Preheat oven to 375°F (190°C).
In a mixing bowl, cream together the butter and sugar until light and fluffy.
Add the eggs and vanilla extract, and beat well.
In a separate bowl, whisk together the flour, cream of tartar, baking soda, and salt.
Gradually add the dry ingredients to the wet ingredients, mixing until combined.
In a small bowl, mix together the sugar and ground cinnamon for the coating.
Shape the dough into 1-inch balls, and roll each ball in the cinnamon-sugar mixture.
Place the coated dough balls onto a baking sheet, about 2 inches apart.
Bake for 8-10 minutes, or until the edges are set and the centers are soft.
Allow the cookies to cool on the baking sheet for a few minutes before transferring to a wire rack to cool completely.