They had things like levels and surveyors tools. With a surveyor’s kit you could get quite accurate – like to within an inch or two or better. Read up on how they built cathedrals. Iron right angles, bubble levels, chalk lines, plumb bobs… the tools of stone masonry and carpentry really haven’t changed much in over a thousand years.
Furthermore, they couldn’t quite build out from the bridge pylons the way they do now with cantilevere’d tower supports etc.. They’d build a pier out in the middle of the river and build it up to a certain level with stone (using my level and surveyors scope from the shoreline to measure the right height). Let it set and remove the wooden caisson they used to dig down to the river bed. Then they’d construct a wodden trestle/framework that spanned from the pier to one shore, while the foundation on that shore is being built. Make both sides of the trestle level is easy enough, then the trestle framework supports the stone until the mortar sets. Rinse/ repeat on the other side, or until you reach the other bank.
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