Tear the basil leaves in half and place in bottom of tall glass. Add sugar and 1/2 ounce of the rum. Coil 12 inches' worth of cucumber julienne in bottom of glass. Add one slice cucumber diced fine. Mash together these ingredients. Squeeze lime and drop in glass. Add remaining rum. Top with soda water.
Leave sugar muddle and lime as undisturbed as possible. As you enjoy the drink, stirring the sugar or prodding the lime with your straw will add tang or sweetness as you like and as you progress through the drink.