60 



KNOAVLEDGE ♦ 



[Jan. 26, 1883. 



roul brncfit to bo obtained from it, nnd no effect at all, excejit what 

 WU8 easily to be accounted for by the temperament of the patient. 

 I tliink the explanation of cases in which it is stated that painless 

 dental operations have been jierfonned without aniesthetics is, that 

 in anticipation the pain is always exaggerated, and when some 

 mysterious agency is employed, it, of course, gets the credit of 

 causing the nnoxpectodly small amount actually endured. 



N. Stevenson. 



A LOGICAL PUZZLE. 



[697] — Porliaps the high numbers you chose for the illustration 

 of this snbjcci; may account in some measure for the difficulty 

 which some of your readers still experience in regard to it. 



The problem is : — 



(1) lor every Z there is an X which is not Y. 



(2) Some Y's are Z's. 



.'. Some X's are not Z's. 

 Suppose there are 10 Z's. Then, by the first premiss, there will 

 be ten or more not -Y X's. Now, a thing cannot be Y' and not -Y' X 

 at the same time ; therefore, if, by the second premiss, even one 'L 

 is a Y, there are only nine loft, which by any possibility can be 

 not -Y X's; but there are ten or more of the latter; therefore, at 

 least one not -Y X cannot be a Z. R. E. 



9iis(U)frs! to Corrcsponlients. 



H. J. BoHTER. No ; Jupiter. Venus not only visible, bnt splendid, 

 in the eastern skies before sunrise. — Tycho. My advice would be to 

 look out for advertisements of a second-hand telescope. But wliat- 

 over you might get from either of the makers yon mention, would 

 be good value for your money, and would always command a good 

 price if you wanted to sell again. — E. C. C. (1) Molecular move- 

 ments exciting ethereal undulations, I suppose. (2) For a time 

 must, in response to other readers, have a little of the more 

 elementary mathematics. — F. Rogers. Sony that your notice 

 arrived so late. On Jan. 10, when it came. Knowledge was 

 already in the press. — A Y'oung Man. Y'our dream was note- 

 worthy. " Only a new-comer ! " That is what science tells each of 

 us. — Peccavi. The idea has been suggested before, but we have 

 little spare space. Thanks for pleasant note. — T. R. Allinsox. Do 

 not wish to discuss that as part of the corset question. The 

 fooleries of fashion are outside " An Observer's " ai'gument, and 

 not worthy of attack, per .sc, from our side. — Algol. When I wrote 

 that it had been but recently noticed that y Cassiopeia; is variable. 

 (2) The colour of Arcturus is vaxiously estimated. I have seen it 

 named among the red stars ; also among the orange ; to me it 

 seems orange yellow ; others think it simply yellow ; you find it 

 sulphur yellow. (3) In some of those ridiculous books on Astro- 

 logy, by Zadkiel (Lt. Morrison), Raphael (Thomas Blodkins), Uriel 

 (Jacob Plugwinkle), and others — to be picked up on bookstalls — 

 you will find the stars of evil influence. — Aurora. We do not 

 usually give advice ; but where the symptoms are so very clear 

 we " break through a rule to oblige an old friend " (as you seem to 

 be from your reference to far back numbers) . For your complaint, 

 the pills known as Cockles' Antibilious ones are invaluable. — W. 

 Grandy. Have no lectm-c engagements at Southampton. Oddly 

 enough, youi- kindly letter came immediately after the most atra- 

 bilious one we have read for a long time. Nine-tenths of our 

 letters, now, however, are of the pleasantest tone. — A. C. MoRASso. 

 Probably the volumes will be half-yearly. As you say, they will 

 be much more convenient. — Canadensis. 1 think difference 

 of temperature, and, consequently, of moisture, in those parts 

 of the ceiling, account for that peculiarity. — R. H. Simpson. 

 Cannot well look out gossip notes in Knowledge for 

 readers. — Lieutenant. The question was entered into rather 

 fully in early numbers. Ice-boats undoubtedly do travel in many 

 cases much faster than the wind which drives them. Y'ou reason 

 out the matter quite correctly. But I think the relative velocity is 

 never (luite so great as you suggest (five or six times the wind's). — 

 R. L. Nichols. Our article touched on the principle laid down by 

 Baron Ilnddleston, not on the evidence.— J. M. Crombie. We 

 hope to use the article shortly —E. P. Sells. Many thanks. 

 Auroras not probably duo to comet, even indirectly; though pos- 

 sibly it may have brushed up the solar sm'roundings in such sort 

 that the earth's magnetic condition was affected. — J. A. S. On re- 

 e.vamining the question I see th.at other solar eclipses mav possibly 

 bo visible in England before 20!)0,— but none satisfactorily. —Upsala. 

 (1) Eccentricity and ellii>ticity are different matters; it was the 

 latter I wanted, and you give me the former. (2) Cannot a line bo 

 drawn from a station to Venus and produced to the sun's surface 

 ■without your proclaiming that the produced line is an illusion ? If 



one, why not two or more ? What occasion for Venus to bo n 

 lens !•" — J. Pescuile. "Miss Edwards an American!" What (.i' 

 have suggested such an idea to you ? Not to know one so widil;. 

 known as a novelist and an Egyptologist argues yourself unknown. 

 — Z. Perhaps I go as far as you do ; but one mast not go too far 

 ahead of the main body. We are in the "forlorn hope" as it 

 is. — E. P. Sells. Thanks. We could hardly have onr " Face <.l 

 the Sky" shown five or six weeks ahead. Subscribers here wuiiM 

 find it inconvenient; and they necessarily form the great body r.i 

 our constituents. — A. Lorr, M.A. Thanks for the information 

 that watches do not "consume" thcii- glasses. " It will go nea i 

 to be thought so, shortly." It is also true, whatever tnidi 

 announcements may affirm, that lead is never " lively," brandy 

 " dull," nor ])otatoe8 " flat," and so forth. Yet, thongli I 

 have noticed these things, it has not occurred to mo tn 

 write to the editors of business papers explaining as much. 

 If I were capable of such an absurdity, I might perhap' 

 be also capable of perpetrating it in the particularly offensivr 

 manner you adopt for yours. " In Heaven's name," you say. 

 "buy a sixpenny dictionary and keep it beside you!" I woul' 

 commend to your attention the cheapness of certain little books 

 on manners. — F. H. I cannot think your way of treating the 

 Corset question as scientific in substance as in form. You 

 assume, contrary to all evidence, that the tendency, as civilisa- 

 tion advances, is towards increase of ornament, and of thosr 

 artificial ways of treating the body of which corset-wearing i.s 

 but one. Compare the fashions of the middle ages with those 

 of two centuries ago. and these in turn with the fashions of our 

 own time, and you will find there has been a steady progress 

 towards simpler and more sensible attire ; though, unfortunately, 

 there is still amj^le room for improvement, especially in feminine 

 costumes. In some of Froissart's pictures you see all the ladies 

 as absurdly pinched about the waist as nowadays only the silliest 

 are. I fancy there is more absurdly tight lacing now- amoug 

 servant-girls out for a holiday than among ladies. 



@ur iflattjtmatical Column. 



TO DETERMINE THE ATTRACTION OF A SPHERICAL 

 SHELL ON AN EXTERNAL PARTICLE. 



Let a P b Q ie a spherical shell of radius r, of thin uniform section 

 r, and of uniform density p ; its centre, E an external particle ,- 

 OE = d. It is required to determine the attraction of the she!, 

 P Q a 0)1 the sphere E. 



Draw aO!)E; tangents E P, EQ; PGQ cutting A E in G. 

 Draw any chord, A G B, through G ; and join O P, B E, O B, O A, 

 and A E. Then, since P E is a right angle, 



0G.GE = PG' = 6G.GA 

 Therefore, a circle will pass through the points A B E. Hence, 

 since chord O A of this circle = chord OB, ^0EB=/0EA; and 



BE : ea::bg : ga. 



Now, if we suppose a conical surface of minute vertical angle 

 2 ^ to have its vertex at G ; it is obvious that the elliptical sections 

 of the sphere at B and A being inclined at equal angles (equal to 

 the angle B A) to the axis, have areas proportioned to the squares 

 of G B, GA. Hence, since their attractions on pai'ticle at E aro 

 directly proportional to the mass, that is, to the squares of GB, 

 G A, and inversely proportional to squares of B E, E A, that is, of 

 G H, G A, these attractions are equal, and their resultant acts in 

 direction of E 0, the bisector of the angle B E A. 



Now, noting that triangle G B E is obviously similar to triangle 



