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1.2Scaling of Area and Volume
Great fleas have lesser fleas
Upon their backs to bite ‘em.
And lesser fleas have lesser still,
And so ad infinitum.
Jonathan Swift
Now how do these conversions of area and volume relate to the ques-
tions I posed about sizes of living things. Well, imagine that you are shrunk
like Alice in Wonderland to the size of an insect. One way of thinking
about the change of scale is that what used to look like a centimeter now
looks like perhaps a meter to you, because you’re so much smaller. If area
and volume scaled according to most people’s intuitive, incorrect expecta-
tions, with 1 m
2
being the same as 100 cm
2
, then there would be no
particular reason why nature should behave any differently on your new,
reduced scale. But nature does behave differently now that you’re small. For
instance, you will find that you can walk on water, and jump to many times
your own height. The physicist Galileo Galilei had the basic insight that the
scaling of area and volume determines how natural phenomena behave
differently on different scales. He first reasoned about mechanical struc-
tures, but later extended his insights to living things, taking the then-radical
point of view that at the fundamental level, a living organism should follow
the same laws of nature as a machine. We will follow his lead by first
discussing machines and then living things.
Galileo on the behavior of nature on large and small scales
One of the world’s most famous pieces of scientific writing is Galileo’s
Dialogues Concerning the Two New Sciences. Galileo was an entertaining
writer who wanted to explain things clearly to laypeople, and he livened up
his work by casting it in the form of a dialogue among three people. Salviati
is really Galileo’s alter ego. Simplicio is the stupid character, and one of the
reasons Galileo got in trouble with the Church was that there were rumors
that Simplicio represented the Pope. Sagredo is the earnest and intelligent
student, with whom the reader is supposed to identify. (The following
excerpts are from the 1914 translation by Crew and de Salvio.)
A boat this large needs to have timbers
that are thicker compared to its size.
The small boat holds up just fine.
A larger boat built with the same
proportions as the small one will
collapse under its own weight.
Galileo Galilei (1564-1642) was a Renaissance Italian who brought the scientific
method to bear on physics, creating the modern version of the science. Coming
from a noble but very poor family, Galileo had to drop out of medical school at
the University of Pisa when he ran out of money. Eventually becoming a lecturer
in mathematics at the same school, he began a career as a notorious
troublemaker by writing a burlesque ridiculing the university’s regulations — he
was forced to resign, but found a new teaching position at Padua. He invented
the pendulum clock, investigated the motion of falling bodies, and discovered
the moons of Jupiter. The thrust of his life’s work was to discredit Aristotle’s
physics by confronting it with contradictory experiments, a program which paved
the way for Newton’s discovery of the relationship between force and motion. In
Chapter 3 we’ll come to the story of Galileo’s ultimate fate at the hands of the
Church.
Section 1.2Scaling of Area and Volume