38
S
ALVIATI
: ...we asked the reason why [shipbuilders] employed stocks,
scaffolding, and bracing of larger dimensions for launching a big vessel than
they do for a small one; and [an old man] answered that they did this in order
to avoid the danger of the ship parting under its own heavy weight, a danger to
which small boats are not subject.
S
AGREDO
: Yes, that is what I mean; and I refer especially to his last assertion
which I have always regarded as false...; namely, that in speaking of these and
other similar machines one cannot argue from the small to the large, because
many devices which succeed on a small scale do not work on a large scale.
Now, since mechanics has its foundations in geometry, where mere size [ is
unimportant], I do not see that the properties of circles, triangles, cylinders,
cones and other solid figures will change with their size. If, therefore, a large
machine be constructed in such a way that its parts bear to one another the
same ratio as in a smaller one, and if the smaller is sufficiently strong for the
purpose for which it is designed, I do not see why the larger should not be able
to withstand any severe and destructive tests to which it may be subjected.
Salviati contradicts Sagredo:
S
ALVIATI
: ...Please observe, gentlemen, how facts which at first seem
improbable will, even on scant explanation, drop the cloak which has hidden
them and stand forth in naked and simple beauty. Who does not know that a
horse falling from a height of three or four cubits will break his bones, while a
dog falling from the same height or a cat from a height of eight or ten cubits will
suffer no injury. Equally harmless would be the fall of a grasshopper from a
tower or the fall of an ant from the distance of the moon.
The point Galileo is making here is that small things are sturdier in
proportion to their size. There are a lot of objections that could be raised,
however. After all, what does it really mean for something to be “strong”, to
be “strong in proportion to its size,” or to be strong “out of proportion to its
size.” Galileo hasn’t spelled out operational definitions of things like
“strength,” i.e. definitions that spell out how to measure them numerically.
Also, a cat is shaped differently from a horse — an enlarged photograph
of a cat would not be mistaken for a horse, even if the photo-doctoring
experts at the National Inquirer made it look like a person was riding on its
back. A grasshopper is not even a mammal, and it has an exoskeleton
instead of an internal skeleton. The whole argument would be a lot more
convincing if we could do some isolation of variables, a scientific term that
means to change only one thing at a time, isolating it from the other
variables that might have an effect. If size is the variable whose effect we’re
Chapter 1Scaling and Order-of-Magnitude Estimates
This plank is the longest it
can be without collapsing
under its own weight. If it
was a hundredth of an inch
longer, it would collapse.
This plank is made out of the
same kind of wood. It is twice
as thick, twice as long, and
twice as wide. It will collapse
under its own weight.
(After Galileo's original drawing.)