Proportional compass (sector)
Two graduated rulers on a hinge that solved proportions, root extraction and gunnery problems with the help of an ordinary pair of dividers.
Galileo Galilei gave the instrument — the sector in English — its commercial form in 1597. Two arms meet at a joint and carry engraved scales: for lengths, areas, regular polygons and the densities of metals. You set the angle, measure across with dividers, and read proportionally, turning multiplication, division and scaling into comparisons of distance rather than arithmetic.
It was the standard tool of the military engineer and the draughtsman for three centuries, lasting into the 1900s until the slide rule displaced it. Galileo also made money from it: he sold the instrument with a printed manual and charged for lessons in how to use it.