Chill a rocks glass with cracked ice. Place the lime quarters in the bottom of a mixing glass, add the syrup, and muddle, extracting the juice and the oil in the skin from the lime quarters. Add the cachaça to the mixture in the mixing glass, dump the ice from the rocks glass into the mixing glass, and shake well. Pour the entire contents of the mixing glass back into the chilled rocks glass and serve.
The Caipirinha came from the countryside, where it was favorite drink of farmers. The word caipira means "country man," and Caipirinha is a little drink with country brandy.