[HM] Bottema's Article on Steiner's Construction of Malfatti Problem

Antreas P. Hatzipolakis (xpolakis@otenet.gr)
Wed, 1 Dec 1999 19:28:32 +0300

Oene Bottema (1901-1992), the great Dutch geometer, was also a good historian
of mathematics. He published several math-history articles, mainly in the
Dutch periodical _Euclides_. See the listing (*) copied below, and taken from:
Jan P. Hogendijk: Bibliography of books and articles in Dutch on history of
mathematics until 1900
http://www.math.ruu.nl/people/hogend/gwbib.html

The very first article is on the Steiner's construction of Malfatti Circles
Problem. This problem is:
<quote>
Given a triangle, to construct three circles mutually tangent to each other,
each touching two sides of the triangle.
Construction:
Let I be the incenter of triangle ABC.
(1) Construct the incircles of the subtriangles IBC, ICA, and IAB.
(2) Construct the external common tangents of each pair of these incircles.
(The incircles of ICA and IAB have IA as a common tangent. Label the other
common tangent Y_1Z_1 with Y_1 on CA and Z_1 on AB respectively.) Likewise
the common tangent of the incircles of IAB and IBC is Z_2X_2 with Z_2 on AB
and X_2 on BC, and that of the incircles of IBC and ICA is X_3Y_3 with X_3
on BC and Y_3 on CA.) These common tangents intersect at a point P.
(3) The incircles of triangles AY_1Z_1, BZ_2X_2 and CX_3Y_3 are the required
Malfatti circles.
</quote>
Paul Yiu: Euclidean Geometry. Preliminary Version.
Fall 1998, pp. 139 - 140

Note that this is NOT the best solution of the *original* Malfatti problem
(marble problem), which says:
<quote>
Given a triangular prism of any sort of material, such as marble, how shall
three circular cylinders of the same height as the prism and of the greatest
possible volume be related to one another in the prism and leave over the
least possible amount of material?
This question reduces to the plane construction problem that we will refer
to here as marble problem: How do you cut three circles from a given triangle
so that the sum of the areas of the three circles is maximized?
</quote>
George E. Martin: Geometric Constructions.
New York (etc): Springer, 1998. p. 92

Steiner published his construction without proof. Two published proofs of
Steiner's construction are:

Zornow's in:
F. G. - M.: Exercices de geometrie comprenant l'expose des methodes
geometriques et 2000 questions resolues.
Cinquieme edition.
Tours: Maison A. Mame et Fils - Paris: J. De Gigord, 1912, pp. 726 - 728

Hart's in:
Julian Lowell Coolidge: A Treatise on the Circle and the Sphere.
Bronx, New York: Chelsea Publishing Company, 1971, pp. 174 - 177.
(Originally published in 1916)

Now, my question to those with access to the Dutch periodical _Euclides_ is:

What does Bottema write about Steiner's construction in the referred article?

* * *

(*) Bottema's Math-History articles:

O. Bottema, Verscheidenheden XXVI. Het vraagstuk van Malfatti.
Euclides 25 (1949-50), pp. 144-149 (ook over J. Steiner).

O. Bottema, Verscheidenheden LI. Over het zijbalkon en over Regiomontanus.
Euclides 37 (1961-2), pp. 325-328.

O. Bottema, Verscheidenheden LXVII. Frans van Schooten aan Christiaan Huygens.
Euclides 42 (1966-7), pp. 204-208, vervolgd
in Euclides 43 (1967-8), pp. 164-166 ("Elementary, dear Watson")

O. Bottema, Verscheidenheden XCV. Brieven van Euler.
Euclides 51 (1975-6), pp. 190-194.

O. Bottema, Verscheidenheden LXXVIII. Euler geometer.
Euclides 47 (1971-2)), pp. 97-101.

O. Bottema, Verscheidenheden LXXVI. Een probleem van Pieter Nieuwland.
Euclides 46 (1970-1), pp. 105-109.

O. Bottema, Verscheidenheden XXVIII. Een probleem van Euler.
Euclides 25 (1949-50), pp. 200-202. (meetkunde)

O. Bottema, Wiskundigen in 1848.
Euclides 23 (1947-8), pp. 121-127.

O. Bottema, Verscheidenheden XXXIX. De straal van de omgeschreven sfeer
aan een simplex. Euclides 34 91958-9), pp. 211-214. (von Staudt)

O. Bottema, Het eeuwfeest van een ongelezen boek, De Gids no. 109
(1946), pp. 160-173, herdrukt in O. Bottema, Steen en Schelp,
Delft 1971, pp. 29-42. (over Grassmann's Ausdehnungslehre).

O. Bottema, Goethe en de wiskunde.
Euclides 58 (1982-3), pp. 41-42.

O. Bottema, Verscheidenheden LXXXVII. Michel Chasles of de tragedie der
goedgelovigheid.
Euclides 48 (1972-3), pp. 349-354.

O. Bottema, Verscheidenheden LII. Een meetkundig vraagstuk van Multatuli.
Euclides 38 (1962-3), pp. 79-82.

O. Bottema (1901-1992): Bibliografie in Nieuw Archief voor Wiskunde 4e
serie, 5 (1987), pp. 254-276. Zie ook: Teun Koetsier, In memoriam Oene
Bottema, Euclides 68 (1993-4), pp. 202-204.

Antreas

Wen dies Werk [= Euklid's Geometrie] in seiner Jugend nicht zu
begeistern vermag, der ist nicht zum theoretischen Forscher geboren.
-- A. Einstein