These are all
dates of historical events!
They are grouped as follows:
* The dates national leaders were born
* The dates explosions(both natural and non-natural) happened
* The dates satellites first were launched in space
* The dates failed assassination attempts occurred
After groupoing these, you'll notice
that these events are all either french, american(USA), japanese, russian. So the next step will be to group these country by country.
Here's an image of my result:
The last square, I obtained from the given clues. Rows(in order): Failed(from failed assassinat.), Beginning(from date of birth), of Heavenly(from satellite), Rapid expansion(from explosion). Columns(in order): Rising(from Rising sun, implying Japan), Mother(from Mother Russia), dream(from American dream), kiss(from French kiss).
Now, looking at this new square's numbers, they seem to represent
The years 806, 1625, 1409, 1712. I have seen some connections between them but nothing that has given me an "AHA!-reaction". Maybe someone in here can finish this last step!
EDIT: Thanks @Prince deepthinker!
- The numbers on the last grid are the exact date of birth and death (8th June 1625 - 14th September 1712) of the Italian mathematician Giovanni Domenico Cassini!. Also, how he is connected to this puzzle beacuase it involves 1) His date of birth 2) He was also an astronomer and has a launched satellite named after him. 3) He worked for King Louis XIV who also happened to have an assassination attempt on him. 4) He's a mathematician and this is a number-puzzle :)
The grid to the left were just simple notations to remember them. Also, I will add/edit more details a bit later when I have more time.