Given how slide rules work, there would have to be a great number of marks, for the same reason that there cannot be only one unique ratio of integers that approximates Pi.

I do not think you have thought that through. You need only one mark for PI on every part of a sliderule to multiply or divide by pi. See for example http://en.wikipedia.org/wiki/File:Slide_rule_cursor.jpg

> I do not think you have thought that through.

My remark was only in the context of the present conversation, where Pi is representable by the ratios of various integers.

I used a lot of slide rules during my years as a NASA engineer, but none of those I owned had marks for physical constants -- too bad, because it's a very good idea.