80 



♦ KNOWLEDGE • 



[JcLY 25, 1884. 



which is not a new relation, the change in our suppositions merely 

 leading to the inversion of tlio former relation. 



In Prop. 10 corresponding suppositions with the interchange of 

 a and b iiive the same results. 



It will bo found that any theorem respecting rectangles may be 

 shown to correspond to an algebraical identity, — and in like manner 

 that any homogeneous algebraical identity of two dimensions may 

 be made to supply one or more geometrical theorems respecting 

 rectangles. 



Let us take as an instance the following identity : — 



(a+b + cy- = a? + h- + c- + 2al + 2hci-2ca ; this resolves itself into 

 the following proposition : — 



Prop. I. Tf a straiijht line AB ie dirided into any three parts in 

 t'le points C and D; then 



the square on A B shall A___ , ^ 



he equal to the aqiiares on 



AC, CD, and D B, together with twice the rectangles contained bij 



A C, C 1), by AC, D B, and by CD, J) B. 



By Euc. II., prop. 4, the square on A B is equal to the squares on 

 A C, C B together with twice the rectangle A C, C B ; that is (again 

 applying Prop. 4) to tlie squares on A C, CD, D B, together with 

 twice the rectangle CD, D B, and (Prop. 1) twice the rectangles 

 AC, CD, and AC, DB. 



Prop. 11 is an important one. It may be enunciated also thus, — 

 To divide a given straight line into tico parts so that the squares on 

 the whole line and on one of the parts mai; be together equal to three 

 times the square on the other part. That this enunciation is equiva- 

 lent to the other follows immediately from Prop. 7. Prop. 11 offers 

 a problem somewhat more difficult than most of those in Euclid. It 

 is made use of by him in Book IV. Prop. 10 ; but when it is required 

 for the solving of Prop. 30, Book VI., he appears to have forgotten 

 that he had already solved it, and adopting a less happy mode of 

 analysing it occupies three long propositions with its solution. We 

 shall note an analogous proposition in our next contribution on this 

 subject. 



{To be continued.) 



(Bm cores Column. 



By Mephisto. 



THE FRESCU DEFENCE. 



1. 



P to K4 



P toK3 



THIS is a perfectly sound and safe defence, and suitable when- 

 ever the second' player wishes to play a clo se game or avoid 

 the more attacking lines of play arising from 1. P to K4 ; White's 



best reply is 2. ?J5_Qi. If now P x P, then P x P followed by 



^ ■' P to Q4 



Kt to KB3, B to Q3, and Castles with a safe game. Wiiite may 



also play 3. 



Kt to QB3 



- — B to Kto would not lead to any favour- 



Kt to KB3 



able result. White now would not gain anything by advanci-sg 

 P to K5, as, after Kt to Q2, Black would vigorously attack the 

 centre by P to QB4, Kt to B3, &c. If 3. P x P, P x P, and we 

 have a normal position which Black follows up by B to K2 and 

 Castles. White has now two lines of play, viz. : — 



B to Q3 and B to KKto. If 



and in this variation White has a 

 good position. It might perhaps 

 have been better to play 5. P to 

 yKt3, followed by B to Kt2, as 

 that would leave the square on 

 B3 open for the Knight. If in- 

 stead 



4. B to KKt5 B to K2 



5. B X Kt B X B 



6. Kt to B3 Castles 

 Whiiji. 7. B to Q3 PxP 



8. B X P Kt to B3 



With two Bishops on the board. Black need not be afraid to have 

 his Pawn doubled, the position being an even one. 



SELECTED PROBLEMS. 

 Black. 



Whitb. 



White to play and mate in three moves. 



Black. 



Whiib. 

 White to play and mate in two moves. 



ANSWERS TO CORRESPONDENTS. 

 ,*, Please address Chess Editor. 



Fraukford.— If 1. Q to K3, P to KtG, 2. Kt to K6(ch), K to Kto, 

 3. Q to Q5, K to R6. 



Geo. W. Thompson. — Solution not quite correct. 1. Kt to B3, 

 2. Kt to Kt3(ch;, q X Kt, 3. Kt to Q7(cli), Kt « Kt. 



George Gouge. — 1. Kt to Bsq, 2. Kc x P, and mates next move. 



Correct solutions received from il. T. Horton and A. Rutherford. 



The Owl. — Part solution correct. 



Contents of No. 142. 



PaGS 



Other Worlds than Ours. By M. 



de Foutenelle. With Notes by 



Richard A. Proctor (Conlimied)... 43 

 Chemistry of Cookery. XXXVIII. 



By W.M. Williams « 



Man and Xature 45 



Optical Recreations, (lllut.) By 



F.R.A.S 46 



Electro-plating. Till. By W. 



Slii}SO 47 



The Entomology of a Pond. By 



E. A. Butler 49 



Superstition 60 



International Health Exhibition. 



VIII. (ntut.) 60 



FA61 



British Seaside Resorts. I. By 



Percv Russell 63 



Zodiacal Maps. (lUut.) By R. A. 



Proctor 64 



The Antarctic Regions By R. A. 



Proctor 64 



The Absolute Capacity of a Con- 

 denser 56 



ne\-iews 56 



The Face of the Sky. By F.R.A.S. 58 



Design for Parlotir Organ. {Iltu*.) 58 



Miscellanea 58 



Correspondence 60 



Our Mathematical Column 62 



Our Chess Column 64 



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