Under ordinary atmospheric conditions the range of the more powerful light-sources used in lighthouses is greater than the range as limited by the curvature of the earth.  For the uncolored illuminants the range in nautical miles appears to be at least equal to the square root of the candle-power.  A real practical limitation which still exists is the curvature of the earth, and the distance an object may be seen by the eye at sea-level depends upon the height of the object.  The relation is approximately expressed thus,—­

Range in nautical miles = 8/7 square root of Height of object in feet.  For example, the top of a tower 100 feet high is visible to an eye at sea-level a distance of 8/7 square root of 100 = 80/7 = 11.43 miles.  Now if the eye is 49 feet above sea-level, a similar computation will show how far away it may be seen by the original eye at sea-level.  This is 8/7 square root of 49 = 8 miles.  Hence an eye 49 feet above sea-level will be able to see the top of the 100-foot tower at a distance of 11.43 + 8 or 19.43 nautical miles.  Under these conditions an imaginary line drawn from the top of the tower to the eye will be just tangent to the spherical surface of the sea at a distance of 8 miles from the eye and 11.43 miles from the tower.

The luminous intensity of a light-source or of the beam of light is directly responsible for the range.  The luminous intensity of the early beacon-fires and oil-lamps was equivalent to a few candles.  The improvements in light-sources and also in reflecting and refracting optical systems have steadily increased the candle-power of the beams, until to-day the beams from gas-lamps have intensities as high as several hundred thousand candle-power.  The beams sent forth by modern lighthouses equipped with electric lamps and enormous light-gathering devices are rated in millions of candle-power.  In fact, Navesink Light at the entrance of New York Bay is rated as high as 60,000,000 candle-power.

Of course, light-production has increased enormously in efficiency in the past century, but without optical devices for gathering the light, the enormous beam intensity would not be obtained.  For example, consider a small source of light possessing a luminous intensity of one candle in all directions.  If all this light which is emitted in all directions is gathered and sent forth in a beam of small angle, say one thousandth of the total angle surrounding a point, the intensity of this beam would be 1000 candles.  It is in this manner that the enormous beam intensities are built up.

There is an interesting point pertaining to short flashes of light.  To the dark-adapted eye a brief flash is registered as of considerably higher intensity than if the light remained constant.  In other words, the lookout on a vessel is adapted to darkness and a flash from a beam of light is much brighter than if the same beam were shining steadily.  This is a physiological phenomenon which operates in favor of the flashing light.