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The Roman Calendar counted backwards. A count down instead of a count up. We count forwards from the first day of the month. They ... didn't.
The first day of a month is called the Kalends. For us day two is the day after that, for them day two is the day before that.
You could look at it as, "How many days are left until X is over?" On the Kalends itself there's one day left till it is over: that day. The day before it there are two days left. The day before that there are three, and so on.
But that's no the only way the Roman Calendar was different from our own. There wasn't just one day in a month that you counted from. There were three.
The Kalends was the first day. The Nones was was the ... thingy day. The Ides was the middle day.
More or less.
The Nones shifted with the Ides. The name comes from the fact that, counting backwards inclusively, it's the ninth day before the Ides.
When the Julian Calendar was implemented extra days in months were added after the Ides so that important festivals wouldn't change.
On June the 13th day of the month is the Ides. That means that today, two days after the Ides, is a day where we count backwards from the Kalends of next month. So the 17th Kalends of July. (16 days left in this month + 1 day for the Kalends of next month.)
Weird, I know.
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