I ignored them, much as I ignored those pages of geometry in the pattern books.
But if you use proportions to determine how big something should be, dividers - or a compass - are how you transfer a dimension from one place to another. Dividing your window into 6 parts to find the width of your casing? Use the divider to transfer that dimension from the window to your piece of wood.
Today we would discuss it this way: "6 inches +?" "6 1/4 inches?" "How about only 6 3/8 inches?" Tricky to figure out, right? The divider is easier.
A compass serve a similar function as a ruler or a measuring tape, but it always refer back to a real thing. Intellectually, inches and feet are abstract numbers, with no relationship to any other thing at all. If you ask, "Why use a 5-1/2 inch casing for that window rather than one that's 6-1/2 inches?", an answer might have to do with cost or personal preference. The answer, "Is the 5-1/2 inch casing a better proportion for the window?" doesn't come automatically. When you get to that question using dividers you have already included the window, that's where you began.
I find I am in uncharted territory. At first I thought I was talking about 'calipers'. Then I found the tool I was thinking about is called 'dividers' or 'a compass' .
So far I haven't found very much written that confirms what I am seeing. I do know that proportions and relationships in medieval construction were often based on the circle and how it can be divided and combined. 17th and 18th century woodworking tool box lists include dividers and compasses. However, the drawing comes from Eric Sloan's Museum of Early American Tools , and he places it among the wheelwright's tools, not with the joiners'.