An interesting exercise is to write a computer program to find the number of days that have inclusively elapsed from one date to another later date.
In the question asking how many days have elapsed from June 12th 1829 until September 22nd 2019, the inclusion means including the days, June 12th and September 22nd.
Every year divisible by four and not ending in two zeros is a leap year. A year ending in two zeros has to be divisible by four hundred to be a leap year. Leap years have 366 days.
An approximate answer is much easier to come by, thus: June 12th 1829 until December 31st 1829 is 22 days to end of June, the 31 days July, 31 days August, 30 days September, 31 October, 30 November, and 31 December of 1829. Adding, there are 206 days remaining in the year 1829.
From December 31st 1829 until December 31st 2018 there are:189 years.Since on average there are 365.25 days in each year, 189 years is 189 x 365.25 days, which is 69032.25 days.
From December 31st 2018 until September 22nd 2019, counting the days of each month plus twenty-two days in September 2019 gives: 31 days in January 2019 + 28 +31 + 30 + 31 + 30 + 31 + 31 + 22 days which is:265 days.
Addition gives:206 + 69032 + 265 days, which is: 69503 days from 20th June 1829 until 22nd September 2019, ignoring 0.25 days.
The day dated 12th June 1829 is not included, but the day dated 22nd September 2019 is included in the answer which is estimated to be 69503 days.
As at today, 11 Sept 2019, 69487 days. To determine this, I tried using Excel to subtract one date from the other, but it didn’t work because Excel doesn’t like dates before 1900. So I shifted the dates forward 4 centuries and got Excel to compute 11/09/2419 minus 12/06/2229 (note the British format dates - that’s because I’m British). Why 4 centuries? Because a year ending 00 is a leap year only every 4th century. (Except when … oh, look it up yourself.)
The switch from Julian dates to Gregorian dates is irrelevant, this happened long before your date range.
The trick in Excel is to enter the dates in cell A1 and A2, them enter in cell A3 “=A1-A2”, and right-click on cell A3 and format it as a number instead of a date.