Jax. 19, 1883.] 



KNOWLEDGE 



45 



<Piir iHatDtmatical Column. 



NOTES OX ELXLID.— No. III. 



By ItlCHARI) A. rKOCTOK. 



(Continued from page 29.) 

 Pbopositio.n XI. — Pbodleu. L(t AB he any straight line; it is 

 required ludivi'te A B into tteo parts, so that the rect. by the iphole 

 and one part shall be equal to the square o/ the other. 



Bisect A B in C, and from A draw AD D 

 perp. to AB, and =AC. Join B D. With 

 D as centre describe circalar arc A K 

 cutting D B in E, and with B as centre 

 describe circular arc E F cutting B A in F. 

 Then shall the rectangle A B, A F, bo 

 equal to the sq. on B F. 

 For 8q. on D A + sq. on A B = sq. on D B = sq. on D E + sq, on E B 



+ 2 rect. DE, EB. 

 .'. sq. on AB = sq. onFB + 2rect. AC, PB 



(•.•DE=AD = AC, andBE = BF). 

 That is, root. A B, AF + rect. AB, FB = sq. on FB + rect. AB, F B. 

 .•. rect. A B, A P = sq. on P B. 



Peoposition XII. Let ABC be an obtuse-angled triangle, AC B 

 being the obtuse angle, and from A let A D be draien perp. to B C 

 produeed ; then the sq. on AB is equal to the sqs. on BC, C A, 

 together tcith eirfcc the rect. B C, C D. 



Sq. on A B = sq. on A D + sq. on B D 

 = Sq. onADi-sq. on C D 

 + sq. on B C + 2 rect. B C, C D. 

 — sq. onAC+sq. on B C 

 + 2 rect. BC, CD 



Proposition XIII. Let ABC be a triangle having angle B acute, 

 attd draw AD perp. to B C, one of the sides containing the acute anr, le : 

 then the sqs. on A B, B C are together equal to (ha sq. on A C, with 

 tu-ice the rect. BD,BC. 



B 



I i.'. 1. Fig. 2. 



first, let D full between B and C (Fig. 1.). 

 Then sq. on A B = sq. on A D + sq. on B D 



sq. on BC=sq. on DC + sq. on BD + 2rect. B D, D C 

 .*. sq. on A B + scj. on B C = sq- on A D + sq. on D C + 2 sq. on B D 

 + 2 rect. BC, DC 

 = sq. on A C + 2 rect. C D, B C. 

 Hett, let D fall on B C pro<luce('. (Fig. 2.). Then 

 Sq. on A B = sq. on A D + sei- on B D 



= Bq. on A D + sq. on D C + sq. on B C + 2 rect. B C, CD. 



/. sq. on AB + sq. on B C = sq. on A C + 2 sq. on B C + 2 rect. B C , C D. 



= Bq. on AC + 2iect. BC, B D. 



Last, the case in which D coincides with C needs no demonstration, 



and has no interest, being to all intents identical with Prop. 47, 



Book I. 



Pbopositiox XIV. To describe a square that shall be equal to a 

 gieen rcclilinctir I'tgure. 



Tliis ))roi)osition cannot bo more briefly dealt with, at tliis place, 

 than as Euclid treats it. Bnt it is not really wanted till after 

 properties have been established in Book III. by which the demon- 

 stration may be shortened. 



©ur aaa&isft column. 



By "Five of Clubs." 



"Revoke" points out, correctly, that tricks 12 and 13 of the 

 game at p. 13, the Nine of Diamonds anti the Ten of Clubs have 

 been inadvertently interchanged. 



r. 



Heart.— 3. 



Spades— A, K, 10, 9, 7, 



6,4. 

 Diamonds — 3, 2. 

 Clubs— 7, 4, 3. 



A. 

 Hearts— Q, Kn, 9, 7. 

 Sj)ades — Q, Kn, 5. 

 Diamonds — Kn, 5. 

 Clubs— Q, Kn, 10, 9 



Hearts— K, 10, 0, 4. 

 Spades — 2. 



Diamonds— K, Q, 10, 4. 

 Clubs— A, 8, 6, 5. 



Z. 



Hearts— A, 8, 5, 2. 

 Spades— 8, 3. 

 Diamonds — A, 9, 8, 7, C. 

 Clubs— K, 2. 



oo] o| ^M4 



PLAYING TO THE SCOUE. 

 The Play. 



Score : — j Y Z =2 



Not 

 trick, I 



1. A plays as if at the score 

 " love all." If his partner has an 

 honour (and the odds are in favour 

 of IS's having one honour at least), 

 A IS only want five tricks to win. 

 A defensive, rather than a forward 

 game, is therefore indicated. 



2. B cannot tell whether A has 

 led from strength or length in 

 Hearts. But Z'a play of the Five 

 shows A had not led from five 

 Hearts. The presumption, as Y 

 renounces, is that A had four 

 Hearts, two honours. 1', regard- 

 ing strength in trumps as declared 

 against his side, discards the 

 penultimate of his long suit. 



3. A is wise, bnt not in time. 



4. Z draws two for one, keeping 

 np the Ace, as he has only one 

 other card of re-entry, and A is 

 presumably strong in plain suits. 

 The Queen, Ten, and Nine of 

 trumps being all unplaycd, Z 

 knows he can gain nothing by 

 leading Ace. 1' seeing Z is not 

 without trump strength, and 

 noting, too, that their case is almost 

 hopeless unless he can bring in 

 his Spades, discards now from A's 

 suit — not from Diamonds, as ho 

 cannot tell whether he may not 

 have to lead them to A more than 

 once. 



5. A B make their fourth trick ; 

 r Z must now make all the rest to 

 save game. 



6. B throws away the game, 

 through sheer inattention (to the 

 score and play both). He knows 

 -4 has tlio winning Clubs, and that 

 Z has but one trump left, for A 

 could not have led trump Seven 

 from Queen, Knave, Seven only. 



7. Z of course draws the remain- 

 ing trumps, and leads his best 

 card in liis partner's suit. Y 

 makes a deep finesse ; not so deep 

 as it looks though, for A, having 

 shown no strength in Diamonds, 

 is almost certain, since he led 

 trumps, to hold Queen, Knave in 

 Spades. Apart from this, it wonld 

 be a question of probability, viz., 

 whether it is more likely that B 



holds Knave or Queen, in which 



case A B win, or whether, if Y plays King, the Ace will draw both 

 Queen and Knave, and unless this happen A B win equally. The 

 finesse comes off, and 



10, 11, 12, 13. Y Z make the remaining tricks and win. 



♦ 44^ ♦ ^ * 



10 



