Groesse is a general term for a size or largeness, and Grassmann uses it in the latin and western sense of magnitude.
Magnitudes crop up all over philosophy and physics and are similarly generally defined. The generality of the notion of magnitude is what Grassmann seeks to change and structure.
This is not his idea, but a long and important idea found in Euclid, but considered over many centuries before him. Grassmann introduces the notion Einheiten in which he bases or founds his notion of a structure to groesse or magnitude, The structure is not continuous but consists of contiguous forms and structures called einheit, or in greek monads.
The notion of monads was fully worked out by Leibniz in a modern treatment of pre-socratic thought and philosophy. In so doing, Leibniz appears to have left out Euclidean philosophy on the subject and engaged in a contre tente with Descartes and his advocate Newton. Newton was a formidable geometer, and what he did not intuitively know about Euclidean geometry is probably not worth knowing-but only probably.
Thus Newton developed a dynamic version of some aspects of Euclidean geometry, while Leibniz developed a more static spatial version called differential geometry. Nevertheless both were extremely interested in motion and Phusis. natural order.
Grassman was thus able to draw on Leibniz, Gauss and Euclid in his thinking, and to be informed by Hamilton's research.
Whereas Hamilton went forward from Gauss's linear algebraic formulation a+b√-1 to isolate number pairs he called couples, that he treated more generally as functions of 2 independent variables, obtaining algebraic rules for conjugate functions as he called them, and showing that the general solution is trigonometric scalar functions of a single variable; Grassmann on the other hand went back to Euclid and recognized that each part of the Gauss form was a monad and the combined form was also another monad that encompassed the 2.
Grassman pondered long over this, not rushing to print but to follow through the Euclidean development f monads. He called his monads einheit, and laid out the structure combinatorially. Apparently this was an idea inspired by his father,but it has a deep root in periodicity, probability, and iterativity.
In any case the structure worked, and enabled Grassmann to develop a taxonomy or ontology of geometrical relations,as per Euclid, and then to see the immediate application to physics as ab applied geometry. One other important thing was he could immediately see how to generalise the taxonomy of geometry to cover as many "Einheten" as he wished, and this in facr was analogous to extending the "dimensions" of geometry.
Alot of other analogies followed across into these extended "Einheiten" and when Grassman "sketched" out his ideas in 1844 Ausdehnungslehre, it was tentatively suggested it may be a new twig/branch of mathematics with applicaions in physics, which would be expounded in a later book.
The year before however , Hamilton had published his work on Quaternions, in which he too mentions a set of n quantities of which he proposed to deal with the quaternions in detail. Hamilton used standard terminology and standard "tricks" to sell his version, Grassmann was excited about the philosophical implications of his work and illustrated any mathematical content very unsatisfactorily, concentrating on a few simple but profound cases of his insight.
The profundity was lost on the great Prussian German elite, who were busy industrialising, and wanted technological advance not classical scholarship. Moebius returned it, Gauss returned it, and others paid it no mind if they did not criticise it for its unintelligibility.
A Monas is a subjective experience defined as "one" Spatially down the singled out thing from a group of things.
Take any group of things. Then i know i "can separate" that group into its consituents (exaston). Each (exastos) constituent thing has therefore been singled out(exaston), set far apart (exateros) from its other group members,and so as a result i perceive that singled out thing (exaston) as only(mone) one (EV) of them. I have the subjective experience of solitary, singled out, cut off from its group, single, representative of its group which i define as "one"
So by looking at a group of forms Euclid and Grassmann were able to identify the monads, but further, they were able to define the related and fractally related monads, the nested and included monads within a given "rank" of monad. By writing these all out as a necessary basis for spanning a space, that is, by recognising that he could only give a complete description of every form in terms of its monads he developed the idea of a "basis" spanning a "space" or form which covered Euler's description of form into faces edges and vertices. It is these vertices which made it crucial to be able to describe how they aggregated if he was going to give a consistent general method of description. Fortunately Moebius had begun this description of geometry in a way consistent with synthetic geometry rather than Cartesian.
In tne light of This Grassman Used his Einheiten concept to describe not a single monas, but a structured monas called a basis for a vector space by some, and a basis by Grassmann, but really encapsulating the general notion of Monas. Thus, his basis was in fact one of many in that the basis represented a field of such basises by which he could describe and measure all and any space geometrically.
A general reference frame is what he had described and encoded. and for any real physical situation a relativistic set of reference frames is necessary.
Thus these general reference frames are a round about description of magnitude, a general notion of and in space.