The 1970 paper of Halmos entitled "how to write mathematics" is a bit old, preposterous to the TeX era, but I think that many of his advices still make sense today. To summarize what he says, "use well chosen words instead of plethora of symbols.".

"Think about the alphabet". And don't hesitate to associate a meaningful symbol to a piece of formula that makes sense by itself, so as to keep your calculation as compact as possible. In particular, when there is some constant at the end of the computation, that comes from an agregation of many constants appearing during the computation, just call it C through the whole calculation, and at the end, give its actual value. "The value of the constant C is..." (I think I read that trick in some paper by Krantz).

"Use words correctly". Explain what is going on. Irrelevancy should be avoided. Writing "Now applying the Cauchy-Schwarz inequality leads to...", and just giving the result, may be better than actually applying it in the middle of the computation without mentioning it.

Beware of too heavy use of formulae notations. Something like "Now applying (32), we get ..." is less helpful than "Let us apply the upper bound on the curvature that was obtained with the help of the Gauss-Bonnet theorem." Give meaningful names to important formulas in your paper instead of refering to them through numbers. The section entitled "resist symbols" in the paper of Halmos gives a few other tricks to replace heavy formalism by well worded sentences.

dotake up more than a couple of lines. However, that's the threshold at which I believe the calculations need to be broken up into conceptual pieces. $\endgroup$ – Victor Miller May 31 '10 at 19:24