June S, 18S3.] 



* KNO^A^LEDGE ♦ 



349 



Agoiu, suppose a problem spoko of a circlo touching a given 

 line in a given point, iiud passing through another given point. 

 Then we should not tlraw a stright line, and, taking tho points, P 



Fig. 2. 



Fig. 3. 



and Q (Fig. 3), attempt to draw a circle through Q to touch A D in 

 P. We should tirst draw tho circle, then draw a tangent, A P B, 

 and take a convenient point, Q, upon the circumference of the 

 circle. 



In like manner if, in a deduction, mention is made of a circle in- 

 scribed within, or circumscribed without a triangle, wo shall obtain 

 a far more satisfactory tigure by drawing the circle tirst, and then 

 forming a triangle round it or within it, respectively, than by 

 drawing the triangle tirst. 



These instauce-s suBice to exhibit the necessity of considering tho 

 order of the constructions needed in our 6gurc. There are some 

 considerations to be attended to, also, respecting the sliapes to be 

 given to different figures, that an examination of tho properties 

 they are meant to illustrate may be made as easy to us as 

 possible. 



It is very important that the different parts of a figure should 

 not exhibit apparent relations not really involved in the problem 

 illnstiated. Lines should not seem to be equal, nor to be at right 

 angles to each other when they are not necessarily so. Triangles 

 should not seem to bo isosceles or right-angled when the problem 

 does not involve such relations. It is well to notice that, in general, 

 the most convenient form of triangle for illustrating general pro- 

 perties is that shown in Fig. 'l ; here the angle A is one of about 

 75°, the angle B one of about 60°, and the angle C one of about 45°. 



Fig. 4. 



Fig. 5. 



When a quadrilateral figure is not necessarily either a parallelogram 

 or a trapesium, it is well to construct it of such a figure as A B C D 

 (Fig. 5), in which the four sides are uneiiual, neither pair of oppo- 

 site sides parallel, and the diagonals AC, B D do not make equal 

 angles with any side. It will be noticed, also, that neither diagonal 

 bisects the other. If we had a problem in which the bisections E 

 and P of tho diagonals were concerned, all that would bo necessary, 

 in order that neither diagonal might bisect the other, would bo to 

 draw the diagonals A C and B D jirst, so that their point of inter- 

 section, G, should be well removed from the bisections, E and P ; 

 then join A B, B C, C D, and D A. 



It is sometimes convenient to draw a part of tho figure in darker 

 lines than the rest. We may distinguish in this way, for instance, 

 between the lines or circles belonging to the enunciation and those 

 belonging to the construction. When we are in donbt as to the 

 necessity of any construction, it may be lightly dotted in. In very 

 complex figures, dark, light, broken, and dotted lines may bo con- 

 veniently employed together. 



Always letter every i>oint of the figure which may have to be 

 referred to as you proceed. It is often as well, when a result has 

 been established which seems to promise to be useful towards the 

 solution of a problem, to re-draw the figure, omitting all lines 

 except those which have served to guide yon to this result. But, 

 except in such instances, or where the figure seems obviously 

 nnsuited to your requirements, or has become overcrowded with 

 constructions, it is well to keep to the same figure as long as 

 possible. The habit of repeatedly re-drawing figures interferes 

 with the concentration of the attention, and the steady progress 

 from result to result, which alone avail toward tho solution of 

 difficult problems. 



(To be continued.) 



^ur €\9ti9 Column. 



By Mephisto. 



PLAYED IN THE SECOND UOUND OX JUNE Isr. 

 Irregular Opening. 

 Bl«ck. Wtite. Black. 



Kt to KB3 

 P toQ4 

 Pto K3 

 Bto K2 

 B takes B 

 Castles 

 P to B4 

 Kt to B3 

 P to QKt3 

 B toQ2 

 R to K sq. (d) 

 P takes P 

 V to K4 

 Kt takes P 

 B takes Kt 

 B takes P 

 K to B sq. 

 Q to B3 

 B to Kt5 

 QRtoQsq.(/) 



Ma 

 PtoQ4 

 Kt to KB3 

 B to Kt5 (o) 

 B takes Kt 

 P to K3 

 B to Q3 

 P to B3 

 QKt to Q2 (fc) 

 K to QB sq. 

 B to Kt sq. (c) 

 P to KR4 

 BP takes P 

 P takes P 

 Kt takes Kt 

 Kt to B3 

 B takes P(ch.) 

 R to QKt sq. 

 B to g3 

 B to K2 (e) 

 Castles 



R (Q6) to Q2 

 Q toB7 

 R to Q3 (i) 

 R to y5 

 Q takes RP(t) 

 Q toB5 

 P to QKt4 

 R takes P(ch.) 

 RtoR3 

 R takes R 

 P to Kt5 



Q to B4 

 B toB6 

 B to Kt2 

 QtoB3 (j> 

 QR to R sq. 

 R takes P 

 R to B7 (0 

 K to Kt sq. (m) 

 R to B6 

 Q takes R 

 B toB3 



E takes Kt 

 B to K4 

 Pto Q5 

 R takes P 



B takes B 

 P to Kt3 

 P takes P 

 Q to Kt3 

 Q to Kt3 (o) B to Kt2 \ 



R to Q6 Q to B4 [ 



KR to Q sq. QK to K sq. (;i) 1 

 B to Q5 Q to B7 I 



B to B4 K to R2 I 



P to Kt6 

 P to Kt7 

 K to Kt sq. 

 QtoQ5 

 K to B sq. 

 QtksP(ch.)(, 

 B takes Q 

 P to Kt3 

 P takes B 

 K to Kt2 

 R to QKt sq 

 K to B3 

 RtoKt5(ch 

 K to Kt4 

 K to Kto 

 R to Kt6 

 R takes P(ch 

 R to QKtC 



K to Kt2 



R to KR sq. 



Bto K4 



B to R7 (ch.> 



Q toB3 

 )Q takes Q 



K takes B (o) 



B takes P 



KtoK3 



R to QKt sq. 



K toB4 



K to Kt4 

 ,) K to B3 



K to Kt2 



KtoR2 



K to Kt2 

 .)K toB2 



Resigns. 



NOTES. 



(a) Which must lead to tho exchange of the Bishop against the 

 Knight, thus leaving the opponent with two Bishops at a very early 

 stage of the game. On the other hand, it must be admitted that 

 the second player obtains thereby some attack. 



(b) If 8 .... P takes P, White recovers tlie Pawn with 9. Q to 

 K2, for Black dare not reply P to QKt4, on account of 10. Kt 

 takes P. 



(e) Making everything ship-shape for an immediate attack 

 against the adverse King's quarters. 



(rf ) For both offensive and defensive purposes, for White threatens 

 now to force the opening of the King's file, and he clears a retreat 

 for his King. 



(c) Tempting, but disastrous would he 19 ... . Kt to Kt5, 20. 

 R takes P (ch.), K to B sq. (best), 21. B takes Q., Kt to R7 (ch.), 

 22. K to K2, Kt takes Q, 23. R takes B, and White remains with a 

 piece ahead. 



(/) 20. B to B4, B to Q3, 21. QB6 (ch.), K to K2 would be in 

 Black's favour. 



(ij) Threatening, R to Q6, or R takes P. 



(h) If 27 B to K4, White wins with 28. R takes P (ch.), 



P takes K, 29. Q takes P (oh.), B to Kt2, 30. B to Q5 (ch.), and 

 Black must sacrifice his Queen, for if 30 .... K to R sq., White 

 would mate in two moves. 



(i) White conducts the attack with much vigour up to this 



