## Section 3.4 Second Meeting

Table 3.4.1 lists the six speakers at the section’s second meeting. Unlike the organizational meeting, this one set the tone for most of the meetings that followed. The program featured six speakers representing six different institutions; all of the speakers were introduced earlier in the chapter.

Speaker | Title |

Crawley | Descartes’ Geometry |

Owens | The Malfatti problem |

Dresden | On matrix equations |

Wilson | Space filling polyhedra |

Fort | Difference equations |

Morris | Positive integral solutions of an indeterminate equation |

At the time of the 1927 meeting the 65-year old Edwin S. Crawley held the distinguished Thomas A. Scott Professorship in Mathematics at the University of Pennsylvania. On this occasion he reflected on the history of mathematics by sketching the contents of The Geometry, the historic work that was one of three influential appendices to Descartes’ monumental Discourse on Reasoning (1637). Crawley described Descartes’ introduction and use of coordinates in the construction of a normal to a curve. In the sense that the topic was on analytic geometry, the first paper at this meeting was similar to the first paper at the organizational meeting.

The second speaker, Frederick Owens, traveled from State College to attend the meeting, probably by train via Lewistown and Harrisburg on the Friday between Thanksgiving and the date of the meeting. In his presentation Owens gave a historical sketch of the Malfatti problem: to construct three circles in a triangle such that each circle is tangent to two sides of the triangle and to the other two circles. Owens concluded with his own solution to the problem, which differed from those usually given.

The strong historical flavor of the talks by Crawley and Owens would enable them to be understood by advanced undergraduate students. The same is true of the presentation by Lehigh’s Tomlinson Fort, who outlined the history and literature of difference equations and stated the principal problems as well. It should be noted that Fort “dwelt upon the advances in difference equations which have been made in America”.

The presentation by Haverford College’s Albert H. Wilson was similar to Fort’s in that his topic has never been part of the mathematical mainstream, and so can be considered as an enrichment topic. Nonetheless, this author wishes that Wilson’s talk could have been taped, for it must have been a sight to behold. He began his lecture by discussing tilings of the plane, a topic that received a major boost in the 1970s with the work of the Dutch artist M. C. Escher. Our own Doris Schattschneider of Moravian College has added immensely to the popularization of this topic and to our understanding and appreciation of it. Unlike Schattschneider, however, Wilson restricted his tilings to polygons, enumerating classes of polygons that tile based on the work of the English mathematician Percy Alexander MacMahon. (Incidentally, George Andrews discussed other results due to Major MacMahon in an invited lecture given 73 years later, in 2000.) Then Wilson moved to tilings of three-dimensional space by prisms, the rhombic dodecahedron, the bees’-cell, and the tetrakaidecahedron, showing models of each and indicating their significance in nature. In the conclusion of what must have been a captivating lecture for the 60 people in attendance, Wilson discussed the work from a related problem in non-Euclidean geometry as well as Euclidean geometry. Models too accompanied this final part, this time four tetrahedra that tile Euclidean space. The question of which tetrahedra tile space is still unsolved, but there are many known families of such tetrahedra.

Lectures by Swarthmore’s Arnold Dresden and Rutgers’ Richard Morris presented recent results at a research level. The mix of undergraduate and graduate levels, with a wide variety of topics, became standard fare at all sectional meetings after this one. The challenge of the Program Committee at that time and the Executive Committee today is to attract a program of capable speakers who reflect this diversity.

Dresden gave a brief report on a method recently developed by W. E. Roth at the University of Wisconsin for determining solutions of the matrix equation \(P(X) = A\) which are expressible as polynomials in \(A\text{,}\) where \(P(\lambda)\) is a polynomial in \(\lambda\) without a constant term, \(A\) is a given matrix of order \(n\text{,}\) and \(X\) is the unknown matrix of order \(n\text{.}\) The fact that Dresden based his talk on work done at Wisconsin combined with Fort’s emphasis on contributions made by American mathematicians reflects a heartfelt pride in achievements by their fellow countrymen.

In the final talk of the day Richard Morris discussed a problem in Diophantine analysis that generalized a problem initially posed in the Monthly. We have been unable to determine more information about the problem