Grassmann: 1844 foreword
.. geometry can in no way be viewed, like arithmetic or the theory of combinations, as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I also had realised that there must be a branch of mathematics which yields in a purely abstract way laws similar to those of geometry, which is limited to space. By means of the new analysis it is possible to form such a purely abstract branch of mathematics; indeed this new analysis, developed without assuming any principles established outside its own domain and proceeding purely by abstraction, was itself this science.
The initial incentive was provided by the consideration of negatives in geometry; I was used to regarding the displacements AB and BA as opposite magnitudes. From this it follows that if A, B, C are points of a straight line, then AB + BC = AC is always true, whether AB and BC are directed similarly or oppositely, that is even if C lies between A and B. In the latter case AB and BC are not interpreted merely as lengths, but rather their directions are simultaneously retained as well, according to which they are precisely oppositely oriented. Thus the distinction was drawn between the sum of lengths and the sum of such displacements in which the directions were taken into account. From this there followed the demand to establish this latter concept of a sum, not only for the case that the displacements were similarly or oppositely directed, but also for all other cases. This can most easily be accomplished if the law AB + BC = AC is imposed even when A, B, C do not lie on a single straight line.
Thus the first step was taken toward an analysis that subsequently led to the new branch of mathematics presented here. However, I did not then recognize the rich and fruitful domain I had reached; rather, that result seemed scarcely worthy of note until it was combined with a related idea.
While I was pursuing the concept of product in geometry as it had been established by my father, I concluded that not only rectangles but also parallelograms in general may be regarded as products of an adjacent pair of their sides, provided one again interprets the product, not as the product of their lengths, but as that of the two displacements with their directions taken into account. When I combined this concept of the product with that previously established for the sum, the most striking harmony resulted; thus whether I multiplied the sum (in the sense just given) of two displacements by a third displacement lying in the same plane, or the individual terms by the same displacement and added the products with due regard for their positive and negative values, the same result obtained, and must always obtain.
This harmony did indeed enable me to perceive that a completely new domain had thus been disclosed, one that could lead to important results. Yet this idea remained dormant for some time since the demands of my job led me to other tasks; also, I was initially perplexed by the remarkable result that, although the laws of ordinary multiplication, including the relation of multiplication to addition, remained valid for this new type of product, one could only interchange factors if one simultaneously changed the sign (i.e. changed + into - and vice versa).
JOC/EFR December 2005
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