224 GROWTH PRINCIPLES AND THEORY 2 



c being a species-specific constant; that is, the cubic root of weight increases 

 proportional to time. Then 



diw\d.t = y]ur'^ (6.2) 



i.e. growth in weight is proportional to a surface, or embryonic growth follows 

 equation (5.24), with catabolism being negligible. 



VII. RELATIVE GROWTH 



{a) The a Home trie equation 



Organic form appears to be that problem in biology which is least amenable 

 to quantitative treatment but it can be approached in several ways. 



Two works have been pioneering in this respect : D'Arcy Thompson's On 

 Growth and Form (1942) introducing the Theory of Transformations, and Huxley's 

 Problems of Relative Growth (1932) which established the Principle of Allometry. 



Morphogenetic changes in a growing animal chiefly take place by relative growth, 

 that is, certain components increase at a higher or lower rate than others or, as also 

 may be said, growth rate is different in different spatial dimensions. In the growth 

 of the organism as a whole as discussed, time and body size are the variables 

 vmder consideration. Relative growth concerns the ratio of growth rates in 

 several components of the organism. A relation found to be of wide applicability 

 is the allometric equation. 



The relationship now termed allometry was first used by Snell (1891) for expressing the 

 ratio between brain size and body size in mammals of different species or individuals of 

 the same species but of different body size. E. Dubois (i8g8) and Lapicque (1898) applied 

 the equation for the same problem in mammals and in birds, respectively. Klatt (1919) 

 was probably the first to realize the general significance of this rule, applying it to the 

 relation between heart size and body size. The equation has found general application 

 following Huxley's work. After some terminological controversies (Huxley, J. Needham and 

 Lerner, 1941; Reeve and Huxley, 1945) the term "allometric growth" was generally 

 accepted. Relations of the same form are also found in social phenomena (Naroll and 

 Bertalanffy, 1956). 



The principle of allometry is expressed by the equation : 



y = bx^, or logjv = log b -\- cf. log x (7-0 



Hence, if a variable {e.g. size of an organ, rate of a physiological process, etc.) 

 is plotted logarithmically against another variable {e.g. size of the organism as 

 a whole), a straight line is obtained. This simple procedure shows whether the 

 allometric relation obtains, and allows estimate of the constants, a is the slope 

 of the allometric line (tangent of the angle it forms with the abscissa; a = tan a) ; 

 b is the value ofj; for the extrapolated value, x = i. If a > i, y grows faster 

 than x, or shows positive allometry. If a = i , both magnitudes grow at the same 

 rate or isometrically. If a < i, the relative increase ofp is smaller than that o^ x, 

 OT y shows negative allometry. 



If the variable^ is expressed per imit size of x, 



yjx = bx""^^ (7.2) 



i.e. the slope of the allometric line is negative. This is the case, for example, in the 



