126 



♦ KNOWLEDGE • 



[Dec. 9, 1881. 



I'vidcncp of Rtiitialics to provo Ihnt cnc-Ii cause must operate to 

 fonu'oxli-nt. It iH prrfeitly dbvioiit", on the ono hand, that itlthoiiffli 

 liiinclrcds of men who would bo held by in.surnnce conii)anies to be 

 *■ bad liveH " may eontruct ninmiigo, yet, on the whole, a principle 

 of selection in in oponition which u\\i»t tend to bring the healthier 

 portion of the mule community into the rankH of the married, and 

 to leave the unheiilthier in the Htiito of ba<'helorliood. A little con- 

 sideration will show, al«o, thnt, on the whole, the members of the 

 loss healthy trades, \ery poor persons, hitbilnni dmnknrds, and 

 others whoso prospects of lontf life nrc unfavoui-able, must (on the 

 uvcrngo of a hir(,'e number) be more likely to remain unmarried 

 than those more favourably situated. Improvident maiTiagcs are 

 undoubtedly numerous, but i)rosperity and adversity have their 

 influence, and that influence not unimportant, on the marriage 

 I'uturns. On the other hand, it is perfectly obvious that the 

 life of a marrietl man is likely to bo more favourable to 

 longevity than that of a bachelor. The mere fact that a 

 man has a wife and family depending upon him will snflice 

 to render him more careful of his health, less ready to un- 

 dertake dangerous employments, and so on ; and there are 

 other reasons which will occur to everyone for considering the 

 life of a married man better (in the sense of the insurance 

 companies) than that of a bachelor. In fact, while we are 

 compelled to reject Dr. Stark's statement, " bachelorhood is 

 more destructive to life than the most unwholesome trades, or than 

 residence in an unwholesome house or district, where there has 

 never been the most distant attempt at smitary improvement of 

 any kind," we may safely accept his opinion that statistics " prove 

 the truth of one of the tirst natural laws jovealed to man, " It is 

 not good that man should live alone." Whether the law required 

 any proof is a question into which we need not enter. 



From the Daily Neivs, Oct. 17, 1868. W. H. Pkrtwee. 



(Bm iWatOfmatiral Column. 



MATUEMATICAL QUE1UE8. 

 [2] — TnK Witch of Agnesi. — Will you kindly furnish a fen- 

 particulars respecting the history, properties, and practical applica- 

 tion (if any) of the above-named curve? If these are accompanied 

 with a tracing of the curve all the better. — E. II. R. 

 [3] — Appakeki Paradox . — 



Let ,T = )j, 



then ,r' = y.r, 



!/' - y' = x'j — >/, 



-•• (•• + ;/) (.<= -v) = y (^- - !')> (-<) 



.T + 1/ = ./, (B) 



V + ij = V, since x = i/ bv Hyp. ; 

 ' 2 ij = V 

 2 = 1. 

 Anyone who kindly explains the fallacy in the reasoning which 

 brings about this impossible result will much oblige — Puzzled. 



[The fault lies in passing from A to 7}. Interpret them, and we 

 see this |at once. A really means that (,r + ;/) times nought is 

 equal to y times nought, which is, of course, true ; just as it is 

 true that twenty times nothing is equal to ten times nothing. But 

 wo can no more infer that ,c + y = y than that 20 = 10. In fact, 

 we cannot divide both sides of an equation by any common factor, 

 unless we are sure that the factor is not equal to nought. In this 

 case we know that it is. — Ei>.] 



[4] — I shall be very much obliged if you will kindly favour rao 

 with a solution of the following problem. Data. JJ ■= 2 j S = v/3 ; 

 1^ = 1; )', r„ r,,, r„ being radii of inscribed and escribed circles : 

 prove that lir, = a', and r^ = )•, = perp. from angle A on side BC 

 of inscribed triangle. — Amici'S. 



d^ur mWn^t Column. 



By "Five of Cluds." 



A '• YAUBOltOUGU" HAND AT WUIST. 



Sir, — I was told the other day that a forn\er Earl of Yaiborough 

 was always ready to wager i;i, 000 to £1 against the occurrence of 

 a hand at Whist in which there should be no cai-d better than a 

 nine. Was the bet a fair one ? ALETnEi's. 



[The bet was decidedly unfair, and if made a great number of 

 limes must have resulted in large gains to the person who made it. 

 It is easy to calculate the odds before the deal, (after the deal, or 

 if the cards are cut and the lowest card is known, the odds are 

 slightly altered). In each suit there are five cards, ace, king, queen, 

 knave, ten, above a nine, or in the pack, 20 cards above a nine. 



From the remaining 32 cardu a hand of 13 cards may lie formed in 



32 • 31 • 30 • 29 ..^^ 20 



1 • 2 • 3 • 4 13 



different ways. The whole pack, however, will form 



52 • r,l ■ GO • 49 40 



T~^ 2 ■ 3 " 4 13 



different hands of thirteen cards. The chance, then, that any hniid 

 taki'n at random will hove no card Ix'ttor than nine is roprescntwl 

 by the i-ntio which the former of these amounts bears to the latter, 

 or bv the fraction 



32 • 31 • 30- 29 20 



52 • 51 ■ 50 • 49... ......40 



32 • 31 ■ 30 • 29 • 28 • 27 



"51 -lO- 47-45 -43 -41 2' 



31 - 15 - 2 9 • 14 - 27 31-29-2-3 



""61 - 40 • 47 - 45 ■ 13 -41 " 17 • 7 ' 47 • 43 • 41 



It will be found, on reducing, that this fraction is rather less than 



1K9S' "" ''""' ^"^ Yarborough, if he had been fair, (assuming 



always that he knew liow to calculate probabilities) should have 



offered rather more than £1,828 to £1 against the occurrence 



of the hand in question. It must be understood, of course, 



thnt he wagered with one of the players against that player 



having a " Yarborough," not against the occurrence of a 



"Yarborough" among the four hands dealt. The chance of this 



latter event is, of course, considerably gi-eater. It might 



sooni at a first view that it was exactly four times as great, 



since there arc fom- hands for each deal, but this is not the cage, 



any more than the chance of the occurrence of a Yarborough in 1,828 



hands amounts to 1,828 times , or to absolute certainty. The 



real chance that a Yarboi-ough will not occur in four hands is thus 

 obtained. The chance that a Yarborough will not occur in any 



1827 . /1S27\' 

 given hand is rrrr^ ; that it will not occur in two hands is I ,;^., I 



1828 /18*^7\^ \i^->/ 

 that il will not occur in three hands is ( 7-^ ) , and that it will not 



/1827\' \1^28/ 



occ lu- ill four hands is I — — I . Tliis is very nearly, but not exactly, 



equal to , " , or to -^- ; so that the chance of a " Yarborough " 



1832 4ob 



occurring in any four hands, taken at random in different deals, is 



Cfinal to about —^ ; nor is the chance different when the four hand.'* 



4o8 

 are iu the same deal. 



Supposing Lord Yarborough offered a wager of £1,000 to £1 to 

 each member of a whist party, for ten deals, on each of 100 nights 

 in each of ten years, he w-ould have cleared about £18,0OO.J 



'■ Mogul " writes to us that there are mistakes, some of them 

 serious, in our Whist Column. It is very likely ; but the principle 

 on which this column, like the rest of Kxowledge, is conducted, is 

 that of free di.ocussion, and the correction of errors as soon as 

 detected and pointed out. " Mogul " o-jly notes one, and there he 

 misapprehends us entirely, lie says wo in effect say that the rules 

 for leading are based on the principle of giving information to your 

 partner. Wo have said nothing of the sort. We have said that 

 the first great principle of the scientific game is to give your 

 partner all the information in your power, consistently with 

 the rules of the game. This is a very different thing. " Mognl " 

 states rightly enough that the primary consideration in se- 

 lecting what card to lead, especially what card to lead from any 

 peculiar combination of cards in a suit, has been the best chance of 

 trick-making. Of course, this is true ; but, as an objection to our 

 statement, "Mogul" might as reasonably have told us that the 

 primary object in Whist was to make tricks. " Mogul" will find, as 

 we proceed, that all questions of leading and play, second-hand or 

 third-hand, are priniai-ily weighed with reference to the chance of 

 making tricks (w-hich, by-the-way, has not yet been fully done, even 

 the ablest Whist players being apt to shirk the mathematical pro- 

 blems involved). But that is not at all inconsistent with the state- 

 ment respecting the distinction between scientific and unscientific 

 Whist, or between what may be called the twenty-six card and the 

 thirteen card games. 



" Mogul " invites our attention to lloyle and Cavendish. The8e> 

 with Matthews (though Uoyle and Matthews are now a little out of 

 date), Pole, Clay, Dmyson, and others are our guides; to this 

 degree, at any rate, thnt we should not depart from their teaching 

 without assigning our reasons and speaking under correction from 

 our readers. But vague corrections, like " Mogul's," arc of little 

 use to us. 



