As I recall, 'old timey' toothpaste consisted of baking soda and pumice.
Unfortunately, pumice wears down the enamel of the teeth, while the baking soda cleanses the teeth.
Frequent brushing of teeth resulted in tooth loss.
The common recipe in the US was baking soda and salt.
This is interesting from another direction, as warm, salty water is regarded as the best general treatment for oral problems.