I was just a short while ago reading Chapter 1 of the book Differential Geometry of Three Dimensions by C. E. Weatherburn, volume 1, and found that there were certain terms being used by the author that were a bit confusing or at least not in conformity with the terms used in more recent books (on calculus).
For example, Weatherburn refers to "consecutive points" on a curve, a term which does not make much mathematical sense (to me at least) in view of the fact that between any two real numbers (and hence between any two points on a continuous curve) there are infinitely many others and thus there is no such thing as two "consecutive" real numbers.
Accordingly, I would like to have a book on differential geometry that covers at least the same material as covered by the above-mentioned book by Weatherburn but does so in a more rigorous and detailed manner.