Updated: Nov 12, 2021
2. The Epistle was written, not by Luke, nor by Barnabas, nor by Apollos, nor by Clement of Rome, but by Saint Paul. Which is confirmed by several arguments.
Concerning the writer or author, the amanuensis of the Holy Spirit in this Epistle, the sense of all is not the same. That it was written by Luke the Evangelist, Origen already of old insisted in Eusebius’ Historia Ecclesiastica, book VI, section 19; and he is followed by Calvin and Grotius. Tertullian, Philastrius, and Cameron attribute authorship to Barnabas: others, to Clement of Rome: finally, Luther and others, to Apollos. Following the ancient Greeks, Justin, Clement of Alexandria, Athanasius, Cyril, Gregory of Nyssa, Nazianzen; and also the Latins, Jerome, Hilary, Ambrose, Augustine, and especially our Helvetic Confession; none of us doubt that this Epistle was written by Saint Paul, both because it is said by Saint Peter, 2 Peter 3:15, 16 that Paul wrote to the Hebrews dwelling or wandering ἐν τῇ διασπορᾷ, in the Diaspora/dispersion, even indeed in some certain Epistle, which he distinguishes from the other Epistles, and concerning the same matters of which Peter also wrote (compare 2 Peter 3:9-16; Hebrews 1:10; 2:3; 3:7; 4:11; 11:25; 12:12, 25; etc.); according to the wisdom given to him by God, and that so heavenly and hidden, wherewith inspired he also wrote the other Epistles; and, finally, δυσνόητά τινα, some things hard to understand, of which sort there are not a few in this Epistle, especially those things δυσερμήνευτα, hard to express, Hebrews 5:11: and because various circumstances demonstrate that Saint Paul is the author, of which sort are, that the Epistle is said to have been written from Italy, Hebrews 13:24, compared with Acts 27; that the writer, in Hebrews 10:34, testifies that he wrote in bonds (for it is evident that he was not yet freed, because he yet hoped for release, Hebrews 13:19), while Saint Paul everywhere in his own Epistles makes express mention of his bonds, and even of those specifically that he wore at Rome in Italy, Philippians 1:13; that the author also brings in Timothy, a sharer in his bonds, and a companion in his journeys, as his brother, Hebrews 13:23, the mention of whom by Saint Paul is common enough in his Epistles; that in Hebrews 13:18 he clears himself by appealing to his conscience as a witness, as Saint Paul also clearly did in Acts 23:1; 2 Corinthians 1:12; that, finally, he implores the prayers of the Hebrews, as it was also customary for Paul, as in Romans 15:30; Colossians 4:3; etc.: and because this Epistle’s matter, form, order of teaching, sentences, phrases, and a great many words closely agree with the rest of the Epistles of Saint Paul, which is able to be demonstrated by induction: and because the scope/goal of this Epistle, the abrogation of the Mosaic law with respect to carnal rites and worship, was also the same scope/goal of the rest of the Epistles of Saint Paul, as especially the Epistles to the Galatains and Colossians plainly do: and, finally, because it is able only with the greatest difficulty to be ascribed to those authors to whom others attribute it. For, the author is not able to be Luke, because he, as a Proselyte and a Gentile (while there is indication that the author of this Epistles was a κοίνησις/kinsman of the Hebrews, Hebrews 2:1; 13:13), was not exerting that kind of authority among the Hebrews; and the style of his Gospel little agrees with the style of his Epistle, since Luke’s is more Grecian than even the Greek of this Epistle, but it abounds in Hebraisms and Syriasms. Nor Barnabas, to whom the circumstances alleged above do not agree, and who was not an Apostle, nor renowned for miracles, and, not being immediately called, did not have that authority among the Hebrews that was required in the author of this Epistle; and whose style in that Greco-Latin Epistle attributed to him (whether rightly or not, we do not here dispute) manifestly disagrees with the style of this Epistle. Nor Clement of Rome, since he, called by Gentiles, was not so conversant in the writings of the Hebrews, and whose diction was not able to spring with Hebraisms. Nor, finally, Apollos, who was not an Apostle, whom the aforementioned circumstances are less favorable.
 Philastrius (died c. 397) was Bishop of Brescia. He participated in the anti-Arian synod of Aquileia held in 381, and wrote Diversarum Hereseon Liber.  Clement of Rome served as Bishop of Rome from 92 to 99.  See the Second Helvetic Confession, chapter XI, section 1b: “…Therefore, with respect to his divinity the Son is coequal and consubstantial with the Father; true God (Philippians 2:11), not only in name or by adoption or by any merit, but in substance and nature, as the apostle John has often said: “This is the true God and eternal life” (1 John 5:20). Paul also says: “He appointed the Son the heir of all things, through whom also he created the world. He reflects the glory of God and bears the very stamp of his nature, upholding all things by his word of power” (Hebrews 1:2, etc.).…”  Peter asserts that Paul also wrote an Epistle to the believing Jews of the Diaspora. See 2 Peter 3:1. 1 Peter 1:1: “Peter, an apostle of Jesus Christ, to the elect sojourners of the diaspora (ἐκλεκτοῖς παρεπιδήμοις διασπορᾶς) of Pontus, Galatia, Cappadocia, Asia, and Bithynia…” See also James 1:1: “James, a servant of God and of the Lord Jesus Christ, to the twelve tribes in the diaspora (ἐν τῇ διασπορᾷ), greeting.”
Dr. Dilday's Lecture: "A Pauline Miscellany, Part 3"