 Unfortunately there used to be a very confused theory about model-theoretic axioms, that also went under the name of implicit definition. By the end of the nineteenth century, mathematical geometry had generally ceased to be a study of space, and it had become the study of classes of structures which satisfy certain ‘geometric’ axioms. Geometric terms like ‘point’, ‘line’ and ‘between’ survived, but only as the primitive symbols in axioms; they no longer had any meaning associated with them. So the old question, whether Euclid’s parallel postulate (as a statement about space) was deducible from Euclid’s other assumptions about space, was no longer interesting to geometers. Instead, geometers showed that if one wrote down an up-to-date version of Euclid’s other assumptions, in the form of a theory \(T\), then it was possible to find models of \(T\) which fail to satisfy the parallel postulate. (See the entry on  geometry in the 19th century  for the contributions of Lobachevski and Klein to this achievement.) In 1899 David Hilbert published a book in which he constructed such models, using exactly the method of interpretation that we have just described.