ABSTRACT The Brooks-Dyar rule predicts a geometric progression in the growth increments of successive instars. Although these increments are readily calculated as ratios, analysis of the relationship between these ratios has been more problematic. In the early twentieth century, the use of logarithms was proposed as a method for calculating the equation of the line. This is as close to a standardized method of analysis that the ratios have come in entomology. It is argued here: (a) that a log scale is a misleading and misunderstood criterion in this context because growth increments do not span multiple orders of magnitude; and (b) a straight-line is an inappropriate mode of representation for growth patterns that are discontinuous. Moreover, the transformation does not provide any means for quantifying the degree of geometric progression. A simple exponential equation can be applied easily to any ratio data set to provide a more realistically curved plot of growth increments. A property of geometric progression sequences (b2 = ac) can be utilized to provide an equation for evaluating the degree of geometric progression in development. Transformations are replaced by greater biological realism and a precise method of quantifying agreement with Brooks-Dyar’s rule.

中文翻译：

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Brooks-Dyar 增长率的几何级数方程

摘要 Brooks-Dyar 规则预测了连续龄期的增长增量的几何级数。尽管这些增量很容易计算为比率，但对这些比率之间关系的分析更成问题。二十世纪初，有人提出使用对数作为计算直线方程的方法。这与昆虫学中的比率一样接近标准化的分析方法。这里有争议：(a) 在这种情况下，对数尺度是一种误导和误解的标准，因为增长增量不会跨越多个数量级；(b) 对于不连续的增长模式，直线是一种不恰当的表示方式。此外，该变换没有提供任何量化几何级数程度的方法。一个简单的指数方程可以很容易地应用于任何比率数据集，以提供更真实的增长增量曲线图。几何级数序列的性质（b2 = ac）可用于提供评估几何级数发展程度的方程。转换被更大的生物现实主义和量化与布鲁克斯 - 迪亚尔规则的一致性的精确方法所取代。