Aug. 17. IX'^:?. 



* i^JNOWLEDGE 



111 



think they fancy they remember they saw, but have carefully 

 noted all observed phenomena. It is hardly necessary to say that 

 in all the multitudinous volumes of Arctic and Antarctic travels 

 extautj nu such absurdities as your old sea captains have told you are 

 mentioned ; but the moon, the guide of the sailor in remote seas, 

 behaves witliin the Ai'ctic circles as she behaves elsewhere ; she is 

 new, half full, gibbous, and full at the times given by the " Nautical 

 Almanac" for the whole world ; and in fact if she has anything at 

 all to say to your sea captains' stories, informs you simply that 

 either they have blundered, or (which is far more iirobable) they 

 have amused themselves by telling old "forecastle yarns" for 

 your edification. — H. Askew. (1.) The geometrical problems willjjro- 

 bably be published. They were submitted, almost in their present 

 form to Messrs. Longmans, in 1868, who submitted them to Pro- 

 fessor Goodeve and Mr. Cock, mathematical lecturers at King's 

 College, who studied them so carefully as to find out that the 

 problems might be more simply solved — as if it were the object of the 

 papers to show how those particular problems might be most briefly 

 solved. (2.) We have unfortunately no space for obituary notices in 

 Knowledge, except in the case of a few very eminent men such as 

 Darwin, J. W. Draper, H. Draper, &c., whose influence on the 

 advance of scientific thought has been very marked. (3.) The 

 Harton Coalmine experiments gave 6'565as the mean density of the 

 earth. Mr. Sketchley probably means that this result raised the 

 average of the best estimates to S'^iSO. — F. M. Di-plock. Pra}' 

 excuse the misspelling of name ; *' I " was not responsible for the 

 mistake, but the fault was as you say with " U " ; only U'say this 

 and I say it with different significance. Thanks for kind inquiries 

 as to effects of railway accident. Have felt the shaking a good 

 deal ; but hope effects will shortly pass off. Though sitting 

 with my back to the engine, only my knees show any marks ! 

 both of them bruised by the counter shock, — in other words I 

 " cushioned " off the seat behind me on to my knees on the seat in 

 front of me, with enough force to mark them both and to bark 

 one. Rather an unusual way of bruising the knees ! — G. S. See 

 solution in our columns. — E. Const. May. Write down the first 

 three odd inimbers in order, each repeated twice, — thus 113355, 

 and divide the latter half of the number thus formed, 355, by the 

 former half, 113. The result is the ratio of the circumference to 

 the diameter of a circle, correct to the fiirst six decimal places : for 

 the quotient is 3'1415y29, &c., which to the sixth decimal 

 place would be written 3'141593 as would the true ratio, which 

 is 3'141592G5, &c. — W. E. Drinkwater. The sun and moon 

 both rule the tides, theii' respecting tide - raising influences 

 being as the numbers two and five. When they combine their 

 influence, the tidal wave bears to the lunar tidal wave about the 

 ratio of seven to five ; while, when they oppose each others' influence, 

 the tidal wave is less than the lunar tidal wave (i.e., the wave which 

 there would be if the sun excited no effect) as three to five. 

 Thus the solar influence produces no tidal wave separate from the 

 lunar wave, but is, as it were, merged in the lunar influence. The 

 tidal period follows the moon, but the variation of the tidal wave in 

 height follows the sun. — T. M. Your theory is presented with 

 assertion only. " The fact should be familiar to all," you 

 say, and go on to give as a fact what is not only not familiar 

 to all, but not a fact at all. As to alluvial and diluvial matter, 

 astronomers find plenty of evidence of that in the moon ; 

 but perhaps this, being a fact, is not familiar to you. — 

 T. C. Fear many I'eaders would not see that you are jesting in 

 finding " the number of the beast "in the name of the great and 

 good man you mention — preposterous though the notion is. Some 

 really do fancy they see the cloven hoof there, — not being quite 

 able to see above his instep. — 11. F. Kekr. You are right in think- 

 ing that I utterly decline to answer your question. I simply cannot 

 tell you what 1 do not know myself. If Knowledge cannot be 

 honestly sold unless the editor tells you what ho understands by 

 what neither he nor any man who has ever lived can under.stand, 

 then Knowledge must continue to be (in your opinion) sold dis- 

 honestly. — J. C. S. That Daily Telegraph ai-ticle about the mid- 

 night sun, earth's inclination, &c., belongs to a type which I had 

 thought played out. It is nearly all nonsense, of course. If there 

 were such a change as supposed, the present forms of vegetation in 

 the temperate and Arctic regions would no longer be so well suited 

 as they are at present. As for sudden change in earth's centre of 

 gravity, tliat is nonsense, too. Adhemar's theory was wild enough 

 in all conscience, without adding that absurdity. — W. Aston. I 

 really do not know how you should set about to get a " cirtificate" 

 to "an able " you to teach and pass pupils. — Geo. IIowakd. I'ardon 

 me, though you may have taken in KiNowLEDGE from the first, you 

 are not an original subscriber. A copy of the index was sent gratis 

 to every subscriber. It most assuredly is not the proprietor's dutv 

 to reprint tho index at heavy cost, because one or two failed to 

 provide themselves with it when it was issued. I differ from you 

 altogether as regards tho Whist Column. It has been discontinued 



dui-ing chess tourneys, but it will be resumed soon. Whist has 

 as good a right to be described as a scientific game ai chess or 

 draughts. 



Sm iJiattjematical Column, 



GEOMETEICAL PEOBLEMS. 

 By Eichakd A. Proctor. 

 PART XII. 



LET us next try the following problem : — 

 Ex. 18. — Determine the locus of the middle points of all the 

 chords of a circle which pass through a fixed point. 



The fixed point may be either within or without the circle. 

 In nearly aU cases of this sort it is well to begin with a point within 

 the circle, trusting to the result thus obtained to guide us in the 

 case of a point without the circle. 



Let P (Fig. 27) be a point within the circle A B C D. We are to 

 draw chords through P, and to bisect them. Draw, first, the 

 diameter A P E C through P. Its bisection, E, is the centre of the 

 circle. This is one point of the required locus. Draw next the 

 chord B P D at right angles to A C. Then the point P is itself 

 the bisection of D B (Euc. III., 3). Therefore P is a point 

 on the required locus. Next draw a chord F P H of through 

 P, and bisect in H. Then H, a point on the locus, is clearly 

 not in the straight line joining P E, so that the locus is 

 not a straight line. It is, therefore, probably a circle. Now 

 we see at once that for every point we get above A C there 

 must be a corresponding point below A C. We see, then, the pro- 

 bability that the required locus is a circle of which P E is the 

 diameter. But even if the student failed to see this at once, he 

 would readily detect it when he had draivn several more chords 

 through P (above and below P C), and bisected them. We de- 

 scribe, therefore, a circle C H P, of which we assume P E to be the 

 diameter, and we look for a proof that a chord drawn as F P H G 

 would be bisected in H where it meets the circle thus drawn. It 

 will clearly be well to join E H. When this is done, one of two well- 

 known properties can hardly fail to occur to our mi nd. We might 

 either remember that the angle in a semicircle being a right angle. 



Fig. 27. 



Fig. 28. 



E H will be at right angles to F G, if P H E really is a semicircle, 

 or we might remember that the line from the centre of a circle to 

 tlie bisection of any chord is at right angles to the cl'ord. so that 

 the angle E H P is a right angle independently of aiv • ..nsideration 

 of the assumed circle P H E. Of course, if we thous;iit of the first 

 property we should be led immediately to the second, and vice versd. 

 The two properties are, in fact, interdependent; and we see at 

 once that their interdependence involves the solution of our 

 problem. 



We now write out the solution in the following form : — 



Let A B C D be the given circle, P tho given point. 



First, let P lie within the circle. Draw any diord, F P H G, and 

 bisect F G in H. Find E the centre of the circle A B C D, and join 

 E II. Then E H is at right angles to F G (Euc. III., 3) ; therefore 

 H is a point on the circle of which P E is a diameter. (Euc. III., 

 31). But F G is any chord through P. Therefore the bisections of 

 all such chords lie on tho oii-cle E H P. Also it is clear that every 

 point on this circle bisects some chord through P ; therefore, this 

 circle is the locus required. 



Next, let P lie without the circle (Fig. 28). Then the proof is the 

 same* up to the words, " therefore the bisections of all chords 



* It is important to notice that in such a case as tho above, by 

 putting tho same letters at corresponding points in both jigures^ th& 

 proof of one case may be made to apply to the other, either witliout 

 change, or with such obvious changes as tho student can have no 

 difficulty in making. 



