In a large mixing bowl beat the butter with an electric mixer on medium to high speed for 30 seconds. Add the 1 1/2 cups sugar; beat until combined. Beat in eggs, cream of tartar, baking soda, vanilla, and salt until combined. Beat in as much of the flour as you can with the mixer. Stir in any remaining flour. Cover and chill for 2 to 3 hours.
Preheat oven to 375 degrees F. Shape dough into 1-inch balls. Roll balls in the 1/4 to 1/3 cup sugar. Place balls 2 inches apart on ungreased cookie sheets.
Bake in the preheated oven for 9 to 12 minutes or until lightly browned. Transfer to wire racks and let cool. Makes 48 cookies