Quote Originally Posted by fusilier View Post
I think what it boils down to is quite simple: the geometrical processes he describes, are Euclid's geometry --> those were the drafting tools available to Europeans at the time (i.e. the straight edge and compass). Johnsson shows, quite convincingly, how those tools could be used to design a sword.
Yup, that's pretty much what I was going to say about it, too. His explanation is definitely great, he did lots of good research, and he deserves credit for being the first to point it out - but the findings are actually far less revolutionary than they're made out to be.
Saying that medieval craftsmen used constructions with compass and straightedge and mystical numerical properties in their design is like saying modern engineers use rulers, rational numbers and the sacred properties of the artifacts that define the metric system stored somewhere in Paris. It's technically true, but completely misses the point.
People use the simplest and most reliable tools for design they have available at the time. And if they do something that requires expensive materials and a lot of work, they'll do a design first and not rush head first into construction.
If I were to make a sword and had to get balance, harmonics etc right, I'd either take one that works, or carve a wooden mockup, and adjust that until I'm happy with its handling. Then I'd measure that thing, and use these measurements as a basis for forging a new one. Otherwise, I might waste lots of materials and work on a faulty design.
If I don't have a ruler to measure it, I'd have to find an (approximate) construction with compass and straightedge to get it right. Any actual sword built from this design will of course have the exact proportions, even if my "perfect" mockup didn't. And as Peter pointed out, using a construction instead of measurements means that I can easily scale my design up or down without using a pocket calculator or risking to get the numbers wrong.
TL;DR: There needn't be a mystical reason for this design approach. It's simply the smart thing to do with the tools they had at the time.