Jan. 5, 1883.] 



KNOWLEDGE 



15 



^iv iHatt)fmatiraI Column. 



NOTES ON EUCLID. 

 Bv Richard A. Proctok. 



I HAVE often wondered that among the v.orious attempts to 

 correct the obvioas defects — for edncational purposes— of tho 

 use of Euclid as a text-book of mathematics, nothing should have 

 yet been done to remodel the book itself. Various writers hiivo 

 published bo<iks of geometry, eacli fondly hoping that his book will 

 not only displace Euclid, but dispose of all rivals. The result has 

 proved rather confusing. A boy who has been at a public school 

 where one of these books has been used, goes, perhaps, afterwards 

 to a private tutor who prefers another text-book of geometry; 

 tlience, perhaps, to a London college, where yet another book is 

 employed ; and finally to one of the Universities, where he finds 

 Euclid still holding the place of honour. 



Now, it Euclid simplified could be put into boys' hands at school, 

 and all other text-books diligently eschewed, there would be a 

 common system at all schools, and no trouble when the old- 

 fashioned Euclid was taken up, at whatever stage of mathematical 

 progress. 



There can be no doubt Euclid perplexes many boys, and disgusts 

 not a few. For my own part, though I was introduced to Euclid 

 in tho absurdest of ways, I loved him from tho beginning. I had 

 been counted rather a dullard at Geometrical Exercises, which I dis- 

 liked, because there were Rules without Reasoning. Just as the 

 foolish arithmetics of those days told us in hard words what 

 to do, but never showed us why, so tho Geometrical books 

 told us to rule this line and describe that circle, in order to 

 bisect lines, set up perpendiculars, and so forth, with no proof 

 that the methods were sound. Besides, while mapping (a mo.st 

 instructive exercise) I had intuitively invented such methods 

 for myself ; so that rules had not even the charm of novelty. At 

 this stage of my progress— or want of it — a preposterous under- 

 master pitchforked me into a higher class where Euclid was rcail, 

 and where, as it chanced, tho IGth Proposition of the First Hook 

 was in hand. It was a new thing to me to find reasoning about 

 matters geometrical. Theoretically, the folly of putting nie at the 

 IGth Proposition jir.^i ought to have made Euclid hateful to me ; 

 but, as a matter of fact, the case proved otherwise : I loved him 

 from tho first. T read, alone, the Definitions, .\xioms, and pre- 

 ceding Props. ; then went along with Props, ahead ; till, before 

 very long, 1 was in the Spider's Web of the last Prop, but one of 

 Book XII. Yet I was by no means a sound, only an eager, student 

 of Geometry ; for I remember devising a new construction for that 

 most delightful of all Props., the 10th of Book I V., which, though 

 much shorter and easier than the original, laboured under the slight 

 defect of being incorrect. 



Still, I think my case was exceptional. Most boys do not take 

 kindly to Euclid, and certainly there is much in his outer appear- 

 ance which is not inviting. In particular, the method of first 

 giving the abstract proposition, and then describing a particular 

 case, tests somewhat painfully the young student's power of atten- 

 tion. It is so much harder to make out what the enunciation 

 means than it would be if each part were explained, as in tho 

 opening words of the proposition, that we cannot wonder if bnys 

 are bewildered and wearied. To the more advanced this is perliapa 

 the real charm of Euclid — to note how each enunciation has the 

 qualities of a good definition in precisely indicating the abstract 

 itlea without any reference to a special ease. But Euclid was not 

 writing for boys. 



Again, there is a charm in the skill mth which Euclid, having 

 adopted a certain method, gets over the difficulties involved in 

 applying that method to i>articular cases. The famous I'ons 

 Asinorum is a case in point. Euclid's plan will not allow him to 

 use the bisector of the angle BAG, because ho has not yet shown 

 how that bisector can be drawn. Nor can he allow himself to sup- 

 pose his initial figure, repeated line- for line, and then applied, .after 

 being turned over, to tho original figure, after the manner 

 already employed in Proposition IV., because he has not yet shown 

 how the " copy " is to be made. Either method would have given 

 him a very simple proof, and as it is certain that there mu.^t be 

 a line bisecting the angle BAG, and again that another figure 

 precisely like that already drawn is conceivable (in the same sense 

 that a straight line or a circle is conceivable from its definition), he 

 was, logically, free to employ either plan. But he h.ad assigned 

 himself certain limits, and he makes out his proof within those 

 limits very ingeniously and prettily — t)hough confusingly to many 

 boys. 



{To he continued.) 



<i^ur WAffiit Column* 



By " FivK OP Clubs." 



r. 



B. 



Hearts — A, 5. 

 Spades— A, Q, Kn, G, 1 

 Diamonds — 9, 8, 7. 

 Clubs— 10, 5, 3. 



A. 

 Jliart.i—q, 9, 7. 

 Spades— K, 10. 

 Diamonds— A, 10, 4, 3 

 Clubs— 8, 6, 4, 2. 



.llT.Koiitor) ^(likl.) BOub-EJ.) illSulC.) 



//,u)-f.,— Kn, 6, 2. 

 Spades— 9, 8. 

 Diamonds — Q, Kn, 2. 

 Clubs— K, Q, Kn, 9, 7. 



H,'(iit.-<—K, 10,8, 4, 3. 

 Spades— 7, 5, 3, 2. 

 Diamonds — K, 6, 5. 

 Clubs— A. 



O 



9 <? 

 <7 <? 



o ^o o o o 

 \ o 



lo^ol I o I lo o 







» 4-1 ys^ \* * 





* * ♦ * t'^t ♦ 



THROWING AWAY WINNING 

 CARD OF PARTNER'S SUIT. 



The Play. 



„ ) A, B,=0 



Score:— j y,Z.=0 



Notes by Fivk of Clubs. 



NoTB. — The card underlined wina thi> 

 trick, and card below leads next round. 



1. A le.ids from his best suit. 



2. Having live trumps, Z leads 

 tlie peimltimatc. 



4. Tho third round is fortunate 

 for Y Z. Y discarding a Club, and 

 A having led Diamonds, Z knows 

 that his partner's suit must be 

 Spades. 



G. A knows this also, and there- 

 fore leads Clubs. It would have 

 been better, ns it turned out, if he- 

 had kept to his own suit. 



7. Z, having four cards of his 

 partner's suit, leads the lowest. Y 

 finesses the Knave, of course. 



8. Z, noting the fall of the cards, 

 perceives that his Seven will be 

 in his partner's way. For neither 

 A nor B have any more (£ cer- 

 tainly not holding the Queen, or 

 ho would not have let l"s Knave 

 make at trick 7). Thus, if Y 

 leads Queen at trick 9, and Z 

 throws his Five, he will have ti> 

 take the fourth trick in Spades, 

 and a trick in Diamonds will go to 

 the enemy. 



9. Z, therefore, throws his Seven 

 to Y's lead of the Queen. But 

 K should not have led the Queen. 

 Ho can count the Spades as well 

 as Z, and knowing tho second best 

 and .a small t'ne was with 2, he 

 should have led the Four, to make 

 his partner's play as simple as 

 possible. Nei-er leave to partner a 

 jt'iint of strategy ivhich you can 

 attend to yourself. 



10. 11, 12, 13. The rest of the 

 game plays itself. 



VORTICOSK. — A leads King; B, 

 his partner, holding Queen and 

 small ones, knows A has led from 

 Ace, King ; if B shows this know- 

 ledge by extending his hand to 

 take up the trick before Z, fourth 

 hand, has played, is there any 

 penalty ? None ; but if iS is in 

 the habit of doing such things, you 

 should avoid playing -mth him. — 

 Five of Clubs. 



