Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature.

Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin.

Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, here referred to as A1, later revised in 1862 as Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet, here referred to as A2.

A year later, he returned to Stettin to teach mathematics, physics, German, Latin, and religious studies at a new school, the Otto Schule.

Over the next four years, Grassmann passed examinations enabling him to teach mathematics, physics, chemistry, and mineralogy at all secondary school levels.

Kummer wrote back saying that Grassmann's 1846 prize essay (see below) contained "commendably good material expressed in a deficient form."

Starting during the political turmoil in Germany, 1848–49, Hermann and his brother Robert published a Stettin newspaper, Deutsche Wochenschrift für Staat, Kirche und Volksleben, calling for German unification under a constitutional monarchy.

After writing a series of articles on constitutional law, Hermann parted company with the newspaper, finding himself increasingly at odds with its political direction.

This essay, first published in the Collected Works of 1894–1911, contains the first known appearance of what is now called linear algebra and the notion of a vector space.

(One should keep in mind that in Grassmann's day, the only axiomatic theory was Euclidean geometry, and the general notion of an abstract algebra had yet to be defined.)

Kummer assured that there were good ideas in it, but found the exposition deficient and advised against giving Grassmann a university position.

[3] In 1862, Grassmann published a thoroughly rewritten second edition of A1, hoping to earn belated recognition for his theory of extension, and containing the definitive exposition of his linear algebra.

The result, Die Ausdehnungslehre: Vollständig und in strenger Form bearbeitet (A2), fared no better than A1, even though A2's manner of exposition anticipates the textbooks of the 20th century.

[4]: 46 Adhémar Jean Claude Barré de Saint-Venant developed a vector calculus similar to that of Grassmann, which he published in 1845.

One of the first mathematicians to appreciate Grassmann's ideas during his lifetime was Hermann Hankel, whose 1867 Theorie der complexen Zahlensysteme.

Felix Klein wrote a negative review of Schlegel's book citing its incompleteness and lack of perspective on Grassmann.

Thirty years after the publication of A1 the publisher wrote to Grassmann: “Your book Die Ausdehnungslehre has been out of print for some time.

His discovery was revolutionary for historical linguistics at the time, as it challenged the widespread notion of Sanskrit as an older predecessor to other Indo-European languages.