How did revolutionary ideals in Europe and Latin America ignite uprisings in the first half of the nineteenth century?

For about 75 years (1775-1850) European countries and their colonies experienced crisis after crisis in their governmental practices.  The revolutionary ideas came from the philosophies of the Enlightenment, which held the radical concept that if you didn't like your government, you could change it.  And change they did.

The American Revolt against England influenced the French Revolution; in the first case, the colonies successfully established their own country; in the second, the dispossessed within the mother country overthrew the government with only moderate success, but the pattern was established.

In the New World, what had happened in North America against Britain began to happen in South America against Spain -- the Spanish Empire, like the British, began to break apart.  Meanwhile, in the Old World, the revolution in France inspired similar movements in what was to become Germany and Eastern Europe.

So Europe underwent its own political transformation, even as its colonies established themselves as independent countries.