1893 - 94 .] Prof. Tait on the Quaternion Method. 
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character ; possibly subjected to “ brave punishments,” the peine 
forte et dure at the very least ! In a word, Hamilton invented the 
Quaternion as Prof. Cayley sees it ; he afterwards discovered the 
Quaternion as I see it. 
If Quaternions are to be compared to a map, at all, they ought 
to be compared to a contoured map or to a model in relief, which 
gives not only all the information which can be derived from the 
ordinary map but something more : — something of the very highest 
importance as regards the features of a country. 
A much more natural and adequate comparison would, it seems 
to me, liken Co-ordinate Geometry (Quadriplanar or ordinary Car- 
tesian) to a steam-hammer, which an expert may employ on any 
destructive or constructive w’ork of one general hind, say the crack- 
ing of an egg-shell, or the welding of an anchor. But you must 
have your expert to manage it, for without him it is useless. He 
has to toil amid the heat, smoke, grime, grease, and perpetual din 
of the suffocating engine-room. The work has to be brought to the 
hammer, for it cannot usually be taken to its work. And it is not 
in general, transferable ; for each expert, as a rule, knows, fully and 
confidently, the working details of his own weapon only. Quatern- 
ions, on the other hand, are like the elephant’s trunk, ready at any 
moment for anything, be it to pick up a crumb or a field-gun, to 
strangle a tiger, or to uproot a tree. Portable in the extreme, 
applicable anywhere : — alike in the trackless jungle and in the 
barrack square : — directed by a little native who requires no special 
skill or training, and who can be transferred from one elephant to 
another without much hesitation. Surely this, which adapts itself 
to its work, is the grander instrument ! But then, it is the natural, 
the other the artificial, one. 
The naturalness of Quaternions is amply proved by what they 
have effected on their first application to well-known, long threshed- 
out, plane problems, such as seemed particularly ill-adapted to treat- 
ment by an essentially space-method. Yet they gave, at a glance, 
the kinematical solution (perfectly obvious, no doubt, when found) of 
that problem of Permat’s which so terribly worried Viviani ! And, 
without them, where would have been even the Circular Hodograph, 
with its marvellous power of simplifying the elementary treatment 
of a planet’s orbit ? I could give many equally striking instances. 
