The Scale Factor In Nature
If you were to load a mouse, a rat, a man and a horse into an airplane and shove them out at 1000 feet, how do you suppose they would fare? Most likely, the mouse would bounce when it hit the ground and scurry off. The rat would stand about a 50-50 chance of survival. The man would definitely be killed, and the horse would splash.
The differences are only one manifestation of nature's scale effect -- particularly the surface-to-volume ratio. An object increases its surface area in proportion to the square of its dimensions, but it increases its volume (and therefore its weight) in proportion to the cube. For example, assume that a young animal is one inch long and one inch tall when it is born. If it grows to be ten times that large (10 inches long and 10 inches tall), its cross-section has increased to 100 square inches, but its volume has increased to 1000 cubic inches. In other words, it has increased its resistance to the air 100 times by growing, but its weight, or driving force as it falls, has been increased by 1000 times. So it splats instead of bounces.
The scale factor becomes very important where surface tension is involved. A water bug is so light in comparison to the bearing area of its feet that it scarcely indents the water surface. On the other hand, the body area of a very small creature is so great in proportion to its volume that surface tension may present dangers to it just as gravity does to us. When we step from a shower, we are dripping with water equivalent to a very small percentage of our body weight, but a wet fly weighs twice as much and is a helpless thing. A small insect can become imprisoned in a drop of water, and a fly with a couple of feet stuck in a water drop may find it hard to get them out.
A famous anecdote relating scale effect with early warfare grew out of a lecture given by historian Garrett Mattingly some years ago. Mattingly was speaking on the Tudor dynasty and Richard III, King of England during the latter part of the 15th century. Richard was noted as a warrior of extreme skill and ferocity. History also records him as being a very small man.
Mattingly stirred up a spirited debate by postulating that Richard's small size may have actually increased his effectiveness on the battlefield. Using an argument that goes back to Galileo, he drew on the fact that while height increases linearly, the corresponding surface of the body increases as the square. Thus, said Mattingly, Richard's armor would have been proportionally much lighter than that of a larger man, leaving him less encumbered in battle.
After the lecture, one listener complained that Mattingly had it all backwards. The correct reasoning, he said, was that physical volume goes up by the cube, while the surface to be protected goes up only by the square. So the big guy should have more strength left over after putting on his armor than the little guy.
Wait a minute, a second listener said. The strength of a muscle is proportional to its cross-section, not its bulk. Therefore, if the cross- section of the muscles increase by the square just like the body area to be protected, it must all even out. From that perspective, it would be the same as if neither combatant were wearing a thing, and it was that aspect which made the outcome look pretty grim for the little guy.
At that point, an engineer interjected that, while muscles may increase their strength proportionally to their cross-section, it's not true that armor increases its weight in proportion to its surface area. To obtain the same strength with a larger area of metal, he explained, the thickness of the metal must also be increased. So maybe little Richard had an advantage after all.
Most likely, in days of olde, victory in battle depended primarily on skill and strength of will. The scale factor in armor could not have been too significant, because armor wasn't really as heavy as most of us have come to believe. A full suit weighed somewhat less than 50 pounds -- about the same weight that a modern soldier carries into combat.
In fact, this is one case in which metalsmiths challenged the versatility of nature. A knight's armor actually weighed less in proportion to his body weight than an armored beetle's.
