40
Salviati address the point specifically:
S
ALVIATI
: ...Take, for example, a cube two inches on a side so that each face
has an area of four square inches and the total area, i.e., the sum of the six
faces, amounts to twenty-four square inches; now imagine this cube to be
sawed through three times [with cuts in three perpendicular planes] so as to
divide it into eight smaller cubes, each one inch on the side, each face one
inch square, and the total surface of each cube six square inches instead of
twenty-four in the case of the larger cube. It is evident therefore, that the
surface of the little cube is only one-fourth that of the larger, namely, the ratio
of six to twenty-four; but the volume of the solid cube itself is only one-eighth;
the volume, and hence also the weight, diminishes therefore much more
rapidly than the surface... You see, therefore, Simplicio, that I was not mistaken
when ... I said that the surface of a small solid is comparatively greater than
that of a large one.
The same reasoning applies to the planks. Even though they are not
cubes, the large one could be sawed into eight small ones, each with half the
length, half the thickness, and half the width. The small plank, therefore,
has more surface area in proportion to its weight, and is therefore able to
support its own weight while the large one breaks.
Scaling of area and volume for irregularly shaped objects
You probably are not going to believe Galileo’s claim that this has deep
implications for all of nature unless you can be convinced that the same is
true for any shape. Every drawing you’ve seen so far has been of squares,
rectangles, and rectangular solids. Clearly the reasoning about sawing things
up into smaller pieces would not prove anything about, say, an egg, which
cannot be cut up into eight smaller egg-shaped objects with half the length.
Is it always true that something half the size has one quarter the surface
area and one eighth the volume, even if it has an irregular shape. Take the
example of a child’s violin. Violins are made for small children in lengths
that are either half or 3/4 of the normal length, accommodating their small
hands. Let’s study the surface area of the front panels of the three violins.
Consider the square in the interior of the panel of the full-size violin. In
the 3/4-size violin, its height and width are both smaller by a factor of 3/4,
so the area of the corresponding, smaller square becomes 3/4x3/4=9/16 of
the original area, not 3/4 of the original area. Similarly, the corresponding
square on the smallest violin has half the height and half the width of the
original one, so its area is 1/4 the original area, not half.
The same reasoning works for parts of the panel near the edge, such as
the part that only partially fills in the other square. The entire square scales
down the same as a square in the interior, and in each violin the same
fraction (about 70%) of the square is full, so the contribution of this part to
the total area scales down just the same.
Since any small square region or any small region covering part of a
square scales down like a square object, the entire surface area of an irregu-
larly shaped object changes in the same manner as the surface area of a
square: scaling it down by 3/4 reduces the area by a factor of 9/16, and so
on.
In general, we can see that any time there are two objects with the same
shape, but different linear dimensions (i.e. one looks like a reduced photo of
the other), the ratio of their areas equals the ratio of the squares of their
linear dimensions:
full size
3/4 size
half size
Chapter 1Scaling and Order-of-Magnitude Estimates