452 



• KNOWLEDGE . 



quantities of rapoiir ; while that the abundant Jurassic fauna could 

 have existed by the h'ght of a diffuse nebula will meet with scant 

 credence indeed among zoologists. — Ed.] 



[Nov. 28, 1884. 



A PROPERTY OF CERTAIN NUMBERS. 



[1523] — Let 1! be ary number prime to 10; that is, any number 

 whose unit figure is 1, 3, 7, or 9. 



Work out the decimal equivalent of _. 



7) 



Let A represent the first figure, n the remainder ; B the second 

 figure, b the remainder ; and so on. Finally, we shall get a figure 

 (say N) giving 1 as remainder. Clear'y the decimal will recur 

 after N. 



Thus, - = .ABCD N, 



71 



And the successive remainders are 



a, hj c, d, 1. 



Now, " = .LiCD NA 



n 



(since gives B, and h remainder, &c) 



n 



And the remainders are h, c, il 1, a. 



Similarly, - = .CD NAB, 



n 



with remainders c, il 1, «, ;.. 



And so on, using each of the remainders in turn as multiplier. 

 This is the whole mystery of the so-called figure puzzles 1398 and 

 1507, and of as many more as there are numbers prime to 10. 



On referring to each of these '" puzzles," it will be seen that in 

 each case the multipliers are the swccidiive remainders in working 



out the decimal equivalent of — or . — 

 7 13 

 I have a8S]imed that all numbers prime to 10 give pure recurring 

 decimals. Readers may like to work out the proof of this for 

 themselves. It is not difficult. W. 



LETTERS RECEIVED AND SHORT ANSWERS. 



K. K. Thanks for your offer, but we are so over-crowded that 

 papers by members of our regular staff are now standing in type 

 because no room can be found for them. — H. F. sends an account 

 of the finding by a gentleman at Rnrutoa, an island in the South 

 Pacific, of a plate belonging to a Royal Mail steamer, no vessel of 

 that company's fleet ever having sailed within thousands of miles 

 of the place. Prior to this the brother of the finder, who was purser 

 on board of the Mexican R.M.S., went down on that vessel; and 

 the suggestion (for no proof is attempted) is that the plate may 

 have belonged to the lost steamer. He also forwards an account 

 of a Mrs. France, the wife of a betting-man in Button-street, 

 Liverpool, who, the night before the Derby was run in 1867, 

 dreamed that Hermit won it in a snowstorm — which, as a 

 matter of fact, he did under those meteorological conditions. 

 The " queer query " might more a]ipropriately go to the Family 

 Herald^ or some similar paper. I cannot insert mere arithmetical 

 puzzles. — E. W. Bakton. I am glad that our Reviews and In- 

 ventors' Column please you so. Lectures, unfortunately, irre- 

 vocably at an end. — Romeike and Curtice. If you conld see, for 

 one brief five minutes, the torrent of matter showered upon the 

 head of the unfortunate Editor of this paper, you would hardly 

 expect him to subscribe for auy addition to it in the shape of news- 

 paper cuttings.— F. W. RuDLER. Received with thanks. — H. 

 PiLLEV. If you will look at a globe, you will see that hour circles 

 all meet at the pole. Hence, the farther you go north or 

 south of the equator, the slower any star or heavenly body 

 seems to move in its path. Hence the reduction. I really 

 cannot define cosine here. Buy Hunter's " Elements of Plane 

 Trigonometry," in Gleig's School Series, published by Longmans. 

 O P is got by simply joining O and P ; directions being given 

 for finding C O and C P. — Starch asks whether either of the years 

 1479 or 2992 indicate some great historical era ? because, if so, the 

 square in letter 1-190 may owe its tremendous powers to the fact. 

 He omits to say whether he refers to the years B.C., Anno Mundi, 

 or what. — W. C. Gurlev. Many thanks. It is both curious and 

 interesting. The penumbra certainly arises from reflection from 

 the back of the plate. — J. Murr.iy. My dear sir, I know that it 

 must be my own stupidity, I feel it painfully ; but I cannot for the 

 life of mo make out how (even supposing, as you say, " a sudden 

 change of temperature were to set in, and making everything vice 

 verse ") our earth could ever become " as gentle at one end as it is 



severe at the other." I am equally unable to grasp the idea of tem- 

 perature and the law of gravitation being "on one scale." While as 

 for drawing "a right angle line through the jierpendicular scale of 

 knowledge " (I) I can't do it — I can't, indeed. — H. Euwarus. Re- 

 ceived. — R. Knight. Light, ^jcr .sc, exercises no rt^pulsive influence. 

 H eat is the operative factor in causing the vanes to move in the radio- 

 meter. — Samuel Kix.vs. The Conductor of this journal is not in 

 England. — W'. Tour general proof will be utilised. In your hyper- 

 critical temper yon quite ignore the fact that a concrete arith- 

 metical example addresses and interests ten people for every 

 one who cares about the formula whence it is derived. — 

 J. Mekcek. I believe that your method (which, by the way, seems 

 to save little or no work) was described in a back volume 

 cf the Eyiglish Mechanic. — Hexky Peter.s. The game you 

 describe has been a popular one in drawing-rooms any time 

 during the last two or three years. Undoubtedly the 

 operators, quite unconsciously and innocently, give indications 

 of their wishes to the subject. Presumably you have read the 

 published accounts of the performances of Mr. Stuart Cumberland, 

 Mr. Edwyns, and Mr. Irving Bishop. — Sam Beook. The thing to 

 which you refer was mentioned as a quasi-fraud. Even if I knew 

 its composition I would not communicate it to anybody ; bnt I do 

 not, and do not want to do. See concluding paragraph (in capital 

 letters) of those which head the Correspondence Column. — Nigei. 

 DoBLE. Yonr dream experiences are of a very common character. — 

 Mark H. Judge. I entirely agree with you that the sooner the 

 British Museum is opened duringa portion, at all events, of Sunday, 

 the better. — Henry George. Politics, equally with theology, arc 

 excluded from these columns. — James Gillespie. I will not write. 

 The gentleman who says that Ursa Major was "right over head" in 

 Australia in June, lies — under some terrible mistake. — Joseph 

 HuDLEY. Received. — Anonymous. Thanks for kindly sending 

 The Christian Life. — W. T. BUBN Callander. I have received 

 your " Morven," but, my goodness mel what does it all mean? 

 — Anonymous. The question of Stores r. Retail Traders 

 utterly foreign to our purpose. — W. A. says, apropos of letter 1503, 

 that his son writes from the southern hemisphere that he has 

 visited places which he had seen years ago in dreams. 

 The coincidence in the case of his own dream is hardly 

 sufficiently striking to deserve record. — John Goeham. A 

 paper descriptive of your remarkably ingenious and efficient 

 pupil-photometer will appear next week, containing the result 

 of personal experiments with it. — H. A. L. S. Forgive me, I 

 really h.ave no space for sums whose solution may be found in every 

 shilling elementary treatise on algebra ever published. — An Anony- 

 mous CoRBFSPONDE.VT Sends me some newspaper cuttings comment- 

 ing on the fact that, while the English "Fiue Art Society" have 

 been refused permission to photograph the pictures in the National 

 G.illery, a German firm, called Brann & Co., have been permitted 

 to do so. This affords but another illustration of the pure and 

 lofty patriotism of Ralph Rackstraw, A.B. of H.M.S. Pinafore, 

 who, " in spite of all temptations to belong to other nations, 

 remains an Englishman." — Enquirer. See p. 235. — John Her- 

 bert asks Mr. T. Foster to inform him (in connection with 

 "The Mystery of Edwin Drood"), first, whether the corre- 

 spondence between Dickens and Miss Hogarth and Mr. Forster 

 has been published or not ? Secondly, Where the engravings were 

 published which were reproduced on p. 401 't Thirdly, What 

 became of Drood's projected visit to Egypt ? And fourthly, Why 

 did the old Opium Woman follow Jasper down to Rochester -^ Mr. 

 Foster is at this moment travelling abroad, but I am pretty sure 

 that K.NowLEncE is posted to him from the office, so that Mr. 

 Herbert's queries will probably reach his eye sooner or later. The 

 pictures appeared on the cover of the monthly numbers as issued. 

 — J. A. KoESON. Tour first method is neither "legitimate nor 

 reliable. Try another example with different figures (say an obtuse- 

 angled triangle), and see where your formula will land you. There 

 is no more simple geometrical method than your second. If you 

 know the angles at the base of your triangle, it is needless to tell 

 yon that the side adjacent to either of such angles x its sine = 

 perpendicular. — Benj. W. Austin. The conductor of this journal 

 has been compelled to make a rigid rule to refuse his autograph : 

 applications for it being too numerous for him to comply with. — 

 Buxtoxa. It is, I think, very doubtful whether the papers by our 

 contributor " Five of Clubs" will appear in a collected form before 

 next year. — Engineer. It is a purely medical question, on which I 

 am incompetent to form an opinion. — Dr. Groth. Your angeronly 

 tends to confirm my determination. I did not seek out your theory 

 to attack or assail it gratuitously. You, as I understood, wished 

 for my unbiassed opinion of it ; and, preoccupied a.i I was, I 

 actually read every word of yonr pamphlet before giving that 

 opinion. Of course, it may be a case of Athanasius contra mundum. 

 You may be right and all the rest of the world wrong, but then 1 

 am only one of the rest of the world. 



