|MadSci Network: Science History|
Len, your question really has two answers.
For thousands of years, people realized the seasons were associated with the varying path of the Sun across the sky throughout the year. Early celebrations of solstices and equinoxes recognized the importantance of these turning points as connected with the changing seasons.
But as to when it was generally accepted that the tilt of Earth's axis caused the seasons--that really had to wait until it was generally accepted that the Earth orbited the Sun. It wasn't until the 16th century that Nicolaus Copernicus proposed that the Earth orbited the Sun instead of the other way around. This idea was not generally accepted, as it conflicted with the accepted notion that the Earth was the center of the universe, surrounded by invisible concentric spheres which carried the Sun, Moon and planets around it.
When Galileo used his telescope to discover the moons of Jupiter, he realized he was seeing a miniature version of the solar system; and if moons orbited Jupiter, then everything in the universe didn't go around the Earth. He was censured by the church for writing about this, and was placed under house arrest for the remainder of his life.
But, Johannes Kepler soon changed that. Writing at time time of religious upheaval, Kepler determined his three laws of planetary motion between 1609 and 1619. These made no sense unless the Sun was the center of the solar system, orbited by the other planets. And the laws clearly predicted the planetary orbits better that any formulation based on an Earth-centric universe.
So, sometime between the publication of Kepler's laws in the early 1600's and Isaac Newton's formulation of the laws of motion almost 200 years later the educated would have accepted that the varying apparant motion of the Sun throughout the year was actually due to the tilt of the Earth's axis.
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