Stadia were awarded as prizes in regimental shooting competitions from the late 1850s through the mid-1880s
as company and regimental prizes. The firing order of the men was determined by drawing lots. Their target was a circular board three feet in diameter, with an eight-inch black bullseye located in the center, and with the rest of the target painted white. Each soldier fired 10 shots from the standing position at a distance of 200 yards [183 meters]. After each shot, the distance from the bullet hole to the center of the target was measured using a length of string, a knot was placed in the string, and the hole was pasted. After each shot, the length of the measuring string increased by a corresponding amount. At the end of the day, the man with the shortest “string” won the prize. Any bullet which struck the target by ricochet was counted as a miss, and each miss added 20 inches [50.8 cm] to the soldier’s string. Ties were broken by firing additional shots. Since this was a standard course, fired on standard targets, over a standard distance, the scores of soldiers at various locations could be easily compared. The highest scoring soldier in each company was to receive a brass stadia, the regimental prize was a silver stadia, and the Army prize was an engraved silver medal. The stadia was a device for measuring distances, which, given the curved trajectory of the Minié ball, was a very useful tool for range estimation. It was expected that in battle the company and regimental prize winners would use their stadia to assist their officers in estimating distances and calling out sight settings to the troops. As an economy measure, the regimental prize winner had to surrender his company prize. If the prize winners did not retain their status during the next year’s target practice season, they were supposed to surrender their prizes to the new winners, since the prizes were company and regimental property. Both the brass and silver stadia were authorized for wear on the soldier’s uniform as a decoration. The Army prize was worn around the winner’s neck on a silver chain. Other than the brief issue of the Badge of Military Merit [Purple Heart] by General George Washington during the Revolutionary War, these were the first decorations of any type authorized for wear on the U.S Army uniform.
Stadia were also issued to artillery battery officers to assist in range estimation.
I bought a nice reproduction brass stadia from the gentleman in Hungary who runs the Capandball.eu YouTube channel.
The stadia was to be held horizontally at a predetermined distance from the viewer's eye. That is to say 90 degress clockwise to how it is shown in the above photo.
The user would look at enemy men through the slit in the sliding middle piece. When the image of the enemy fit precisely from top to bottom in the viewing slit a range measurement in yards could be read off the ruled lines engraved on the stadia. There was a scale to measure men on foot and a second scale for measuring men on horseback.
Here's a video by the gentleman who made the repro stadia with a better explanation of how they work.