I am not a Spear and Jackson history expert but I understand that the Mermaid saws were carpenter’s saws. A photo of the saw would help us but you should be able to tell from the basic tooth form and number of teeth per inch. If the teeth are relatively large (for a carpenter or panel saw) and if the front face of the tooth is generally perpendicular to the blade then it originally was a rip saw. Backsaws are a little less straightforward because of the probability of finer teeth in both saw types but if the teeth really are that worn (that you can’t tell) then you would really need to re-tooth the saw. That means you can put whatever tooth form you want on it.