The existence of these cross-bred varieties does not, of itself, explain how new species can arise of their own accord. Varieties are not species; moreover, the guiding hand of the breeder is evident. But varieties do make it clear that there must be plenty of variability within a species. In fact, the variability is so great that one can readily imagine selective breeding leading to entirely new species, given enough time. And the avoidance of hybrids can maintain varieties from one generation to the next, so their characters (biologese for the features that distinguish them) are heritable (biologese for `able to be passed from one generation to the next'). So Darwin has his first ingredient: heritable variability.
The next ingredient was easier (though still controversial in some quarters). It was time. Oodles and oodles of time, the Deep Time of geologists. Not a few thousand years, but millions, tens of millions ... billions, in fact, though that was further than the Victorians were willing to go. Deep Time, as we've previously observed, is contrary to the biblical chronology of Bishop Ussher, which is why the idea remains controversial among certain Christian fundamentalists, who have bizarrely chosen to fight their corner on the weakest of grounds, completely needlessly. Deep Time is supported by so much evidence that a truly committed fundamentalist has to believe that his God is deliberately trying to fool him. Worse, if we can't trust the evidence of our own eyes, then we can't trust the apparent element of `design' in living creatures either. We can't trust anything.
Lyell realised that the age of the Earth must be many millions of years, when he looked at sedimentary rocks. These are rocks like limestone or sandstone which form in layers, and have been deposited either underwater, as muddy sediments, or in deserts, as accumulating sand. (Independent evidence for these processes comes from the fossils found in such rocks.) By studying the rate at which modern sediments accumulate, and comparing that with the thickness of known beds of sedimentary rock, Lyell could estimate the time it had taken for the layers of rock to be deposited. Something in the range 1000-10,000 years would produce a layer about a metre thick. But the chalk cliffs of the south coast, around Dover, are hundreds of metres thick. So that's several hundred thousand years of deposition, and we've only dealt with one of the numerous layers of rock that make up the geological column - the historical sequence of different rocks.
We now have many other kinds of evidence for the great age of our planet. The rate of decay of radioactive elements, which we can measure today and extrapolate backwards, is in general agreement with the evidence of the rock layers. The rate of movement of the continents, when combined with the distances they have moved, is again consistent with other estimates. We've seen that India was once attached to Africa, but about 200 million years ago it broke off, and by 40 million years ago it had moved all the way to its current position, butting up against Asia and pushing up the Himalayas.
When continents move apart - as Africa and South America, or Europe and North America, are doing now - new material forms on the ocean floor, flowing out from the mantle beneath to form huge mid-ocean ridges. The rocks in the ridges contain a record of the changes in the Earth's magnetic field, `frozen in' as the rock cooled. They show a long series of repeated reversals of the field polarity. Sometimes the `north' magnetic pole is at the northern end of the Earth, as now, but every so often the polarity flips, so that the magnetic pole near the northern end is the `south' one. Mathematical models of the Earth's magnetic field predict that such reversals occur roughly once every five million years. Count the number of reversals in the ocean-ridge rocks, multiply by five million ... again, the numbers fit reasonably well, and careful comparisons and a lot of disputation by experts lead to revised numbers that fit even better.
The Grand Canyon is a deep gash through layers of rock one mile (1.6km) thick. You have a choice. You can understand what the record of the rocks is telling you here: it took a very long time to lay down those rocks, and quite a long time - though less - for flashflooding in the Colorado river to erode them again. Or you can follow one book that until recently was displayed in the `science' section of the Grand Canyon bookstore, until a lot of scientists complained, and assert that the Grand Canyon is evidence for Noah's flood. The first choice fits huge amounts of evidence and geological understanding. The second is an excellent test of faith, because it fits absolutely nothing. A flood that lasted only 40 days could never have produced that kind of geological formation. A miracle? In that case, the Sahara desert could equally well be hailed as evidence for Noah's flood, miraculously not forming a deep canyon. Once you admit miracles, you can't pursue a logical thread.
Anyway, that's the second ingredient - Deep Time. It takes huge amounts of time to change organisms into entirely new species, if all you can do - as Darwin believed - is make very gradual changes. But even Deep Time, when combined with heritable variation, is not enough to lead to the kind of organised, coherent changes that are needed to create new species. There has to be a reason for such changes to occur, as well as opportunity and time. Darwin, as we've seen, found his reason in Malthus's contention that the unchecked growth of organisms is exponential, whereas that of resources is linear. In the long run, exponential growth always wins.
The first assertion is pretty much correct, the second highly debatable. The qualifier `unchecked' is crucial, and real populations only grow exponentially if there are plenty of resources available. Typically, the growth starts exponentially with a small population and then levels off as the population size increases. But in most species, two parents (let's think sexual species here) produce some larger number of offspring. A breeding female starling lays about 16 eggs in her life, and with `unchecked' growth, the starling population would multiply by 8 every lifetime. It would not be long before the planet was knee-deep in starlings. So, of necessity, 14 of those 16 offspring (on average) fail to breed - usually because something eats them. Just two become parents in their turn. A female frog may lay 10,000 eggs in her life, and nearly all die in various grotesque ways to achieve each two parents; a female cod contributes forty million or thereabouts of her offspring to planktonic food chains, for each two that breed. Here the multiplier, with `unchecked' growth, would be 20 million per cod-lifetime. Unchecked growth simply doesn't bear thinking about as a realistic prospect.
We suspect that Malthus plumped for linear growth of resources for a slightly silly reason. Victorian school-textbook mathematics distinguished two main types of sequence: geometric (exponential) and arithmetic (linear). There were plenty of other possibilities, but they didn't get into the textbooks. Having already assigned geometric growth to organisms, Malthus was left with arithmetic growth for resources. His main point doesn't depend on the actual growth rate, in any case, as long as it is less than exponential. As the starling example shows, most offspring die before breeding, and that's the main point here.
Given that most young starlings cannot possibly become parents, the question arises: which ones will? Darwin felt that the ones that survived to breed would be the ones best suited for survival, which makes sense. If one starling is better at finding food, or hanging on to it, than another one, then it's clear which one is more likely to do best if food supplies become limited. The better one might be unlucky and get eaten by a hawk; but across the population, starlings that are better equipped to survive are generally the ones that do survive.