When Dr. Pickerbaugh of Nautilus decides to hold a public health fair, one of his main displays is a group of parents and children known as "The Eugenic Family." The word "eugenic" in this case means that the family has "clean genes" that haven't been corrupted by any family history of disease or mental disability. As the narrator tells us,
"None of these novelties was so stirring as the Eugenic Family, who had volunteered to give, for a mere forty dollars a day, an example of the benefits of healthful practices." (23.1.4)
Sinclair Lewis can't help but point out the hypocrisy of a family that promotes a good moral life and then charges a bunch of money to do so.
To make things even more hypocritical, Pickerbaugh doesn't even fire the family when he finds out from the police chief that they aren't a family at all. Rather,
"… they're the Holton gang. The man and the woman ain't married, and only one of the kids is theirs. They've done time for selling licker to the Indians." (23.1.14)
For Pickerbaugh, the Eugenic Family still serves its purpose as long as people think they're healthy and good. The Eugenic Family is heavily symbolic of the hypocrisy that comes with public office: there is no kind of truth to the Eugenic Family. They're not healthy, they're not upstanding, and they're not related to each other.
And for an idealist like Martin Arrowsmith, politics mixed with science is very much like the Eugenic Family. Politics mixed with science isn't healthy, because it compromises medical advances. It's not upstanding, because it degrades scientific truth. And they're definitely not related to each other. For Martin, politics and science are about as closely related as a daffodil and an elephant.