I use both the Common calendar and the Roman calendar every day, chiefly in the daily entries on the "What's New" page of my web site. I use both of them on my Facebook page, too, where I announce changes I've made to the T́uliǹgrai Etymological Dictionary, and new stories, books, and poems that I've published on my web site.
In using both calendars every day, I've noticed that the number of the day in the Common calendar, and the number which is part of the name of the day in the Roman calendar, occasionally coincide. For instance, yesterday was the 4th of July, 2019 CE (Common Era). In the Roman calendar, it was the 4th day before the Nones of July, 2772 A.U.C. (Anno Urbis Conditae, "Year of the Founding of the City", i.e., Rome). I've coined the word coincidate for these temporal coincidences.
There are lots of calendars, actually. Many ancient cultures had their own native calendars, such as the ancient Egyptians, the Jews, the Chinese, and Islam. Most people around the world use the Common calendar and the Christian calendar interchangeably. I use both the Common and the Roman calendar, as do the people in my stories and books who speak Latin.
The Common calendar is the Christian calendar, secularized; it uses the same months, and the same dates, but all the Liturgical seasons, feast days of saints, and events of the Liturgical year have been purged. Great Christian feasts, like Christmas and Easter, may be marked as comments on the day they occur. Purely astronomical events, like the solstices, the equinoxes, and the phases of the moon, are noted the same way; and each country's secular holidays, such as Independence Day (in the United States, the 4th of July), victories in war (in Mexico, Cinco de Mayo), and other occasions, such as May 6, Children's Day, in Korea, and May 1, for the international Labor movement, in many countries.
The Christian calendar appends A. D. to its years. A. D. stands for "Anno Domini", or in English "Year of the Lord". The Christian religion was adopted as the official religion of the Roman Empire, and the Christian calendar is based on the ancient calendar of Rome, using the same months in the same order. The Church used not just the Roman months, but Roman dates as well, at least in priests' breviaries, until 1962, and the Vatican II council.
For the almost entirely illiterate laity of post-Classical Rome and the Middle Ages, however, they simply numbered the days of the month 1, 2, 3, etc. Even that dumbed-down system of marking the days was buried under a flood-tide of feast days; a day was at least as likely to be referred to as "St. Stephen's Day", or "The Feast of Stephen", or even "Stephenmas", as "December 26th".
The days of the month, in the original Roman calendar, are no robotic sequence of numbers. The first day of every month is a holiday, the Kalends. The third holiday is the Ides, which occur ("fall") on the 13th day of most months, but the 15th day of four of them. In between the Kalends and the Ides are the Nones, which come on the ninth day inclusive before the Ides, thus on the 5th day of eight months, but the 7th day of the others.
So every Roman month has its Kalends on the first day, its Nones on the 5th or the 7th, and the Ides on the 13th or 15th. What about the other days? I'm so glad you asked. Click here to see the present month in the Roman calendar. Today's date is shown in red, and the Kalends, Nones, and Ides are shown in blue.
As you can see, in any Roman month, you have the Kalends, then you count down to the Nones, then you have the Nones, then you count down to the Ides, then the Ides occur, and then you spend roughly half the month counting down to the Kalends of the next month. The number of days in each section depends on what day the Nones fall on, what day the Ides fall on, and how many days there are in a month, which can be 31, 30, 29, or 28.
So, a "coincidate" occurs when one of the sections of a Roman month, counting down to the next Nones, Ides, or Kalends, happens to hit the same number as the Common month ticks off while counting up from 1.
So now that we're clear on what the Common calendar is, and how it works; and what the Roman calendar is, and how it works, how often does a "coincidate" occur? How many times in a year will the two calendars come up with the same number? Let's find out! Follow along on pages xvi and xvii of Cassell's Latin Dictionary, if you like. (If you don't have a copy, your local library ought to.) Be careful not to wander into the wrong column of the chart, the way I did when I first added up these numbers!
Let's get the oddballs out of the way first. In exactly four months of the Roman year, the Ides occur on the 15th day of the month: March, May, July, and October. All four of these months have 31 days. The 4th of the month is the 4th day before the Nones, and the 8th day of the month is the 8th day before the Ides, so that's two coincidates in each of these four months, or 8 altogether. (There are no coincidates in the second half of these months, because they just miss each other; the 16th day of these months is the 17th day before the Kalends of the next month, and the 17th day of the month is the 16th day before the Kalends. Ooh, they came that close!)
February has the Ides on the 13th, and 28 or 29 days. It has 3 coincidates: the 3rd is the 3rd day before the Nones, the 7th is the 7th day before the Ides, and the 15th is the 15th day before the Kalends of March. This is the same whether it's a leap year or not, because in the Roman calendar, after the 6th day before the Kalends of March, the Roman February has another 6th day before the Kalends, in a leap year. If the Romans had added the extra day to the end of February, the way the Christian and Common calendars do, there would only be two coincidates in February, in a leap year!
Moving on, four months (April, June, September, and November) have 30 days, and the Ides are on the 13th. These months have three coincidates apiece: the 3rd is the 3rd day before the Nones, the 7th is the 7th day before the Ides, and the 16th is the 16th day before the Kalends. 3 coincidates times four months is 12 coincidates for these months.
Finally, three months have the ides on the 13th, and 31 days: January, August, and December. These three months have two coincidates each, when the 3rd of the month is the 3rd day before the Nones, and the 7th of the month is the 7th day before the Ides. 2 coincidates times 3 months makes 6 coincidates.
So, how many coincidates in a year? 8 + 3 + 12 + 6 = 29. I had guessed 24, estimating two coincidates per month. That was 5 too low, as it turns out. 36, on the other hand, would've been 7 too high.
1. "The Roman Calendar", pp. xv-xvii, in Cassell's Latin Dictionary by D. P. Simpson, Wiley Publishing, Inc., New York NY, 1968.
2. "Roman Dates and Times", pp. 230-231, in Pocket Oxford Latin Dictionary, edited by James Morwood, Oxford University Press, Oxford, revised edition, 2005.