The whites of their eyes

It's Friday, and this is the last of my blogs on famous phrases and math.

The quote until you see the whites of their eyes has been attributed to a number of military commanders. The original source is unclear. It might have been:

• Lieutenant-Colonel Agnew to the Royal Scots Fusiliers at Dettingen in 1743
  
• Prince Charles of Prussia in 1745
• Frederick the Great in 1755
• General Wolfe on the battlefield of the Plains of Abraham in 1759
• Revolutionary War commanders Putnam, Stark, Prescott or Gridley at Bunker Hill in 1775

The meaning of this phrase is clear - wait until they are close enough before engaging in battle. But how close is that? Is there a mathematical way to convert see the whites of their eyes to feet and inches?

Prescott said in his battle report that they engaged at 30 yards. Others have put the distance at 10 yards. Still others have said the distance was irrelevant as long as they didn't miss a shot.

Hollywood producers know the importance of the right distance; a battle filmed without the enemy in view is pretty boring. Great whites of their eyes were provided in the spaghetti westerns of Sergio Leone!

For example, this view shows the whole scene, but is clearly too far back:

This is the right distance to see the whites of their eyes:

 

 

The same holds true in space battles (as this article makes clear). How entertaining would it be for a Klingon battle cruiser to fire when it's two galaxies away? You want to see everything on one screen:

Despite these movie portrayals, in real life we are seeing less of the whites of other people's eyes.

• More people are looking at Facebook and not real faces
• Sales of sunglasses keep rising
• More cars and homes have tinted windows for privacy and energy savings

If you feel the need for a few more numbers than I've provided today, examine these facts:

1. All window glass filters out ultraviolet B and transmits UV A and visible light
2. Some glass provides ultraviolet A protection without loss of visible light (tint)
3. In California, vehicle windows with tint must have light transmittance of 70% or more, the tint alone must have a minimum light transmittance of 88%, and UV A must be reduced
4. Many sunglasses have plastic lenses and don't automatically reduce ultraviolet ( "The degree to which sunglasses will attenuate sunlight and block UV varies with the physical, chemical and optical properties of the lenses")
5. European sunglasses are divided into 5 categories:

Filter category	Description	Range of luminous transmittance
		Above (per cent)	To (per cent)
0	Clear or very light tint	80	100
1	Light tint	43	80
2	Medium tint	18	43
3	Dark tint	8	18
4	Very dark tint	3	8

I have been unable to ascertain the percentage of luminous transmittance that allows the whites of their eyes to be seen.