**Otto Hesse** worked on the development of the theory algebraic functions and the theory of invariants. He is remembered particularly for introducing the Hessian determinant.

- Born in Königsberg, Otto Hesse grew up in the famous city where he attended the Old City Gymnasium.
- 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.
- 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 Ⓣ(The eight points of intersection of the second order).
- 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.
- 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.
- We mentioned the famous students who undertook doctoral studies under Hesse's supervision at Königsberg.
- Hesse's main work was in the development of the theory of algebraic functions and the theory of invariants.
- 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.
- Hesse introduced the 'Hessian determinant' in a paper in 1842 during an investigation of cubic and quadratic curves.
- Some recent research has suggested that Hesse did much more than improve the presentation of certain results by Jacobi.
- In 1857 Hesse published another presentation of the theory, which has been regarded merely as an improved exposition of Jacobi's results.
- Indeed, Hesse shifts from an algorithmic approach to the calculus of variations to an emphasis on its analytical character.
- 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.
- 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.
- 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 Ⓣ(Lectures on the analytical geometry of space: in particular on surfaces of the second order) (1861) and Vorlesungen über analytische Geometrie der geraden Linie, des Punktes und des Kreises in der Ebene Ⓣ(Lectures on the analytical geometry of the straight line, the point and the circle in the plane) (1865).
- 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.
- 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.

Born 22 April 1811, Königsberg, Prussia (now Kaliningrad, Russia). Died 4 August 1874, Munich, Germany.

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Origin Russia

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive