So where is ASL from anyway?
Schroeder is right in that we have no record of what ASL looked like in the early- to mid- 1800s, neither taped nor written, so it is difficult if not impossible to determine what it looked like. We have no way of knowing what the pidgin mix of LSF and Martha's Vineyard signs looked like.
However, I believe plenty of evidence, both linguistic and historical, exists that clearly demonstrates beyond a doubt that ASL and LSF are linguistic relatives.
Sign "with."
Really. Go ahead and sign "with."
Okay. Look at your hands. What handshape are you using to sign "with?"
That's right. A.
And the French word for "with" is...
*drumroll*
avec.
And the LSF sign for "with" is a cognate of ours.
There are more examples. Sign "see" as in "I see something."
What handshape are you using? V. The French word for to see? voir
Want another? Sure... sign "search" or "look for." Handshape? C. French word? chercher
And all of these signs have LSF cognates (if I remember my LSF right... my LSF book isn't handy at the moment). There are many ASL-LSF cognates that do not have English-French cognates, such as the sign for "excuse me," which is a near-perfect cognate of the LSF sign that corresponds to "pardon."
Another factor pointing to the idea that LSF existed first and was one of the progenitors of ASL is that there was a deaf school in Paris for close to a century before there was a deaf community in America, not counting Martha's Vineyard. The Institut National des Jeunes Sourds is where Jean Massieu and Laurent Clerc went to school and later taught. This is where Gallaudet was taken once he had decided to check out the French way of teaching deaf children. (aside here: if you ever get to visit the school, head to the right rear stairwell. There's this stone engraved passage talking about how Gallaudet came to the school and brought Clerc to America, and started a school for the deaf in America. In their definitive French deaf history book, Les Pouvoir des Signes, or The "Can" of Signs, they speak of Gallaudet and American deaf education in very reverent tones.) LSF was already in existence and in use when Gallaudet talked Clerc into leaving everything he'd ever known for the American wilderness. There was a book of LSF in existence as early as 1779, written by Pierre Desloges. (The site is in French, but the info is in the first paragraph.) In Nora Ellen Groce's book (by the by, Schroeder's source quotes from this book as well), it discusses the differences between ASL and Islander signs. One of the eldest islanders said in an interview that he could never understand young deaf sign language because the signs used on the island were so different. Ergo, Islander sign language is probably not the most significant source of ASL. We've established that LSF existed before ASL as much as possible and that it is substantially different from Martha's Vineyard signs without concrete evidence of what ASL looked like a century and a half ago.
Another clear indicator is the grammatical construction found in ASL and in spoken/written French. One oft-quoted example of ASL grammar is CAR RED, in which the adjectives occur after the noun, unlike English, where the adjectives precede the noun, as in red car. Care to hazard a guess which construction spoken/written French uses? Yup, you got it. Same as ASL - la voiture rouge. The linguistic similarities are much deeper than simple word order, however. There's an idiom in English which reads, "It's clear as mud." If this were to be translated into French, it would read "Il est clair comme boue." If we translate word-for-word back from French into English, it would be "It is clear how mud." Feel familiar? It should if you use ASL, because this is a construction that often comes up in ASL, but never ever in English. As a Francophone, I've found I have a much easier time translating between written French and ASL because their grammatical constructions are more similar than those of English and French. There are more examples, but you get the idea.
While this is probably not sufficient evidence to satisfy everyone out there, it is way more than enough to satisfy me that ASL is directly descended from LSF, with other influences thrown in and time and distance have helped to separate the two languges as well.