The time of origin of archaeocetes calculated here takes nonuniform aspects of the distribution of potential fossils into account. The resulting estimate at ca. 51.64 Ma, is a little more than 2 m.y. before the first fossil archaeocetes appeared in the fossil record at ca. 49.50 Ma. This is a substantial 16% increase in the estimated temporal range of archaeocetes compared to their observed temporal range (15.64/13.50 = 1.16), but it is a rather small 4% increase in the estimated temporal range of cetaceans as a whole (51.64/49.50 = 1.04). The fossil record of early cetaceans is not yet adequate to answer all questions we ask of it, but it is adequate to constrain the time of origin of Cetacea. Further, the early Eocene time of origin we calculate here is consistent with the very primitive transitionally-aquatic remains of early Eocene fossil cetaceans found in recent years.
The temporal range of mesonychians, from at least
63 to 33 Ma (middle Paleocene through early Oligocene) and possibly from 66.7 Ma
to about 33 Ma (latest Cretaceous through early Oligocene), precedes and
overlaps the origin of archaeocetes in the early Eocene. The temporal range of
arctocyonians plus artiodactyls, from at least 67 to 0 Ma (latest Cretaceous to
the present), coincides with or overlaps the origin of mesonychians in the
latest Cretaceous or early Paleocene. These temporal relationships, with times
of origin of Archaeoceti and Mesonychia constrained to a range of reasonable
likelihoods, are shown in Figure
8. 
Phylogenetic
relationships shown in Figure 8 are the same as those outlined in a cladistic
analysis by Geisler
and O'Leary (1997). Mesonychia are
monophyletic in the sense that all are thought to be descendants of a single
common ancestor, but paraphyletic in the sense that some descendants
(principally those in Cetacea) are not included.
Claims that whales originated or whale ancestors separated from artiodactyls 80-90 million years ago, based on cladistic analyses (Novacek 1992; Archibald 1996) and/or molecular clocks (Hedges et al. 1996, Kumar and Hedges 1998), conflict with our results. It may be possible to shape the cladograms in question to conform more closely to the fossil record (since cladistic analyses are rarely constrained by geological time or the age of fossils in any case), but there appears to be no way to reconcile our results with early divergence times hypothesized from molecular studies. The many conflicting hypotheses of cetacean relationship to and within Artiodactyla in the current literature suggest that times of divergence based on present molecular 'clocks' cannot be taken seriously either. From the point of view of the fossil record and geological time scale, the idea that whales might be found in the mid-Cretaceous involves vanishingly small likelihoods (on the order of 1 to 10 in a billion). This does not prove that a mid-Cretaceous origin of whales is impossible, but shows the conjecture to be beyond reasonable expectation given what we know about the fossil record of whales (in the same way that 28 heads and no tails is beyond reasonable expectation when a coin is tossed 28 times).
A distinction is sometimes drawn between the beginning of an estimated stratigraphic or temporal range and the 'true' time of origin of a taxon, and it is possible that there is a slight difference. However the geological time scale that is almost universally used as a context for discussion of animal history is itself based on the fossil record, and it necessarily takes account of whatever this slight difference might be, meaning, possibly, that the whole Phanerozoic time scale should be inflated by some small percentage. However, this would affect all times calibrated from geological evidence uniformly, it would affect all inferences concerning evolutionary time and rates uniformly, and the difference, consequently, is negligible.
It will undoubtedly be possible to discover and quantify additional factors influencing estimation of the time of origin of Cetacea, but these are not likely to add more than possibly another 0.5 to 1.0 m.y. to the estimate given here. There is a high probability that additional fossil whales will be found intermediate in time between known records, and there is a reasonable probability that the fossil record can be extended earlier than any record known at present. There is of course some chance that a fossil whale could be found at any time in the geological past, but this chance diminishes with predictable rapidity farther back in time. In Figure 5 we provide quantitative estimates of how likely we are to find fossil whales older than any known at present. The importance of earlier discoveries would be inversely proportional to such vanishing likelihoods (that is, of great importance), and we hope skeptics will channel their energy and attention into a search for earlier fossil whales.