Combine shortening, 1 1/2 cups sugar, and eggs in a large mixing bowl, beating well.
Sift together flour, cream of tartar, soda, and salt in a small mixing bowl. Add flour mixture and vanilla to sugar mixture, stirring well.
Combine remaining sugar and cinnamon in a small mixing bowl, stirring well; set aside.
Shape dough into 1-inch balls; roll in reserved sugar-cinnamon mixture. Place on lightly greased cookie sheets; bake at 400° for 6 minutes or until lightly browned. Remove from cookie sheets, and cool on wire racks.