More Than One Way to Size a Fish
There is often more than one way to solve a puzzle. When I was a cocky teen-ager, blissfully unaware of Randall's work determining the size of Megalodon, I tried to solve the same puzzle from a different perspective. Everything I had read on Megalodon stated that it was closely related and probably very similar to the extant White Shark in over-all morphology. I reasoned that,
1) the White Shark is a gill-breathing vertebrate,
2) gas exchange takes place across the surface of gill membranes at a finite rate that varies depending on temperature and concentration gradient,
3) surface area increases as a square of dimension, but body volume increases as a cubic dimension, and thus,
4) there had to be a cut-off point at which gill surface area would be insufficient to supply the oxygen demands of the corresponding tissue volume.
This cut-off point, would therefore give me a theoretical maximum size for megalodon.
Unfortunately, no one had measured oxygen diffusion rates across the gill membranes of a living White Shark. As an added complication, the white shark was suspected to be warm-bodied like its relatives the Shortfin Mako and Porbeagle sharks, but this had not yet been measured. So I had to make some semi-educated guesses. I picked a hypothetical water temperature of 50 degrees F (10 degrees C) and looked up oxygen diffusion rates for trout gills at that temperature. I used an average of published oxygen consumption rates per unit of body mass for the spiny dogfish (Squalus acanthias) and multiplied that figure by a fudge factor of five times, to allow for mild warm-bodiedness in the white shark. I used average tissue densities of several sharks to determine the relationship between body mass and volume, and then submerged teensy plastecine White Shark models in a graduated beaker to relate body volume to total length. Then I plotted two curves on a sheet of graph paper, total length versus body mass and oxygen demand versus body mass. The curves bisected at 15.1 metres, suggesting that was the maximum length Megalodon could be. At the time, I had no idea whether or not I was correct - or even in the right ballpark.
When I finally discovered Randall's paper on the size of Megalodon, I was gratified by how close my figure was to his (only 13% too large). At the time, I congratulated myself for obviously having made some astute guesses; in retrospect, I probably made some lucky guesses and derived a figure that happened to be close to Randall's. (The water temperature I chose was probably too cold, and the body temperature too warm; so the effects may have more-or-less canceled out). Of course, there is a fundamental difference between largest measured and largest possible. The estimated size of Megalodon based on the largest of its fossil teeth is not the same thing as maximum theoretical size. But at least my little experiment was a very interesting exercise in functional morphology.