Question

what hypothesis did darwin investigate by studying the breeding of pigeons with distinctive traits?
new species can appear gradually through small changes in an ancestral species.
the amount of variation in a population is related to the type of environment it lives in.
organisms have only the number of offspring that are supported by the environment.
some traits appear in each generation, while others seem to disappear in some generations.

Explanation:

Darwin's work on pigeon breeding was part of his exploration of how new species could arise. By observing the gradual changes in traits through selective breeding (akin to natural selection in nature), he was looking at the idea that new species can form from ancestral species via small, incremental changes. The other options are not as directly related to the key concept he was investigating with pigeon - breeding. For example, the amount of variation related to the environment (Option 2) is more about population genetics in a specific ecological context rather than species - formation. Option 3 is incorrect as organisms can have more offspring than the environment can support (a key point in his work on natural selection, but not the focus of pigeon - breeding studies for species - formation). Option 4 is about trait inheritance patterns in generations, which is a part of genetics but not the main hypothesis of species - formation through gradual change.

Answer:

New species can appear gradually through small changes in an ancestral species.