Mathematics at Work

Mathematics at Work is a book that challenges our assumptions about our subject. This is the fourth edition of a book devoted to

Practical Applications of Arithmetic, Algebra, Geometry, Trigonometry, and Logarithms to the Step-by-Step Solutions of Mechanical Problems, with Formulas Commonly Used in Engineering Practice and a Concise Review of Basic Mathematical Principles.

 Browsing through the book reveals an amazing mix of topics, from very elementary stuff about divisibility to rather sophisticated chapters on plane and spatial geometry. Many pages contain elaborate tables of results on trigonometric functions, solving triangles, factoring expressions, and so on. Many specific construction problems are considered too, such as (to choose an example at random) how to find the "radius of a circle tangent to a given circle and to two lines at a given angle." There is a whole section on approximate formulas which includes a rather sophisticated discussion of where such formulas come from and how one might decide whether it is OK to use them. There is a chapter on "gear ratio problems", which turns out to be about continued fractions. At the back, there are dozens of tables giving all sorts of interesting and useful functions (one wonders why, in an age of calculators, one would need tables of common logarithms or trigonometric functions, but there they are).

I find this a fascinating artifact. On the one hand, it can be an excellent source of "real world" problems for those of us who teach elementary mathematics. Many of these problems are quite interesting, and some are quite difficult; in fact, I'd probably have trouble solving some of them on my own. On the other hand, it reflects a perspective on what mathematics is all about that I find deeply alien. This mathematics is about vast collections of apparently unrelated facts that one looks up as needed. (So, for example, there are not only tables indicating how the sines and cosines of complementary and supplementary angles are related, but also similar tables for tangents, cotangents, secants and cosecants!) Here is mathematics as a tool, but perhaps also as less than a science.