Jan. II, 1872J 



NATURE 



203 



point for four right angles, i.e., that they have our notion, but 

 misapply it ; then it follows that they have our conclusion, 

 that the angles of a triangle together equal two right angles ; 

 and their misapplying does not avail anything, seeing that 

 the geometrical conclusion (the universality of which is here 

 disputed) does not propose to deal with facts, but with sup- 

 positions only. The supposed rectilinear figures of these beings 

 are (though wanting all physical counterparts) the very figures 

 of Euclid. 



Now, first, the fallacy lies m what the late Professor John 

 Grote called the "pseudo- psychology," the confusion of thought 

 and thing, of the psychical and the physical. For the question 

 is here of geometry, the science which regards (say) all the sup- 

 posed or postulated rectilinear angles about a point as equal to 

 four right angles : the question is not of the physical science which 

 discovers "more or less " exactly what angular or other qualities 

 may belong to any physical object ; and so true is this, that 

 geometry is not conversant with right and left hand, nor with 

 above and belo.v. And, secondly, the fallacy is concealed by an 

 ambiguous use of terms in the statement, " with them, the angles 

 of a triangle would always, more or less, exceed two right angles." 

 The "with them " may mean with them in imagination, or with 

 them in fact ; and, but for this ambiguity, the fallacy mast have 

 exposed itself; for, first, it is obvious that two angles which they 

 imagined right ones would, in their imagination, equal, and not 

 be "exceeded by," the angles of a triangle they imagined recti- 

 linear; we could not have said otherwise than this, with the case 

 clearly stated And, secondly, we could never have said (dis- 

 tinctly) that the physical fact being one way or another, could 

 affect the universality of a geometrical position which does not 

 aftirm anything of physical facts ; but we should have perceived 

 that we were only combating a statement that the angles of a 

 physical triangle supposed to be, though not really, rectilinear, 

 are together really equal to two right angles ; a statement ob- 

 viously not tme, and as ob\'iously not geometrical. 



In mathematical argument, anything I should bring in aid of 

 Prof Jevons's able comments would be equally presumptuous 

 and useless ; and it is only because I feel that his reasonings are 

 not quite so unassailable on the psychological side that I venture 

 any additional evidence. Prof. Jevons asks (I think needlessly), 

 " Could the dwellers in a spherical world appreciate the truth of 

 the 32nd proposition of Euclid's first book ? I feel sure that, if 

 in possession of human powers of intellect, they could. In large 

 angles the proposition would altogether fail to be verified; but they 

 could hardly help perceiving that, as smaller and smaller angles 

 were examined, the spherical excess of the angles decreased, so 

 that the nature of a rectilineal triangle would present itself to them 

 under the form of a limit." Now the terms " spherical excess " 

 here mean the quantum by which all the angles of their triangle 

 would, to the knowledge of these beings, exceed two bond fide 

 right angles. They therefore know already (by Prof. Jevons's 

 supposition) what a rectilinear angle is, and, thence, what a 

 rectilinear triangle is with all its geometrical properties (as above 

 shown), for it is admitted that we require no objective experience 

 beyond that of a rectilinear angle in order to deduce said pro- 

 perties, and these beings, having our intellectual powers and our 

 data, can deduce the same. I would only suggest here that, after 

 this, to suppose any experimental evidence necessary to " verify " 

 the proposition is very much like conceding the hypothesis that 

 geometrical conclusions are not independent of experience. 



Another point not directly met by Prof. Jevons is ingenious, 

 but amounts to the assertion that, if we could not actually 

 :iynui a straight line, we should not be able to define it as " the 

 shortest distance between two points ;" for these imagined beings, 

 who cannot possess a physical straight line, will have "an infi- 

 nite number of shortest lines between any two diametrically 

 opposite points in their sphere. " An argument, interesting only 

 so far as it illustrates to what lengths of ingenuity a sophism 

 may be cirried ; for have we not to prove that our geometrical 

 conception or definition depends upon our physical experience, 

 and are we not here advancing for proof, that beings without this 

 experience cannot have the geometrical conception, and that they 

 cannot have it because — we cannot have it? If anything could 

 convince us of the inherent impotence of these experimental 

 /■ypolJieses, it should be this inevitable appearance of the ' ' circle " 

 jist when proof is called for. And again, " shortest distance " 

 he-e has two senses. First it means the shortest path available 

 to ►he imagined bemgs, and then (in order to invalidate the 

 defi.iition of a straight line) it means the shortest path con- 

 ceivible. 



Ir this case it appears then (as I proposed to show) that, while 



the geometrical certainties have been questioned, the logical code 

 has been violated, and all logical certainty confounded by an 

 ambiguous use of terms. I have here attempted no demonstra- 

 tion of the opposite theory ; but I think if the eminent sup- 

 porters of the hypothesis just e.xamined would be content to affirm 

 roundly that all our notions, conclusions, and beliefs are mere 

 resultants of intellectual action plus given e.xperience, and to for- 

 bear any hypothetic deductions till this thesis is made good, they 

 would find that the essence of the question is distinctly psycho- 

 logical, and that any experiments with hypothetical physics are so 

 many attempts to get out of a complex thing that which is simply 

 not in it. J. L. Tjpper 



Meteorological Phenomena 

 On the loth of November, a Uttle after 4 p.m., the sun 

 was behind a bank of thick stratus clouds, on the upper 

 edge of which, attached to it, about 10' above the sun's 

 position, and 15° to 20° to the north of it, I, with two other 

 persons, observed a small irregularly-shaped cloud, about 2° in 

 apparent diameter, which exhibited the colours of the least 

 refrangible portion of the spectrum, commencing with the red 

 on the south end nearest the sun, succeeded by orange, yellow, 

 and pale greenish yellow, fading into white on the north edge, 

 the rays being perpendicular. This appearance continued for 

 about five minutes or upwards while we viewed it, and then faded 

 away. Though the phenomenon appears simple, the light cloud 

 merely refracting the sun's rays, it is not evident uhy the com- 

 plementary colours of the more refrangible portion of the spec- 

 trum should not have been visible ; and, as far as I am aware, 

 a similar appearance has not been recorded before. G. F. D. 



In Nature of August 31 there is a note headed, " A Rare 

 Phenomenon," from Magdeburg. Your correspondent, I think, 

 evidently refers to what in India, or at any rate in Ceylon, is 

 called " Buddhu's Rays," an appearance in the sky very com- 

 monly observed here, and for which I have never heard any 

 scientific explanation attempted. I regret to say that hitherto I 

 have never taken any exact notes of the position of these rays. 

 They generally occur, I think, when the sun is low, sometimes in 

 the west at sunset, but also occasionally in the east. The ap- 

 pearance presented is that of alternate broad streaks of rose 

 colour and blue radiating from one point on the horizon, and 

 extending, I should say, for about thirty or forty degrees. I will, 

 whenever I see them in future, take exact notes of their position, 

 &c. At present I can only say that I certainly think that dust in 

 the atmosphere can take no part in their production. 



Colombo, October 1871 Boyd Moss 



Crannogs in the South of Scotland 

 It may interest some readers of Nature to learn that a con- 

 siderable number of crannogs, various articles of the New Stone 

 Period, and some "kitchen-middens" have been discovered in 

 connection with the small lochs which stud the surface of Wig- 

 tonshire and Dumfriesshire. Dowalton Loch, Machermore Loch, 

 and the lochs which surround Castle Kennedy in Wigtonshire, 

 have been examined within tlie last few years, and have disclosed 

 ancient lake-dwellings. The Black Loch of Sanquhar and Loch- 

 maben Loch in Dumfriesshire contain platforms of wood and 

 stone. In some cases canoes and causeways connecting the arti- 

 ficial islands with the adjacent shores have been traced. Sir 

 William Jardine, in his presidential address to the Dumfries 

 Natural History Society, 1864-5, gives an interesting account of 

 the crannog discovered at Sanquhar Black Loch ; and recently 

 the Rev. Geo. Wilson, Glenluce, read a detailed description 01 

 the crannogs in his vicinity to the Scottish Antiquarian Society. 



J. Shaw 



Freshwater Lakes without Outlet 

 In your notice of Morelet's "Central America" (Nature, 

 December 28, 1S71) you speak of the water of the lake of Peten 

 as fresh, though without an outlet. This is uncommon, but not 

 unexampled. The lake of Araqua in Venezuela, described by 

 Humboldt, is of this kind, and so are the lakes near Damascus, 

 into which the Abana and Pharpha respectively discharge. The 

 best account of these latter is, I believe, in Mr. Macgregor's 

 work, " The Jiol> Koy on the Jordan." 



Joseph John Murphy 

 Old Forge, Dunmurry, Co. Antrim, Jan. i 



