What was the next step that required math?

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I am guessing that you are referring to the discovery of the idea of trajectories and how to calculate them?

Because the boys had been accused of starting a forest fire which led to a moratorium on their launches, Miss Riley shows Homer a book including instructions on how to figure out trajectories.  Homer works out the details and discovers that it is impossible that their rocket caused the fire.  They prove it by finding their rocket right around where they expect to and quite a distance away from the site of the fire.  It must have been a flare from the airport, they decide, which allows them to return to their work.

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