Aug. 10, 1883.] 



♦ KNOWLEDGE ♦ 



85 



Consider the geometry of the matter. It is simple enough : 

 — Let A, B, C, lie, Fig. 5 represent the same points 

 as the same letters in Fig. 3. Draw I h' c' square to < A 

 producing B h and c to meet I c' in U and c' respectively. 

 Also join t h, and produce to meet C c in c". Then we 

 know that the angle c tc' \s equal to the angle in the alter- 

 nate segment of the large circle of which the is a part, 



The loser of the wager asserts that the signal c (not the 

 central point c) was on the line h h\ though admitting that 

 b was above c. Thus if the centre of c was on li h' the 

 centre of h was above h It ; and if the centre of b was on 

 h Id the centre of c was below /( h'. Either result would 

 suffice to show that the explanation based on the straight- 

 ness of the line t b c was certainly erroneous. Both b and c 



and therefore to twice the angle b tc on the arc b c (which 

 is half of t b c). Hence, the lines B h', C c' being approxi- 

 mately parallel, we see that 



hli =kb =6 ft. approximately ; 



cc"=c'z" ^j'k=V2ii. approximately; 

 and (■ c' = 2c c" = 24 ft. approximately. 



Now when the telescope at t is truly levelled it is so 

 directed that its optical axis is in the straight line t U c', — 

 so that as the disc c lies in the direction t c, while b lies in 

 the direction t h c", we ought if the centre of the field were 

 exactly marked to see 6 below that centre by a certain 

 small distance and c twice as far below that centre. 

 Regarding these as small dots, we should see what is shown 

 in Fig. 6 where a marks the intersection of the cross lines, b 

 the centre of disc b, and c the centre of the remoter disc c. 



All this is nearly as obvious as what we found in the 

 other test, only not quite so simple. If our telescope were 

 perfectly levelled, the horizontal cross hair lih' absolutely 

 central, b removed, and c were found to be exactly on li h', 

 then — why then — Well, if the sky were to fall we should 

 catch larks. The earth-tlatteners should hardly expect 

 astronomers even then to admit that the earth is Hat, simply 

 because there are a hundred other absolutely overwhelming 

 proofs that the earth is a globe. It would be diliicult to 

 explain such an anomaly as would thus be indicated. 

 Either it would appear that the water surface ABC 

 unlike the ocean surface, or any other liquid surface, was 

 absolutely plane, perhaps through some abnormal local 

 attractions — or else it would appear that some almormal 

 refractive action of the atmosphere had raised c up to the 

 true horizon of t. Not one jot would or could the faith of 

 astronomers in the rotundity of the earth, proved a hundred 

 ways, waver before such evidence. But, of course, no such I 

 evidence has ever been obtained. 



were not on h h', as they must both have been had the 

 straight line from t to c been in the real horizon. 



We know then already that either the observation on 

 the strength of which c was supposed to be on h h' was but 

 rough, or else the instrumental adjustments were imperfect, 

 the telescope not truly levelled, or h K not truly across the 

 centre of the field. 



But let us see how great the distance a c in Fig. 6 should 

 have been, supposing the adjustments all perfect. Suppose 

 the magnifying power of the telescope to have been 12, 

 that being about the utmost magnifying power likely to be 

 used in a levelling experiment of the kind — for long tele- 

 scopes are never used with exact adjustments to bring the 

 optical axis level. Then we have e (i (Fig. 5) = 24 feet 

 But atmospheric refraction practically reduces cc' by 

 about a fourth leaving 18 ft. only. Thus the angle cie' 

 as observed with the naked eye {i.e., as distinguished 

 from the geometrical angle ctd) is the angle subtended by 

 18 ft. at a distance of six miles or of eighteen times 1700 ft 

 This is the angle sulitended by 1 ft at a distance of 1760 ft, 

 or by 1 inch at a distance of 17 GO inches, or by the 

 hundredth of an inch at a distance of nearly 1 S inches. As 

 the telescopic magnifying power increases this angle twelve- 

 fold, we have the angle actually observed with the telescope 

 equal to that subtended by the hundredth of an inch at a 

 distance of 1 i in. It is exactly one-tenth of the angle 

 B O A in Fig. 7, and is fairly represented by the dark line 

 B pointed at end 0, 



Fig. 7. 



Is it likely that an unpractised observer (all the 

 observers in the Bedford experiment were unpractised) 

 would detect the depression of the distant signal c below 

 h h' by this small angle — the signal not being a point but 

 a disc — even if the cross hair h It had been precisely ad- 

 justed to correspond with the real horizon ] But it is ex- 

 ceedingly unlikely that /( It was properly adjusted. If not, 

 and there are fi-w nicer adjustments in practical survey- 

 ing, it is certain that none of the observers employed could 

 have even detected, far less have corrected, the error. 



This particular test, then, under the conditions existing, 

 was practically worthless, — the appearance of c below 6 

 was a hundred times better as a test, and indeed of itself 

 proved (if the loser's statement about c being on the cross 

 hairs can be trusted) that the levelling of the telescope and 

 the adjustment of the horizontal cross hair were inexact 



