Before discussing various methods and the conclusions drawn from such methods within the context of gospel origins; that is as they are understood by modern academic conception, I'd like to trace, albeit simplistically, the reasons why such an endeavor became possible even necessary.
The advent of a modern academic understanding emerged from 19th century German scholarship. Through a fairly large body of literature much of which is often contentious, reactive, and controversial these scholars reshaped the contours of N.T. studies with respect to gospel origins. But first, for the sake of symmetry, a quick word about the traditional or orthodox conception of gospel origins.
In the 5th century St. Augustine mounted a detailed and formal defense of Christianity in his "On the Harmony of the Evangelists." This in response to the Pagan philosopher Porphyry's work "Adversus Christianos" which was a list of "contradictions and implausibilities within the gospels" (Kloppenborg, 2000:295) Augustine's harmonization model became the preferred conception of gospel origins for Roman Catholics and later Protestants. Culling information for various early church fathers (Tatian, Justin, Papias etc) he proposed the gospels were written by disciples of Jesus (as in the case of Matthew) or by close associates of the disciples (as in the case of Mark).
All four wrote in canonical order, that is as they appear in the N.T., and each wrote aware of their predecessor's work. Each author chose to focus their gospel on various aspects of Christ: MT on royalty, MK on humanity, LK on priesthood, and JN on divinity. With respect to chronology, all four were written within the lifetimes of those first eyewitnesses, say prior to 70 C.E. Problems of contradiction, disagreement, and incongruities were rendered moot by an appeal to the natural fallibility of memory. Thus we have, in this model, the gospels we read today. Historicity is to be had within the framework of compound or multiple attestation, that is where 2, 3, or all 4 gospels agree. Harmony, as it were, equated historical fact.
The problematic nature of this understanding has been well established. "We do not...know the original wording of any of the gospels, for their autographs have long since perished. What we posses are about 6000 manuscripts-none from the 1st century, small fragments from the 2nd, portions of individual NT books from the 3rd and complete bibles only after the 4th century." (Kloppenborg, 2000:12) To further complicate the matter, the names ascribed a given gospel MT, MK, LK, JN are not original to the text but were added to them as late as the early 3rd century. The texts themselves are anonymous (Ehrman, 1998:10). Given this rather bleak situation, and other complicating factors available elsewhere, by the dawn on the 19th century, German scholarship began to doubt the legitimacy of the "Augustinian solution."
Already by the late 18th century, Hermann S. Reimarus (1694-1768) in a burst of minimalist bravado mounted a full scale attack on the historicity of the gospels viewing them not as "expression of Jesus intentions but as fraudulent and fantastic accounts of miracles, prophecies, angels, the resurrection, and the Parousia." (Kloppenborg 2000:275) The wedges that Reimarus had driven, notes Kloppenborg, among the gospels and between the gospels and Jesus would prove impossible to extract from all future scholarly discussion. In fact the next 100 years in NT study in relation to gospel origins would be either in assent or dissent from the Reimarus indictment.
The end of the 19th century saw a shift in the contours of understanding the gospels. By now, the rather disjunctive nature of the first three gospels (MT,MK,LK) as compared to JN became clear. An acceptance of the so-called synoptic problem meant there were two separate and distinct issues within gospel origins for which one had to account. "Synoptic" ,by the way, refers to the relative ease with which once can read the first three gospels "at a glance" because of their similar structure and internal content. The realization of the synoptic problem coupled with a rejection of the Augustinian harmonization model emphasized a dichotomy between the texts of the gospels and the dogma or theology that emerged from them. Add to this the rejection, as well, of various post-apostolic church fathers (Eusebius,Papias, Irenaeus) as reliable sources of information about gospel origins and you have the fertile environment from which the modern academic understanding of the gospels emerged.
The advent of this new understanding flourished in the text-dogma dichotomy. It allowed (required?) for novel approaches, bold conjectures, and the use of new methods of inquiry to account for various textual phenomena and literary problems. In the first place, exterior evidences of authentication were subsumed to internal literary analysis of wording and sequence, patterns of agreement and disagreement; literary criticism which employed various methods to discern source relationships, issues of dependence and independence became the vehicle through which various questions could be addressed. It was from such a 19th century context that two enduring hypotheses saw the light of the 20th century: the Greisbach hypothesis (GH) and the Two Document hypothesis (2DH)
Johann J. Greisbach became an early advocate of these new methods of inquiry. In 1789-90 he published a comprehensive outline of his hypothesis. His solution to the synoptic problem would prove significant for modern standards of study. He left Jn's gospel aside to focus on a close comparison of the synoptic threesome. That procedure is now set: how do the synoptics relate to each other and how does Jn's gospel relate to them? Greisbach concurred with Augustine in terms of Matthean chronological priority but argued that it must have been Mk, not Lk, who wrote last.
He based his conclusion on various Markan omissions of key Mt and Lk texts, like the Sermon on the Mount. In 1826, W.M.L. de Wette boulstered Greisbach's claim of Mk posteriority by attempting to show how Mk has alternated use of both Mt and Lk. A key example was his analysis of "double expressions" found in Mark. A typical example of this is Mk 1:32 ("when it was evening, when the sun had set"). For Greisbach and de Wette, it seemed as though this was a conflation of two separate texts from Mt and Lk. Mt 8:16 reads ("when it was evening") and Lk 4:40 reads ("when the sun had set"). Were this the case, clearly Mk would have been subsequent to both Mt and Lk.
Against this interpretation, Bernard Weiss (1827-1918) argued that Mk's double expression were neither tautological and secondary but necessary and original since for Mk the incident occurred on a sabbath (1:21) it was necessary to stipulate the sun had set so that the fact that the sick came to Jesus was explicable. Lk, who also dates the incident to a sabbath preserved the mention of the setting sun but in a form that betrays Mk's genetive absolute (when it was evening). In Mt, however, the healings don't occur on a sabbath and so it was not necessary for him to preserve the precise temporal reference. (Weiss 1861:683 in Kloppenborg 2000:290)
H.J. Holtzmann (1832-1910) a contemporary of Weiss argued that if Mk used Mt and Lk for his composition, one would expect to find the favorite vocabulary of his two sources within Mk. However, it lacks Mt's ("kingdom of the heavens"), ("it has been said"), ("at that time") etc and numerous Lukanisms. In response to de Wette's proposal, Holtzmann emphasized that such editing by Mk would have required "unimaginable procedures."
The works of B. Weiss and H.J. Holtzmann in critique of the G.H. paved the way for Markan priority. With this shift, the traditional Augustinian model, even with respect to Mt priority, lost all credibility within academic discourse. In the same generation of scholars, Karl Lachmann (1793-1851) made some significant insights. whereas Greisbach focused on Markan omissions, Lachmann payed special attention to disagreements among the synoptics. His conclusion? Kloppenborg sums up: "when one compared the order...in the synoptics, the greatest degree of disagreement was registered between Mt and Lk" (2000:297) Or again, with respect to order and sequence of specific units or pericopae, Mt-Mk-Lk have substantial agreement between them (the Triple Tradition) several cases where Mt-Mk agree against Lk or where Mk-Lk agree against Mt but rarely is it the case where Mt-Lk agree against Mk. When it occurs the Markan sequence is followed. Where Mt-Lk agree in content they differ in wording and sequence. Hence, they can only be independent of one another. (Kloppenborg, 2000:297)
The priority of Mk and the establishment of the independence of Mt and Lk provide the first two tiers in the architecture of the second enduring postulate of gospel origins from the 19th century. It remains the most widely accepted theory among scholars though its critics are not few. It involves a third tier as well but first a second case to establish further both the priority of Mk and the independence of Mt and Lk. Perhaps the best argued case has been made by J. D. Crossan. He cites several cases of "Markan literary fingerprints" present within both Mt and Lk. He notes, "one of the most peculiar and distinctive Markan compositional devices has been called intercalation or sandwich...basically, Event (A1) begins, then Event (B) begins and ends then finally, Even (A2) finishes." (Crossan, 1998:106) There is a fairly wide consensus on at least six cases of intercalations in Mk's gospel: Mk 3:20-35(event A), 3:22-30(event B to end), 3:31-35(event A ends) & Mk 5:21-24(event A), 5:25-34 (event B to end), 5:35-43(event A ends) are but two examples. It is the presence of intercalations within both Mt and Lk that establish Mk priority most securely, as Crossan emphasizes, those intercalations are peculiarly if not uniquely Markan. The independence of the two can be seen in the varying ways in which both Mt and Lk appropriate the "sandwich" device into their gospels. This divergence clues us into the trajectory of the device, namely from Mk into both Mt and Lk, though independently of one another.
Recall the three tiered architecture of the 2DH. Mk priority was the primary tier with the independence of Mt and Lk the secondary one. The third tier, then, is a logical inference based upon the preceding two conclusions. In 1890 Johannes Weiss, echoing the work of F. Schleiermacher, proposed a source behind Mt and Lk other than Mk. The word for source in German is "quelle" from which this hypothetical document derives its name, 'Q' now called the sayings source Q or quite simply the "Q" gospel. It must be emphasized that this document is a purely hypothetical postulate. No evidence of its existence occurs outside scholarly conception. We do not have that source available to us like we do with Mark. The same reasoning goes into this supposition as was used for Mk as a source, namely verbal agreement.
A classic example of such a possibility is to be found in Mt 3:7-10=Lk 3:7-9. Crossan explains, "That indictment by John the Baptist, over 60 words in Greek, is verbatim the same...those are not twin versions of a common oral matrix but the very...faithful reproduction of a common source." (1998:105) The "Q" hypothesis, then, is a function of the independence of Mt and Lk. If they are dependent on one another there would be no need to posit a common source.
I hope it's clear that the main contribution of that 19th century German scholarship was the introduction of and emphasis upon source-theory and literary criticism and the solutions they provide; as Kloppenborg and Crossan emphasize: they provide the most "economical accounting of the evidence than other available theories." Critics of such methods abound, usually among the more conservative scholars like N.T. Wright, L.T. Johnson, B. Witherington III to name a few. This shape of understanding the gospels leads to further conclusions that can be built atop that foundational layer.
