1-705529 

 The student will notice how the negative characteristics are dealt 

 with in such problems. We must always add enough to the nega- 

 tive characteristic to make it exactly divisible by our divisor, 

 treating the number thus added as a positive characteristic for the 

 rest of the division. 



G 



©uv CftfSs Column. 



AME recently played at Simp.son's Divan between Mr. A. P. 

 Barnes, of New York, and Jlr. Gunsberg. 



■While. Black. 



Mr. BamcB. >[r. Gonsbeig. 

 Qucou'* Gomt.it Jeclini-d. 



1. P. to Q.4. P. to Q.4. 



2. P. to Q.B.4. P. to K.3. 



3. P. to K.3. Kt. to K.B.3. 



4. P. to Q.R.3. (•) P. to Q.B.4. ('') 



5. P. takes P. B. takes P. 



6. P. to Q.Kt.4. B. to K.2. (') 



7. P. to Q.B.5. Castles. (") 



8. Kt. to K.B.3. P. to Q.E.4. 



9. B. toQ.Kt.2. P. toQ.Kt.3. 



10. P. takes Kt.P. (') P. takes P. 



11. B. to K.2. V. takes P. 



12. B. takes P. B. takes B. 



13. R. takes B. K. takes E. 



14. Kt. takes K. Q. takes P. 



15. Castles. B. to R.3. 



10. Q.Kt. to Q.Kt.5. (0 Kt. to K.5. («) 



17. Q. to Kt.3. Kt. to Q.2. 



18. K. to Kt.sq. R. to Kt.sq. 



19. Q. to R.3. Kt. to Q.3. ('■) 



20. Q. to Kt..J. B. takes Kt. 



21. B. takes B. Q. takes B. 



White resigns. 



NOTES BY "MEPUISTO." 



(") This is prepaiatory to advancing tlie Pawns on the Queen's 

 wing. We cannot approve of such a course mth all the White 

 pieces yet undeveloped. 



C") In most openings, w-liere the first player opens up the Queen'g 

 wing first P.io Q.B.4 (to be followed, if feasible, by Kt. to Q.B.3) 

 will bo found effective, as it attacks the centre pawns, which 

 threaten to domineer over Black's game. 



(') The Bishop is sometimes withdrawn to Q.B.2 i-i'd Kt.3, the 

 idea being that on B.2 he is available for attack on the King's side 

 (this is problcmationl). We prefer B. to K.2, for, in the first 

 instance, it affords some protection against B. to Q. Kt.2. Secondly, 

 w-e consider the hostile Queen's wing weakened, and from K.2 the 

 Bishop will render assistance in attacking the Wliite Pawns success- 

 fully (this is positive). 



(") Necessary before begiuning the attack. Many good 

 games arc often thrown away through rashness in attack and 

 insnflicieut reg.ard for one's own safety. 



P takes E.P., then 



(') There is nothing bolter : if 10. 



10. 1'. takes B. P., and the Rook's Pawn is lost; or if 10. 

 P. takes B. P. 



Q. to B.: 



P.takesB.P. 



11 



Kt. to K.6. 



(') If B. takes Kt., then Q. to B .3. would w-in the piece back. 

 White intended to bring his Knight to Q.4, but it w-ould have 

 been much simpler to have brought him ii'a B.2. 



