Hermann Günther Grassmann | Biography

Hermann Günther Grassmann was a gifted German thinker whose work spanned the fields of mathematics and linguistics, theology, and botany. His decision to focus on mathematics came when he was 31, but he abandoned the field 20 years later when his formulation of a geometric calculus did not receive the recognition it deserved. Grassmann's conception of n-dimensional vector spaceand multi-linear algebra, laid out in his monumental work Die Lineale Ausdehnungslehre, were ahead of his time but had great impact once they were grasped by late 19th- and early 20th-century mathematicians.

Grassmann was born on April 15, 1809, in Stettin, Prussia (present-day Szczecin, Poland), the third of 12 children. His father, Justus Günther Grassmann, taught mathematics and physics at the local gymnasium and wrote several basic-level mathematics textbooks. The Grassmann family was a religious one: Justus had briefly served as a Protestant minister before becoming a teacher, and Grassmann's mother, Johanne Medenwald, was the daughter of a minister.

Studies in Several Fields

Grassmann was schooled first by his mother and at a private academy before enrolling at the Stettin Gymnasium. He was a fine student, earning the second highest score on the final secondary school examination. During his three years at the University of Berlin, Grassmann focused his studies on theology and classical languages and literature. His work on mathematics and physics was done on his own once he returned to Stettin in 1830.

Over the next decade Grassmann took a series of examinations in order to secure a job in the scholastic community. He passed an examination in December of 1831 that allowed him to teach only at the elementary school level. The following spring, Grassmann took a position teaching at the Stettin Gymnasium. In 1834, he passed the first level of theological examinations administered by the local Lutheran church, but instead of pursuing a religious career, he took a job as senior master at the Gewerbeschule in Berlin. Grassmann changed jobs a year later to take a teaching post at the Otto Schule in Stettin. By 1840, he had completed his round of tests, passing the second-level theology examination and the mathematics examination that allowed him to teach at the secondary school level.

Introduces Geometric Calculus

It was during this last mathematics examination that Grassmann first applied his geometric calculus to solve a problem on the theory of the tides. After this 1840 examination, Grassmann decided to devote his energy to mathematics, particularly to the development of the geometric calculus he had been working on since 1832. In 1844, Die Lineale Ausdehnungslehrewas published, presenting Grassmann's geometric calculus as a combination of synthetic geometry's treatment of points (and not numbers) with analytic geometry's use of calculations. Grassmann introduced n-dimensional vector space and multi-linear algebra, concepts that paved the way for the creation of exterior algebra.

Despite containing profoundly revolutionary ideas, Ausdehnungslehre was largely ignored by Grassmann's contemporaries because of the work's abstract nature and unreadable style. With mathematicians such as Julius Plücker and August Ferdinand Möbius refusing to write a review about the book, the professional community generally disregarded the work. Grassmann used the concept of connectivity he established in Ausdehnungslehre in an 1845 paper in which he revised Ampere's fundamental law for the reciprocal effect of two infinitely small currents. Again, Grassmann's poor writing style obscured a scientifically important paper. When Grassmann applied for a position as a university professor in 1847, E. E. Kummer's critique of Grassmann's 1845 paper, which stated that it contained "commendably good material expressed in a deficient form," prevented Grassmann from landing the job.

Grassmann married Marie Therese Knappe on April 12, 1849, and the couple had 11 children, two of whom died as young children. Convinced of the importance of his geometric calculus, Grassmann revised his Ausdehnungslehre for republication in 1862. Although Grassmann tried a new approach in explaining his methodology, the second version met with the same reception as the first. By the mid-1860s, Grassmann was frustrated by the lack of recognition for his mathematical contributions, so he turned his full attention to his studies in linguistics and other sciences.

Contributes to Linguistics and Other Fields

As early as 1849, Grassmann began studying Sanskrit, followed by Lithuanian, Russian, and older forms of Prussian and Persian. In 1854 he developed a theory about the tonal components of vowels, and his 1863 theory about aspirates and the sound shift in Germanic languages became the linguistic law that bears his name. By 1860, Grassmann had taken interest in the Hindu literary masterpiece, the Rig-Veda. Grassmann compiled a glossary and composed a translation of the Rig-Veda during the 1870s, finding the instant acclaim in linguistics that he was never awarded in mathematics.

Grassmann also tried his hand at a variety of other projects during his lifetime, including a study of colors in his 1853 Zur Theorie der Farbenmischung and the renaming, using German etymological roots, of plant species native to German-speaking areas in his 1870 Deutsche Pflanzennamen . Writing for a political newspaper in 1848, Grassmann penned a series of articles supporting a Germany united under constitutional monarchy. Grassmann even wrote folk songs in which he harmonized up to three voices.

Returns to Mathematics

After his 1871 election to the Göttingen Academy of Sciences, Grassmann returned to mathematics. He published several papers before he died on September 26, 1877, of heart failure. The following year a third version of Ausdehnungslehre, prepared before his death, was published. An appreciation of Grassmann, appearing a year after his death in the Schulprogrammof the Stettin Marienstifts gymnasium, said of Grassmann, "... only a quite independent spirit could dare to break his own paths in mathematics, on which others followed him only after decades ...." Indeed, it was only after his death that mathematicians began drawing off Grassmann's Ausdehnungslehre and crediting his discoveries with the later development of linear matrix algebra.