3.2 



♦ KNOWLEDGE ♦ 



[July 13 1883. 



is made — by the double process above described — to approach 0. 

 Thus we are rjuite certain that O is the required point. Instead, 

 however, of proving this by facing through the necessary steps of 

 the above process — which mAiId be a sufficient proof — the student 

 should give the proof in the following form, obviously suggested by 

 the process he had before followed : — 



Draw the diagonals A C, B D meeting in O ; then is the required 

 point. For, let P be any other point, and therefore not on both 

 diagonals — say not on B D. Then B P and P D are greater than 

 B D (Euc. I., 20), and A P and P C are not less than A C (greater 

 than A C if P do not lie in A C) ; hence P A, P B, P C, and P D 

 are together greater than O A, B, C, and O D together. 



We have given the process determining the solution, in the form 

 which would most probably snggest itself. The double process is 

 also veiy instructive and suggestive. But the practised geometri- 

 cian would probably notice at once that the approach of P towards 

 0, in a straight line, diminishes at once the sum of P B, PD, and 

 that of P A, P C. Hence we would argue — in presenting the proof 

 — must be the point we seek ; for, let any other point give a 

 tninimum sum, then, by taking a point nearer O, we obtain a less 

 sum — that is, said point does not give a minimum : which is absurd. 

 (To be continued.) 



&nv Cljtss Column. 



By Mephisto. 



USEFUL END GAME. 



IN a game between Blackburne and English in the late tourna- 

 ment, the former player effected a draw in an ingenious manner, 

 although being at the time a Pawn behind. 



E.VGLISH. 



Black. 



It will be seen that we were trj-ing to get a position which might 

 give us a similar chance for a draw, and in spite of the fact of our 

 opponent having three good parvus, we think a draw can be forced. 

 We hope the preceding example will enable our readers to work 

 out this very interesting position. We shall give our analysis next 

 week. 



Whitb. 



BLACKEfP.N'E. 



In this position Blackburne drew by playing 



K to B4 R to B7 (ch) 



K to Q5 (If Black checks on Kt4, White also 



plays K to Q5) E to B6 



K to BG R takes QP 



B to Kt8, and Black could not avoid the draw, 



for if K to E2 the E checks on Kt7 and Kt8. 



The ingenious idea of placing the K on B6 and threatening a 

 mate or a perpetual check with the Rook, led us to ai-rive at the 

 following position when playing an actual game : — 



White to play and draw. 



PROBLEM Xo. i 

 By a. J. Ma.\s. 

 Black. 



White to play and mate in three moves. 



REVIEW. 

 Chess Blossoms. 



Miss F. F. Beechy, an experienced problem composer, has 

 published a collection of her two-move problems. These composi- 

 tions are of more than average merit, as proved by the fact of 

 several of them having won prizes. The prize problems of English 

 problem competitions for 1882 are appended, together with some 

 readable (chessy) verses and poetry. This is a pleasant little book 

 and is published by the authoress at Matlock, Bath. 



ANSWERS TO CORRESPONDENTS. 



•»* Please address Chess Editor. 



R. v. Tyley. — Dran-n games are sometimes counted half to each 

 player. 



W. Teebili,. — Could you kindly send a copy ? 



G. W. Thompsox. — If P moves there is no mate. 



W. — Solutions of Ending and Problems No. 86 and 87 correct. 



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