Earth and Sky's 'Skywatching' feature begins thusly:

The ancient eye test for those wishing to join the Roman army was administered using stars in the handle of our modern-day Big Dipper. If you passed, you got a job as an archer. If you failed, you had to serve in another capacity … perhaps as a cook. It’s said that sultans of the past also tested their soldiers’ eyesight in this way.

You can take this ancient eye test, too. Go outside around 9 p.m. You should see the Big Dipper just off the northeast horizon. The middle star in the “tail” of the question mark, is Mizar. If you look for a couple seconds longer, you may see a little starry point right next to Mizar. This star is called Alcor. If you had lived in the time of the early Romans, and you had seen Alcor, you would have been eligible to be an archer in the Roman army. If not, you would have to have served as cook or in another capacity for the Caesar.

... the belief is prolific over around the web, where it the 'eye test' is generally ascribed to some combination of Greek, Roman, and or Arabic culture/recruiting office. Can't seem to track down an ancient source, though ...