78 



♦ KNOAVLEDGE ♦ 



[July 25, 1884. 



read) from my sister's writing-table, almost the first thing that 

 met my eye was a reprint of this very poem. It would have been 

 the last place I should have thought of looking in. 



Ardennes. 



PERSPECTIVE. 



[1347] — Some of your corre.=!pondents seem to have got into a 

 puzzle over their perspective. They seem to find a ditiiculty in 

 understanding how parallel lines perpendicular to the line of sight 

 — that is, lying in planes parallel to the plane of the picture, can be 

 correctly presented to the eye of the spectator by parallel lines on 

 the plane of the picture. 



Your correspondent " C. E. Bell " seems to forget that, although 

 the upper and lower edges of the cube may be shown by parallel 

 lines in the picture, yet the angle subtended with the eye by the 

 perpendicular edges of the cube varies with the distance to right 

 or left, as the case may be, while tlie edges are from the centre of 

 the picture or line of sight. 



Thus, if P P in the annexed diagram bo the plane of the picture 

 shown in plan, S, the spectator's eye, and SA, the line of sight, 

 it is obviously seen that a given dimension M on the plane of the 

 picture at A will subtend a larger angle with the eye than the 

 same dimension M at B. 



Let " R. Jones," " C. E. Bell," and others simply examine the 

 photograph of any rectangular object taken with the plate of the 

 camera parallel to one face of the object, and they will find that all 

 the horizontal lines in the object which arc in planes parallel to 

 that of the plate will appear as parallel lines in the jiietvire. 



In connection with this subject I woitld remark that artists are 

 in the habit of showing the sea-line as a straight horizontal line. 

 Now this is not strictly correct, and especially when the sea is 

 viewed from a great height. 



The spectator's eye being a point outside a sphere, may be re- 

 gsurded as the apex of a flat cone, the base of which is a circle 

 whose radius is (approximately), the distance of the spectator from 

 the horizon. 



If a straight-edge be held up to the eye when looking towards 

 the sea-line from a lofty cliff on the sea-shore, the curvature of the 

 horizon becomes plainly visible. I commend this experiment to 

 crazy Zetetics, though, doubtless, they can explain this as easily 

 as every other fact that tells against them. E. W. Young. 



[What is " a flat cone " ?— Ed.] 



SHIPS' LIGHTS. 



[1348] — Apropos of your propositions respecting ships' lights in 

 " Sent to the Bottom," in a former number of Knowledge, the 

 reduction of the number of lights being most desirable, may I 

 venture to offer for your consideration the following alteration, 

 namely, that tim lights on each side only be adopted, in lieu of three 

 side and two end lights. Thus, let the foremost light be white on 

 each side, and visible from right ahead to right astern, and placed 

 15 (?) feet from the stcmmost, and 5 (?) feet above that light — the 

 sternmost light to be particoloured — white showing ahead to well 

 abeam, but say red astern to nearly abeatn ; steamships carrying 

 a fifth masthead light as at present. A sketch may make my sug- 

 gestion clearer : — 



The opening apart of the lights from a perpendicular line, as 

 when seen from " right ahead," to their greatest distance apart, as 

 when seen from " abeam," and their closing again, the lower be- 

 coming red, to their original position, as when seen from " right 

 astern," would indicate the required information as to course of 

 observed ship. The reijulation interval between the lights being 



invariably adopted, would become by habit easily and surely recog- 

 nisable as representing by their varied proximity the direction and 

 distance of the observed ship. 



I leave this suggestion with you, as you may in a few moments 

 run over an imaginary series of positions as depicted in your excel- 

 lent magazine, substituting the two lights alone in lien of the five 

 there shown, bearing in mind, at the same time, that ships at sea 

 are not, as a rule, on an even keel or even beam, so to put it. A 

 ship, say 30' — nay, 40° — out of the perpendicular, as often hap- 

 pens, and at a time when lights are most requisite and important, 

 would materially alter the relative length of the perpendicular side 

 of your triangle, and that of each of the other two. I recognise 

 that that upright side forms the gauge by which the apparent 

 lengths of the other two are measured, and so the course of the 

 vessel estimated. What, therefore, are we to do to obtain that 

 information when our gauge or foot-rule is liable to considerable 

 variation from this heeling over of the ship at sea? Let it also be 

 remembered that perpendicular space available for effectively show- 

 ing lights from the side of a ship is very limited. What would our 

 lee triangle be like with our bottom light in danger of entire sub- 

 mersion. The side lights now are carried almost invariably several 

 feet above the top rail of bulwarks for this very reason, and if 

 placed much higher would be eclipsed by the sails. All this, and 

 more, I am sure, will present itself to your mind, and I need en- 

 large no further. My best thanks are due to you for many most 

 instructive and enjoyable hours spent in company with yon in 

 Knowledge from Xo. 1. — I am, sir, &c., Cuas. Rice. 



TESTS OF DIVISIBILITY. 



[1349]— In letter 1334 (p. 39) I stated that 999 ... to (m-l) 

 figures can be divided by n without remainder whenever n is prime 

 to 10. I find it is not so. 



When n is a prime number it will divide 999 .... to (n — 1) 

 figures. 



When n is composed of uneqiial factors a, b, c, &c., the number 

 of 9's required is (a — 1) (i> — 1) (c — 1) . . . . , or some factor of 

 this. 



When n contains eqttal factors (= say, o''b'c'' •••-)! the number 

 is 



a''-'(a-l)i'-'(6-l)c'-'(c-l) 



or a factor of this. 



Thus the number of 9's required can never exceed (n — 1). 



W. 



LETTERS RECEIVED AND SHORT ANSWERS. 



FitzGeeen. The word "aperture," in the reply to which you 

 refer, had reference to a refractor, but2i inches was a misprint for 

 2i inches. This would have an approximate focal length of 30 

 inches, and would cost between £7 and £8. An instrument fur- 

 nished with a 2i-inch object-glass would be of about 3 feet focus. 

 Either of these would bear a power of 200 on close double stars ; 

 the latter even a somewhat higher one. No reflector of less tliau 

 3i inches aperture is of the slightest use. — Castor. The cause to 

 which you refer, viz., the production in the offspring in an aggra- 

 vated form of diseases common to their consanguineous parents, is 

 the generally-accepted reason given against marriage between 

 cousins. On the other hand, as you say, two people standing in 

 that relation, each healthy and of exceptional intellectual power, 

 might fairly be expected to beget a high type of children. Still, 

 every agriculturist knows the evil of " breeding in and in " ; and 

 what is so well established in the case of the bfute creation may 

 reasonably be held to apply to man in his animal relations. — 

 A. CD. C. suggests, in connection with a statement on p. 23 as to 

 the origin of a rite held as sacred by Christians, " that Mr. 

 Clodd should show that a similar one was observed in any 

 religious system previous to the introduction of Christianity." He 

 also finds fault with Mr. C.'s assertion that " the eastward position 

 is the undoubted relic of worship of the rising sun," with which 1 

 am considerably more surprised, inasmuch as I imagined that tl 15 

 was admitted as an indisputable fact by all save a very few bigoted 

 and ignorant people indeed. — W. A. Leonard. Thanks, but 1 have 

 already declined a series of articles on the same subject from a 

 thoroughly competent member of our own staff. — -J. Mubb.ay. The 

 apology is sufficient. • — Sigma. Your reasoning is utterly falla- 

 cious. You deal with infinity as a quantity which can form a 

 member of an equation ! Were this possible, you might 

 multiphj it; but you can no more talk of five times in- 

 finity than you can of nothing-. An "infinite circle" is 



