we mentioned above an error in galileo's theory of motion as he set it out in de motu around 1590.

How would the previous sentence be correctly capitalized?
Answer: We mentioned above an error in Galileo's theory of motion as he set it out in De Motu around 1590.
this monday was all about relaxing and hanging out with my friend, linea, and for that i wore a super comfy and fun outfit, if i must say so myself.

How would the previous sentence be correctly capitalized?
Answer: This Monday was all about relaxing and hanging out with my friend, Linea, and for that I wore a super comfy and fun outfit, if I must say so myself.
through the next half century the geometry text underwent thirty-nine editions and the algebra text twenty-four.

How would the previous sentence be correctly capitalized?
Answer: Through the next half century the geometry text underwent thirty-nine editions and the algebra text twenty-four.
netsh ras aaaa show authentication: displays the current authentication provider.

How would the previous sentence be correctly capitalized?
Answer: netsh ras aaaa show authentication: Displays the current authentication provider.