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interested in seeing, then we don’t really want to compare things that are
different in size but also different in other ways.
Also, Galileo is doing something that would be frowned on in modern
science: he is mixing experiments whose results he has actually observed
(building boats of different sizes), with experiments that he could not
possibly have done (dropping an ant from the height of the moon).
After this entertaining but not scientifically rigorous beginning, Galileo
starts to do something worthwhile by modern standards. He simplifies
everything by considering the strength of a wooden plank. The variables
involved can then be narrowed down to the type of wood, the width, the
thickness, and the length. He also gives an operational definition of what it
means for the plank to have a certain strength “in proportion to its size,” by
introducing the concept of a plank that is the longest one that would not
snap under its own weight if supported at one end. If you increased its
length by the slightest amount, without increasing its width or thickness, it
would break. He says that if one plank is the same shape as another but a
different size, appearing like a reduced or enlarged photograph of the other,
then the planks would be strong “in proportion to their sizes” if both were
just barely able to support their own weight.
He now relates how he has done actual experiments with such planks,
and found that, according to this operational definition, they are not strong
in proportion to their sizes. The larger one breaks. He makes sure to tell the
reader how important the result is, via Sagredo’s astonished response:
S
AGREDO
: My brain already reels. My mind, like a cloud momentarily illuminated
by a lightning flash, is for an instant filled with an unusual light, which now
beckons to me and which now suddenly mingles and obscures strange, crude
ideas. From what you have said it appears to me impossible to build two
similar structures of the same material, but of different sizes and have them
proportionately strong.
In other words, this specific experiment, using things like wooden
planks that have no intrinsic scientific interest, has very wide implications
because it points out a general principle, that nature acts differently on
different scales.
To finish the discussion, Galileo gives an explanation. He says that the
strength of a plank (defined as, say, the weight of the heaviest boulder you
could put on the end without breaking it) is proportional to its cross-
sectional area, that is, the surface area of the fresh wood that would be
exposed if you sawed through it in the middle. Its weight, however, is
proportional to its volume.*
How do the volume and cross-sectional area of the longer plank
compare with those of the shorter plank. We have already seen, while
discussing conversions of the units of area and volume, that these quantities
don’t act the way most people naively expect. You might think that the
volume and area of the longer plank would both be doubled compared to
the shorter plank, so they would increase in proportion to each other, and
the longer plank would be equally able to support its weight. You would be
wrong, but Galileo knows that this is a common misconception, so he has
Section 1.2Scaling of Area and Volume
Galileo discusses planks made of
wood, but the concept may be easier
to imagine with clay. All three clay rods
in the figure were originally the same
shape. The medium-sized one was
twice the height, twice the length, and
twice the width of the small one, and
similarly the large one was twice as
big as the medium one in all its linear
dimensions. The big one has four
times the linear dimensions of the
small one, 16 times the cross-sectional
area when cut perpendicular to the
page, and 64 times the volume. That
means that the big one has 64 times
the weight to support, but only 16 times
the strength compared to the smallest
one.
*Galileo makes a slightly more complicated argument, taking into account the effect of leverage (torque). The result I’m
referring to comes out the same regardless of this effect.