Heron triangles are those with integer sides and integer area. It is well known how to construct them as the union or difference of a pair of integersided right-angled triangles with a common side. For example, the triangles with sides 8, 15, 17 and 20, 15, 25 may be united, with common side 15, to form an acute Heron triangle with altitude 15, base 28 and other two sides 17 and 25. Its area is 210. Alternatively, their difference may be formed to create an obtuse Heron triangle with altitude 15, base 12 and other two sides 17 and 25. Its area is 90. Triangles with rational sides and rational area may be enlarged to have integer sides and integer area, and so may be classed as Heron triangles also. There have been many articles about Heron triangles in recent times, both in the Gazette [1,2] and elsewhere [3,4, 5], to mention just a few. This is not surprising as the number theory involved has a direct and pleasing application.

The content of this article seems not only to have appeal for, but also to be well within the problem-solving capabilities of, many students of elementary geometry. No new results are cited (see [1]) but the concepts are presented differently, providing a viable and concrete means of explaining elementary partitions of space.

Considering the pertinent remarks on “Approximations” in The Teaching of Arithmetic in Schools, the steady change of treatment of the subject in text-books and the recent article by Mr. Inman in the Gazette, it seems that considerable independent investigation of this matter has been going on. Also few people will disagree with Mr. Inman when he says that the teaching of the subject leaves room for improvement. That here is matter worth spending time on in the classroom is clear—but the same may be said of many more topics than we can ever have time to deal with. It is in the belief that the subject of the accuracy of figures, or approximations, is one we really ought not to omit that I venture further remarks.

The reader should first try the following exercise, which motivates the later work.

Exercise: Prove that the set of real matrices

forms a group under the usual matrix multiplication.

This exercise comes with a guarantee that it will provoke at least a doubletake when first encountered. The usual reaction is to protest that these matrices are obviously singular, or to complain that the identity is missing from the set. Of course those familiar with this example can feel suitably smug in their rapid dismissal of these criticisms, but what was your reaction the first time you saw it?

In a previous article I wrote about the remarkable theorem of Holditch, about areas generated by a sliding or sweeping motion. While thinking further on this topic I discovered some relations concerning the areas within closed curves and the lengths of the perimeters of curves.

The rank of the coefficient matrix plays a dominant role in the theory of linear algebraic equations. It is not surprising, therefore, that a test for the rank of a matrix, that was a by-product of some work in dimensional analysis, proves to be an an admirable tool in this theory With its aid the consistency requirement assumes a simple and effective form, and the solution of both homogeneous and non-homogeneous systems is given explicitly in terms of submatrices.

The subject of this address is Technical Education. I wish to examine its essential nature and also its relation to a liberal education. Such an enquiry may help us to realise the conditions for the successful working of a national system of technical training. It is also a very burning question among mathematical teachers; for mathematics is included in most technical courses.