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The 7-string guitar had an extra "floating" bass string above the fretboard. It was only utilized for the open drop D, occasionally tuned to C.
Much of Napoleon Coste's music was written for 7-string guitar. Their music can be played on a normal 6-string guitar, but the low bass notes must be transposed up an octave. The extra low bass also allows many drop-D pieces to be played without de-tuning, or to occasionally drop the octave of normal D's for added resonance or emphasis.
The Coste-style 7-string Lacote below is in the private collection of Bernhard Kresse. The pictures below are of a French guitar from Mirecourt restored by Bernhard Kresse and owned by Raphaella Smits, and played on her new CD. Original period 7-string guitars are exceedingly rare.
According to Kresse, none of the early 19th century extra-string guitars have wider necks. Always just the first 6 strings are playable and the additional strings are beside the fingerboard. Even on Lacote's 10-string guitar with a wider neck, the frets stop after the sixth string.
The 1820 7-string shown beow is from the Musee de la Musique - another example of the Coste-style 7-string guitar with the normal 6 strings plus an added drop D in the bass.
It is difficult to determine if this guitar was originally built as a 7-string, or if it was later converted. The body shape and dimensions conform to 6-string models. The bridge is not centered; it is longer on the bass end to accommodate the extra bass string. The headstock appears to have been built normally, but with the side attachment added. The 7th string is a "floater".
The Russian variant of the romantic guitar was often a 7-string instrument with different tuning (tuned DGBdgbd') than the modern guitar, or what we think of as Early Romantic Guitar - which was actually a Western European invention.
The best person to contact for information about the Russian 7-string guitar is Oleg Timofeyev. Oleg is a specialist in the Russian guitar -- more about his bio on Mel Bay Authors and TalismanMusic. Oleg has recorded probably the only CD in existence on this instrument. The amount of repertoire is vast, but not readily available in the West.
Oleg collects original Russian guitars from the 19th century from the point of view of a performer and almost all the time has one for sale. Oleg carries the modern 7-string guitar by Vladimir Azhikulov, the only Muscovite maker specializing in Russian seven-string guitars (around $1500 with case).
Editions Orphee carries "The Russian Collection" - these are transcriptions of 7-string Russian guitar music for the modern 6-string guitar.