Today historians can use technology to prove historical facts false or accurate, they can test out theories better.
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The most obvious way that European explorers affected history was discovery. European explorers discovered a lot of the world like Australia, the Antarctic, America... for example. Without them we wouldn't know about the world. They would have helped with trade and discoveries as well. Like for trade they would discover new ways to get to different places so that people could trade with each other, like ways from getting from Europe to Asia faster by ship. They would help with discoveries too. If the Galapagos Islands hadn't been discovered would Darwin have been able to make his observations that led to the theory of evolution.
A Discourcs of a Discoveries for a new Passage to Cataia.
History, our past, does not change. However, every new day we add to history by doing new things.
Washington Irving
The history of math started with sir Williams rowan hamilton. The history of math is almost as long as the history of man . Indeed one can fill a book with just the greatly condensed highlights. The related link to the wikipedia article is a good summary and gives references for further reading. Even the history of just one branch of math is long and complicated. Today there are dozens of journals publishing hundreds of new mathematical discoveries being made daily by thousands of mathematicians all over the world.