The Statistical Paleontology of Charles Lyell and the Coupon Problem
McCready, Thomas A., Schwertman, Neil C., The American Statistician
Lyell, a founder of the science of geology, used statistical models to describe the changes that had occurred in the earth and its environment. From this model he attempted to establish a time frame for each epoch. This article shows that Lyell's model is equivalent to the classic coupon problem included in many probability texts. Furthermore, it is shown that the time frame deduced by Lyell is inconsistent with the model he was using. The proper time frame consistent with the model is provided. A second model that was considered by Lyell is also investigated.
KEY WORDS: Geology; Geometric distribution.
Charles Lyell's monumental three-volume work, Principles of Geology (1830-1833), is "the book that most geologists regard as the founding document of their discipline's modern era" (Gould 1987, p. 101). Principles is perhaps best known for its impact on Charles Darwin, who read at least the first two volumes during his celebrated voyage aboard the Beagle, and who certainly also read the remaining volume (Mayr 1982, p. 398 and p. 427). The impact of Lyell's work on Darwin's developing theory of organic evolution, cannot be overstated; in fact, it has been said that "Lyell's Principles of Geology was Darwin's 'bible' as far as the problem of evolution was concerned" (Mayr 1982, p. 406). Nevertheless, Lyell in the 1830s did not believe in evolution and is best described as a creationist: he believed that species came into being, and were essentially unchanged during their periods of existence (Gould 1987, p. 124, p. 147; Mayr 1982, p. 375). This belief was the basis of a technique for dating Tertiary deposits, a method that has been referred to as the "statistical paleontology" or the "geochronological model" of Lyell (Rudwick 1978; Tasch 1977). (It should be noted that in later years Lyell strongly supported Darwin's evolution theory).
The nature of the Tertiary deposits made them difficult to date. The older, (Primary and Secondary) strata were deposited in layers which could be traced over large areas of Western Europe, thus making possible the ordering, and thus the dating, of these strata. By contrast, the younger Tertiary deposits tended to be found in small, disconnected basins, making place-to-place correlations difficult (Gould 1987, pp. 155-157).
Lyell suggested an ingenious method, based on fossil snail shells in an existing collection, to date these Tertiary deposits (Gould 1987, pp. 155-161). Specifically, to infer the ages of a deposit, Lyell suggested a linear relationship between the number of species that became extinct and age. For example, a deposit where 90% of the species had become extinct was determined to be nine times as old as a deposit where the extinction rate was 10%.
In his book, Time's Arrow, Time's Cycle, Stephen Jay Gould eloquently described Lyell's method. In Gould's description, a bean bag contains, at all times, 1,000 beans. At regular intervals (every two minutes), one bean is removed at random and replaced by a new bean. The beans carry no information concerning their time of entry into the bag. On the fifth day of the experiment, four X-rays are taken at midnight, 6:00 a.m., noon, and 6:00 p.m., but not identified by time taken, and the task is to arrange them in proper temporal order by comparison with the final bean configuration at midnight.
As Gould noted, this model overlooks the fact that "species don't arise at equally spaced intervals, and ... total numbers of species are not truly constant." Nevertheless, it yielded a method of estimating the time elapsed since the four named "snapshots," for which Lyell invented the names Eocene, Miocene, Older Pliocene, and Newer Pliocene. Each period was characterized by both the types of fossil shells, and the percentage now extinct, with the Eocene being the oldest, and the Newer Pliocene the youngest. Lyell acknowledged the somewhat arbitrary nature of his divisions (Gould 1987, pp. …