Talking with my brother he gave me some good thoughts,
Square Rule carpentry is an idea that seems very typical of the Enlightenment era, which is precisely when it is supposed to have been invented. The idea governing this method is to seek the theoretical perfect timber that exists within each imperfect timber, an idea very much in the spirit of the times.
The numerical, mathematical, whatever you want to call it system on the other hand views each timber once it reaches the layout floor as a unique piece. It is what it is, and rather than forcing it to conform to our ideas of perfection we choose instead to work with it just as it is. Whereas the former could be viewed as an enlightenment idea, this could be seen as a Romantic approach to the same problem.
Both systems, I think, are examples of a building culture's response to the Industrial Revolution first and foremost. Both cultures started from pretty much the same point -skilled carpenters with scribe systems- and moved in the same direction, industrialized framing methods. Americans moved through scribe rule into balloon framing. The Swiss moved from scribed timber frames to a different type of timber frame scribed in a different way, then to a system of mapping and calculating the joints.
The American system is very democratic. Let's make carpentry accessible and easy to understand, something anyone can grasp. The Swiss system is very... Swiss...
Let's make our frames efficient, strong, but perfect. The Swiss carpenter I don't think would ever invent the square rule system, because a joint that has to be reduced to work would be considered poor and unsightly. A housed or reduced joint, you see, would be viewed as an unforgivable violation against the nature of the wood and the harmony of the frame (though they would never word it quite that way, just say, it doesn't look good)
Moving beyond this, My mind has been thinking in terms of mapped scribe systems all day and considering the possibilities. I want to do some experiments, I can see how you could accomplish the scribing of some pretty complex forms without the need of a scribe floor with this principle...
