This theory, known as ``neo-Darwinism'' or ``the modern synthesis,'' was founded by three men: Ronald Fisher, Sewall Wright, and J. B. S. Haldane. Of the three, Haldane was by far the most interesting. He was the son of J. S. Haldane, eminent in his own right as a physiologist and an Idealist philosopher; he took first honors in both mathematics and classics at Oxford; was an infantry officer in France during the First World War, and was probably the only person to publicly claim he enjoyed the experience; he was for many years a devoted Communist, writing a science column for the British Communist paper, the Daily Worker (he abandoned Communism somewhat after the Lysenko affair); in addition to his work on evolutionary theory, he was an excellent biochemist, and his last years were spent in India, where he did biochemical research, became an Indian citizen, and wrote amusing poems about the growth of his tumors.
Haldane was also by far the best writer of the three; a good writer, period. His works of popular science are models of their kind, and his speculations (such as the short book Dædalus) inspired some of the best science fiction of the century, including John Brunner's Stand on Zanzibar. (A selection of both sorts of essays, edited by his pupil John Maynard Smith, has been brought back into printed by Oxford University Press under the title On Being the Right Size.) Most of The Causes of Evolution is written at this level, that of skillful verbal argument: those in search of the mathematical hard core will find full frontal integral signs in an appendix.
The situation of evolutionary biology in 1932, when Haldane wrote was as follows. In 1858, Darwin and Wallace had proposed a mechanism, natural selection, for shaping up spontaneous, inherited variation into adaptations. Of the mechanisms of variation and heredity they knew nothing. Like most (all?) of their contemporaries, they accepted that, at least to some degree, the effects of use and dis-use on various organs, and acquired characteristics generally, would be transmitted from parent to child. They also supposed, again in line with their contemporaries, that characters were inherited in a ``blending'' manner, so that (say) the result of crossing a very tall plant with a very short one would be a plant of medium height. Together with merely emotional difficulties, these two difficulties helped put Darwinism into eclipse by the end of the nineteenth century.
The inheritance of acquired characteristics, somewhat inaccurately called Lamarckism, is an obvious rival source of adaptations, but it's perfectly possible to accept that both it and natural selection are at work; Darwin and Wallace did, after all. The problem with heredity is more serious. If the characteristics of organisms are some sort of blend or average of those of their parents, it is very hard to see how any reasonable intensity of selection could maintain and encourage favorable variations, at least in sexually-reproducing species. Fortunately for Uncle Charles, both of these suppositions were wrong.
Already in 1864, Mendel had shown that at least some traits are not inherited in this way, but are determined by discrete, particulate factors. His classic experiments were conducted on peas, and perhaps the most striking result has to do with their height. He had two true-breeding varieties of pea plant, one much taller than the other, which he then crossed. If height was determined by some average of the parents, the crosses should have been intermediate in height, but in fact they were all tall. He then crossed these first-generation hybrids with each other, and obtained a mix of tall and short plants, three-quarters of the plants being tall, one quarter short. Of the tall plants, a third bred true, and the rest produced mixed off-spring in the same proportions as the original hybrids. So the result of crossing first-generation hybrids with each other were 1/4 true-breeding tall, 1/4 true-breeding short, and 1/2 hybrid. The simplest explanation (which Mendel seems to have thought of before actually doing the experiments, like a good follower of Karl Popper --- or Claude Bernard) was that height was controlled by a pair of ``factors'' in the organism, and that each parent contributed one or the other of its factors to its off-spring at random. The plants in the true-breeding lines therefore must have both of their factors the same (in modern terms, they are homozygotes); call them aa (for short) and AA (for tall). When these plants were crossed, each descendant plant got one A factor and one a, so all of the first crosses were Aa. Since these plants were all tall, it follows that the A factor dominates a, and so it's called a dominant or a Mendelian dominant. When these mixed plants (heterozygotes) were crossed among each other, getting a homozygote, either tall or short, means getting the same factor from both parents, and the probability of that would be 1/2 x 1/2 = 1/4; but to get another mixed plant, you could either get A from the male and a from the female (probability 1/2 x 1/2 = 1/4), or vice versa, so the total probability of being a heterozygote was 1/2. Q.E.D.
In further experiments, Mendel showed that other traits, like the color or shape of the peas, were also inherited in this manner, and independently of each other, so that (say) tallness and yellowness are not linked. Today we'd say that in each case there were several alleles of a gene which controlled height, or color, or whatnot. We also know today that not all genes are strictly independent, but the degree of linkage between them can be determined experimentally. Needless to say, Mendel's laws apply to more than just peas: as far as we know, all sexually-reproducing species enjoy Mendelian heredity (though some have more than two sets of genes). There aren't many easily visible traits in human beings which are determined by a single gene; the most striking is polydactyly, the condition of having six fingers on each hand --- which is actually dominant over the usual number!
Mendel published his results, replete with algebra, in the journal of a fairly obscure horticultural society, and unsurprisingly was utterly ignored. The laws themselves were re-discovered, independently, about 1900, and Mendel's work was unearthed shortly thereafter. There was at first a great vogue for supposing that all sorts of things were determined by a single Mendelian gene, and it was actually supposed that the new genetics constituted yet another alternative to natural selection.
By the twenties, thanks largely to the work of Morgan and his school of fly-breeders, it was known that genes were somehow or other located on actual physical bits of the chromosomes, that Mendel's laws had to be corrected to take account of linkages, that genes sometimes mutated spontaneously and that the rate of mutation could be greatly enhanced by radiation and certain chemicals, and that many traits were partially controlled by many genes, which were inherited discretely but made incremental contributions to the trait. What genes were made of, and how they worked, were quite unknown. (It's still not exactly crystal-clear how one gets from a change in a single gene to having an extra finger!) Still, the stage was set for Haldane & co. to show that, however genes work, provided they obey Mendel's laws (subject to the usual caveats) and that they affect the form and behavior of organisms, then natural selection is competent to produce adaptations from the variation thrown up by mutation, recombination of genes, and the like, and is almost certainly stronger than any Lamarckian inheritance or orthogenesis (inherent tendency to evolve in a certain direction) which may exist. (Haldane actually thought that he could show how apparent orthogenesis arose from genetics and selection, but as Leigh points out in his afterword, the argument is flawed.)
The basic argument runs as follows.
As I said, most of Haldane's book is devoted to setting forth the problem, verbally explaining the neo-Darwinian solution, and following out some of its consequences, particularly in deflating various optimistic notions about what evolution says about Man's Place in Nature and Universal Progress and the like. The proper mathematical demonstrations live in an appendix, which should be read in close conjunction with Leigh's afterword, since the latter corrects a number of mistakes and closes a number of gaps in the former.
The Causes of Evolution is both a milestone in the evolution of evolutionary theory, and a work which can be read with profit and pleasure by almost any intelligent person; the publishers deserve to be congratulated for bringing it back into print after more than half a century.