Packard (Packard, 2012) restudied Huxley's measurements of Uca pugnax and presented not only a two-parameter power function but also a three-parameter power function; this three-parameter model ‘is better than the two-parameter model for describing the observations’. The two-parameter model has a biological meaning, as explained by Huxley (Huxley, 1924); the three-parameter model has no biological meaning.
I have shown (Geraert, 2004) that there is a constant change in the relationship (and not a constant relationship) between a small amount of growth of body part y compared with that of body part x; mathematically speaking, the ‘second difference’ is constant. This second difference is the growth rate and is present in the quadratic factor of a quadratic equation; the other factors in the equation have no biological meaning but are necessary to position the quadratic curve in a diagram.
In my study (Geraert, 2004), growth is followed from the new-born stage to the adult; in Huxley's study on the fiddler crab, a comparison is made among adult males; as these adults show a very large variation in the development of the claw, Huxley (Huxley, 1924; Huxley, 1932) interpreted this also as ‘growth’. An attempt is made to see whether a quadratic parabola can also be used to describe variation in adults, called here ‘comparative’ growth.
In the case of Uca pugnax, a quadratic curve describes the comparative growth in males in a satisfactory way; moreover, it also has a biological meaning. Power curves with two parameters and three parameters (Packard, 2012) describe the phenomenon as well, but the three-parameter curve has no biological meaning. In my study (Geraert, 2004), the quadratic curve describes real growth in such an impressive way that fare-reaching conclusions could be made; it seems not necessary to introduce similar conclusions for comparative growth.
In a study on comparative growth in adults, if a curved line is obtained it is interesting to evaluate whether a quadratic curve is appropriate to describe the phenomenon; if the match is good enough, the ‘growth’ rate, indicated by the quadratic factor, could help us to make assumptions about larger and/or smaller values. The term allometry can be continued, although it can be mathematically formulated by a quadratic curve instead of a power curve.