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There were so many different significant cultural and intellectual developments in antiquity that it is somewhat difficult to summarize them, especially as "antiquity" or "the ancient world" includes a period of 3500 years and covers most of the globe.
Important intellectual, as opposed to technical developments included mathematics, which became the foundation for science. Both arithmetic and geometry were well developed, and it was possible to predict eclipses using observational data and calculations. Math was also important for architecture and building. Medicine was important in Mesopotamia, and many of the principles of medical practice and theory and ethics continued to develop in Greece and Rome. Other important developments included philosophy, law codes, visual arts (especially sculpture, painting, and pottery), literature (poetry and drama), history, and biology.
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