Edward Sang, on The Theory of Commensurables

 

Edward Sang is best remembered, when he is remembered at all, as " probably the greatest calculator of logarithms of the 19th century" in the words of Dennis Rogel.   Wikipedia adds that he is, "best known for having computed large tables of logarithms, with the help of two of his daughters. (They did not mention the names of the daughters, but in Rogel's paper he cites Flora and Jane.) These tables went beyond the tables of Henry Briggs, Adriaan Vlacq, and Gaspard de Prony." 

I first became acquainted with Sang while researching the history of near equilateral triangles, like 3,4,5 or 13,14,15, or 51,52,53.  Sang found a Pell equation for the square root of three gave an infinite number of even middle sides for  such triangles. He was looking at these triangles with a special focus, were the areas commensurable numbers, what he called a theory of commensurables.  We could form a triangle with any three consecutive integers, but were the areas also integers (commensurable with the nits of the sides).

Sang even listed a 1,2, 3 triangle with area of zero.  For the triangles above, the areas are all integers, 12, 84, and 1170.

Here is Sang's list:

But this was just one of the questions about commensurables.  Sang begins his inquiry thinking about the incommensurables that might have sparked interest in early mathematicians about this problem.  Consider the simple forms of geometry.  The equilateral triangle with integer sides  has an altitude that is not an integer.  The square with integral sides has diagonals that are not integers. Regular hexagons, because they are made of  equilateral triangles have their major diagonal as integers, twice the side length.  Sang imagined exploring areas, and other features of each shape.  

Sang opens his paper with this general statement, "The general proposition in the theory of commensurables is to determine the conditions under which lines, surfaces, or solidities, connected with prescribed figures or forms, may have their ratios expressible by integer numbers."

"Many problems of a similar nature may be proposed: thus we may require that the three sides of a trigon, and the line bisecting one of the angles, be all commensurable ; that the four sides and the two diagonals of a tetragon be all expressed by integer numbers, and so on.The methods of solving such problems may be said to constitute the theory of commensurables."

His first study is upon right triangles and what we today call primitive Pythagorean triangles, or PPT, and he proceeds to a number of interesting explorations.  I think the article would be a great exploration for teacher and student alike. 

The complete paper is here.