62 



KNOWLEDGE 



[Nov. 18, 1881. 



"Iiown in the tables, bnt in reality they are implied. Thun, the 

 tiil>lc8 Rive — 



the loKiiiitlini of .3191 us 5039208; 

 tlilH reiilly mcnnn that — 



tl.o lo),nmtlini of 3 191 is 5U39268. (A.) 

 (The Btiuleiit should here tHrn to liis tables and boo for himself how 

 tho matter Btnnds ; ho should also ask hiniBolf what slatement A. 

 really means. On one side we have the number 3191 ; on the other, 

 a dceinml fraction slightly cxcoodin({ J, which wo are told is the 

 lo);aritlini of 3191. Now, remember that tho logarithm of a number 

 is that |iower to which 10 must ho raised to give the number, and 

 we see at once how A is to be iuterprotod. For 10 to power 4 is the 

 s;ime aa tho R<iunre root of 10, which we know to bo rather greater 

 than 3, so that in the equation 



3 + sonio small fraction = v/10 or 10* 

 we have a rough njiproach, near enough to illustrate its meaning to 

 tho statement — 



the logarithm of 3-191 - 50392C8. 



A few inquiries of this sort, with a book of logarithms in hand, will 



very soon give tho intelligent reader a good idea of their meauing 



and use. Take one other C4i8e. We find from tho tables that 



tho logarithm of 2 is -3010300. 



30103 . , , 



This mcnns that 10 raised to the power , qqqqq 's equal (very 



approximatelj-) to 2. Now let ns see whether taking the simple 



3 3 



fraction ^ w-e find 10 to the pow-er .^ approach in value to 2, 



though, of course, not quite so closely as would 10 to the power 

 3010300. We know that 10,'„- is the same as the 10th root of 10' 

 or of 1000 ; and we know that this is not very far from 2, for 2 

 raised to the tenth power gives 1024.) 



Bnt the tables are not the less complete that they only give the 

 logarithms of such numbers as 3-191, 9874156, and so forth. For, 

 since shifting a decimal ])oint one place to the right means multi- 

 plying by 10, while shifting the point one place to the left means 

 dividing by 10, and since the logarithm of 10 is 1, we have only to 

 add or subtract 1, 2, 3, 4, 5, &c., to the logarithm given in the 

 table, to get the true logarithm of any number whatever within 

 the limits ranged over by tho table. Thus — 

 Logarithm of 3191 = 0-5039268 

 Logarithm of 31-91, or of 10 x 3191 = log. 10 x log. 3191 



= 1-5039268 

 Logarithm of 319-1, or of W x 3191 = 2-5039268 

 Logarithm of 3191, or of 10= x 3191 =3 5039268 

 and so forth, the whole number in tho logarithm being always 1 

 less than the number of digits in the integral part of the original 

 number (that is. of digits on the left of tho decimal point). 



Again, logarithm of 03191, or of 3- 191 +10 = 0-5039268-1, which 

 for convenience and symmetry is written — 



Logarithm 0-3191 =1-5039268 

 Logarithm of 03191, or of 3191 + 10^ = 25039268 

 Logarithm of 0003191, or of 3191 4-10' = 35039268 

 and so forth, tho whole number under tho minus sign being always 

 1 more than the number of cyphers following the decimal point of 

 the number. 



it will be observed that in the case of numbers less than 1 we do 

 not follow what might, at first sight, seem the obvious course, of 

 subtracting the tabular number from 1, 2, 3, or whatever is the 

 number under the ini)tns sign, leaving the logarithm negative, 

 llius we write 1 5039208 short for - 1 -^ 05039268 ; wo do not write 

 — 0-1960732. Tho reason is that it is found convenient to have all 

 the (|uantities on the right of the decimal point positive thi-oughout 

 all computations. 



Mathematical Qi-e.stio.n.— Is there any solution of either or both 

 the following sets of simultaneous equations ? 



x' + ijz-'a.'' (i) ,t' + xy + v' = a' (') 



i/' + :.f = 1' (ii) and l/'-^ i/:-Hs' = 6' (ii) 

 ^-^a■y = c' (iii) i»+:x + x' = c» (iii). A.N.J. 



Wo know of no solution to tho former set ; that is, it cannot be 

 made, so far as we know, to produce a quadratic equation. The 

 second set may be solved thus ("A. N. J." will probably only 

 need to be shown the steps) : — 



Subtracting (ii) from (i) gives (x + y + z) (,r — 2)=a' — b' (iv). 

 Subtracting the sum of the squares of (i) (ii) and (iii) from twice 

 the sum of the products (i) x (ii), (ii) x (iii ), and (iii) x (i i), we obtain 

 rij + yz + xz=^i [aa'b'-' -t- 26V -h cW — a' — b* = c* 

 •=P' (say) (v) 

 adding (i) (ii) (iii) and 3 (v) we got 



x + y + z=^i{a.' + b' + c'' + 3p'=iii (say) (vi ) 

 From (iv) and (■vi) m {z — t)=a'—h', 



and similarly, m (y — *)— b' — <:• 



whence, subtracting, m (y + s — 2x) — 21'— a'— c' 



= CT(m-3i)-26*-o'-c' 



or a- = — (m»-fo'+e'-2l') 



3m 



^(m• + l'■^a'-20; : - i-(m'-t- a'-f !,'-LV, 



By symmetry 



(!Pui- ©abisJt Column. 



1!y " Five of Clubs." 



WK give this weik a simple whist game, showing the inferences 

 which can be drawn from the play by one of tho players 

 (the leader), and making notes also on the jday as it proceeds. Tho 

 inferences are all of the simplest kind, supposing the game to be 

 conducted according to the accepted principles for sound play. 

 This will appear as we proceed, in later papers, to develope these 

 principles. 



The Hands. 

 Dtumoiids— Q, 9, 7, 2. 

 Spades— Q, Kn, 10, 3. 

 Hearts— Q, 6. 

 Clubs— K"-, 0, 4. 



Diamonds — 5, 3. 

 Spades — A, Kg, 6. 

 Hearts— A, 7, 5, 3. 

 Clubs— A, Kn, 8, 2. 

 Score : — A B = ; Y Z = 3. 



Note. — The underlined card i 

 A T B Z 



3 trick, and card below it leads next. 



A's INFERENCES. 



1. Either 3C and 4C are both 



with B, or else T or Z is signalling 



for trumps. B has not tho Queen. 



Note to tbick 1. — Uavinp five trumps, 

 one huiiour, and his partner haTing an 

 honour, Tnuuld be justified in si)^allinf; 

 for trumps were the score low, but not t 



the 





« ♦ ♦ ♦ ♦ 

 ♦ ♦ ♦ ♦ ♦ 



2. BhasKnS, 10 S, and probably 

 one or more small Spades. Z is 

 not signalling for trumps, and 

 therefore has neither 3 C nor 4 C. 



XoTB TO TRICK 2. — A does well to take 

 the trick and return ibe Ave, thus leaTing 

 B the command of the suit. 



3. Y has signalled, and therefore 

 has either 4 C or 3 C ; the other 

 being with B. As Z tamed an 

 honour, T and Z are probably two 

 by honours, in which case A Ii 

 must make five by tricks to save 

 the game. 



4. The last Spade is with B (the 

 Knave). 



5. B has 40, but no more Clubs. 



KoTS TO TBICK 5. — B rct ums t he highest 

 of two cards. 



6. Z has not the King of Hearts ; 

 B has not the Knave (Uuorts 

 must be Z's best suit,'trumps being 



NoTS TO TRICE 6.— Under the circiun- 

 Btancea T ehould have played the King. 

 It is his beat chauce of getting a lend. 



