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3 Evolution by the Numbers: Mathematical Arguments in The Origin of Species

I have deeply regretted that I did not proceed far enough at least to understand something of the great and leading principles of mathematics; for men thus endowed seem to have an extra sense.

—Charles Darwin

So every idea of Darwin—variation, natural selection, sexual selection, inheritance, prepotency, reversion—seems at once to fit itself to mathematical definition and to demand statistical analysis.

—Karl Pearson

Because Darwin’s impact on the social and scientific developments in the nineteenth and twentieth century has been so large, his work, particularly The Origin of Species, has garnered a lot of attention from scholars in a variety of fields, including rhetoric, history, philosophy, and literature. In rhetoric alone, his persuasive efforts have been the focus of papers by authors such as: John Angus Campbell, Jeanne Fahnestock, Alan Gross, and Carolyn Miller (Campbell, “Perspective,” “Polemical,” “Rhetorician,” “Invisible,” “Believed;” Fahnestock, “Series Reasoning;” Gross, “Taxonomy;” and Miller and Halloran). Their works explore different aspects of his rhetorical strategy in The Origin of the Species, including the use of analogy between the human breeder and nature, to help his audience understand the operation of natural selection and the importance of the rhetorical figures incrementum and gradatio in making the argument about variation and diversity among groups of organisms.

Despite the wide range of topics and issues in argument covered by rhetorical scholars, there has as yet been no substantive discussion about the rhetorical importance of mathematics in making arguments in The Origin of the Species. The purpose of this chapter is to offer arguments and analyses that suggest that Darwin relies heavily on mathematical elements such as quantification and basic arithmetical operations for support and invention of his arguments for dynamic variation, relation by descent, and the principle of divergence of character in The Origin of Species. It will also make the case that, by following the best practices of quantitative induction, Darwin hoped to establish an ethos of precision and rigor for his work which was commensurate with the rising importance of quantification to the study of biological phenomena in the middle of the nineteenth century.

Mathematical Darwin?

Though historians and philosophers of science have expended considerable effort tracing the development of different mathematical fields and examining their political, cultural, and even rhetorical influence (e.g., Cullen; Patriarca), they have not, with rare exceptions, taken up investigations into the role of mathematics in Darwin’s arguments. A survey of eleven books and nine articles published by historians, philosophers, and rhetoricians of science, most published in the last twenty-five years, reveals that few texts associate Darwin’s arguments with mathematical reasoning (Appendix A). Those texts that do associate the two predominantly comment either on the lack of mathematical reasoning in the text, or on Darwin’s inability to use mathematics to make his case (Ghislen; Hull; Gale; Depew and Weber). Only four assign any real importance to mathematics in Darwin’s arguments (Browne; Schweber; Parshall; Bowler).

Mathematics in The Origin of Species

The previous examination of selected books and articles in the history, philosophy, and rhetoric of science suggests that many modern scholars do not believe or have not considered mathematical argument as an important facet of Darwin’s persuasive strategy in The Origin of Species. These results raise the question, “If mathematics plays such an important role in Darwin’s argument, why is it that so few scholars in rhetoric and history bothered to write about it?”

A cursory review of the text itself reveals that there are very few places where mathematical symbols, numbers, tables, equations, etc. are used. This scarcity of mathematical notation is puzzling even to those who argue in favor of the importance of mathematics in The Origin of the Species, like historian Janet Browne, who comments on the scarcity of mathematics in the text:

That Darwin’s botanical arithmetic has been neglected by historians is partly his own fault. In On the Origin of Species, he barely referred to his botanical statistics or the long sequence of calculations which he had undertaken from 1854 to 1858. He compressed and simplified these into a few meager paragraphs, giving his reader only six pages of statistical data to fill out the discussion of “variation of nature” in Chapter II. (53)

Despite its absence in the actual text, a brief review of Darwin’s notebooks, letters, the published manuscript of his “big species book,” and Variation of Plants and Animals under Domestication, reveals the extent to which mathematics influenced the development of his theories.1 In these publications, Darwin supplies his readers not only with lists of quantitative evidence and calculations, but also with occasional glimpses of the degree to which these data and calculations helped him formulate his conclusions.

The existence of precisely quantified data and calculations in these extrinsic sources, however, still does not explain why, if they were so important to Darwin’s argument, the majority of them were left out of his text. The answer to this query is provided by Darwin himself in the introduction to The Origin of the Species.

I can here give only the general conclusions at which I have arrived, with a few facts in illustration, but which, I hope, in most cases will suffice. No one can feel more sensible than I do of the necessity of hereafter publishing in detail all the facts, with references, on which my conclusions have been grounded; and I hope in a future work to do this. (4)

Here Darwin explains that he is able to give only a general outline of his theory, and as a result, has to forgo presenting all of the data and discussion that he might have otherwise provided. The reason for this brevity is that he has been rushed into publication by the emergence of Alfred Russel Wallace’s theory of evolution which, for all intents and purposes, offered the same conclusions as his own. Additionally, Darwin’s lack of specific, quantitative detail may have been a strategy to make his work accessible to a wider readership for whom a text dense with quantitative data and arithmetical calculations would have seemed too formidable (Beer viii).

Besides the infrequency of quantified data and operations in The Origin of the Species, influential historians discussing the arguments in the text, notably David Hull, have made the case that Darwin could never have integrated mathematical reasoning into his arguments because this type of reasoning was deductive and could not be brought into the service of an inductive theory. In his book, Darwin and His Critics, Hull takes the position that because Darwin was developing arguments in the non-physical sciences, deductive mathematical reasoning could not aid him in prosecuting his argument:

Darwin could not help but know the crucial role which mathematics had played in physics, since Herschel had repeatedly emphasized it in his Discourse, but it did not seem to be in the least useful in his own work in biology. . . . For Darwin, mathematics consisted of deductive reasoning, and he distrusted greatly “deductive reasoning in the mixed sciences.” In his own work, he seldom was presented with a situation in which he could use such deductive reasoning. He was constantly forced to deal in probabilities, and no one could tell him how to compute and combine such probabilities. (12–13)

Hull’s assessment of the impossibility of Darwin’s use of mathematics rests on the assumption that Darwin believed that mathematical reasoning was deductive, and therefore could not be used in inductive arguments. Like Hull, rhetorical theorists most likely also miss the rhetorical dimension of the text because they assume that mathematical warrants are deductive. Though this does not preclude them necessarily from functioning in Darwin’s text, it does remove them, in the minds of most rhetorical analysts, from being the focus of a rhetorical investigation. To my knowledge there has been no explicit statement such as Hull’s that this consideration has kept rhetoricians from examining the mathematical aspects of Darwin’s arguments. However, this restriction is articulated in influential theoretical texts such as Philip Davis and Rueben Hersh’s “Rhetoric and Mathematics,” in which they write that, in the minds of most rhetorical scholars, “If rhetoric is the art of persuasion, then mathematics seems to be its antithesis. This is believed, not because mathematics does not persuade, but rather that it seemingly needs no art to perform its persuasion” (53).

Philosophers, historians, and rhetoricians of science have not recognized an important role for mathematics in Darwin’s arguments. However, careful examinations of early nineteenth century botany and geology, a detailed investigation of Darwin’s ideas in his notebooks and letters, and a close textual analysis of the arguments in The Origin of the Species, reveal that quantification and basic mathematics were important to his work. They show that mathematics played a central role in Darwin’s formulation and defense of his arguments, including his rhetorical efforts to establish an ethos of precision and rigor for his work.

Keeping Count: The Rise of Statistics in the Nineteenth Century

One of the fundamental characteristics of robust science in both modern and Victorian characterizations is quantification. Without the ability to “translate” natural phenomena into the language of numbers, induction leading to the formation of empirical laws could not commence. It was, therefore, the first step in the formation of any science to discover the method or system of measure on which quantitative induction could be founded.

Although various attempts had been made in the eighteenth century by Linnaeus and others to quantify certain aspects of biological research (such as the classification of leaves and reproductive organs in plants), they were all considered, at least by nineteenth century standards, artificial, and therefore not sufficient for the basis of a quantitative science. At the beginning of the nineteenth century, however, two important developments afforded new opportunities for advancing quantitative investigations of organic phenomena. The first was the increased interest in and use of statistics. The second was the discovery of fossils of extinct organisms, whose forms were completely alien from existing flora and fauna, which focused attention on questions about the origin and dispersion of organic forms. In his search of evidence and arguments for The Origin of Species, Darwin was influenced by both of these developments, which inspired him to cultivate quantitative evidence and mathematical arguments to support his theories of variation and evolution.

Though vital statistics (numbers of births and deaths) had been collected since the seventeenth century by religious and political organizations, the number of investigations and degree of attention to their results was limited to very small audiences.2 At the beginning of the nineteenth century, however, social, political, and economic contingencies converged to create what statistical historian Harald Westergaard dubs the “The Era of Enthusiasm” for statistics (136), and Ian Hacking calls a period with “a professional lust for measurement” (5).

There is no single, agreed upon cause for this sudden interest in and collection of statistics. Some historians attribute it to the need to for precise measurement required by the Industrial Revolution, which gathered momentum at the beginning of the nineteenth century (Hacking 5). Others argue that it was the result of a sudden increase in the availability of statistical information that followed the end of the Napoleonic wars (Chatterjee 267). Yet others contend, perhaps most convincingly, that the supply of statistical data increased to meet a greater demand by governments who required quantitative data in order to make better informed political decisions and more persuasive policy arguments (Westergaard 141; Cullen 19–20; Patriarca 13–14). In particular, governments required statistics on birth and death rates as well as the resources of their domain and the domains of other nations to make rational economic policy decisions.

The statistical fever that had grabbed hold of politicians and moral philosophers in the early decades of the nineteenth century also infected geologists and botanists who were exploring the vast biodiversity of the Americas and Australia. Like political economists, they began in earnest to gather quantitative data on organic populations and the conditions under which they thrived. However, unlike their counterparts, the ends for their statistical efforts were affected by important scientific questions raised by new geological theories which assumed a dramatically older earth and grappled with new fossil evidence of organisms unlike any flora or fauna known to Victorian science. These discoveries, which challenged the tenants of the Christian doctrine of creation, encouraged investigations attempting to reconcile, to some extent, the scientific evidence with religious doctrine.

These efforts gave rise to a new field of biogeography, whose aim was to answer fundamental questions about organic populations, including: “What causes influence the thriving or extinction of particular species?”; “What is the distribution of species and genera upon the globe?”; “What is the population of any given species?”; “How can we account for the appearance of new species throughout geological time?”; and “What are the laws by which plants and animals of different parts of the earth differ?”3

Part of the spirit of this new field was that these questions needed to be answered not through classification of organisms and minerals, but rather through the juxtaposition of quantitative facts about climate, population, and location. Alexander von Humboldt, one of the early founders of biogeography, proclaims this goal in Aspect of Nature in Different Lands and Different Climates (1849):

Terrestrial physics have their numerical element, as has the system of the universe, or celestial physics, and by the united labors of botanical travelers we may expect to arrive gradually at a true knowledge of the laws which determine the geographical and climactic distribution of vegetable forms. (108)

The path towards a new biogeographical physics was laid down in works such as Alphonse de Candolle’s “Essai Elementaire de Geographie Botanique” (Elementary Essay on Botanical Geography) (1820), Robert Brown’s General Remarks, Geographic and Systemical, on the Botany of Terra Australis (1814), and Joseph Hooker’s Botany of the Antarctic Voyage, Vol. 2 (1853).4 In their texts, statistics on temperature, elevation, size of organic populations, and the size and distribution of genera and species were used to make arguments bearing on the questions of distribution, variation, origination, etc. of plants and animals.

Robert Brown’s work exemplifies the biogeographer’s efforts to use basic, arithmetical operations and quantified data to compare and make arguments about variation in organic phenomena and the relationship of this variation to geographic and climactic conditions. In the Botany of Terra Australis, for example, Brown tests the assertion commonly held by nineteenth century botanists that dicotyledonous plants outnumber monocotyledonous plants by examining whether climate affects the numbers of either type in the general botanical population: 5

With a view to determine how far the relative propositions of these two classes [dicotyledons and monocotyledons] are influenced by climate, I have examined all the local catalogues or Floras which appear most to be depended on. . . . The general results of this examination are that from the equator to about 30° of latitude, in the northern hemisphere at least, the species of dicotyledonous plants are to monocotyledonous plants as about 5 to 1 . . . and that in the higher latitudes a gradual diminution of dicotyledonous takes place, until at about 60° N. lat. and 55° S. lat. they scarcely equal half their intratropical proportion. (Miscellaneous Botanical Works 8)

In the passage Brown draws on quantitative descriptions of location and previously tabulated statistics on the number of species of each sort of plant as well as calculated ratios to describe the limits of the geographical distribution of dicotyledonous and monocotyledonous plants. He concludes, based on the data and calculations, that dicotyledonous plants are abundant near the equator but become less abundant in northern latitudes. This conclusion provides precise quantitative detail supporting what was otherwise an anecdotal assumption about the difference in dicotyledonous and monocotyledonous plants in the general population of flora. It also supplies new information about the relationship between location and the thriving of dicotyledonous plants, which was previously unknown.

Darwin and the Biogeographers

The quantitative data and mathematical methods used by Brown and other biogeographers inspired Darwin to develop quantitative/mathematical arguments about the phenomena of variation and evolution. Evidence that Darwin was inspired by their methods can be found in his reading habits, in the private thoughts he recorded in his notebooks, and in his letters discussing the construction of his arguments for The Origin of the Species.

A brief assessment of Darwin’s reading habits during the development of The Origin of the Species reveals that Darwin was familiar with the central works of biogeography and read them as he was developing his ideas for his masterwork. He was most likely introduced to them by his mentor John Henslow, who presented him, before he left Cambridge, with Humboldt’s Personal Narrative of Travels to the Equinoctial Regions of the New Continent during the Years 1799–1804 (1814–1829) in which the explorer discusses the importance of quantification in the discovery of natural laws and offers statistics on temperature, altitude, longitude and latitude, etc. for various regions and flora in South and Central America (Schweber 205). It is clear that Darwin read this book and found it important because his copy contains extensive notes. According to the compiler of Darwin’s marginalia, Mario A. Di Gregorio. Humboldt got Darwin “thinking about the distribution and the relation of organism to organism in the context of isolation, extinction, and the breeding of wild and domesticated animals” (xxxv). In addition to Humboldt, Darwin also owned multiple editions of Lyell’s Principles of Geology (1st, 5th, 6th, 7th, 9th, 10th, and 11th) (Di Gregorio 530–44). In the fifth edition, volume three, chapter five, “Laws which Regulate the Distribution of the Species,” discusses at length the important figures of biogeography and their work. Like Humboldt’s Personal Narrative, Darwin marked and commented heavily on the text, providing evidence for his interest in their ideas (Di Gregorio 535–36).

When Darwin returned home and began developing the ideas that would be presented in The Origin of the Species, he continued to show interest in the work of biogeographers, as references to their works and ideas in his notebooks testify. At the end of Notebook C, in a section titled, “To be Read,” for example, Darwin lists “Brown at end of Flinders & at end of Congo voyage, Decandolle. Philosophie, or Geographical Distribution. <<in Dict. Sciences. Nat. in Geolog Soc.>>” as texts which he believed would be beneficial to the development of his ideas (268).6 Of these two works, Alphonse de Candolle’s “Essai Elementaire de Geographie Botanique” was of particular importance to developing his ideas in The Origin of the Species. Of all the books listed in Gregorio’s Charles Darwin’s Marginalia, Darwin’s copy of “Gèographie Botanique” has the most marginalia, prompting Di Gregorio to write that the work “seems to be a catalyst for much thinking around distribution, the struggle for existence, isolation, and consequently selection” (xxxiv).

Not surprisingly, the writings in the notebooks themselves echo the biogeographers’ sentiments about the importance of developing a quantitative/mathematical approach to the study of plant and animal life. In Notebook D, for example, Darwin echoes Humbolt’s sentiment that investigators of organic phenomena could and should emulate the quantitative inductive methods used in celestial and terrestrial physics:

Astronomers might formerly have said that God ordered each planet to move in its particular destiny.—In the same manner God orders each animal created with certain form in certain country, but how much more simple, & sublime power let attraction act according to certain laws such are inevitable consequent let animal be created, then by fixed laws of generation, such will be their successors.—let the powers of transportal be such and so will be the form of one country to another.—let geological changes go at such a rate, so will be the numbers & distribution of the species!! (Notebook B, 101–102)

In this passage, Darwin reveals a clear link in his thinking between the methods of astronomy and the study of organic phenomena ascribed to by biogeographers. Just as astronomers create general laws by quantifying the period of the revolution of celestial bodies, their distance from one another, the amount of space they sweep out in a given period, etc., the biogeographer could arrive at quantitative laws describing the number and distribution of the species by determining the rate of generation of organisms, the rate in change of geological conditions, and the power of transport.

Darwin’s reading habits and his notebooks supply evidence that he was considering the idea that natural laws might be discovered by working inductively from quantified evidence as he was gathering his thoughts for what would become The Origin of Species. However, it was not until the mid-1850s, when he began writing the manuscript of Natural Selection (his “big species book”), that there is evidence that Darwin began to put these ideas into practice by gathering statistical data and making arithmetical calculations to test his theory of evolution and build his arguments.

In 1855 Darwin began lengthy correspondences with Asa Gray and H. C. Watson in which he asked the botanists to supply him with information about genera and species, and discussed with them calculations of the ratios of varieties to genera in large and small genera. The quantified data and mathematical calculations discussed in these letters serve as sources of invention and argument for a relationship of descent and the principle of divergence of character in The Origin of the Species.

In his second letter to Asa Gray, on August 24, 1855, for example, Darwin hints to Gray that if he could get a reliable systematist to help him identify “close species,” (i.e., species that closely resemble one another) he could calculate whether there was a propensity for larger genera to have more of these types of species than smaller genera:

It occurred to me that if I could get some good systematists . . . to mark (without the object being known) the close species in a list; then if I counted the average number of the species in such genera, & compared it with the general average . . . of the species to the genera in the same country; it would, to a certain extent, tell whether on average the close species occurred in the larger genera. (Darwin to Gray, August 24, 1855)

Darwin makes a similar request to Watson, who obliges him by marking close species in his catalogues of English plants (Watson to Darwin August 17, 1855). Using Watson, Gray, and other botanists’ marked compendiums of species, Darwin searched for proof of a general pattern in the ratios of closely related species in large and small genera. He hoped to find that larger genera had a greater number of close species, while smaller genera had fewer, more distinct species. He reasoned that if this pattern did exist it would support his theory of variation and relation by descent because larger, more successful genera would be producing new species that would be recent and closely related. Conversely, unsuccessful genera would not be producing newer species, and some of the older species they had produced would have died out, leaving gaps between the existing species and making them appear less closely related (Watson to Darwin November 19, 1854 n2). Evidence of this strategy appears in the draft of the big species book from which The Origin of the Species was abstracted (Table 1).

The Origin of Species: A General Overview of the Argument

An examination of Darwin’s letters, notebooks, and reading lists reveals that he was familiar with ideas and methods of biogeography and that he adopted them in his search for evidence and arguments to support his theories of variation and evolution. In order to make the case that mathematics is an important aspect of Darwin’s argument in The Origin of the Species, however, it is imperative to show that mathematical argumentation exists in the text and to understand what role it plays in supporting Darwin’s conclusions. Towards this end, the next two sections examine the arguments in chapters two and four of text as well as the peripheral documents associated with them. This investigation reveals that Darwin employs mathematical argument in the book, and that this mathematical argumentation plays an important role in helping him invent and/or support his arguments for the existence of a process of dynamic variation and the principle of divergence of character.

To understand the significance of the arguments in chapters two and four in the overall scheme of Darwin’s argument, it is useful to review the primary conclusions he attempts to establish in the text. The basic arguments in are: (1) that organisms are highly plastic and can be made to vary to a great degree, (2) that variation accumulates over time, resulting in populations of organisms that were once related becoming physically distinct, (3) that the spread of variation in nature is the result of natural selection, and (4) that the more diversity in a species or genera the more likely it is that its members will successfully reproduce.

These primary arguments are introduced and developed in the first four chapters of the book. The other chapters of the text are concerned with presenting qualitative evidence from geology, animal behavior, comparative anatomy, and other areas of knowledge that support Darwin’s four main arguments and his efforts to address possible disputations of his position.

Chapter II: Variation under Nature

In the second chapter of The Origin of the Species, “Variation under Nature,” Darwin argues for the possibility of selection without human intervention through the process of natural selection. He achieves this goal with the help of quantified comparisons using arithmetical operations that prove that not only are taxonomic categories of species fuzzy, but also that this fuzziness can be accounted for by conceiving of diversity as the result of the dynamic process of continual variation, revealing a relationship of descent between the different levels of the taxonomic hierarchy.

In the first portion of chapter two, Darwin sets up his argument by refuting the position of special creationists who believed that each species identified in the taxonomic hierarchy marked a unique creation that was readily identifiable by the existence of an indelible set of features. He argues that if this position is correct, then there should be a definite consensus about which organisms belong in a particular category. In order to test the veracity of this assumption, Darwin quantitatively compares the categorization statistics made by experts in the field, including H.C. Watson:

Compare the several floras of Great Britain, of France, or of the United States, drawn up by different botanists, and see what a surprising number of forms have been ranked by one botanist as a good species, and by another as mere varieties. Mr. H.C. Watson, to whom I lie under deep obligation for assistance of all kinds, has marked for me 182 British plants, which are generally considered as varieties, but which have all been ranked by botanists as species. . . . Under genera, including the most polymorphic forms, Mr. Babington gives 251 species, whereas Mr. Bentham gives only 112, - a difference of 139 doubtful forms! (41)

Using their own data, Darwin reveals that even experts in plant identification and categorization come to astonishingly little agreement about which organisms should be ranked as varieties and which as separate species. By casting doubt on the fixity of taxonomic categories, he creates an opportunity to present his own theories of evolution and natural selection, which he believes more adequately account for the data.

He opens the second portion of the chapter by clearly laying out his position:

Hence I look at individual differences, though of small interest to the systematist, as of high importance for us, as being the first step towards such slight varieties. . . . And I look at varieties which are in any degree more distinct and permanent, as steps leading to more strongly marked and more permanent varieties, and at these latter, as leading to sub-species and to species. . . . I attribute the passage of a variety, from a state in which it differs very slightly from its parent to one in which it differs more, to the action of natural selection in accumulating . . . differences of structure in certain definite directions. (44).

In these lines, Darwin presents a vision of diversity in nature as a dynamic process rather than as a static condition. He argues that the small differences observed in individual organisms can spread by descent throughout successive generations, making the offspring of those individuals slightly different from the general population from which they originated. These differences can widen through the continued accumulation of variation and eventually transform distinct varieties into distinct species. This dynamic process, Darwin argues, can be attributed to natural selection, which he defines as, “the preservation of favorable variations and the rejection of injurious variations” (68).

Once he has established his position on the source and character of diversity, Darwin presents arguments to attempt to link the size and range of a group of organisms at a particular level of the taxonomic hierarchy to the number of subordinate categories of organisms associated with that group. In order to do this, he depends both on arithmetical calculations using quantitative data and on the rhetorical/logical commonplace (topos) of the more and the less with which he establishes a connection between size/range of a population and the characteristic of diversity. In Book II of the Topics, Aristotle explains this strategy of argument: 7

Moreover, argue from greater and less degrees. There are four commonplace rules. One is: see whether a greater degree of the predicate follows a greater degree of the subject. . . . For if an increase of the accident follows an increase of the subject, as we have said, clearly the accident belongs; while if it does not follow, the accident does not belong. You should establish this by induction. (The Complete Works, 114b 35–115a 6)

Using statistics from available botanical compendia, Darwin calculates the number of botanical varieties belonging to species with the greatest estimated population sizes and ranges in hopes of discovering some general pattern in, or connection between, these species: “I thought that some interesting results might be obtained in regard to the nature and relations of the species which vary most, by tabulating all the varieties in several well-worked floras” (Origin 45).

To make his calculations, Darwin first divides the species in the compendia into large and small species according to the author’s size designations. He then divides the number of species in the large and small groups by the number of varieties that are connected with them to produce an average of the number of varieties for each species, large and small. The results of these tabulations and comparisons reveal that there is a correlation between the size and range of a species’ population and the number of varieties recorded for that species (Table 2).

Table 2. Arithmetical Comparisons of the Ratios between Species and Varieties in Large and Small Genera. Reprinted from Charles Darwin, Charles Darwin’s Natural Selection, Ed. R.C. Stauffer, p. 150. © 1975. Used by permission of the publisher, Cambridge University Press.

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 importance of variety to the evolutionary success of organic populations.

In a letter to Hooker in August of 1857, Darwin’s belief in the importance of these calculations to the development of his theory is evident.

I intend dividing the varieties into two classes, as Asa Gray and Henslow give the materials, and, further, A. Gray and H.C. Watson have marked for me the forms, which they consider real species, but yet are very close to others; and it will be curious to compare results. If it will all hold good it is very important for me; for it explains, as I think, all classification, i.e. the quasi-branching and sub-branching of forms, as if from one root, big genera increasing and splitting up, etc., as you will perceive. But then comes in, also, what I call a principle of divergence, which I think I can explain. (Darwin to Hooker, August 22, 1857)

This shift of attention towards the importance of the breadth of variation to evolutionary success inspired Darwin to think more carefully about why variation might matter so much in the process of selection. These investigations also led him to develop his principle of divergence of character, which responds to a question he and his critics considered a major obstacle to any theory of variation: “How do the small differences that are observable between populations of closely related species and varieties grow into the large differences that we see between genera, families, etc.?” (Browne 74).

Because the calculated ratios showed that greater variety was a hallmark of species and genera with larger populations and ranges, Darwin felt encouraged to explain evolutionary success in terms of variation. He reasoned that when a species reaches equilibrium between its numbers and resources, the only way it could continue to grow is through diversification. Evidence from The Origin of the Species suggests this line of thinking:

Take the case of a carnivorous quadruped, of which the number that can be supported in any country has long ago arrived at its full average. If its natural powers of increase be allowed to act, it can succeed in increasing . . . only by its varying descendants seizing on places at the present occupied by other animals: some of them, for instance, being able to feed on new kinds of prey . . . some inhabiting new stations. . . . The more diversified in habits and structure the descendents of our carnivorous animal became, the more places they would be enabled to occupy. (93)

Using the hypothetical case of the “carnivorous quadruped,” Darwin argues that a species’ survival and replication can be improved if some of its members can expand, through variation, into ecological niches where there is less competition for resources with other conspecifics. In a separate passage, he extends this reasoning, concluding that because larger genera and species produce greater degrees of variation, they will be better off in the struggle for existence.

In a large genus it is probable that more than one species would vary. . . . In each [large] genus, the species, which are already extremely different in character, will generally tend to produce the greatest number of modified descendents; for these will have the best chance of filling new and widely different places in the polity of nature. (99)

In addition to making the case that bigger is better, Darwin argues by implication that the breadth of diversity of larger genera and species promotes further diversity and population expansion. This means that larger genera and species will most likely continue to grow and outstrip their smaller counterparts, unless checked by some unforeseen circumstances. The supposed correlation between size and variation assumed in this passage suggests that, for Darwin, bigger genera and species are bigger in the first place because they are more diverse. Thus, diversity itself becomes a crucial factor in evolutionary success.