Quantitative genetics is relevant to evolution because it has to do with heredity and understanding how heredity works. Darwin didn't understand how heredity worked; he just knew that it did — sort of. Back in those days, folks understood that the heritability of some traits was more predictable than others. For some things, such as hemophilia, researchers could trace the ailment back to a single factor that they could map onto a family tree. (Today, scientists know that this factor is a particular DNA sequence.)
Other traits, such as how much milk a cow made, weren't so clear cut. True, researchers in Darwin's day knew that things like how much milk a cow made had a heritable component, but figuring out what that heritable component was wasn't as simple as determining the heritable component of eye color or a disease like hemophilia.
Fast-forward several generations. Scientists now can explain some of the things Darwin and his contemporaries could only wonder about. We now know that a character is often the result of several genes rather than a single gene.
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