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ment. Whitehead applied it to the theory of poles and polars with respect to quadrics
determined by the corresponding inner product (see [20,§§108 124]).
Following Grassmann, Whitehead eventually uses the notation AB for both regressive
and progressive products, depending on context to remove the ambiguity. The regressive
product AB of two blades A, B of step r, s with r + s > n is defined in terms of the
progressive product of |A with |B by
|AB =(|A|B) . (A3)
If r + s
expressed by
|(AB) =|A|B . (A4)
For r + s = n both interpretations are allowed.
In our notation (A3) and (A4) correspond to the following relations
(A (" B) I =(A I) '"(B I) if r + s >n ,
(A'"B) I =(A I) ("(B I) if r + s
(A '" B) I =(A("B) I =[AB] if r + s = n.
Whitehead expands regressive products using his extended rule of the middle factor
[20, §103] which is equivalent to using (2.16) to expand (2.17) with B = I.
Multiple products in Grassmann s notation must be read from left to right. The rules
given above uniquely determine whether a given product between two blades is progressive
or regressive. Grassmann avoids parentheses by introducing periods as markers. As
examples showing how, take a, b, c, d, x as points and A, B, C, D, X as lines in a plane,
and consider the translations
x = ABa · bc =((A("B) '"a) ("(b '"c),
X= ABaCbDc = (((((A (" B) '" a) (" C) '" b) (" D) '" c,
(xaAB)(xcAb · ad · xc)
= {((x '" a) (" A) '" B}{((((x '" c) (" A) '" b) (" (a '" d)) '" (x '" c)}
= [((x '" a) (" A)B] ((x '" c) (" A) '" b a '" d x '" c .
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36
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