Final answer:
Darwin applied Malthus' principles of population growth and artificial selection in developing his theory of natural selection, while Lyell's theory of uniformitarianism provided evidence for gradual changes over time.
Step-by-step explanation:
Darwin applied Malthus' principles of population growth and artificial selection in developing his theory of natural selection by recognizing that in any given population, more individuals are born than can survive due to limited resources. This creates a struggle for existence, where individuals with favorable variations are more likely to survive and reproduce, passing on their traits to the next generation. Similarly, artificial selection, which involves humans selectively breeding organisms with desired traits, helped Darwin understand how natural selection could occur in the wild. Lyell's theory of uniformitarianism was important to the theory of natural selection because it provided evidence for the gradual changes in the Earth's surface over long periods of time. This supported Darwin's idea of gradual change in species over time. By recognizing that the Earth has a long history, Lyell's theory provided the necessary framework for understanding how natural selection could lead to the diversity of life on Earth.