Sept. 28, 1883.] 



♦ KNOWLEDGE ♦ 



205 



the Theory of Numbers very remarkable instances of pro- 

 jjositions observed to hold good for very long series of 

 numbers, and which are nevertheless untrue. 

 ,. I pass in review certain mathematical theories. In 

 arithmetic and algebra, or say in analysis, the numbers or 

 magnitudes which we represent by symbols are, in the first 

 instance, ordinary (that is, positive) numbers or magni- 

 tudes. We have also in analysis and in analytical geometry 

 negative magnitudes ; there has been in regard to these 

 plenty of philosophical discussion, and I might refer to 

 Kant's paper, " Ueber die negativen Grossen in die Welt- 

 weisheit" (1763) ; but the notion of a negative magnitude 

 has become quite a familiar one, and has extended itself 

 into common phraseology. I may remark that it is used in 

 a very refined manner in bookkeeping by double entry. 



But it is far otherwise with the notion which is really 

 the fundamental one (and I cannot too strongly emphasize 

 the assertion) underlying and pervading the whole of 

 modern analysis and geometry — that of imaginary magni- 

 tude in analysis and of imaginary space (or space as a 

 locvs in quo of imaginary points and figures) in geometry. 

 I use in each case the word "imaginary " as including real. 

 This has not been, so far as I am aware, a subject of philo- 

 sophical discussion or inquiry. As regards the older meta- 

 physical writers this would be quite accounted for by 

 saying that they knew nothing, and were not bound to 

 know anything, about it ; but at present, and considering 

 the prominent position which the notion occupies, say even 

 that the conclusion were that the notion belongs to mere 

 technical mathematics, or has reference to nonentities in 

 regard to which no science is possible, still it seems to nie 

 that (as a subject of phOosophical discussion) the notion 

 ■ought not to be thus ignored ; it should at least be shown 

 that there is a right to ignore it. 



Although in logical order I should perhaps now speak of 

 the notion just referred to, it will be convenient to speak 

 first of some other quasi-geometrical notions ; those of 

 more-thanthreedimensional space, and of non-Euclidian 

 two- and three-dimensional space, and also of the generalised 

 notion of distance. It is in connection with these that 

 Eiemann considered that our notion of space is founded on 

 experience, or rather that it is only by experience that we 

 know that our space is Euclidian space. It is well known 

 that Euclid's twelfth axiom, even in Playfair's form of it, 

 has been considered as needing demonstration ; and that 

 Lobatschewsky constructed a perfectly consistent theory, 

 wherein this axiom was assumed not to hold good, or say a 

 system of non-Euclidian plane geometry. There is a like 

 system of non-Euclidian solid geometry. 



My own view is that Euclid's twelfth axiom in Playfair's 

 form of it does not need demonstration ; that it is part of 

 our notion of sp.ace, of the physical space of our experience 

 — the space, that is, which we become acquainted with by 

 experience, but which is the representation lying at the 

 foundation of all external experience. Kiemann's view 

 before referred to may, I think, be said to be that, having 

 in intel/eclii a more general notion of space — in fact, a 

 notion of non-Eucludian space, we learn by experience that 

 space (the physical space of our experience) is, if not 

 exactly, at least to the highest degree of approximation, 

 Euclidian space. But suppose the physical space of our 

 experience to lie thus only approximately Euclidian space, 

 what is the consequence which follows'! Not that the 

 propositions of geometry are only approximately true, but 

 that they remain absolutely true in regard to that 

 Euclidian space which has been so long regarded as being 

 the physical space of our experience. 



COMMAS AND COLONS. 



THERE is a further argument against the Brighton 

 theory of parenthetic commas round adverbs, which 

 will reveal itself if you merely write instead of " He 

 insulted me, and therefore I went away," " He insulted 

 me, and I went away in consequence," or " therefore." 

 Who would think of putting a comma before " therefore " 

 Ln that case 1 And in either position who would think of 

 speaking it commaticalli/, which is often, if not always, the 

 best rule for punctuation 1 



I by no means disregard the distinction between colons 

 and semicolons myself ; but I cannot shut my eyes to the 

 fact that much better writers than I am are far from 

 consistent in their use of them ; and I cannot find, or 

 hardly imagine, a case where the substitution of one for 

 the other would make any difTerence in either the real or 

 the apparent sense. On the other hand, cases are con- 

 stantly occurring where one would like a smaller comma — 

 or a larger one, as the case may be. 



I thank you for your protection as to the e^■idently 

 missing comma in my former letter, which I was sure that 

 some " F. R." would be down upon. His criticism on my 

 ambiguous " it " proves that he is one of those who think 

 that if you can put a wrong or absurd meaning into some- 

 body else's writing the somebody else must have meant it. 



Edm. Beckett. 



THE FACE OF THE SKY. 



Feom Sept. 28 to Oct. 12. 

 By F.E.A.S. 



THE sun will be examined, as usual, for indications of disturb- 

 ance on every clear day, and the Zodiacal liglit may be looked 

 for in tbe East before sunrise. Map X. of "The Stars in their 

 Seasons," gives a picture of the Kight Sky during our prescribed 

 period. Mercury sets after the sun to-night, but comes into inferior 

 conjunction with him on October 7, after which Mercury is a morning 

 star. He is most indifferently placed for the observer. Venus is 

 practically invisible. Mars still souths during the morning hours, 

 but sooner and sooner every night, between the X.E. and tho 

 E.N.E. points of the horizon. He still presents the appearance of 

 an exaggerated red star. Jupiter now rises before midnight. He is 

 pretty close to Mars in the sky. No phenomena of his satellites 

 occur at times suited to the ordinary observer during the next four- 

 teen days. Saturn, some 3° or -4° North of Aldebaran, rises soon 

 after 8 p.m., and is getting nightly into a more favourable position 

 for the observer. "The remarks which we made a fortnight ago 

 with regard to Uranus and Neptune are still applicable. The 

 moon is -09 days old at noon to-day (September 28), and 

 03 days old on October 1 ; her age being quite evidently 11'3 days 

 at the same hour on October 12. Hence she will be fairly weU 

 placed for the observer during a good deal of tho time to which 

 these notes refer. No occultations of stars happen during the 

 period which they cover. The moon is in Sextans at noon to-day, 

 re-entering Leo about midnight. She quits this constellation for 

 Virgo about 4 a.m. on tho 30th. She occupies until about 2 p.m. 

 on Oct. 3 in crossing Virgo, and at that hour enters Libra, which 

 she takes until between 5 and 6 p.m. on the 5th to traverse. Then 

 she arrives at the narrow northern part of Scorpio, over which she 

 jiasses before -1 o'clock the ne.^t morning. From that hour until 

 nearly midnight on tho 7th she is traversing the southern part of 

 Ophiuchus ; but she then crosses its boundary into Sagittarius. 

 Her path through the last-named constellation takes her until 

 11 a.m. on tho 10th to describe. About 3 o'clock on the morniug of 

 the 4th sho crosses into Aquarius. There we leave her. 



{rurt 11. in OKI- 



r(.) 



NouTu American Indians as Railway Navvies. — 

 From fifty to seventy Winnebago Indians are employed on 

 the Omaha Railway as labourers. They are eager to obtain 

 the work, to which they consider they have a claim, and 

 make excellent labourers, doing a better day's work than 

 the Chinese. — Engineering. 



