How technological advanced a civilization would need to be to prove the existance of the shadow Earth?

In a short sci-fi story i am writing, Earth has a shadow twin. A planet with the same chemical/geological characteristics as Earth and wildlife very similar (but different species because evolution and mutations are random).

The humanity in Earth-prime XXI century knows of the shadow twin existance, but I am having difficulty in pin-pointing when it became common knowledge.

Same planetary masses. The Shadow-Earth, Earth and the Sun orbit each other using the Figure-Eight orbit. This is not tagged hard-science, so lets do a minor hand-wave on the orbital issues, and assume the figure eight three body orbit really works.

Also, if you know of any works of fiction dealing with this setup, please state in your answer or leave a comment. thanks.

Keeping the suspension of disbelief as low as possible, in a sci-fi setting on our Earth, when is the historical earlier I could place the "discovery" of the existance of Earth-Shadow? Renaissance? Ancient Egypt/Greece? Space Race? Victorian period?

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How technological advanced a civilization would need to be to prove the existance of the shadow Earth?

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