An opisometer is a device for measuring the length of a curve. The usefulness of the opisometer is in the fact that the line being measured does not have to be straight, it doesn't even have to be piecewise straight. It is typically used on maps or similar diagrams.
The central component of the opisometer is of a thin bar with a spiral groove in it like a screw. (The groove is wound very tightly, there are several revolutions of the thread per millimeter.) Mounted on the bar is a thin wheel which has a groove on the inside that exactly matches the thread of the screw. The opisometer also has a handle, which is affixed to the ends of the bar - this serves the dual purpose of allowing the instrument to be used comfortably and keeping the wheel from falling off either end of the bar.
The opisometer is zeroed by rolling the wheel all the way to one end of the bar. Then, the wheel is placed on one endpoint of the path to be measured. The wheel is rolled along the curve to the other endpoint, then lifted off the map. It is placed on the scale of the map, or simply on a ruler, and rolled in a straight line in the opposite direction - so that the wheel goes back to the same end of the bar it started at. The distance the opisometer rolls along the scale or ruler is the length of the path.
It is important to keep the handle of the opisometer at a constant angle during the whole process. For example, if two points are very close together, the handle of the opisometer could be swung back and forth so the wheel rolls on a straight line between them without moving along the bar at all. The amount the wheel moves along the bar is really the measure of how far the wheel has rolled, so the handle (and hence the bar) must be kept at a constant angle to accurately capture the number of rotations of the wheel.