Strictly speaking, the origins of the gospels as a strictly literary question was not a concern for ancient and medieval Christians. Thus, any information about their literary origins tended to be incidental and, as a result, rather vague.
Eusebius quoted Papias as stating that Mark had been the interpreter of Peter and that Matthew put together the "logia" in a Hebrew manner of speech. Interpretation of these obscure fragments would dominate discussion of the origins of the gospels for centuries.
In his defense of the apostolic and chronological priority of the four canonical gospels over heretical works, Irenaeus laid out the following sequence: Matthew was written first in Hebrew when Peter and Paul were founding the church in Rome (i.e., the 60s), that Mark and Luke handed down their preaching in writting after their departure (i.e, after 66), and that John was written last. Interestingly, except for Irenaeus's supposition of a Hebrew Matthew written in the 60s, his summary is consistent with modern, critical datings and sequence for Mark, Luke, and John.
c.200 Clement of Alexandria
Eusebius quoted Clement as: "The Gospels with the genealogies [Matthew and Luke?] were 'written before'." Commonly thought to be a statement of Mark's posteriority to Matthew and Luke, Clement's comments are better interpreted as a reference to the public nature of their publication in contrast to a more private dissemination of Mark.
In his defense of the consistency of the gospel accounts, Augustine assumed that each evangelist wrote with knowledge of the previous Gospels, not out of any interest in their literary origins (a modern interest) but only at the beginning of his work to present the evangelists as a model of sympathetic collaboration with their predecessors. Thus, silence in other gospels should not be interpreted as disapproval or ignorance, and differences in the gospels should be understood as differing emphases or style. Throughout the rest of his work, however, the literary origins of the gospels played no role in his explanations, often using the terminology of omission and addition without any regard to their chronological order.
Perhaps one of the earliest to break away from pure utilization solution consistent with Augustine's remarks, LeClerc argued that the evangelists could have used earlier sources. (LeClerc 1712: 429; accord Michaelis 1777: ¶ 125; both according to Farmer 1964/1976: 9 n.11).
Henry Owen was the first to articulate was is now called the Griesbach or Two-Gospel Hypothesis: that Mark used Luke as well as Matt. (Owen 1764: 53-75, according to Farmer 1964/1976: 7 n.8, Farmer 1990: 147-156)
Another early source critic is Anton Büsching would also teld that Mark used Luke as well as Matt, but felt that Luke was first. (Büsching 1766: 109-119 according to Farmer 1964/76: 7 n.8)
J. J. Griesbach can be credited with the publication of the first literary synopsis. (Farmer 1964/76: 3, 5-6; Greeven 1978: 26-28; Reicke 1978b: 69-70)
1778 Lessing (pub. 1784)
Aramaic Ur-Gospel source (Gospel of the Nazarenes) used independently by Mt, Mk, and Lk.
Marcus non epitomator Matthaei [Mark not the abridger of Matthew] because Mark deviated in order from Mt but which Lk supports over the apostle. Gospels compiled from Greek and Hebrew fragments.
Successive dependence with order Mt, Lk, Mk. Mk is a conflation of Mt and Lk, not an epitome of Mt.
Storr is the earliest proponent of Mark's priority, arguing against Koppe based on Papias's testimony that the authoritativeness of Mark comes from Peter.
Original Aramaic Gospel in four revisions. A - basis for Mt; B - basis for Lk; C from A and B - basis for Mk; and D - used by both Matt. and Luke but not Mark.
Oral gospel (Lk1:2), and Mk similar to Peter's words in Ac1:22.
1798 Marsh (pub. 1802)
Two Greek, common sources :- an Ur-Gospel (Aleph) which contains the Triple Tradition, and a primitive sayings source (Beth), both independently used by Evangelists.
The Development of the Markan Hypothesis
Fragmentary Hypothesis, gospels are a collection of various written materials. Papias' Mark is a gospel written by Mark, but not the canonical Mk. Papias' "Logia" is a collection of Jesus' sayings that formed a "Proto-Matthew" to which narrative material was added. but not by Luke who used Mt. (Canonical Mk is a later gospel as in GH.)
Order of the fragments in oral tradition best reflected in Mark.
1835 Tübingen School, D.F.Strauss & F.C.Baur
Greisbach Hypothesis adherents with late date (c. 130) for Mt. Mt is not eyewitness but myth. Lasted until 1864 with the appointment of Weizsäcker.
Markan priority, with both Matt. and Luke using Mark. The Double Tradition is from Matt.'s use of Luke.
Both Matt. and Luke combined Papias' Mark and the Logia. Papias' Mark was first identified as the canonical Mk, thus making the first formulation of the Two-Source Hypothesis.
1842 De Wette/1862 Bleek
Mk a conflation of Mt and Lk, but both Mt and Lk independently based on an Ur-Gospel.
1848 Ewald: 9-Source Hypothesis
(1) Ur-Gospel, (2) Logia, (3) Mk on 1+2, (4) "book of higher history", (5) Mt on 1+2+3+4, (6) (7), (8), and (9) Lk on 1+2+3+4+6+7+8, not Mt!
Retreats from his identification of Papias' Mark (and hence Mt & Lk's source) with our Mk, but with an Ur-Markus, containing the shared narrative between Mt & Lk.
Two-Source Hypothesis: (1) Alpha (Ur-Marcus), an early gospel similar to Mk and (2) Lambda [later called Q by J.Weiss], a sayings source used by Mt and Lk. Mt and Lk are independent. Both sources are thought of in terms of Papias.
1880 Simons & Holtzmann
Luke also used Matthew as subsidiary source. Proposal basically went unnoticed.
Attributes Mt and Lk's shared narrative to Q also, allowing Ur-Markus to be identified with canonical Mk.
The Refinement of the Two-Source Hypothesis
brings Holtzmann's synthesis to Oxford.
Encyclopaedia Britannica article. Argument from wording against 2GH: there is an almost total lack of additional matter in common between Mt and Lk in Triple Tradition contexts. Minor agreements accounted for by independent editorial changes.
1884 Abbott & Rushbrooke
Another argument from wording against 2GH, Mark must have carefully underlined the words common to Mt and Lk and include all those and still be a natural composition.
Applying Lachmann's argument from order, Ur-Marcus is substantially the canonical Mk. Four objections to Griesbach: (1) argument from omission, (2) Mark's selections are inscrutible, (3) argument from order, and (4) Griesbach does not explain relationship between Mt and Lk as well as 2SH.
Minor agreements are a problem to the 2SH, perhaps to be cleared up by textual criticism.
Explains how copyist errors can create minor agreements.
In his Horae Synopticae, Hawkins argues that many of the Minor Agreements are independent editing by Matthew and Luke of Ur-Marcus, rather than by the final revisor of Mark. Ur-Markus, then, is almost identical to Mark.
Minor agreements now best explained by a recension of Mk which was used by Mt and Lk.
If Ur-Marcus existed, then either Mk copies Ur-Marcus exactly or whenever Mark didn't, either Mt or Lk also did not.
1911 Oxford Studies with Streeter
Mark/Q overlaps: Mk knew Q.
The Four Gospels, "Fundamental Solution" Mt and Lk dependent on canonical Mk, not Ur-Marcus. Mk did not know Q. Minor agreements explained by (1) Irrelevant Agreements: compression of details and improvements in the Greek; (2) Deceptive Agreements: coincidental stylistic improvements; (3) Mark/Q overlaps; (4) Textual Corruptions.
The Four Gospels, proposes proto-Luke = Q + L.
Minor agreements is positive evidence for Lk's use of Mt.
Argument from order is invalid if Gospels are directly dependent.
In a publication of a series of his lectures, James Hardy Ropes noted that Q would be unnecessary if Luke used Matthew.
Defense of Augustinian Hypothesis, literary evidence against 2SH, reiterates Jameson's criticisms of 2SH.
Streeter committed the "Lachmann Fallacy," which is a misapplication of Lachmann's argument from order to directly dependent gospels. Lachmann's argument really means that Mark is some kind of a "middle term."
Challenges Markan priority by merging Mark and "M" into a proto-Matthew, which is the source for Matthew and Mark.
Dispensing with Q with Luke's use of Matthew.
Accepts Butler's critique of the Lachmann Fallacy, but defends Marcan priority on other grounds (by literary phenomena).
Revives the Griesbach Hypothesis. Lachmann's argument also supports Mk being a conflation. Eschews reliance on hypothetical sources (i.e., Q). Calls attention to the pattern of alternating agreement over Mark's order between Mt and Lk, in addition to much noticed absence of agreement between Mt and Lk over Mk's order.
Posits Luke subsidary use of Matthew based on statistics.
Argues for Matthew's used of Mark without Q.
Posits Luke subsidary use of Matthew based on Matthean foreign bodies.
The pattern of alternating agreement is also mathematically possible under the 2SH.
Q was composed in stages.
Argues that a Matthean Vocabulary Fallacy and a Lukan Priority Fallacy is used in favor of Q.
The Case Against Q