Preheat oven to 350 degrees F (175 degrees C). Grease and flour six 9 inch round pans. Sift together the flour, baking powder and salt; set aside.
In a large bowl, cream together the butter and sugar until light and fluffy. Beat in the eggs one at a time, then stir in the vanilla. Alternately beat in the flour mixture and the milk, mixing just until incorporated.
Divide batter into six 9 inch pans. Bake in the preheated oven for 15 to 20 minutes, or until a toothpick inserted into the center of the cake comes out clean.