Ernst Reinhold Eduard Hoppe (November 18, 1816 – May 7, 1900) was a German mathematician who worked as a professor at the
University of Berlin
Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o ...
.
Education and career
Hoppe was a student of Johann August Grunert at the
University of Greifswald
The University of Greifswald (; german: Universität Greifswald), formerly also known as “Ernst-Moritz-Arndt University of Greifswald“, is a public research university located in Greifswald, Germany, in the state of Mecklenburg-Western Pom ...
,
graduating in 1842 and becoming an English and mathematics teacher. He completed his doctorate in 1850 in Halle and his
habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a ...
in mathematics in 1853 in Berlin under
Peter Gustav Lejeune Dirichlet
Johann Peter Gustav Lejeune Dirichlet (; 13 February 1805 – 5 May 1859) was a German mathematician who made deep contributions to number theory (including creating the field of analytic number theory), and to the theory of Fourier series and ...
. He also tried to obtain a habilitation in philosophy at the same time, but was denied until a later re-application in 1871. He worked at Berlin as a
privatdozent
''Privatdozent'' (for men) or ''Privatdozentin'' (for women), abbreviated PD, P.D. or Priv.-Doz., is an academic title conferred at some European universities, especially in German-speaking countries, to someone who holds certain formal qualific ...
, and then after 1870 as a professor, but with few students and little remuneration.
When Grunert died in 1872, Hoppe took over the editorship of the mathematical journal founded by Grunert, the ''Archiv der Mathematik und Physik''. Hoppe in turn continued as editor until his own death, in 1900.
[. See in particula]
pp. 435–437
In 1890, Hoppe was one of the 31 founding members of the
German Mathematical Society
The German Mathematical Society (german: Deutsche Mathematiker-Vereinigung, DMV) is the main professional society of German mathematicians and represents German mathematics within the European Mathematical Society (EMS) and the International Mathe ...
.
Contributions
Hoppe wrote over 250 scientific publications, including one of the first textbooks on
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
.
His accomplishments in geometry include rediscovering the higher-dimensional
regular polytope
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, f ...
s (previously discovered by
Ludwig Schläfli
Ludwig Schläfli (15 January 1814 – 20 March 1895) was a Swiss mathematician, specialising in geometry and complex analysis (at the time called function theory) who was one of the key figures in developing the notion of higher-dimensional space ...
),
and coining the term "polytope". In 1880 he published a
closed-form expression
In mathematics, a closed-form expression is a mathematical expression that uses a finite number of standard operations. It may contain constants, variables, certain well-known operations (e.g., + − × ÷), and functions (e.g., ''n''th roo ...
for all triangles with consecutive integer sides and rational area, also known as
almost-equilateral Heronian triangles. He is sometimes credited with having proven
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, theologian, and author (described in his time as a "natural philosopher"), widely recognised as one of the grea ...
's
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 19 ...
on the
kissing number problem
In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement of ...
, that at most twelve congruent balls can touch a central ball of the same radius, but his proof was incorrect, and a valid proof was not found until 1953.
Hoppe published several works on a formula for the ''m''-fold
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
of a
composition of functions
In mathematics, function composition is an operation that takes two functions and , and produces a function such that . In this operation, the function is applied to the result of applying the function to . That is, the functions and ...
. The formula, now known as "Hoppe's formula", is a variation of
Faà di Bruno's formula
Faà di Bruno's formula is an identity in mathematics generalizing the chain rule to higher derivatives. It is named after , although he was not the first to state or prove the formula. In 1800, more than 50 years before Faà di Bruno, the French ...
. Hoppe's publication of his formula in 1845 predates Faà di Bruno's in 1852, but is later than some other independent discoveries of equivalent formulas.
In his work on
special functions
Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by ...
, Hoppe belonged to the Königsburg school of thought, led by
Carl Jacobi.
He also published research in
fluid mechanics
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and bio ...
.
[. See in particula]
p. 71
Awards and honors
He was elected to the
Academy of Sciences Leopoldina
The German National Academy of Sciences Leopoldina (german: Deutsche Akademie der Naturforscher Leopoldina – Nationale Akademie der Wissenschaften), short Leopoldina, is the national academy of Germany, and is located in Halle (Saale). Founded ...
in 1890.
[.]
Books
*
Theorie Der Independenten Darstellung Der Höhern Differentialquotienten' (Leipzig: Joh. Ambr. Barth, 1845)
*
Zulänglichkeit Des Empirismus In Der Philosophie' (Berlin: Wilhelm Thome, 1852)
*
Lehrbuch Der Differentialrechnung Und Reihentheorie Mit Strenger Begründung' (Berlin: G. F. Otto Müller, 1865)
*
Principien Der Flächentheorie' (Leipzig: C. A. Koch, 1876)
*
Tafeln Zur Dreissigstelligen Logarithmischen Rechnung' (Leipzig: C. A. Koch, 1876)
*
Lehrbuch Der Analytischen Geometrie' (Leipzig: C. A. Koch, 1880)
References
{{DEFAULTSORT:Hoppe, Reinhold
1816 births
1900 deaths
University of Greifswald alumni
19th-century German mathematicians