Survival of the fittest is the basis of these ideas, but there is also natural selection. Natural selection is when Nature chooses which species will reproduce and survive. So it may not be entirely that a horses strong legs will be passed down, but rather that it will be able to reproduce and keep that breed of horses alive for as long as nature will let them.

I wouldn't say that Eugenics are completely influenced by the environment but i agree that Darwins concepts have contributed to Eugenics. He based his theory on what he saw on his voyage. The slaves were being treated horribly and he concluded that they were of a lesser being and it also influenced scientists to use eugenics to create stronger and better slaves.

During this time in history the slave trade was occurring and when Darwin would travel he would see the slaves. He would believe that they were lesser beings because of the quality of which they were being treated in. Also because they resembled a lesser race because of their color. This inspired Darwin to make his theories.

The difference between these two ideas is that Darwinism is the idea of natural selection, a process in which nature decides which organisms are best fit to reproduce and fill the environment with their offspring and Social Darwinism is the idea of "SURVIVAL OF THE FITTEST". The idea is based on the general concept of which organisms are the fittest, meaning can they survive long enough to reproduce? It means whether they are able to collect food better than other animals and also whether or not they can find a mate faster than the other animals.

Darwins concepts had to do with change over time with organisms which can also be based on the quality of the environment like whether it is wet, dry, mountain, or just like a forest. The organism adapts to its surroundings with results in the change over time. In eugenics, it is chosen with what qualities the next generation will have. This can be done by either mating two organisms who have the desired qualities, or kill the organisms that don't possess these qualities. What can be done is choosing organisms from different environments and mate them which can make a well rounded offspring. This is where the two ideas intertwine with the environment concept.

During Darwin's travels he witnessed the treatment and the quality of the slaves during the slave trade. He indeed thought that they were lesser beings mainly because of their appearance. What scientists were thinking of at the time was how to increase the quality of the slaves. For instance they would take the male slave who could lift the heaviest objects and they would take the strongest female slave and they would make them mate. The result would drive the scientists to continue in this if the offspring was strong. The treatment of these slaves were completely horrible considering that the scientists were acting more like lesser beings than the slaves were. Also, Charles Darwin's cousin Sir Francis Galton actually came up the general idea of eugenics.