Cut the limes in half, and then cut each half into 8 pieces. Place 12 pieces of lime in each of 4 double old-fashioned glasses (or other 12-ounce glasses.) Sprinkle 2 tablespoons of sugar into each glass. Using a muddler or the butt end of a large wooden spoon, mash the limes into the base of the glass. Try to dissolve the sugar as you do this, and stir occasionally. Fill each glass with 1 1/2 cups of ice and 1/4 cup of cachaca. Pour the contents of the glass into a shaker, and invert the glass into the shaker to act as a lid. Shake the glass and shaker to combine the sugar and cachaca for several seconds. Lift the glass from the shaker, and pour the contents back into the glass. Repeat the process with the remaining 3 glasses.