I am planning on building a roof over my hot tub and ran across a simple "wooden umbrella" in a book called "Build Your Own Outdoor Structures in Wood". Basically it's a roof on four posts. The roof has radial timbers (2x4s in the plan) which seem to be supported by a pair of cross beams stretching across diagonally from post to post and connected with a lap joint. A post sits above the lap and supports the roof like an umbrella (hence the name). Cover strips of 1x are used to fill in the roof. Overall it looks to be a fairly light structure.

Here's my question. The plans are 6'6" on a side from corner to corner. I would like to scale it up to somewhere from 9-10 feet on a side which would give a 14 foot span diagonally.

What do you think I should scale the cross beams up to? 2x8? Double up 2x6s? Actual beams?

I could wing it, but an answer with some knowledge would always be helpful. I'm familiar with framing (put myself through undergrad this way), but it's been many years and I'm not sure I know the engineering behind such a calculation.