Sept. 5, 1884.] 



• KNOWLEDGE ♦ 



195 



rig. 3. 



former one. But this he does not find to be the case. 



When he has got to C (Fig. 3), 



making B 0, equal to A B, he finds 



that the pole of the heavens has 



Tisen by exactly as much as it was 



raised by his former journey. Now, 



if we draw a line from C to P, we 



get an angle C P B, which a very 



brief consideration will show to be 



larger than the angle A P B — that 



is, we get P C N larger than our observer actually finds it, 



so that, as seen from 0, the pole has another direction, as 



■C P' P", giving the angle C P" B, equal to the angle 



AP B. 



But which of the three points P, P', and P", is the real 

 polel They cannot all three be. Yet, if our traveller 

 takes the observations made at A and B, he gets the point 

 P by purely geometrical reasoning ; if he takes the obser- 

 vations made at B aud C, he gets the point P" with equal 

 reason ; and if he takes the observations made at A and C, 

 he gets the point P'. 



He is forced, then, by this second line of argument, also, 

 to abandon the notion that he has been travelling in [a 

 straight line. 



He has now to consider what his observations require 

 from him. He has first, to get the axis, B P, in Fig. 1, 

 -coincident in direction with A P. 



He draws A N (Fig. 4) to repre- 

 sent the line on which he set out 

 from A, and he sets up A P at an 

 angle of 51^° to A N. Then he 

 draws at a convenient distance P' B 

 parallel to A P, and, of course, 

 meeting AN at an angle of 51.^''. 

 Now, if he can draw an arc A B, 

 touching A N at A, and having at ^ig. 4. 



B a direction (B N') making an angle of 57' with B P' 

 then he knows that A B represents very satisfactoriU' the 

 -curved path he must have followed when journeying from 

 A to B. Then again he draws a third parallel, C P", at 

 the same distance from B P' that B P' is from A P ; and 

 he makes the arc B C (just as he made the arc A B) so 

 that C N" may be inclined about 62i° to C P". This con- 

 struction makes ABC the arc of a circle, because the 

 circle is the only curve whose tangents (as A N, B N', 

 C N") change equally in direction for points separated (as 

 A, B, and C) by equal arcs. 



The observer is guided, therefore, to the belief that the 



section of the earth's surface along which he is travelling 



forms a circle. And he can roughly 



tell what the size of that circle is. 



For he has found that A (Fig. 5) 



being about 7 GO miles, the tangent 



C N" has fallen away, in direction, 



about 1 1° from A N. But if AC L 



represent the circle of which A C 



is a part, he knows that lines O A 



and O C from the centre include 



the same angle between them as 



that by which C N" is inclined to 



AN.* Hence AC is an arc of about 11°. And therefore 



the complete circumference of this circle A C L is about 



360 „ . 



^pp X 760 miles, or in round numbers, some 25,000 miles. 



* This will perhaps be obvious at once to my readers ; but if 

 not, they will easily see its truth by drawing K C M parallel to 

 A N. Then, since C N" is a right angle, the angle II C X" is the 

 complement of K C 0. But K O C is also the complement of K C 0. 

 Therefore, K C is equal to M C S". 



Fig. 5. 



This gives to the circle a diameter of about 7,900 

 miles. 



In order, however, to test the justice of this view, our 

 observer first travels farther north. He finds the slow 

 change of elevation of the pole continuing quite uniformly 

 as he continues journejing in that direction. After going 

 as far north as he can, at which time he finds that the pole 

 of the heavens is nearly overhead, he is satisfied that in 

 this direction, at any rate, the circular figure of the path 

 he has followed continues unchanged. 



He next returns to A, and commences a series of similar 

 journeys towards the south. These confirm in every respect 

 his theory that the section ho is traversing is either actually 

 circular, or does not differ sensibly (so far as his instru- 

 mental means are concerned) from the circular form. For 

 every successive distance he travels there appears to be a 

 proportionate depression of the pole. At length the pole 

 sinks to the very horizoir, and the relations now presented 

 by the heavens are so interesting that we shall have to 

 consider them attentively. 



But first, we must exhibit the general results of our 

 observer's voyages up to this point. 



We see from Fig. G, that even if our traveller's first 

 observations were doubtful on account of the comparative 

 smallness of the changes noticed, yet 

 the great change he now notices — the 

 apparent coincidence of the pole with 

 the horizon — is altogether inexplic- 

 able on the assumption that he has 

 been traveUing in a straight line. For 

 though undoubtedly by going very far indeed in direction 

 A E he would bring the pole very low down, yet in the 

 first place he would have to travel very much farther 

 than he has actually done, in order that E P might seem 

 horizontal, and in the second, it 

 is absolutely impossible that the 

 heavens should seem to revolve 

 about A P when he was at A, 

 and about a nearly horizontal line 

 E P, when he is at E. On the 

 other hand, we see from Fig. 7, 

 how his uniform progression round 

 the circular arc C B A E would 

 lead to a uniform depression of the polar axis, until at E 

 that axis appeared to lie (as Ep) in the horizon-plane of 

 the observer at E. 



And then two things served very strikingly to confirm 

 the views of the traveller. We see from Fig. 6 that on 

 the assumption of the earth being plane, the pole of the 

 heavens P has a distance from points on E C which is com- 

 parable with the distances passed over by the observer. 

 This being so, the stars which lay round the axis A P 

 ought not only to change appreciably in brightnes.^ as the 

 observer varies his position along E C, but they ought to 

 change in relative position. The polar constellations, for 

 example, could not possibly present the same aspect when 

 viewed from A as when viewtd from E. But the observer 

 can detect not the slightest change in the aspect of any one 

 of the constellations he had become familiar with when at 

 A, either in the brightness of the component stars or in 

 their relative position. Now, Fig. 7 accounts perfectly for 

 this. We see all the lines representing the polar axis 

 parallel in position ; in other words, the pole is removed to 

 a distance indefinitely great compared with the distances 

 A B, B C, E A, ic. This makes the sphere of the fixed 

 stars very large indeed compared with the arc E C ; and 

 while this has been the direct result of geometrical con- 

 siderations of another sort, it explains at once and simply, 

 the striking fact that, let the traveller journey as he may 



