Greenleaf begins his book by arguing for the need to suspend prejudices and to be open to conviction, "to follow the truth wherever it may lead us" (p. 1). He cites Bishop Daniel Wilson's Evidences by stating that Christianity does not "bring irresistible evidence" but offers sufficient evidences for "the serious inquirer" (p. 2). He limits the scope of his book to an inquiry "to the testimony of the Four Evangelists, bringing their narratives to the tests to which other evidence is subjected in human tribunals" (p. 2). His specific inquiry is concerned with testing "the veracity of these witnesses by the same rules and means" employed in human tribunals (p. 3). Greenleaf argues the case by first inquiring as to the genuineness of the four gospels as ancient writings. Here he applies what is known in law as the ancient documents rule, stating that "Every document, apparently ancient, coming from the proper repository or custody, and bearing on its face no evident marks of forgery, the law presumes to be genuine, and devolves on the opposing party the burden of proving it to be otherwise" (p. 7). Greenleaf maintains that the Four Gospels do not bear any marks of being forgeries and the oldest extant copies may be received into court as genuine documents.
Greenleaf proceeds to argue that "In matters of public and general interest, all persons must be presumed to be conversant, on the principle that individuals are presumed to be conversant with their own affairs" (p. 9). On the basis of this legal rule, Greenleaf briefly profiles those traditionally attributed as authors of the Four Gospels, Matthew, Mark, Luke and John, concerning (in the case of John and Matthew) their firsthand knowledge of the life of Jesus of Nazareth and (in the case of Mark and Luke) their intimate personal links with Jesus' original band of disciples.
Greenleaf then builds a cumulative case by claiming to cross-examine the oral testimony of the evangelists in their accounts of the death and resurrection of Jesus. Greenleaf develops his case on the basis of the following tests:
"The credit due to the testimony of witnesses depends upon, firstly, their honesty; secondly, their ability; thirdly, their number and the consistency of their testimony; fourthly, the conformity of their testimony with experience; and fifthly, the coincidence of their testimony with collateral circumstances" (p. 28).
Greenleaf then argues that the gospel writers can be shown to be honest in their character and do not show any motives to falsify their testimony (pp. 28–31). He claims that keen observations and meticulous details are related by Matthew and Luke, and he concludes this demonstrates their ability (pp. 31–32). Greenleaf notes that there are parallel accounts from the evangelists concerning the central events of Jesus' life and that these accounts are not verbally identical. He maintains that discrepancies in their accounts are evidence that the writers are not guilty of collusion, and that the discrepancies in their respective accounts can be resolved or harmonized upon careful cross-examination and comparison of the details (pp 32–35). Greenleaf argues against the scepticism of the Scottish empirical philosopher David Hume concerning reports of miracles. He finds fault with Hume's position about "immutable laws from the uniform course of human experience" (p. 36), and goes on to assert that it is a fallacy because "it excludes all knowledge derived by inference or deduction from facts, confining us to what we derive from experience alone" (pp. 37–38). Greenleaf takes as his own assumption that as God exists then such a being is capable of performing miracles. He then argues that the various miracles reported in Jesus' ministry occurred in open or public contexts where friend and foe alike were witnesses (pp 39–42). Lastly, Greenleaf examines the problem of uniform testimony among false and genuine witnesses, and finds there is sufficient circumstantial evidence to support the accounts of the Four Evangelists.
Greenleaf sums up his argument with the following plea:
"All that Christianity asks of men on this subject, is, that they would be consistent with themselves; that they would treat its evidences as they treat the evidence of other things; and that they would try and judge its actors and witnesses, as they deal with their fellow men, when testifying to human affairs and actions, in human tribunals. Let the witnesses be compared with themselves, with each other, and with the surrounding facts and circumstances; and let their testimony be sifted, as if it were given in a court of justice, on the side of the adverse party, the witnesses being subjected to a rigorous cross-examination. The result, it is confidently believed, will be an undoubting conviction of their integrity, ability and truth ... Either the men of Galilee were men of superlative wisdom, and extensive knowledge and experience, and of deeper skill in the arts of deception, than any and all others, before or after them, or they have truly stated the astonishing things which they saw and heard" (pp. 46 & 53).

Question: How does Greenleaf  get the Four Gospels received into court?
Answer: He explains how they do not bear any marks of forgery. 
Alternative answer: ancient documents rule