For a detailed discussion of Grassmann's work see :
and see page 186 for the discussion of the formula :
After basically applying the conceptual constructions from the first section of Extension Theory in a more general way to elementary magnitudes in the second section, Graßmann introduces, after dealing with the problem of coordinates, a new form of generating products with these vector magnitudes. In quite difficult constructions, characterized by an excessive striving for extreme abstraction, he obtains a conjunctive product which is contrasted as a dual form of the exterior product and which he calls “eingewandtes Produkt” (a conjunction which later, in the 1861 revised edition of Extension Theory, is termed the “regressive product” and comprehensively developed there).
Graßmann connects his reflections to dimensional considerations on the generating system of two multivectors $A$ and $B$ entering into a multiplicative conjunction. Thus he obtains the fundamental relation:
$dim(A) + dim(B) = dim(A + B) + dim(A ∩ B)$,
that is, the sum of the dimensions of the generating systems of $A$ and $B$ is equal to the sum of the dimension of the sum-space (“umfassendes Gebiet”) and the dimension of the intersection space (“gemeinschaftliches Gebiet”) of the generating systems of $A$ and $B$ [ ref to : Linear Extension Theory. In: A new branch of mathematics. The Ausdehnungslehre of 1844 and Other Works by Hermann Graßmann. Translated by Lloyd C. Kannenberg. Chicago and La Salle: Open
Court 1995, p. 202 ]. If one now equates the dimension of the initial space (“Hauptgebiet”) to $n$, then the exterior product of the multivectors $A$ and $B$ will only be unequal to zero if
$dim(A ∩ B) = 0$ and $dim(A) + dim(B) = dim(A + B) ≤ n$,
that is, when “$A$ is completely outside the domain of $B$” (hence the term “exterior product”). For the product $AB$ we can then say:
$dim(AB) = dim(A) + dim(B) = dim(A + B)$.
Of course, we have to check on the original German text : Drittes Kapitel : Das eingewandte Produkt, §125-on, to find the "best approximation" to the said formula.
See footnote page 185, with the discussion of the example of two lines (zweiter Stufe = second order):
System zweier Linien, so wird, da jene als Elementarsystem von dritter, diese von zweiter Stufe sind, das gemeinschaftliche System von $(2+2-3)$ter, d.h. von erster Stufe sein, und somit entweder durch einen Punkt [...]
