In a medium bowl, cream together butter and sugar.
Add egg and vanilla; continue to cream until smooth. Sift together the flour, baking soda, and cream of tartar; stir into the creamed mixture until well blended. Refrigerate dough for at least 1 hour.
Preheat oven to 375 degrees F ( 190 degrees C ).
Roll dough into 1 inch balls and place onto a cookie sheet. Cookies should be at least 2 inches apart. Lightly grease the bottom of a glass, dip it into the granulated sugar, and press the cookies down slightly.
Bake for 8 to 10 minutes, until golden brown.
Remove from the baking sheet to cool on wire racks.