Cream butter in a large mixing bowl; gradually add sugar, beating until light and fluffy. Add egg and vanilla, beating until well blended.
Sift together flour, baking powder, and salt in a medium mixing bowl; add to creamed mixture, mixing well.
Shape dough into a long roll, 1 1/2 inches in diameter; wrap roll in waxed paper, and refrigerate 2 hours or until thoroughly chilled.
Cut into 1/4-inch slices. Place 3 inches apart on lightly greased cookie sheets; bake at 400° for 8 minutes. Remove from cookie sheets, and cool on wire racks.