Question:
how did the classification of quadrilaterals come into being? Was there a single major contributor who coined terms like "rectangle", "square", "trapez/ium/oid", "kite", "deltoid", etc. or were these definitions contributed by several individuals over a longer stretch of time?
what was the motivation for adding quadrilaterals with specific properties to the taxonomy; is it because the validity of certain theorems is restricted to them or is the motivation something else and what is it then?
why are by far fewer kinds of deltoids classified than convex quadrilaterals?
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asked Feb 23, 2019 at 17:06
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4$\begingroup$ This might be more appropriate at the history of math stackexchange. $\endgroup$– Noah SchweberCommented Feb 23, 2019 at 17:45
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1$\begingroup$ When I was in school, I recall that what the British called a "trapezium" the Americans called a "trapezoid" (and vice versa, I think). I'm not sure if that's still the case, but at any rate, I've often wondered how the divergence in terminology came about. $\endgroup$– Timothy ChowCommented Feb 23, 2019 at 18:40
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$\begingroup$ @TimothyChow --- there seems to be a 1795 source for this confusion (I've added the reference to the answer), but why the US and the UK followed different path ways remains a mystery. $\endgroup$– Carlo BeenakkerCommented Feb 23, 2019 at 18:48
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1 Answer
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Q1. If one wishes to single out one main early contributor to the systematic classification of convex quadrilaterals it would have to be the 9th century Indian mathematician Mahavira, who divided quadrilaterals into five classes: those with unequal sides (mainly cyclic quadrilaterals, having a circumcircle), all sides equal (rhombus and square), opposite sides equal (parallelogram and rectangle), two opposite sides equal (isosceles trapezium), and three sides equal (trapezium with three equal sides). See for example Mathematical Achievements of Pre-modern Indian Mathematicians.
The names used by Mahavira (e.g. "sama" for a square and "dvidsama" for a rectangle) were of course not the Latin names we use today, but that seems of less significance. Some of these Latin names have a remarkably late date of first use, it seems "rhombus" was not used until the 16th century (source).
In response to Timothy Chow's question: the two different definitions of trapezium versus trapezoid appear to originate from the 1795 Mathematical and Philosophical Dictionary by Charles Hutton, in which the definitions of the two terms were reversed. The US followed that reversed definition, the UK did not (source).
Q2. In a fun and inspiring posting, Ben Hambrecht argues for "length-angle" duality as the key organizing principle that motivates the taxonomy.
answered Feb 23, 2019 at 18:27
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2$\begingroup$ those with unequal sides (cyclic quadrilaterals) Uh? Not all quadrilateral with unequal sides are cyclic. $\endgroup$ Commented Feb 23, 2019 at 18:46
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$\begingroup$ thanks, Mahavira only considered the cyclic ones, I should specify that. $\endgroup$ Commented Feb 23, 2019 at 18:46
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$\begingroup$ @CarloBeenakker nice and complete answer as always. $\endgroup$ Commented Feb 23, 2019 at 19:48