Based on this arithmetical comparison on the size of Genera and number of species, Darwin argues:
In any limited country, the species which are most common, that is abound most in individuals, and the species which are most widely diffused within their own country . . . often give rise to varieties sufficiently well-marked to have been recorded in botanical works. Hence it is the most flourishing, or, as they may be called, the dominant species . . . which oftenest produce well-marked varieties. (45–46)
What Darwin discovers, or confirms, as the result of his calculations, is that the more populous species tend to have a greater number of identified varieties associated with them. This correlation is accounted for by his dynamic theory of natural diversity because a correlation between the size of a population and the development of sub-populations would be expected as larger populations would have more offspring, and, therefore, a greater number of variations to be selected.
The correlation between the calculated size of a species and the number of varieties associated with it supports Darwin’s argument that diversity in nature is the result of the production of variations. In order to strengthen the conviction of the audience that this relationship is legitimate and to make the case that the relationship exists at all levels of the taxonomic hierarchy, Darwin predicts that the same relationship will be found between genera and species. To validate this prediction he conducts further calculations and comparisons to assess whether or not the principle holds true at the taxonomic level of genera. He walks his readers through his process of calculating and explains his results:
If the plants inhabiting a country and described in any Flora be divided into two equal masses, all those in the larger genera being placed on one side, and all those in the smaller genera on the other side, a somewhat larger number of the very common and much diffused or dominant species will be found on the side of the larger genera. (46)
Here, Darwin affirms his quantitative prediction that the same correlation between population size and variation which exists between species and varieties also exists between genera and species.
Strengthened by the predictive power of his model and the accumulating evidence, Darwin makes a point to emphasize the success of his theory in accounting for the patterns revealed by his calculations:
From looking at species as only strongly-marked and well-defined varieties, I was led to anticipate that the species of the larger genera in each country would oftener present varieties, than the species of the smaller genera; for wherever many closely related species . . . have been formed, many varieties or incipient species ought, as a general rule, to be now forming. . . . On the other hand, if we look at each species as a special act of creation, there is no apparent reason why more varieties should occur in a group having many species, than in one having few. (46–47)
In addition to playing up the success of his prediction, Darwin also challenges special creationists to account for the same results. If the different taxonomic categories did in fact represent unique populations of organisms that shared no relationship with other populations, then what would account for the correlations his comparisons reveal? Though opponents of his theory might argue that these correlations are coincidental, Darwin suggests here that the fit between the patterns he describes in the quantitative data and the process of variation that he proposes in his theory is too good to be coincidental (47).
An analysis of the arguments in the second chapter of The Origin of the Species reveals that Darwin employed quantitative comparison and basic arithmetical operations to support his theories of dynamic variation and relation by descent between the different levels of the taxonomic hierarchy. In the opening portion of the chapter, he uses quantitative comparison with precise numerical values to challenge the veracity of the existing paradigm of special creation by revealing that, among experts, there is no clear consensus on the categorization of organisms in nature. Once he has cast doubt on the theory of his opponents and offered his own, he accumulates evidence to support it. With evidence from calculated averages and precise numerical comparison and argument from the commonplace of the more and the less, he reasons that a correlation exists between the size of populations and the development of recognized variations within related subpopulations. These connections suggest that there is a relationship of descent between different levels of the taxonomic hierarchy wherein varieties associated with a given species are actually variations of a common ancestor, and so forth, up the taxonomic hierarchy.
Without the support of overt, quantitative comparisons and behind-the-scenes mathematical operations, Darwin’s argument for the existence of relation by descent would have been purely speculative. But by using mathematical comparison and a quantitative commonplace, Darwin hoped to establish an ethos of precision and rigor commensurate with the conventions for robust scientific argumentation prescribed by Herschel and Whewell and the values of his target audience of geologists, botanists, and zoologists who were caught up, like himself, in the biogeographical revolution.8
Chapter IV: Natural Selection and Calculating Diversity
With the evidence and arguments in place that diversity in nature is the result of the spread of variations through organic populations and that a struggle for existence takes place in nature, Darwin proceeds to describe the details of species formation by natural selection.9 In the fourth chapter of The Origin of Species, arithmetical computation and comparison of ratios help Darwin discover and support new lines of argument about selection and species formation, namely: (1) that the more diversified a group of organisms are the better they will do in their struggle for existence, and (2) that the success of highly divergent organisms in part explains how great degrees of difference come to exist between related species.
In the initial stages of his calculations of the ratios of varieties to species, Darwin divided the total number of organisms he was investigating into “large” and “small” groups and calculated the average number of varieties for each species and species for each genus in these size categories. He then compared the average number varieties calculated for the large and small categories of species and genera to determine whether there was a correlation between the size of a species or genre, and the amount of variation produced. (This is the argument strategy explained in the detailed discussion of chapter two of The Origin of the Species.)
These calculations revealed that those genera and species designated “large” had more species and varieties. This evidence supported his conclusion that there was a relationship between the size and range of a population and the number of variations it had. A communication from Sir John Lubbock, the son of Darwin’s neighbor at Down in the summer of 1857, however, apprised Darwin that building his case on assumed average estimates of size created problems in establishing rationally defensible comparisons of the relative degree to which larger genera might be better producers of species and varieties. Lubbock suggested that instead of averages, Darwin should calculate the ratios of varieties to species in large genera and then use them to predict the expected ratio of varieties to species in small genera. Although Darwin was initially skeptical about Lubbock’s suggestion, he reworked his estimates with good results:
I have divided the New Zealand Flora as you suggested, there are 339 species in genera of 4 [species] and upwards, and 323 in genera of 3 [species] and less. The 339 species have 51 species presenting one or more varieties. The 323 species have only 37: proportionately (339:323 :: 51 : 48.5) they ought to have had 48 1/2 species presenting vars.–So that the case goes as I want it, but not strong enough, without it be general, for me to have much confidence in. / I am quite convinced yours is the right way; I had thought of it, but should never have done it had it not been for my most fortunate conversation with you. (Darwin to Lubbock, July 14, 1857)
What Darwin discovered in working out the projected ratios of varieties to species in large and small genera was that, in the case of small genera, there are fewer varieties than expected, or, in the case of larger genera, more varieties than expected. The results of these calculations shifted Darwin’s attention away from the correlation between the size and range of a group of organisms at a particular level of the taxonomic hierarchy and the number of subordinate categories of organisms associated with that group, and towards the