Preheat oven to 375 degrees F (190 degrees C). Sift together the flour, cream of tartar, baking soda and salt. Set aside.
In a large bowl, cream together the butter and sugar until smooth. Beat in the eggs one at a time, then stir in the vanilla. Gradually blend in the sifted ingredients.
Roll rounded teaspoonfuls of dough into balls, and roll the balls in the remaining cinnamon-sugar mixture. Place cookies 2 inches apart on ungreased cookie sheets and flatten slightly.
Bake 8 to 10 minutes in preheated oven. Allow cookies to cool on baking sheet for 5 minutes before removing to a wire rack to cool completely.