In article <vyhqqahgm1ei@legacy>, JCAlbornoz@aol.com (Julio Albornoz) wrote:
> .... In reference to rectangles and squares, weather a square can be > defined as a "kind of rectangle" remains to be mathematically proven. > ....
George Ivey has already pointed out that it's not a question for mathematical proof.
However, the word "oblong" gives a perfectly good solution to this particular problem. In the definitions in Euclid Book I, English translators have used "oblong" for the Greek "heteromekes" from Simson (18th century) to Heath (the standard 20th century translation). Heath says: "Of quadrilateral figures, a square is that which is both equilateral and right-angled; an oblong that which is right-angled but not equilateral; ...."
Our modern word "rectangle" can most conveniently encompass both cases, so every rectangle is either a square or an oblong.
This is regularly overlooked in two other contexts. The four-group is very often called the group of a rectangle, ignoring the fact that some rectangles are square (so have a bigger group). I always tell my students that it's the group of an oblong. Even more blatant is the term "rectangular matrix" which people use for a matrix which isn't square. All matrices are rectangular. A non-square one is an oblong matrix.