12 



♦ KNOW^LEDGE ♦ 



[November 1, 1887. 



angles, lying now in the same direction as at first, with 

 changed ends. Hence the three angles, which are together 

 equal to the angle through which the rod has pivoted in all, 

 amount together to two right angles. 



With regard to axiom 12, and the treatment of the whole 

 subject of parallelism and non-parallelism in Euclid, the 

 trouble seems to arise from two causes : First, the use of a 

 negative definition of parallel lines, instead of a positive 

 definition from which the negative property could be 

 deduced ; and secondly, from the wish to prove certain 

 matters really axiomatic — for instance, that opposite or 

 vertical angles are equal. The axiomatic ideas underlying 

 axiom 12, which .as presented in Euclid (with its converse, 

 Euc. I. 16, as a jyroposltion and in company with pro- 

 positiuns relating to parallels) is not axiomatic at all, may 

 be thus dealt with : — 



If the straight lines ab, de, crossing in c, be supposed 

 shifted so that the point c falls at c. in ab, ab remaining 

 unchanged in position, and de falling into the position fgh, 

 FH can nowhere cut de. For any argument showing that 

 GF produced towards f would cut cd produced towards d. 



would equally show that ce produced towards e would cut 

 GH produced towards h. And if both intersections occurred, 

 the two straight lines de and fh produced far enough would 

 enclose a space. 



This, however, is not axiomatic, unless we take it for 

 granted that the angle fgb, which (we see from the way in 

 which it was obtained) is the same as the angle dcb, is 

 equal to cgii, which (for a like reason) is the same as the 

 angle ace ; for the obviousness of the argument depends 

 on the fact that we have the same arrangement precisely 

 when we look at the figure as it stands that we have 

 when we reverse it. 



Now it is not one whit more obvious that two straight 

 lines will not enclose a space than it is that opposite angles 

 are equal, or that two straight lines ca7i in any way be so 

 drawn as never to meet. 



After it is seen that fgh obtained by moving acd to 

 position agf will not cut dce, it ought to be equally obvious 

 that any straight line q/ througli a on the side of gf 

 towards d must cut CD, no matter how small the angle /gf 

 may be. For the point _/' Ls some distance from gf, and by 

 doubling of this distance will be doubled (this may be 

 proved by superposition), and the doubled distance will in 

 turn be doubled in the same way (by quadrupling g/'), and 

 so on as long as we please : hence we must at length by 

 successive doublings obtain a distance less than the dis- 

 tance of any point in ED from fh ; in other words, c./ pro- 

 duced far enough must pass to the side of ef remote from 

 FH — that is, G/'will eventually cut ED. 



Or we may put the matter in the form of a general 

 axiom, — Through a given point (g), outside a given straight 

 line (dce), only one parallel can be drawn to that line. 



These properties, whether presented as axioms or reasoned 

 out, serve all purposes. But in reality, if such matters are 



to be reasoned out, then the statement that two straight 

 lines cannot enclose a space should be reasoned out too ; 

 and I do not envy any one vvho makes the attempt. In 

 fact, it leads directly to the discussion of non-Euclidean 

 geometry, a part — and a useless part — of Dream INIathe- 

 matics. 



Of the following three propositions, winch, dealt with 

 as a subject for reasoning, would be the more difficult 1 — 



First, Through a given poitit there cannot be drawn a 

 straight line in the same plane as another straight line, 

 irhich, being jn'odiioed sufficiently far both ways, shall meet 

 the latter straight line on both sides of the given point. 



Secondly, Through a given point there can be drawn a 

 straight line in the same plane as another straight line, 

 which, however far it be produced cither icay, shall not meet 

 the latter straight line on either side of the given point. 



Thirdly. Through a given point there can only be drawn 

 one straight line in the same plane as another straight line, 

 which, however far it be produced either >vay, shall not meet 

 the latter straight line on cither side of the given jyoint. 



A FIVE-FOLD COMET. 



HE figure illustrating this article is taken 

 from L'Astronornie, and represents the re- 

 markable Southern Comet of January last, 

 as drawn on successive days by Mr. Finlay, 

 of Cape Town. 



The comet was first seen by a farmer and 

 a fisherman of Blauwberg, near Cape Town, 

 on the night of January 18-19. The same night it was 

 seen at the Cordoba Observatory by M. Thome. On the 

 next night Mr. Todd discovered it indejiendently at the 

 Adelaide Observatory, and watched it till the 27th. On 

 the 22nd Mr. Finlay detected the comet, and was able to 

 watch it till the 29th. At Rio de Janeiro M. Cruls 

 observed it from the 23rd to the 2.5th ; and at Windsor, 

 New South Wales, Mr. Tebbutt observed the comet on the 

 2sth and 30th. Moonlight interfered with further observa- 

 tions. 



The comet's appearance was remarkable. Its tail, long 

 and straight, extended over an arc of 30 degrees, but there 

 was no appreciable condensation which could be called the 

 comet's head. The long train of light, described as nearly 

 equal in brightness to the Magellanic clouds, seemed to be 

 simply cut off at that end where in most comets a nucleus 

 and coma are shown. 



This comet has helped to throw light on one of the most 

 perplexing of all the ])uzzles which those most perplexing 

 of all the heavenly bodies, comets, have presented to 

 astronomers. 



In the year 1668^a comet was seen in the southern skies 

 which attracted very little notice at the time, and would 

 probably have been little thought of since had not attention 

 been directed to it by the appearance and behaviour of 

 certain comets seen during the last half-century. Visible 

 for about three weeks, and discovered after it had already 

 passed the point of its nearest approach to the sun, the 

 comet of 1068 was not ob.served so satisfactorily that its 

 orbit could be precisely determined. In fact, two entirely 

 different oibits would satisfy the observations fairly, though 

 one only could be regarded as satisfying them well. 



This orbit, however, was so remarkable that astronomers 

 were led to prefer the other, less satisfacforj- though it was 

 in explaining the observed motions of the comet. For the 

 orbit which best explained the comet's movements carried 

 the comet so close to the sun as actually to graze his visible 

 surface. 



