62 



♦ KNOAA/LEDGE ♦ 



[July 27, 18b3. 



LETTERS RECEIVED, AND SHORT ANSWERS. 

 Jos. Clayton. Yes ; the planetary aspects were evil, in the 

 astrological sense, when those particular catastrophes happened. 

 But it would be hard for all the catastrophes to happen when the 

 planets were more smilingly arranged. If you know anything of 

 astrology, yon know that about one-third of the time the aspects 

 are evil ; for another third they are good ; for the rest of the time 

 they are both good and evil. Consequently, for two-thirds of the 

 time you can indicate evil planetary aspects. Thus, in the long 

 run, two-thirds of all the catastrophes will occur when the planets, 

 according to astrologers, threaten evil. Leare the other catas- 

 trophes out of consideration, and — there you are : astrology 

 is shown to be a true science. Only as a mere detail, so 

 to speak, it may be noticed that if the good aspects were 

 regarded as the erU and vice versi, astrology would be just as 

 satisfactorily established. — Sexex. (I) Unfortunately, pictures 

 of spots could only appear after the spots had gone out of view. 

 Then what use would they be ? They would be meant, I suppose, 

 for telescopists, and the telescopist can see them for himself. 

 (2) It often happens that the penumbra of a spot looks widest 

 towards the nearer edge of the sun. The circumstance is considered 

 and illustrated in my book on the sun. It does not militate against 

 the theory of spots being cavities, but shows that the evidence 

 which has proved them to be cavities is not always presented. — 

 J. B. Ask yotir plain question ; you omit this trifling detail in 

 yotir last letter. Dictionary meanings of such words as " level," 

 "horizontal," &c., are, nine times out of ten, wrong. — W. Lalakde. 

 Nothing personally. — S. Ixgham, F. Q L. See Mr. Theobald's 

 advertisements. — Ax Ixquiker wishes to learn the name and author 

 of any eshatistive work on the history, manners, character, 

 &c., of the Chinese. — Tresilliax, IIoke Light. It would 

 afford a very inconvenient precedent to do what you 

 wish, or I would, gladly. — Ax Ixterested Reader of 

 " Kxowledge " would like to see a protest here against 

 the wholesale destruction of butterflies for stocking coUections ; 

 also to learn if the leaves of the garden rhubarb are poisonous, and 

 if water in which potatoes have been boiled is unwholesome. An 

 "I. R. of K." mentions also an amusing instance of ignorance in a 

 charwoman, who, having accidentally run the nib of a rusty pen 

 into her thumb, covered the nil mth grease and put it carefully 

 away on a shelf, to prevent her tlntinh from festering. — A Lover of 

 THE Stars. If you were on a world travelling round Alpha 

 Centauri, and could see both the Sun and Neptune, you would find 

 the distance between them equal to about 32 seconds of arc, or, in 

 other words, they would appear as a rather wide double star, not 

 like Gamma Yu'ginis. — Jas. SwiXEtjRXE. Thanks. — Hugh Dalziel. 

 Rather more stiitable to The Field than to these columns.— 

 Eev. W. Hardmax, LL.D. Considering the natiue of the sub- 

 ject, may we not admit on both sides that some problems in 

 psychology, which in former times were explained as due to 

 spiritual agencies external to our personality, may be rationally 

 explained nozc as dne to processes simply natural? If the "some" 

 of several be "all," while the "some" of other some are few, 

 the contest over thai difference should be waged elsewhere. Mr. 

 Clodd, you may be sure, meant no scorn for accepted beliefs about 

 external agencies ; nor does he definitely reject them for all cases. 

 But you must admit that many cases of supposed demoniac influence 

 have had an internal, not an external origin. — Leoxakd Rodway. 

 The indestructibility of energy is really against your views. If the 

 earth is constantly radiating heat into space, she cannot retain 

 it unchanged in amount within her own globe. So long as the crust 

 is drawing on the internal supplies, it is natural that there should 

 be little apparent change of temperature. — W. G. Mortimer. Do 

 not know the address of the Ladies' Dress Reform Association. — J. 

 Homes. Such an article might be of interest. Must see it before I 

 can say. — C.^. Meetkeeke. Thanks, but paper not suitable; shall 

 it be returned ? — J. Lawsox. In early numbers of Kxcwiedge a 

 great number of magic squares were given. See Mr. Sutton's square 

 of 9, shortly. — J. M. Gladstox"e. — The figure simply gives the lengths 

 corresponding to vibrations for various notes. — JI. Griffiths. The 

 question is scarcely an optical one. People do not really estimate the 

 size of the sun when they say it looks as large as a cheese-plate, a six- 

 pence, a round table, or so forth. Its apparent size is the same for 

 all, and not a matter for estimation, except for those who wish to 

 learn precisely how many seconds of arc the disc's diameter really 

 subtends. When anyone says the sun looks as large as, say, a 

 halfpenny, he really asserts that it seems to him about nine feet 

 away, for at that distance a one inch globe (a halfpenny is exactly 

 one inch in diameter) would look as large as the sun. — W. T. 

 SouTHWABD. I can only say that I differ from you. If science said 

 dogmatically there is no such thing as a disembodied spirit, there is 

 no future life, and so forth, your objections would be sound 

 enough. But that science should not be permitted to dis- 

 cuss the ascertained origin of many of the beliefs men have 



formed and formulated respecting matters spiritual, seems to 

 me altogether absurd. If you are right, I am not a recent 

 offender, or officially so only (as editor, to wit) against your 

 doctrine. In the Essay on Ghosts and Goblins, in my " Border- 

 land of Science," I consider certain cases of supposed apparitions, 

 and show how they were to be explained. This, of course, is not 

 the same as asserting that ghosts and goblins never can appear or 

 have appeared ; yet you would say that I had no business even 

 go so far, and that you consider science is " degraded by attempts 

 to explain," &c. It is absolutely certain that many races have 

 based doctrines about disembodied spirits, a future life, heaven, 

 hell, lie, on mistaken notions about shadows, dreams, optical 

 phenomena, the sky, subterranean fires, and so forth. To show 

 this is not to assert that there is no future life, no disembodied 

 spirit, no spiritual Heaven or spiritual Hades ; and undoubtedly 

 science has the fullest possible right to consider the origin, progress, 

 and decay of mistaken interpretations of shadows, di-eams, the 

 arched sky -ivith its celestial phenomena, sulphurous volcanic 

 regions, uiid so weiter. Whether ideas about disembodied spirits, 

 heaven, hell, &c., are in themselves right or "wrong, makes no 

 difference in this respect. If they are wrong, there is nothing to 

 be said for them ; if they are right, they cannot be too soon 

 dissociated from erroneous explanations. 



<^ur iilatt)£inatical Column* 



GEOMETRICAL PROBLEMS. 



By Richard A. Psoctob. 



PART IX. 



WE may remark in passing that there is no absolute necessity 

 for restricting ourselves in aU respects to Euclid's manner. 

 Take as an instance his treatment of the famous pons asinorum. 

 In dealing with this, as with all other propositions, he confines him- 

 self entirely to constructions which he has shown to be possible. 

 Therefore, the following proof of the first part of the proposition 

 would not be in his manner, though it would be difficult to find 

 any flaw in the reasoning. 



There must be some line which divides BAG (Fig. 17) into two 

 equal angles.* Let A E represent this line. Then in the triangles 

 B A E, C A E, B A is equal to A C {hi/p.) ; A E is common ; and the 

 angle B A E is equal to the angle C"a E. Therefore (by I., 4) the 

 angle A B E is equal to the angle A C E. 



Again, the following proof of both parts of the proposition is 

 complete, though not in Euclid's manner: — 



Conceive that the figure foi-med by the Unes F K, F L, and G H, 

 Fig. 18, is one that would coincide exactly with the figure formed 

 by the lines A D, A E, and B C ; F K coinciding with A D (Fig. 18), 

 FL withAE, andGH with BC. Now conceive the figureFKL to be 

 turned face downwards, and so applied to the figure A D E that F K 

 may coincide with A E ; then since the angle G F H is equal to the angle 

 CAB, F L coincides with A D. Also since A B, A C are equal to 

 each other, and also to F G, F H, the points G and H coincide with 

 the points C and B, and G H with C B. Thus the angle ABC 

 coincides with and is equal to the angle F H G. But by our sup- 

 position the angle A C B is equal to the angle F H G. Therefore 

 the angle A B C is equal to the angle A C B. In like manner DBG 

 coincides with G H L ;t but, by our supposition, B C E is equal to 

 G H L. Therefore D B C is equal to B CK. 



* The assumption here is precisely the same in character as that 

 made in defining a riglit angle. 



t Here we assume as axiomatic the property which Simpson has 

 attempted to prove in the corollary he has added to I., 12. He 

 forgot, apparently, that Euclid had already (in prop. 4 and else- 

 where) assumed "the property as self-evident, and that prop. 12 

 itself cannot be solved on any other assumption. 



