Aug. 31, 1883.] 



♦ KNOWLEDGE • 



139 



PRETTY PEOOFS OF THE EARTH'S 

 ROTUNDITY. 



CHIEFLY FOR THE SEASIDE. 

 By Richard A. Proctor. 



{Continued from page 117.) 



BUT now, returninn; to the mirror proof of the earth's 

 rotundity, porae readers of these paper.«, comparing 

 together Fig. <S siid Fig. 9 may be led to ask why, ia 

 looking into a mirror as in Fig. 9, we do not recognise the 

 water surface c (/ as curved like the surface A B. Ought 

 it not they may say to curve downwards on either side of 

 the vertical centre line across the sweetly pretty face in 

 Fig. 9, just as it does on one side of the point A 1 A little 

 consideration will show that this is precisely the same 

 difEcultj' which is described in the Paradox Column at 

 p. 77 by the anti-Copernican carpenter Mr. Hardy. He 

 thinks or rather is certain that if the curvature of the earth 

 can be recognised in the direction of the line of sight it 

 ought also to be recognisable at right angles to that 

 direction. If a ship twenty miles away is hull down, 

 owing to the earth's rotundity, a range of twenty miles of 

 sea horizon ought to show well-marked convexity. If our 

 pretty observer at p. 101 can really see the sea-horizon 

 depressed owing to the rotundity of the earth as illustrated 

 in Fig. 8, he ought to see the horizon curve downwards to 

 right and left of him. 



It may be worth while to mention at the same time 

 another difficulty. In Fig. 8, a^ in Fig. 3 p. 69, and 

 Fig. 5 p. 85 we have an upright on the right which is not 

 really upright, but parallel to the true upright at A ; 

 would not the argument be altered, and especially the part 

 of the proof which is geometrically given, were the uprights 

 c in Fig. 3 and Fig. 5 and B b in Fig. 8, made really 

 square to the surface on which they are supposed to stand ? 



Taking the latter diiliculty first, let it be noted that C c 

 in Fig. 3 may be set vertical without in the slightest degree 

 affecting the argument. In the argument at p. 85, C c 

 is described only as approximately parallel to A a ; while 

 B 6 in the argument at p. 101 is spoken of as "appreciably 

 parallel." In reality the upright B 6 in Fig. 8, the case 

 where the departure from parallelism is greatest, is so nearly 

 parallel to A a, that though the eye might just be able to 

 recognise the departure from parallelism, the real difference 

 between A // and 'i b would be far too small to be noticed 

 on the scale of the figure. The curvature of the surface 

 A»iB is in reality monstrously exaggerated — though ne- 

 cessarily ; and A a which is supposed to represent a height 

 of only 200 ft. really represents a height of many miles. 

 There had to be exaggeration somewhere, or // // and b' B 

 would have been invisible ; putting Bb at right angles to 

 the surface at B would not have made matters much worse, 

 but it would have suggested the additional error that b is 

 much farther from a than b' is from A. The argument 

 would have lieen in no way afi'ected. 



Let us, however, inquire what the true proportions of 

 the lines in Fig. 8 should have been, or the true lengths if 

 any given length is assigned. Suppose a b to represent 

 17 miles, being in actual length 38 inches. Then since 

 B b should represent 100 feet, and 100 feet are contained 

 about 22.5 times in 17 miles, the length of lib in the figure 

 should be the 225th part of 3-8 inches, or considerably less 

 than the fiftieth of an inch. If the student will draw a 

 line as A //, .3 8 inches long, take A a, b' b, and // B, each less 

 than the hundredth of an inch, and fill in a curved surface 

 A B touching A A' at A and « B at B, he will get a good 



ea at once of the slightness of the real curvature of A B 



and of the utter insignificance of any difference between 

 the uprightness of the tiny line B b and the parallelism of 

 that line to A a. 



Or we may take B b the same in length as in Fig. 8 to 

 represent 400 feet, and inquire what should be the length 

 of n b to represent 17 miles. The actual length of B 6 is 

 about seven-tenths of an inch. ISIultiply this by 225 and 

 we get 157^ inches or more than 13 feet. Imagine B b and 

 A a unaltered in length but set close on 4i yards apart, A b' 

 and « B drawn, and AmB carried with a circular sweep 

 to touch these lines at A and B respectively. It would 

 require a radius of aViout 1,000 yards or considerably more 

 than half-a-mile This is the kind of curvature which some 

 think we ought to recognise in looking (as in Fig. 9) into 

 a mirror half a-foot, perhaps, square. As for the displace- 

 ment of b, owing to B b being parallel to A a instead of per- 

 pendicular to the surface at b, this displacement even on 

 the enormous scale just considered (in which Bi represents 

 400 feet, and the radius of A m B is more than half-a-mile 

 long) corresponds only to setting b to the left of its true 

 place by less than the 200th part of the length B b, or by 

 about l-320th part of an inch. 



Take we now the other difficulty, though it has indeed 

 been already disposed of by the reasoning just run 

 through. 



Why should not the sea-horizon, whether viewed directly 

 or by reflection in a mirror, seem curved 1 



Suppose an observer whose eye is 200 ft. above the .sea- 

 level looks at the long horizontal roof-ridge of a house, 

 beyond which lies a sea horizon, and that he brings the 

 middle of the ridge just below the sea-horizon exactly in 

 front of him — ought he not as liis eye ranges to right and 

 left along the ridge, to lose the sea-horizon through its 

 curving down below the ridge 1 Or, if he brought the ends of 

 the ridge exactly level with the sea-horizon, ought not the 

 sea-line to stand visibly above the middle of the ridge's 



Theoretically it ought and it does : practically the ques- 

 tion is one of degree, and our inquiry must be, how much 

 does it curve] 



Suppose the ridge to be 50 ft. long, and its middle point 

 25 feet from the eye, so that in sweeping along the ridge 

 the eye ranges over a right angle. Suppose also a fixed 

 point set near the eye to guide it, for otherwise the observa- 

 tion would be altogether inexact. Every line of sight must 

 be taken athwart this fixed point, 25 feet from the centre 

 of the perfectly horizontal ridge line, to different parts of 

 this line ; and what we want to find is how much the lines 

 of sight to either end of the ridge line pass above a line of 

 sight to the sea-horizon there, when a line of sight to the 

 middle of the ridge line just touches the sea-horizon. 



Let a, Fig. 13, be the fixed point athwart which the lines 

 of sight are taken, E B F the roof ridge, 50 feet long, B 

 its middle point ; a B square to E F ; and a B=E B=B F 

 = 25 ft. ' et a vertical plane through E B F cut the true 

 horizonta plane through a in ebf, and let E' B' F' repre- 

 sent the sea-horizon as supposed to be seen on this plane ; 

 e E E', i B, and /F F' being vertical lines. We want to 

 find the length of E E' and F F'. 



Now obviously B ri b in Fig. 13 represents the same 

 angle as 15 a b in Fig 8 as dealt with at p. 101. For, pro- 

 duced far enough — really to some 17^ miles — aB would 

 meet the sea surface and lie the a B of Fig. 8 ; a b of Fig. 

 1 3 would then be the a b of Fig. 8 ; and 6 B would there- 

 fore be 400 ft. The proportions of the triangle B a b would 

 be precisely the same in both cases (because we are dealing 

 with a point a in each case 200 ft. above the sea-lerel). 

 In each case B ?> (as shown at p. 101) is l-228th part of 

 a b. But ab in Fig. 13 represents 25 ft. Therefore B 6 

 represents 25 ft. -f- 228, or about 1 \ in. 



