Figured bass written when it was a living notation is often not so neatly formalized as modern figured bass used as a teaching aid for learning common practice harmony.
Also, most music editions published in this era have plenty of typos, so if something seems totally incomprehensible, it may just be nonsense!
The horizontal lines are continuation lines, but not positioned in a systematic way as in modern figured bass. You have to work out what note in the previous chord is intended to be tied, bearing in mind that the relevant note might not have been indicated by a figure at all.
In the first example, the two chords are G Bb D E and A D F, so the two D's should be tied together to make one long note. That restricts the possibilities for voicing the chords, since the two Ds can not be in different voices, or an octave apart.
In the second example the first chord, with no figures, implies 8 5 3. In the second chord, the 5 is tied over, the 8 falls to 7, and the 3 rises to 3#.
In the first example on the second line, the only common note is the octave of the bass D, which (like the bass note itself) would be tied.
In the final example the common note is the doubled A in the first chord, and the rest of the voice leading would be from C# to C natural, and E to F#.
If you want to study this text in detail, download the "supplement" from IMSLP (which I assume is where you found the main book.) The preface to the supplement contains some explanation about completing Geminiani's figuring - for example 6 4# (which is a dissonant chord) implies 6 4# 2, though of course 6 4 (which is consonant) is complete as written (or it means 8 6 4 if you want to be pedantic.)