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
