March 13, 1884] 



NA TURE 



453 



body of a small young shark. It was about eighteen inches long. 

 I secured this as evidence. This locality is about 1 70 miles from 

 the mouth of the river. 



During the six months we were in the country, the bed of the 

 river, which varies from 50 to Soo yards in width, was almost 

 dry, with the exception of deep pools at intervals connected with 

 each other by a narrow stream, often very shallow, running under 

 the high banks. In the summer time the river is deeply flooded, 

 the water rising ten to twenty feet (as .shown by drift wood in 

 trees) above the banks, in many places from forty to fifty feet 

 high. The force of the flood might at its height prevent fish 

 going up, but they could easily ascend in the intermediate season. 

 In some cases the fish must have lived months in the upper 

 waters, for portions of the Margaret, at least, are absolutely dry 

 in the winter season. May to November usually. 



I am not aware that such a circumstance has ever been noted 

 before. If not, the fact is sufficiently interesting in itself. It is 

 also important from a geological point of view, as showing that 

 some caution must be observ'ed in the classification of strata as 

 freshwater or marine on the evidence of fish alone. No doubt 

 many of these remains are embedded in the river detritus, and if 

 discovered at some future time when the physical geology of the 

 country has altered, might lead to the conclusion that these 

 deposits were of marine origin. 



Edward F. Hardman, 

 H.M. Geological Survey, Government Geologist 



Perth, Western Australia, January 28 



The Zodiacal Light 



One of the members of the staff of this establishment, Mr. 

 E, G. Constable, observ'ed a brilliant appearance of the 

 zodiacal light at about 7 p.m. on the evening of Wednesday 

 the sth inst., the cone of light being exceedingly well defined. 

 The phenomenon was not visible long, having completely 

 disappe.ared by 7.20 p.m. G. M. Whipple 



Kew Observatory, Richmond, Surrey, March 7 



THE AXIOMS OF GEOMETRY 



Q INCE the time when Riemann and Helmholtz began 

 *-' their investigations on the axioms of geometry so 

 much has been written on this subject in learned papers 

 and in a more or less popular form that it might have 

 appeared superfluous again to call the attention of writers 

 on, and teachers of. elementary geometry to it, had it 

 not been for the publication a year or two ago of a new 

 edition of the first si.x books of Euclid's " Elements," with 

 annotations and notes, by Prof. Casey. I hope the eminent 

 author of this in many respects excellent book will excuse 

 me for criticising some points in it, and making them the 

 opportunity for again returning to the question about the 

 axioms in geometry. 



The points I object to besides his treatment of Book V., 

 of which I may possibly say a few words on another 

 occasion, is contained in Note B at the end of the book. 

 Here Prof. Casey gives Legendre's and Hamilton's proofs 

 of I. 32, that the sum of the interior angles of any triangle 

 is equal to two right angles, implying, of course, that he 

 considers these proofs valid, proofs which are independent 

 of the theory of parallels. The theorem in question de- 

 pends in Euclid upon Axiom XH., and all depends upon 

 the question whether this axiom is necessary. For the 

 two propositions in this axiom and in Theorem I. 32 stand 

 in such a relation that either is a consequence of the 

 other. Hence if I. 32 can be proved independently, the 

 Axiom XII. changes into a theorem. But the investiga- 

 tions above referred to show that it is this axiom which 

 tells us what kind of a surface the plane really is, and 

 that until this axiom is introduced all propositions apply 

 equally well to the spherical and to the plane surface. 



I select for discussion the " quaternion proof '" given 

 by Sir William Hamilton, this being the easiest of the 

 two. But that by Legendre can be treated in exactly 

 the same way. 



Hamilton's proof consists in the following : — 



One side A B of the triangle a B c is turned about the 

 point B till it lies in the continuation of b c ; next, the line 

 B C is made to slide along B C till B comes to C, and is then 

 turned about c till it comes to lie in the continuation of 

 A C. It is now again made to slide along C A till the point 

 B comes to a, and is turned about A till it lies in the line 

 A B. Hence it follows, sinci: rotation is iiidepi-iidciit of 

 translation, that the line has performed a whole revolu- 

 tion, that is, it has been turned through four right angles. 

 But it has also described in succession the three exterior 

 angles of the triangle, hence these are together equal to 

 four right angles, and from this follows at once that the 

 interior angles are equal to two right angles. 



To show how erroneous this reasoning is — in spite of 

 Sir William Hamilton and in spite of quaternions — I 

 need only point out that it holds exactly in the same 

 manner for a triangle on the surface of the sphere, from 

 which it would follow that the sum of the angles in a 

 spherical triangle equals two right angles, whilst this sum 

 is knovvn to be always greater than two right angles. The 

 proof depends only on the fact, that any line can be made 

 to coincide with any other line, that two lines do so coin- 

 cide when they have two points in common, and further, 

 that a line may be turned about any point in it without 

 leaving the surface. But if instead of the plane we take 

 a spherical surface, and instead of a line a great circle 

 on the sphere, all these conditions are again satisfied. 



The reasoning emplojed must therefore be fallacious, 

 and the error lies in the words printed in italics ; for these 

 words contain an assumption which has not been proved. 

 In fact they contain an axiom which completely replaces 

 Euclid's Axiom XII., viz. it expresses that property of a 

 plane which differentiates it from the sphere. 



On the sphere it is, of course, not true that rotation is 

 independent of translation, simply because every transla- 

 tion — sliding along a great circle — is a rotation about the 

 poles of the great circle. 



From this it might be said to follow that the calculus 

 of quaternions must be wrong. But this again is not 

 correct. The fact is that the celebrated author of this 

 calculus had built it up with the full knowledge of the 

 fundamental space properties in his mind, and making 

 full use of them. Afterwards, on reasoning backwards, he 

 got these space properties out of his formulae, forgetting 

 that they were exactly the facts with which he started. 

 The process is, as far as logic is concerned, not very dif- 

 ferent from that practised by some alchemists, who pre- 

 tended to make gold, and actually did produce gold out 

 of their crucibles, but only as much as they had them- 

 selves put in. 



The following considerations may help to clear up this 

 point still further : — 



Prof. Sylvester once conceived, in illustration of some 

 points connected w-ith our subject, an infinitely thin book- 

 worm living in a surface, and consequently limited in its 

 space conceptions to the geometry on such surface. In a 

 similar manner we may imagine an intelligent being con- 

 sisting merely of an eye occupying a fixed point in space, 

 but capable of perceiving rays of light in every direction. 

 For such a being space would have two dimensions only, 

 but in this space it could conceive figures for which most 

 of Euclid's definitions and all axioms with the exception 

 of the twelfth, and therefore all propositions up to the 

 twenty-sixth in the first book, would hold. Only the 

 names point, line, angle, Sec, would stand for objects 

 difterent to those which they represent to our mind. 

 Nothing can put the vagueness of Euclid's definitions and 

 the real nature of his axioms, viz. that they contain the 

 real logical definitions of the geometrical entities, in a 

 clearer light than the fact that it is possible to use these 

 so-called definitions for objects quite different from those 

 to which Euclid applied them. 



To return to our imaginary being : let us suppose it 

 capable of studying Euclid. A ray of light, that is, a line, 



