http://bpeers.com/blog/?itemid=1001I chose W00t because he goes through the expected Demonstration of the quaternion as an exponential and logarithmic form.

This is Mathemythics!

The Thought Pattern is: there must be a way of showing that this is consistent with standard arithmetical processes especially differentiation ones.

The student and the teacher are left in awe snd somewhat bemused for lack of understanding.

The whole demonstration begins with product deign! Thus it is not a natural consequence of everyday arithmetic, it is a designed use of standard arithmetical forms but the forms are analogies. They are analogies by design. And it is because they are analogies that we call the process symbolic arithmetic aka Algebra!

In designing these analogies the combinatorial structures are important , but the internal semantics are different. So we replace the arithmetical operations by combinatorial and compositional ones.

Algebra since Hamilton Boole and Grassman et al has pushed this formal analogy to the limit.

The label for the exponential form is just that. e

^{ix} is a label or function form . Somewhat we put in its argument is what we express as defined.

The design ofnthebquatenions product rules and addition rules ( composition and combination) are dealtbwithnextensivly by Grassmann in his paper on " where to find Hamiltons Quaternions in the Ausdehnungslehre"! It is a very sophisticated read of a very sophisticated thinker ( Hamilton) . Hermann dispels the Mathmythics by clearly demonstrating the constructed nature of these combinatorial forms.

The geometrical implementation/ motivation/ interpretation is then rather obvious . Where the obscurantism lies is in conforming the processes to the expectations and intuitions of arithmetical mathematicians!

For most Mathematicians it is precisely this " mind fluff!" of symbols that confuse 95% of the rest of humanity that is such a drug! We love to tortuously work through the twist and turns of any demonstration to get a result! But it has to be the "correct" result! It has to Bethesda right answer.

Hermann and other Algebraists of his time recognised the constructive psychology of mathematical behaviour and gave an exposition of it.

Math,sticians I'd not like this, it was too revealing of what they were actually doing! But philosophers were saying: don't be Luddites! You can simplify and improve and extend your expertise by understanding what you are actually doing!!!

The exponential label came after the logarithmic process was clarified by Napier and Briggs and Bruges(?). Mercator worked out he Logarithmic series which Newton Wallis et al confirmed. These were difference processes for interpolation something Mathematicians had been engaged in for centuries in calculating the Sine or Chord tables for different arc segmnts/ angles. The establishing of a serviceable labelling convention was part of the so called Function revolution. Mathematicians got so bogged down in detail they lost sight of what they were doing psychologically and philosophically, and to a certain extent theologically.

If you read Newton, Euler and Grassmann you find a clear philosophicl,and psychological adjustment to what they were about. As Berkely pointed out it was as nebulous as any religious discusdions or definition or debate of their time! Those who pursued this kind of constructive thought patterning usually had some expertise to apply it with. Thus a master craftsman might develop these kinds of notions over years of studying nd applying his/ her trade.

Newtons trade was Mechanics Astrology and Alchemy, all of which he applied throughout his lifetime.

So dear reader do not get the impression that quaternions are strange otherworldly entities, as Hmilton poetically inclined to believe. Realise that Grassmann explained it as a constructed analogous method or system for labelling and expressing ordinary spatial experince of dynmic entities like moving and shaping a block of wood !

Of course they have more sophisticated applications beyond spatial even temporal experinces, but you need to grasp Grassmanns analysis of extensive and intensive magnitudes to appreciate how a line segment and a step rise can encode those distinctions.

There is one point about commutativity that is prominent in W00ts explanation. The issue is not about arithmetic or counting numbers. These are compositional expressions and in general commutativity of compositions has to be defined or constructed . Nothing can be assumed .

If we define the Composition of two exponential quaternions we have to be aware of th conjugate something Hamilton was not initially clear about in his original presentation of the rules of composition of ijk. Indeed he was not clear about a lot of things regarding i j k ! For that matter Grassmann struggled in his early work on rotation trying to grasp the way to cope with the immense freedom line segments or rays had in the Raum!

The solution is that i,j,k specify planes of circular arc that are orthogonal. This was a olution based on Justus Grassmans work on specifying a system for points planes etc in the Raum.