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MR. 0. HEAVISIDE ON THE FORCES, STRESSES, AND 
of notation, and this may be considered fortunate, for whilst I can fully appreciate and 
(from practical experience) endorse the anti-quaternionic argument, I am unable to 
appreciate his notation, and think that of Hamilton and Tait is, in some respects, 
preferable, though very inconvenient in others. 
In Hamilton’s system the quaternion is the fundamental idea, and everything 
revolves round it. This is exceedingly unfortunate, as it renders the establishment 
of the algebra of vectors without metaphysics a very difficult matter, and in its 
application to mathematical analysis there is a tendency for the algebra to get more 
and more complex as the ideas concerned get simpler, and the quaternionic basis forms 
a real difficulty of a substantial kind in attempting to work in harmony with ordinary 
Cartesian methods. 
Now, I can confidently recommend, as a really practical working system, the 
modification I have made. It has many advantages, and not the least amongst them is 
the fact that the quaternion does not appear in it at all (though it may, without 
much advantage, be brought in sometimes), and also that the notation is arranged so 
as to harmonise with Cartesian mathematics. It rests entirely upon a few definitions, 
and may be regarded (from one point of view) as a systematically abbreviated 
Cartesian method of investigation, and be understood and practically used by any one 
accustomed to Cartesians, without any study of the difficult science of Quaternions. 
It is simply the elements of Quaternions without the quaternions, with the notation 
simplified to the uttermost, and with the very inconvenient minus sign before scalar 
products done away with.'" 
Tait’s in vol. 43, pp. 535, 608. Tins rather one-sided discussion arose out of Professor Tait stigmatising 
Professor Gibbs as “ a retarder of quaternionic progress.” This may be very true; but Professor Gibbs 
is anything hut a retarder of progress in vector analysis and its application to physics. 
* §§ 7, 8, 9 contain an introduction to vector-analysis (without the quaternion), which is sufficient 
for the purposes of the present paper, and, I may add, for general use in mathematical physics. It is an 
expansion of that given in my paper “ On the Electromagnetic Wave Surface,” ‘ Phil. Mag.,’ June, 1885. 
The algebra and notation are substantially those employed in all my papers, especially in “ Electromag¬ 
netic Induction and its Propagation,” ‘ The Electrician,’ 1885. 
Professor Gibbs’s vectorial work is scarcely known, and deserves to be well known. In June, 1888, I 
received from him a little book of 85 pages, bearing the singular imprint Not Published. Newhaven, 
1881-4. It is indeed odd that the author should not have published what he had been at the trouble of 
having printed. His treatment of the linear vector operator is specially deserving of notice. Although 
“ for the use of students in physics,” I am bound to say that I think the work much too condensed for a 
first introduction to the subject. 
In ‘The Electrician’ for Nov. 13, 1891, p. 27, I commenced a few articles on elementary vector- 
algebra and analysis, specially meant to explain to readers of my papers how to work vectors. I am 
given to understand that the earlier ones, on the algebra, were much appreciated; the later ones, 
however, are found difficult. But the vector-algebra is identically the same in both, and is of quite a 
rudimentary kind. The difference is, that the later ones are concerned with analysis, with varying 
vectors; it is the same as the difference between common algebra and differential calculus. The 
difficulty, whether real or not, does not indicate any difficulty in the vector-algebra. I mention this on 
account of the great prejudice which exists against vector-algebra. 
