Re: [HM] Grebe

Julio Gonzalez Cabillon (jgc@adinet.com.uy)
Mon, 13 Dec 1999 21:20:43 -0200

Re: "I am wondering who was Grebe of the Grebe Point."

Dear Antreas,

For what it is worth, let me say that Ernst Wilhelm Grebe was a
German teacher of mathematics [more precisely _Oberlehrer am Gym-
nasium_] at Kassel, and was born on August 30, 1804 -- same birth
date (Aug. 30) for Joseph Serret, Carle Runge, Olga Taussky-Todd,
among others.

Ernst Grebe is remembered only for a thoughtful paper appeared in
1847 [2] concerning some interesting properties of the triangle:

If on each side of a given (arbitrary) triangle ABC one describes
a square ( exterior to ABC ), then the extended outside sides of
the squares, thus obtained, form a similar triangle A'B'C'. The
center of similarity of both triangles is the meeting point of
the straight lines AA', BB', CC'. In German this point was first
called _Grebe'schen Punkt_ [Grebe's point], a TERM which seems to
have been first referred to by E. Hain as early as 1875, in his
paper "Ueber den Grebe'schen Punkt" [ _Archiv der Mathematik und
Physik_ (= Grunert's _Archive_) volume LVIII (1876), pp. 84-89 ].
Afterwards, the term _Grebe'schen Punkt_ appeared many times in
the _Jahrbuch ueber die Fortschritte der Mathematik_ by reviewers
such as Dr. Schemmel (Berlin, 1875), Prof. Mansion (Gent, 1881),
Prof. Lampe (Berlin, 1881), Dr. Lange (Berlin, 1885), et cetera.

The first math contribution of Grebe that I am aware of is a book
titled _De Linea Helice_ [1], in Latin, and published in Marburg.
As you may read below, Grebe wrote many articles and books mainly
on Geometry.

Grebe died in Kassel, on January 14, 1874 (same date, January 14,
for Nicolaus Mercator, Edmond Halley, George Berkeley, Charles
Bossut, Charles Dodgson (= Lewis Carroll), Charles Hermite, Kurt
Go"del, William Feller, among many others).

Grebe's books and papers:

[1] De linea helice ejusque projectionibus orthographicis commen-
tatio, quam ad summos in philosophia honores rite Capessendos
Amplissimo Philosophorum Marburgensium Ordini. 59 pages. Marburg,
1829.

[2] Das geradlinige Dreieck in Bezug auf die Quadrate der Per-
pendikel, die man von einem Punkte seiner Ebene auf seine Seiten
faellen kann, 9 pages, Grunert's _Archiv_ 9 (1847).

[3] Ueber die Verwandlung der Wurzeln quadratischer Gleichungen
in Kettenbrueche, Cassel, 48 pages, 1847.

[4] Aufloesung reiner Gleichungen, insbesond solcher des 3. Gra-
des durch Kettenbrueche 99 pages, Grunert's _Archiv_ 10 (1847) &
16 (1851).

[5] Eroerterung einer Spielerei durch d. Wahrscheinlichtkeit-
Rechnung, 2 pages, Grunert's _Archiv_ 11 (1848).

[6] Beweis einer Formel fuer \pi, 6 pages, Grunert's _Archiv_ 12
(1849).

[7] Rationalmachen von Nennern mit unbestimmt vielen irrationalen
Gliedern, 5 pages, Grunert's _Archiv_ 13 (1849).

[8] Theilung eines ebenen Dreiecks durch 2 sich innerhalb dessel-
ben schneidende Geraden in 4 gleiche Flaechenstuecke, 3 pages,
Grunert's _Archiv_ 13 (1849).

[9] Ausdruecke, welche fuer Wurzeln hoeheren Grades mit
(B+ A sqra)(B - A sqr a) analog sind, 5 pages, Grunert's _Archiv_
13 (1849).

[10] Ueber das Auffinden von Dreiecken, deren Seiten sich gleich-
zeitig mit den Halbierungs-Linien durch ganze Zahlen ausdruecke
lassen, 11 pages, Grunert's _Archiv_ 17 (1851).

[11] Vergleich zwischen dem arithmetischen, geometrischen & har-
monischen Mittel, Schloemilch Journal, 1/2 page, 3 (1858).

[12] Das prismatoid, 4 pages, Grunert's _Archiv_ 39 (1862).

[13] Formeln der sphaerischen Trigonometrie, 3 pages, Grunert's
_Archiv_ 39 (1862).

[14] Beitraege zur Lehre von dem geradlinigen Dreieck, Cassel:
Doell & Schaefer, 16 pages, 1862.

[15] Ueber einen Satz der Geometrie (Jubilaeumsschrift des Dr
Gerling), Kassel, 1862.

[16] Lehrsatz der Geometrie, 2 pages, Schloemilch's Journal 8
(1863).

[17] Zusammenstellung von Stuecken rationaler ebener Dreiecke,
Halle: Schmidt, 248 pages, 1864.

[18] Fuenfzig Aufgaben ueber das geradlinige Dreieck trigonome-
trisch geloest, Cassel: Doell & Schaefer, 13 pages, 1865.

Re: "Also, why is it symbolized with K?"

At the time, the choice of "K" came as the first 'unused' letter
of the alphabet in Lemoine's scheme, since the first letters A,
B, C, ... were (and still are) used for naming the vertices of a
triangle, and the letters F, G, H, I had already a standard and
somehow accepted meaning:

F: [F]euerbach point (= center of the 9-point circle)
G: [G]ravity center (= Barycenter = Median point = Centroid)
H: [H]oehen(schnitt) punkt (= Orthocenter)
I: [I]ncenter

Therefore, within Lemoine's scheme, there was:

J: used for other purposes (especially, J_a, J_b, & J_c)
K: available

And many thanks for many 'thinks' ( = thoughts & things) of those
that you always do not seem to forget!

Greetings from warm and friendly Montevideo,

Julio Gonzalez Cabillon