At the university Hesse studied mathematics and natural sciences where his lectureres included Jacobi, Bessel, Carl Neumann, and F J Richelot. If it had not been for the inspired teachings of Jacobi, Hesse may have chosen to specialise in a science subject other than mathematics. However Hesse graduated in 1837 with a qualification which allowed him to teach mathematics, physics and chemistry in secondary schools, and then he spent a year as a probationary teacher at the Kneiphof Gymnasium in Königsberg. In the summer of 1838 he travelled through Germany and Italy, furthering his education. Back in Königsberg by the start of the autumn school term he took up a post teaching physics and chemistry at a trade school there.
Hesse had continued to study for his doctorate under Jacobi's supervision and he was awarded the degree from Königsberg in 1840 after submitting his thesis De octo punctis intersectionis trium superficium secundi ordinis Ⓣ. In 1841 he submitted his habilitation thesis to Königsberg and was appointed as a privatdozent. At this point he resigned his teaching position at the trade school. In the same year Hesse married Marie Sophie Emilie Dulk, the daughter of Friedrich Philipp Dulk (1788-1852) who was the Professor of Chemistry at Königsberg; they had one son and five daughters.
In 1845 Hesse was promoted to extraordinary professor at Königsberg and spent his most productive years there publishing most of his work in Crelle's Journal. Many famous mathematicians did their doctoral studies under Hesse's supervision. These doctoral include Gustav Kirchhoff and Carl Neumann at Königsberg, but he also lectured to several other students there who would go on to become exceptional mathematicians including Siegfried Aronhold, Alfred Clebsch and Rudolph Lipschitz. In 1855 Hesse was appointed as an ordinary professor at Halle, but he only held this post for one year since when he was offered the chair at Heidelberg, to succeed Ferdinand Schweins, he was eager to accept and join his former students Kirchhoff and Bunsen there. In 1856 he took up the appointment in Heidelberg, officially taking up the position from 9 September, and remained there until 1868 when he took up a post in the new Munich Polytechnic. We mentioned the famous students who undertook doctoral studies under Hesse's supervision at Königsberg. At the Ruprecht-Karls University of Heidelberg he had an even longer list of famous research students including Adolph Mayer, Ernst Schröder, Heinrich Weber, Olaus Henrici, and Max Noether.
Hesse's main work was in the development of the theory of algebraic functions and the theory of invariants. Haas writes in :-
His achievements can be evaluated, however, only in close connection with those of his contemporaries. Hesse was indebted to Jacobi's investigations on the linear transformation of quadratic forms for the inspiration and starting point of his initial works on the theory of quadratic curves and planes. For proof (again influenced by Jacobi) he used the newly developed determinants which allowed his presentation to reach an elegance not previously attained.In fact Hesse introduced the 'Hessian determinant' in a paper in 1842 during an investigation of cubic and quadratic curves. Subsequently this concept has been widely applied in algebraic geometry.
Some recent research has suggested that Hesse did much more than improve the presentation of certain results by Jacobi. For example in  Fraser presents a strong argument that Hesse's work was more fundamental than many have considered. He examines Jacobi's 1837 result on the calculus of variations and Hesse's reformulation in 1857:-
Jacobi's result does not have much visibility in current texts on the calculus of variations and even less so Hesse's. Both have major contributions to this field, which might be considered the doorstep of functional analysis. After the important breakthrough of Euler and Lagrange, it was a natural step to study the second variation. Jacobi in 1837 proposed a theory of the second variation which was generally well received. In 1857 Hesse published another presentation of the theory, which has been regarded merely as an improved exposition of Jacobi's results. The author [of ] challenges these views, attributing to Hesse an effectively distinct presentation of the theory. Indeed, Hesse shifts from an algorithmic approach to the calculus of variations to an emphasis on its analytical character. This was the line of research adopted in the method of fields of extremals, which characterizes progress in the calculus of variations in the late nineteenth century. Hesse might be considered a precursor of these developments.Another result by Hesse which has proved particularly influential is the 'principle of transfer' which he gave in 1866 during his work on projective geometry. How this influenced many different areas of mathematics is studied by Hawkins in the interesting paper . Wilhelm Meyer gave a general form of Hesse's principle of transfer in 1883 which in turn was used by Cartan in 1913 to construct all irreducible representations of a complex semisimple Lie algebra.
Hesse's work was also influenced by Steiner, particularly work he did on the geometrical interpretation of algebraic transformations. Plücker and Poncelet had also made major contributions which Hesse built on. His student Aronhold showed that some of Hesse's results here were best possible. Hesse worked on some topics that Cayley was also working on and both produced a theory of homogeneous forms which they published at the same time.
Haas writes in  of Hesse's contributions as a teacher:-
Hesse's teaching was also influential. In his long years as a lecturer, he continually showed his enthusiasm for mathematics, and his textbooks on analytical geometry must be seen in this context. the special forms of linear equation and of planar equation that Hesse used in these books are called Hesse's normal form of the linear equation and of the planar equation in all modern textbooks on the discipline.The two textbooks which Hesse wrote during his years in Heidelberg are Vorlesungen über analytische Geometrie des Raumes: insbesondere über Oberflächen zweiter Ordnung Ⓣ (1861) and Vorlesungen über analytische Geometrie der geraden Linie, des Punktes und des Kreises in der Ebene Ⓣ (1865). Let us also mention his important paper Sieben Vorlesungen aus der analytischen Geometrie der Kegelschnitte Ⓣ which appeared in Zeitschrift für Mathematik und Physik in 1874.
Many academies honoured Hesse with membership including the Berlin Academy of Sciences and the Göttingen Academy of Sciences (Königliche Gesellschaft der Wissenschaften) in 1856 and the Bavarian Academy of Sciences (Königlich Bayerischen Akademie der Wissenschaften) in Munich in 1869. In 1871 he became an honorary foreign member of the London Mathematical Society. He was also honoured with the award of the Steiner prize of the Berlin Academy of Sciences in 1872.
Hesse died in Munich from a liver problem but was buried in Heidelberg at his request since he always felt that city to be his second home. His complete works was published by the Bavarian Academy of Sciences in 1897 with a foreword by Walther von Dyck, S Gundelfinger, Jacob Lüroth and Max Noether. The 731 page book was reprinted by the Chelsea Publishing Company in 1972 (this is the book ).
Article by: J J O'Connor and E F Robertson