It makes sense that Achelis wrote to Catholic authorities; the calendar most countries use today was created by them. In 1582, Pope Gregory XIII introduced what became known as the Gregorian Calendar, which replaced the Julian Calendar. The problem for any calendar designer comes from the way we divide time. Years are defined by the length of time it takes for the earth to travel around the sun once, months are defined by the phases of the moon, and days are defined by the amount of time it takes for the earth to rotate on its axis. Unfortunately, these aren't wrapped up in a nice mathematical bow. There are 365.2422 days in a year. If our calendar only accounted for the length of a day, over time the beginning of the year would drift through the seasons.
To address this, Julius Caesar added an extra day every four years: a leap year. However, that system still allowed for a gain of three days every 400 years. For Catholics, the issue was that the date of Easter was calculated based on the spring equinox, so eventually, it would be held at a time of year that was not intended. Pope Gregory XIII fixed the problem by making it so that years evenly divisible by 100 would not be leap years unless they were evenly divisible by 400.
However, no system is perfect, and people found issues with the Gregorian calendar. Critics point out that because days of the week don't line up consistently with dates, new calendars need to be printed every year, and holidays and schedules are always different. In 1930, Achelis proposed the World Calendar, which would always start on Sunday, January 1. It would still have 12 months, but each quarter would have three months of 31, 30, and 30 days. An extra, un-numbered day would be added outside the calendar after December 30 as a world holiday to account for the earth's trip around the sun, and in leap years a day would be added after June 30.
Despite some support and the continued existence of a group promoting it, the World Calendar has not yet caught on. The World Calendar Association - International is pushing for adoption in 2023, since that is the next year in which January 1 falls on a Sunday.
For more information about the fascinating and complicated history of the calendar, check out some of these sites:
Official Site of the World Calendar Association - International
Calendars through the Ages
The Home Page for Calendar Reform