October 4

On October 4 1582, to adjust the inaccuracy in the date caused by an extra day per century in the Julian calendar, Pope Gregory XIII ordered ten days to be subtracted from October of 1582. The calendar jumped from October 4 to October 15, and the new Gregorian Calendar, which we use today, was devised.

From : http://historymedren.about.com/library/date/bl104.htm.

On October 4 1830, the Belgian Provisional Authority proclaimed the independence which was later confirmed by the National Congress that issued a  liberal constitution and declared the new state a constitutional monarchy under the house of Saxe-Coburg.

See : http://en.wikipedia.org/wiki/List_of_national_independence_days.