Cream margarine and sugar in mixer, beating well until light and fluffy. Add egg and beat well. Add vanilla until combined. In a separate bowl, sift flour, baking soda and cream of tartar three times. Add to creamed mixture. Beat until combined. Form dough into a ball, cover and chill at least one hour.
Heat oven to 350 degrees. Place a clean dishcloth on the work surface and lightly flour the cloth. Place dough on cloth and roll thin, 1/4 to 1/8 of an inch. Cut into circles or other desired shapes. Carefully transfer cookies to baking sheet. Sprinkle with sugar. (Will need to bake in several batches.) Bake 7 to 10 minutes or just until brown around the edges. Cool a few minutes on baking sheet, then remove to rack to cool completely.
Be sure to sift the flour before measuring.