It should be a part of mandatory training for graphic designers that keep drawing these gears that can't work. Lookup for "gears icon", it's appalling.

The simple of it: Gear teeth are made in such a way so that when their faces touch as the gear rotates, they "roll" against echother faces rather than slide.

Reading this kind of stuff always makes me admire Victorian engineers even more, who had to do all the math by hand, to say nothing of the machine designs to cut and create these types of gears and mechanical systems.

I suppose they learned a lot from prior tech, like clockmaking, but even there it would seem that fundamental problems would have needed solutions from scratch (eg shear failure probably isnt much of a worry in most clocks but would be catastrophic in a steam engine).

Circle involute is one of my favorite curves. It has a particularly nice Cesàro equation: κ = c√s. It also has the unusual property of being its own parallel curve. Thus, if you were to draw a curve using piecewise circle involutes, you'd have an exact mathematical offset.

It'll show up in a blog post soon, once I get back to having enough time to write.