The amount of light an instrument can gather depends on its area. Let's compare the area of an eye to the area of one element of binoculars (since there are 2 eyes and 2 elements). Area(eye) pi * R(eye)^2 R(eye)^2 -------------- = ---------------- = ------------ Area(binoc) pi * R(binoc)^2 R(binoc)^2 (6/2 mm)^2 = ------------- = 0.0294 (35/2 mm)^2 So the eyes only gather 0.0294 times as much light as a pair of binoculars; in other words, the binoculars gather about 34 times as much light. So, a person may see a star through binoculars which is about 34 times fainter than a star he can see with his eyes alone. Convert that ratio to a magnitude difference. m(eye) - m(binoc) = - 2.5*log10 ( 0.0294 ) = - 2.5 * (-1.53) = 3.83 mag Thus, through binoculars, a person might see a star of magnitude 6.0 + 3.83 = 9.83