political events in france meant that cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to paris to do so he lost all his positions there. coriolis was offered cauchy 's position at the École polytechnique but by this time he was highly involved in his research and decided not to take on any further teaching duties.

How would the previous sentence be correctly capitalized?
Answer: Political events in France meant that Cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to Paris to do so he lost all his positions there. Coriolis was offered Cauchy 's position at the École Polytechnique but by this time he was highly involved in his research and decided not to take on any further teaching duties.
high blood pressure, meanwhile, may cause sudden blindness.

How would the previous sentence be correctly capitalized?
Answer: High blood pressure, meanwhile, may cause sudden blindness.
on board, guests learn many secrets about fine dining, fine wines and creative cocktails:

How would the previous sentence be correctly capitalized?
Answer: On board, guests learn many secrets about fine dining, fine wines and creative cocktails:
as a child, he noticed the habits of animals and what they did when they were sick or injured.

How would the previous sentence be correctly capitalized?
Answer: As a child, he noticed the habits of animals and what they did when they were sick or injured.