Why is October the tenth month
of the year and not the eigth?
A couple of answers here came close, but no one has got the answer exactly right yet.
People mistakenly believe that July and August were ADDED to the Roman calendar in honor of Julius and Augustus Caesar. What happened was that the names of the fifth month, Quintilius, and the sixth month, Sextilius, were CHANGED to Julius and Augustus to honor the Caesars.
The original Roman calendar had just 10 months, starting with Martius (became March), and then after December came an indeterminate "winter period" of about 61 days that were not assigned to any month. The original months were: Martius, Aprilis, Maius, Junius, Quintilis, Sextilis, September (7th), October (8th), November (9th) and December (10th). Note that the Latin names for the later months actually do correspond with their numbered positions.
The last two months added to the Roman Calendar were Januarius (now January) and Februarius (now February). This pushed all the other months forward two numbers when later people came to regard January as the "First" month. Probably because of the winter solstice, January became regarded as a time of "renewal" for the sun, and hence the start of a new solar cycle.