EXAMPLE 12 The Power of Growth
How do we arrive at those slopes of 1.0 and 1.2 in Figure 18.14?
You could use statistical software or your calculator to find the equation of the least- squares regression line (as discussed in Section 6.4 on page 260) through the points on the log-log plots. Here, we find approximate values for the slope a for each line from the coordinates of the points at the ends of the lines, for ages 0.42, 0.75, and 25.75. The observations and the corresponding logarithms are as follows:
| Age | Height | Log (Height) | Arm Length | Log (Arm Length) |
|---|---|---|---|---|
| 0.42 | 30.0 | 1.48 | 10.7 | 1.03 |
| 0.75 | 60.4 | 1.78 | 25.1 | 1.40 |
| 25.75 | 180.8 | 2.26 | 76.9 | 1.89 |
The slope for the line from age 0.42 to age 0.75 is the vertical change over the horizontal change in terms of log units:
The slope for the line from age 0.75 to age 25.75 is
So up to 9 months, and after 9 months. Up to 9 months, arm length grows according to . After 9 months, arm length grows according to , and we get , which is a linear relationship describing proportional growth—that is, growth according to geometric similarity. on ordinary graph paper, proportional growth appears as a straight line and allometric growth as a curve. on log-log paper, both patterns appear as straight lines.
Allometry was used by paleontologists to determine that all specimens (just six!) of the earliest bird, 4rchaeopteryx, are indeed of the same species, and that the puzzling minute fossil fish Palaeospondylus (found only in Scotland) was probably just the larval stage of a better-known fish.