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adult, is at about the lower size limit for mammals. It must eat continually,
consuming many times its body weight each day to survive.
Changes in shape to accommodate changes in size
Large mammals, such as elephants, have a small ratio of surface area to
volume, and have problems getting rid of their heat fast enough. An
elephant cannot simply eat small enough amounts to keep from producing
excessive heat, because cells need to have a certain minimum metabolic rate
to run their internal machinery. Hence the elephant’s large ears, which add
to its surface area and help it to cool itself. Previously, we have seen several
examples of data within a given species that were consistent with a fixed
shape, scaled up and down in the cases of individual specimens. The
elephant’s ears are an example of a change in shape necessitated by a change
in scale.
Large animals also must be able to support their own weight. Returning
to the example of the strengths of planks of different sizes, we can see that if
the strength of the plank depends on area while its weight depends on
volume, then the ratio of strength to weight goes as follows:
strength/weight
.
A/V
.
1/L.
Thus, the ability of objects to support their own weights decreases
inversely in proportion to their linear dimensions. If an object is to be just
barely able to support its own weight, then a larger version will have to be
proportioned differently, with a different shape.
Since the data on the cockroaches seemed to be consistent with roughly
similar shapes within the species, it appears that the ability to support its
own weight was not the tightest design constraint that Nature was working
under when she designed them. For large animals, structural strength is
important. Galileo was the first to quantify this reasoning and to explain
why, for instance, a large animal must have bones that are thicker in
proportion to their length. Consider a roughly cylindrical bone such as a leg
bone or a vertebra. The length of the bone, L, is dictated by the overall
linear size of the animal, since the animal’s skeleton must reach the animal’s
whole length. We expect the animal’s mass to scale as L
3
, so the strength of
the bone must also scale as L
3
. Strength is proportional to cross-sectional
area, as with the wooden planks, so if the diameter of the bone is d, then
d
2
.
L
3
or
d
.
L
3/2
.
If the shape stayed the same regardless of size, then all linear dimensions,
including d and L, would be proportional to one another. If our reasoning
holds, then the fact that d is proportional to L
3/2
, not L, implies a change in
proportions of the bone. As shown in the graph on the previous page, the
vertebrae of African Bovidae follow the rule d
.
L
3/2
fairly well. The
vertebrae of the giant eland are as chunky as a coffee mug, while those of a
Gunther’s dik-dik are as slender as the cap of a pen.
Galileo’s original drawing, showing
how larger animals’ bones must be
greater in diameter compared to their
lengths.
Chapter 1Scaling and Order-of-Magnitude Estimates