Sept. 15, 1882. 



• KNOWLEDGE • 



209 



&nv iWatDematiral Column, 



I SEND Tou two questions -nhicli I would be glad of a reply to. 

 The first was intended to be given at e'camination for licensed 

 surveyors in this colony (New Zealand). Your papers on proba- 

 bilities suggested No. 2. — Toowoomb.4. 



Prob. 1. — Prove the follov.'iiij method of drav.-in'j tangents to a 

 circle from a point O. 



Drmv any tioo chor.h OAB, and OCD. let AD n,id BC meet in 

 Y, and CA and DB i« X. Let XY meet the circle in P and Q, then 

 OP and OQ are the tangents required. 



This property is true if ABDC is any conic section, and is most 

 simjily ]iroved as follows : — OAB and OCD being chords from a 

 point "(), CA and DB intersect on the polar of O, and so also do AD, 

 BC. Therefore XY is the polar of 0, and P, Q, in which XI' 

 cuts the conic, are the points in which tangents from O meets the 

 conic. 



But Toowoomba's letter suggests that the solution of the problem 

 might probably have been required to be independent of the theory 

 of polars. 



Try then the following, — 



Take OB, OD as axes of y and .f, and let OC =« ; OD = n'; 0A = /?; 

 and OB=/3'. 



Then the equations to CA, BD, CB, and AD, are respectively— 



a ^ ir 



a'^ fy 



Hence the equation to XY will be of both the forms 



(i), 



and 



.. (i.f-^ )=. 



since (i) is the equation of a lino passing through X, and (ii) is the 

 equation of a line passing through Y. Obviously the only equation 

 which is of both these forms, and therefore the equation to XY, is 

 obtained by putting »i = m = l. Hence equation to XY is 



Now let the equation to the conic ABDC bo 



fi.t' + hxij + cif + dj(! + eij + f=0 

 Put y = 0, ami divide by x', giving 



(iii). 

 (iv) 



The two vaUies of - obtained by solving this qu.idratic are, of 



cour.sc, . — and cr — and — ; and we know that tlvia sum 



OC OD o a' 



= -<!. 'Jims, 



L + X =-d, and similurlv — + 4 = " '^• 

 a „■ • li li' 



Hence, equation (iii) may bo written 



<!x + ey + 2/=0 (v) 



Now let ./, 1/' 1 e the co-ordinates of P, and ,i ', y" those of Q, 

 the points in which tangents from meet the cui'vcs ABDC. 



Th'Mi we know that the equations to the tangents at P and Q may 



{2ax' + by' + d)x+{2ry'-tW + e)y + dx:' + cy' + 2f=0 

 (Zax" + by" + d)x + (2cy" + bx" + e)yi dx" + ey" + 2f=0. _ 

 Since these both pass tlirongh the origin, they must be satisfied 

 when a; = Oand i/ = ; that is, we must have 



dx' + ey' + 2f = and dx" -H £i/" -I- 2/= ; 

 Or both the points "P {x'y') and Q (x", y") lie on (v), that is on the 

 straight lino XY. Q. E. D. 



[I have given the proof more fully than is necessary. For 

 equation (v) is known to bo the equation to the chord through the 

 points in which the tangents from the origin meet the conic (iv)]. 



Problem 2. — Having iron the first game at whist, it is nsual to lay 

 the odd:', 5 to 2 on winning the rubber. How are these odds calcu- 

 lated? 



The odds are'incorrect. Independently of the deal, it is an equal 

 chance that the second game is lost, and also an equal chance 

 that the third game is lost ; so that the chance is i x i or i that 

 the rubber is lost by the wieners of the first game. The odds are, 

 therefore, 3 to 1 in favour of their winning. With the deal they 

 are rather more than 3 to 1 ; without the deal the odds are rather 

 less than 3 to 1.— En. 



(J^ur ClifSs Column. 



By Mepiiisto. 



SOME GENERAL REMAIiKS UPON THE STRATEGY OF 



THE GAME. 



The First Move. 



MANY of our readers who know the rudiments of Chess play, 

 cannot follow the analysis of an ordinary game without feeling 

 somewhat disappointed. Such a game will present to them many 

 unintelligible points, which they cannot explain themselves. They 

 wish to know the precise reason for every move. We will endeavour 

 this time to satisfy— and, let us hope, improve— this weaker class 

 of om- Chess patrons. 



1. P to K4. 

 We have a lively recollection of the despair of a young man who, 

 when playing with a young lady, was, with great perseverance, 

 asked why people always played 1. P to K4, and why not any other 

 move. The only answer he could give was, everybody begins with 

 that move. This, however, did not satisfy the inquirer at all, who 

 wanted reasons. In our opinion there are various reasons, based on 

 the theoretical foundation of the game. Firstly, the Pawn occupies 

 as nearly as possible the centre of the board, and is intended to 

 form the nucleus of a force which, being centrally posted, has the 

 greatest command over the enemy's camp, both to the right and 

 left. A good position is the first step towards the attack which 

 leads to victory. Secondly, P to Ki liberates two very important 

 pieces, namely, the Queen and the King's Bishop. The more 

 squares your pieces command, the more you limit your opiionent's 

 possible moves, and as the game proceeds, if you succeed in con- 

 centrating the commanding action of your pieces on one of your 

 opponont'.s Iii. (■•■.=, \nu will cut ..ft' the defence or retreat of such a 

 piece, and ciipl HI r'ii : ;iih1 m .M:itc is nothing but the successful 

 posting of VMiii- pi. < (^ in Ml, !i ;i c.ninianding position as to cut off 

 tho defence nr rem :ii ..f tli.' li..>tik' King. Like every otherpotent 

 fact, this proof can also be mathematically demonstrated. By 

 moving 1. P to K4, your Bishop has the command over five squares 

 and the Queen over'four— viz., B to RC and Q to R5. This gives 

 nine moves, and it is imi)Ossiblo by any other first move ou the 

 board to obtain tho command over nine squares by tho pieces 

 liberated. 1. P to Q-t only gives the command over seven squares 

 by the Queen aud the Queen's Bishop. This undoubted mathema- 

 tical inferioritv is, however, counterbalanced by the fact that in 

 reiily to P to Q-i your opponent cannot reply with the superior 

 move of P to K4, as then the Pawn could be captured, which neces- 

 sitates P to Q4 in reply to 1. P to Q4. Thirdly, the King's Bishop's 

 Pawn is the weakest point in an opponent's camp, it being only 

 jirotectod bv the King, and 1. P to K4 enables B to B4 to follow, 

 also Kt to KU3. Both these pieces indirectly threaten the Pawn 

 on KB7, the Bisliop from B4, and tho Knight from Kl5, where it 

 might be plavcd to, or the Queen from, R5, where it also might be 

 played prior 'to or after moving tho Knight. Of course. Black has 

 a ready defence, but while he is defending. White develops lus 

 forces, and is enabled soon to bring his King into safety by 

 Castling. 



1. P to K4. 



For the student this is, no doubt, the best reply, althcu^h one 



