Ryle gives the following history of the differences in the books of Esdras as represented in the Septuagint and the Vulgate:
In the lists of the Old Testament which include the Apocryphal books, an element of confusion is caused by the Apocryphal ‘Ezra,’ our First Book of Esdras. In the LXX Version, the Old Latin, and the Syriac, this Apocryphal Greek Book was placed, out of regard probably for chronology, before the Hebrew Ezra, and was called the First of Ezra…while our Ezra and Nehemiah appeared as one book, with the title of the Second of Ezra. In his translation of the Vulgate, Jerome did not recognize the Canonicity of the Apocryphal Books. He translated the Hebrew Ezra (our Ezra and Nehemiah) as one book with the title of Ezra; but he acquiesced in the division of the Canonical Ezra into two books, for he speaks of the Apocryphal books as the third and fourth of Ezra…In the Vulgate, accordingly, Ezra and Nehemiah were called the First and Second of Ezra; the Apocryphal Greek Ezra was called the Third of Ezra; the Apocalyptic work, the Fourth of Ezra…The influence of the Vulgate caused the names applied in the books in that version to be generally adopted in the West. At the Council of Trent, Ezra and Nehemiah are called 'the first book of Ezra and the second of Ezra which is called Nehemiah.'88
These references make it clear that the Septuagint version of 1 and 2 Esdras was different from the one decreed by Trent. The New Catholic Encyclopedia (quoted above) states that for the first five centuries many fathers of the Church regarded 1 Esdras of the Septuagint to be canonical because they followed the Septuagint. Jerome was the first to separate Ezra and Nehemiah into separate books and to assign the title of I Esdras to Ezra and 2 Esdras to Nehemiah in order to conform to the Hebrew canon. The Septuagint version of 1 Esdras is quoted, for example, by Justin Martyr, Origen, Clement of Alexandria, Eusebius, Athanasius, Gregory of Nazianzus, Ephrem Syrus, Basil the Great, Chrysostom, Cyprian, Ambrose, Theophilus of Antioch, Dionysius of Alexandria, Augustine and Prosper of Aquaitaine.89 Augustine quoted from the book of III Esdras (I Esdras in the Septuagint) in his work The City of God.90 Thus, when the Council of Carthage gave its list of canonical books for the Old Testament it followed the Septuagint translation. In referring to Esdras as comprising two books they were referring to I and II Esdras of the Septuagint. And when Carthage sent these decrees to Rome for confirmation, it was these books which were confirmed as canonical. Innocent I affirmed this in his letter to Exuperius91 and they were later included in the decrees of Popes Gelasius and Hormisdas. B.F. Westcott confirms these facts:
The enlarged canon of Augustine, which was, as it will be seen, wholly unsupported by any Greek authority, was adopted at the Council of Carthage (A.D. 397?), though with a reservation (Can. 47, De confirmando ist Canone transmarine ecclesia consulatur), and afterwards published in the decretals which bear the name of Innocent, Damasus, and Gelasius…and it recurs in many later writers.92
This contradicts the decree passed by Trent which followed Jerome in assigning I and II Esdras to the canonical Hebrew books of Ezra and Nehemiah respectively. Therefore, Trent declared uncanonical what the Council of Carthage and the bishops of Rome, in the fourth, fifth and sixth centuries, declared to be canonical. Clearly, then, Carthage did not authoritatively establish the canon for the Church universally.