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“As I was reading the extract from your paper in the geometric sum and difference… I was struck by the marvelous similarity between your results and those discoveries which I made even as early as 1832…
I conceived the first idea of the geometric sum and difference of two or more lines and also of the geometric product of two or three lines in that year (1832). This idea is in all ways identical to that presented in your paper. But since I was for a long time occupied with entirely different pursuits, I could not develop this idea. It was only in 1839 that I was led back to that idea and pursued this geometrical analysis up to the point where it ought to be applicable to all mechanics. It was possible for me to apply this method of analysis to the theory of tides, and in this I was astounded by the simplicity of the calculations resulting from this method.”

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Hermann Grassmann

Letter to Saint-Venant (1845) as quoted by Michael J. Crowe, A History of Vector Analysis: The Evolution of the Idea of a Vectorial System (1967)

“From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a proceeding which even on the part of his commentators was no longer understood, and likewise with modern mathematicians, unfortunately for science, has met with little imitation. As a matter of fact, the abstract methods of mathematical science know no axioms at all.”

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Hermann Grassmann

As quoted in "Diverse Topics: The Origin of Thought Forms," The Monist (1892) Vol. 2 https://books.google.com/books?id=8akLAAAAIAAJ&pg=PA120 ed., Paul Carus, citing The Open Court Vol. II. No. 77. A Flaw in the Foundation of Geometry by Hermann Grassmann, translated from his Ausdehnungslehre

“A work on tidal theory… led me to Lagrange's Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the new analysis in such a simple way that the calculations often came out more than ten times shorter than in Lagrange's work.”

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Hermann Grassmann

the "ideas of analysis" to which he returned, are those quoted above.
Ausdehnungslehre (1844)

“Geometry can in no way be viewed… as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I… realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry.”

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Hermann Grassmann

Forward, as quoted by Mario Livio, Is God a Mathematician? (2009)
Ausdehnungslehre (1844)

“I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results.”

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Hermann Grassmann

Ausdehnungslehre (1844)

“The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept.”

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Hermann Grassmann

Ausdehnungslehre (1844)

“The first impulse came from the consideration of negatives in geometry; I was accustomed to viewing the distances AB and BA as opposite magnitudes. Arising from this idea was the conclusion that if A, B, and C are points of a straight line, then in all cases AB + BC = AC, this being true whether AB and BC are directed in the same direction or in opposite directions (where C lies between A and B). In the latter case AB and BC were not viewed as merely lengths, but simultaneously their considered since they were oppositely directed, Thus dawned the distinction between the sum of lengths and the sum of distances which were fixed in direction. From this resulted the requirement for establishing this latter concept of sum, not simply for the case where the distances were directed in the same or opposite directions, but also for any other case. This could be done in the most simple manner, since the law that AB + BC = AC remains valid when A, B, and C do not lie on a straight line.
This then was the first step which led to a new branch of mathematics… I did not however realize how fruitful and how rich was the field that I had opened up; rather that result seemed scarcely worthy of note until it was combined with a related idea.”

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Hermann Grassmann

Ausdehnungslehre (1844)

“The concept of centroid as sum led me to examine Möbius' Barycentrische Calcul, a work of which until then I knew only the title; and I was not little pleased to find here the same concept of the summation of points to which I had been led in the course of the development. This was the first, and… the only point of contact which my new system of analysis had with the one that was already known.”

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Hermann Grassmann

Ausdehnungslehre (1844)

“I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative.”

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Hermann Grassmann

Zero can never be a unit.
Definition 3. Part 1. The Elementary Conjunctions of Extensive Magnitudes. Ch. 1. Addition, Subtraction, Multiples and Fractions of Extensive Magnitudes. 1. Concepts and laws of calculation. Extension Theory Hermann Grassman, History of Mathematics (2000) Vol. 19 Tr. Lloyd C. Kannenberg, American Mathematical Society, London Mathematical Society
Ausdehnungslehre (1844)

“While I was pursuing the concept of geometrical product, as this idea was established by my father… I concluded that not only rectangles, but also parallelograms, may be viewed as products of two adjacent sides, provided that the sides are viewed not merely as lengths, but rather as directed magnitudes. When I joined this concept of geometrical product with the previously established idea of geometrical sum the most striking harmony resulted. Thus when I multiplied the sum of two vectors by a third coplaner vector, the result coincided (and must always coincide) with the result obtained by multiplying separately each of the two original vectors by the third… and adding together (with due attention to positive and negative values) the two products. [Thus A(B + C) = AB + AC. ]
From this harmony I came to see a whole new area of analysis was opening up which could lead to important results.”

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Hermann Grassmann

Ausdehnungslehre (1844)