464 



NATURE 



[March 16, 189^ 



in the presentation of the subject. Of^course, in some sense 

 and to some extent it is and must be true. Whatever is special, 

 accidental, and individual, will die, as it should ; but that which 

 is universal and essential should remain as an organic part of 

 the whole intellectual acquisition. If that which is essential 

 dies with the accidental, it must be because the accidental has 

 been given the prominence which belongs to the essential. For 

 myself, I should preach no such doctrine to those whom I wish 

 to convert to the true faith. 



In Italy, they say, all roads lead to Rome. In mechanics, 

 kinematics, astronomy, physics, all study leads to the considera- 

 tion of certain relations and operations. These are the capital 

 notions ; these should have the leading ^parts in any analysis 

 suited to the subject. 



If I wished to attract the student of any of these sciences to 

 an algebra for vectors, I should tell him that the fundamental 

 notions of this algebra were exactly those with which he was 

 daily conversant. I should tell him that a vector algebra is so 

 far from being any one man's production that half a century 

 ago several were already working toward an algebra which 

 should be primarily geometrical and not arithmetical, and that 

 there is a remarkable similarity in the results to which these 

 efforts led (see Proc. A.A. A.S. for 1886, pp. 37, ff.). I should 

 call his attention to the fact that Lagrange and Gauss used the 

 notation (aySy) to denote precisely the same as Hamilton by his 

 8(0/87), except that Lagrange limited the expression to unit 

 vectors, and Gauss to vectors of which the length is the secant 

 of the latitude, and I should show him that we have only to 

 give up these limitations, and the expression (in connection with 

 the notion of geometrical addition) is endowed with an immense 

 wealth of transformations. I should call his attention to the 

 fact that the notation \i\r^, universal in the theory of orbits, is 

 identical with Hamilton's V(piP2), except that Hamilton takes 

 the area as a vector, i.e. includes the notion of the direction of 

 the normal to the plane of the triangle, and that with this 

 simple modification (and with the notion of geometrical addi- 

 tion of surfaces as well as of lines) this expression becomes 

 closely connected with the first-mentioned, and is not only 

 endowed with a similar capability for transformation, but en- 

 riches the first with new capabilities. In fact, I should tell him 

 that the notions which we use in vector analysis are those which 

 he who reads between the lines will meet on every page of the 

 great masters of analysis, or of those who have probed deepest 

 the secrets of nature, the only difference being that the vector 

 analyst, having regard to the weakness of the human intellect, 

 does as the early painters who wrote beneath their pictures 

 " This is a tree," " This is a horse." 



I cannot attach quite so much importance as Mr. McAulay to 

 uniformity of notation. That very uniformity, if it existed 

 among those who use a vector analysis, would rather obscure 

 than reveal their connection with the general course of 

 modern thought in mathematics and physics. There 

 are two ways in which we may measure the progress of 

 any reform. The one consists in counting those who have 

 adopted the shibboleth of the reformers ; the other measure is the 

 degree in which the community is imbued with the essential 

 principles of the reform. I should apply the broader measure 

 to the present case, and do not find it quite so bad as Mr. 

 McAulay does. 



Yet the question of notations, although not the vital question, 

 is certainly important, and I assure Mr. McAulay that reluc- 

 tance to make unnecessary innovations in notation has been a 

 very powerful motive in restraining me from publication. Indeed 

 my pamphlet on "Vector Analysis," which has excited the 

 animadversion of quaternionists, was never formally published, 

 although rather widely distributed, so long as I had copies to 

 distribute, among those who I thought might be interested in 

 the subject. I may siy, however, since I am called upon to 

 defend my position, that I have found the notations of that 

 pamphlet more flexible than those generally used. Mr. McAulay, 

 at least, will understand what I mean by this, if I say that some 

 of the relations which he has thought of sufficient importance to 

 express by means of special devices (see Proc. R. S. E., for 

 1890-91), may be expressed at least as briefly in the notations 

 which I have used, and without special devices. But I should 

 not have been satisfied for the purposes of my pamphlet with 

 any notation which should suggest even to the careless reader 

 any connection with the notion of the quaternion. For I con- 

 fess that one of my objects was to show that a system of vector 

 analysis does not require any support from the notion of the 

 quaternion, or, I may add, of the imaginary in algebra. 

 NO. 1 220, VOL. 47] 



I should hardly dare to express myself with so much freedom,, 

 if I could not shelter myself behind an authority which will not 

 be questioned. 



I do not see that I have done anything very different from 

 what the eminent mathematician upon whom Hamilton's 

 mantle has fallen has been doing, it would seem, unconsciously. 

 Contrast the system of quaternions, which he has described in 

 his sketch of Hamilton's life and work in the North British 

 Review for September, 1866, with the system which he urges 

 upon the attention of physicists in the Philosophical Magazine 

 in 1890. In 1866 we have a great deal about imaginaries, and 

 nearly as much about the quaternion. In 1890 we have nothing 

 about imaginaries, and little about the quaternion. Prof. Tait 

 has spoken of the calculus of quaternions as throwing off in the 

 course of years its early Cartesian trammels. I wonder that he 

 does not see how well the progress in which he has led may be 

 described as throwing off the yoke of the quaternion. A 

 characteristic example is seen in the use of the symbol V. 

 Hamilton applies this to a vector to form a quaternion, Tait ta 

 form a linear vector function. But while breathing a new life 

 into the formulae of quaternions, Prof. Tait stands stoutly by the 

 letter. 



Now I appreciate and admire the generous loyalty toward 

 one whom he regards as his master, which has always led Prof. 

 Tait to minimise the originality of his own work in regard to 

 quaternions, and write as if everything was contained in the 

 ideas which flashed into the mind of Hamilton at the classic 

 Brougham Bridge. But not to speak of other claims of 

 historical justice, we owe duties to our scholars as well as to our 

 teachers, and the world is too large, and the current of modern 

 thought is too broad, to be confined by the ipse dixit even of a 

 Hamilton. J. Willard Gibbs. 



Glacial Drift of the Irish Channel. 



It seems of interest to record that the eurite or microgranite 

 containing blue amphibole (Riebeckite), the rock noticed by 

 Mr. P. F. Kendall in the drifts of the Isle of Man and Caernar- 

 vonshire, occurs abundantly in the form of small pebbles on the 

 shore at Killiney, co. Dublin, doubtless derived from the 

 " glacial gravels " of the coast. I have also found a pebble in 

 the raised beach at Greenore, co. Down. 



Mr. Teall's description of the rock of Ailsa Craig {Mineral- 

 ogical Magazine, vol. ix. p. 219) enabled the very characteristic 

 pebbles collected by Mr. Kendall to be referred to that mass as 

 a source, or to formerly existing bosses south of or adjacent to it. 

 As far as I am aware, all the material is in the form of pebbles, 

 often only an inch in diameter. This is hardly likely to be its 

 original condition, if removed by ice from Ailsa Craig, and is 

 only one of many points that indicate a redistribution of our so- 

 called "glacial" beds by subsequent action of rivers or other 

 waters. Grenville A, J. Cole. 



Royal College of Science for Ireland, Dublin, 

 March 12. 



THE SACRED NILE. 



THAT Egypt is the gift of the Nile is a remark we owe 

 to the father of history, who referred not only to the 

 fertilising influence of the stream, but to the fact that 

 the presence of the Nile and its phenomena are the 

 conditions upon which the habitability of Egypt alto- 

 gether depends. That that part of Egyptian archzeology 

 and myth which chiefly interests astronomers is also the 

 gift of the Nile is equally true. 



The heliacal rising of Sirius and other stars at the time 

 of the commencement of the inundations each year ; all 

 the myths which grew out of the various symbols of the 

 stars so used, are so many evidences of the large share 

 the river, with its various water levels at different times, 

 had in the national hfe. It was, in fact, the true and unique 

 basis of the national life. 



In this the Nile had a compeer, or even compeers. 

 What the Nile was to Egypt the Euphrates and Tigris 

 were to a large region of Western Asia, where also we 

 find the annual flood to have been in ancient times a 

 source of fertility over an enormous area which is now 



