Abstract. At a given moment, the concept of polite mathematics emerged, the underlying idea of which is that, even if you have only 60 readers, it pays to spend an hour if by doing so you can save your average reader a minute. By inventing an idealized ‘average reader’, we could translate most of the lofty, human goal of politeness into more or less formal criteria we could apply to our texts. This note is devoted to the resulting notational and stylistic conventions that were adopted as the years went by. We don't want to baffle or puzzle our reader, in particular it should be clear what has to be done to check our argument and it should be possible to do so without pencil and paper. This dictates small, explicit steps. On the other hand it is well known that brevity is the leading characteristic of mathematical elegance, and some fear that this ideal excludes the small, explicit steps, but one of the joys of my professional life has been the discovery that this fear is unfounded, for brevity can be achieved without committing the sin of omission. I should point out that my ideal of crisp clarity is not universally shared. Some consider the puzzles that are created by their omissions as spicy challenges, without which their texts would be boring; others shun clarity lest their work is considered trivial.

A multiprogramming system is described in which all activities are divided over a number of sequential processes. These sequential processes are placed at various hierarchical levels, in each of which one or more independent abstractions have been implemented. The hierarchical structure proved to be vital for the verification of the logical soundness of the design and the correctness of its implementation.

This paper describes the circumstances under which the ALGOL 60 translator of the Mathematical Centre, Amsterdam, has been constructed. It describes its main features, virtues and short-comings. Furthermore, it tells how the translator has shown itself to be an inspiring tool for a varied group of users.