According to the flesh.

Ancient instances of the phrase in Romans 1.3.

Christ, who had not yet taken vpon him mans nature, neither was the sonne of David according to the flesh, as he was afterward: but appeared then in a figure, and that in the cloudes, that is, being separate from the common sort of men by manifest signes of his divinitie.
—Geneva Bible Annotations, Daniel 7.13.

Refer also to the use of the expression born of a woman in ancient literature, and also to the use of the term flesh in the Pauline epistles.

Jeffrey Gibson has kindly provided a .pdf file containing various instances of this Greek phrase in antiquity.

Aristotle, History of Animals 17:

Of the viscera the liver in some animals becomes fatty, as among fishes is the case with the selachia, by the melting of whose livers an oil is manufactured. These cartilaginous fish themselves have no free fat at all in connection with the flesh [κατα σαρκα] or with the stomach. The suet in fish is fatty, and does not solidify or congeal. All animals are furnished with fat, either intermingled with their flesh [κατα σαρκα] or apart.

George K. Stroduch translates Epicurus, Principal Doctrines 4 (Ratae Sententiae, fragment 4) as follows:

Bodily pain does not last continuously. The peak is present for a very brief period, and pain that barely exceeds the state of bodily pleasure [or pleasure according to the flesh, κατα σαρκα] does not continue for many days. On the other hand, protracted illnesses show a balance of bodily pleasure over pain.

Source: A post on the IIDB by Peter Kirby, based on one of his Thesaurus Linguae Graecae searches.

Epicurus also apparently uses the phrase κατα σαρκα in Deperditorum Librorum Reliquiae, treatise 31, fragment 23, and in fragment 66 of Epistularum Fragmenta.

Josephus, War 2.8.11 §154-155, writing of the Essenes:

Και γαρ ερρωται παρ αυτοις ηδε η δοξα, φθαρτα μεν ειναι τα σωματα και την υλην ου μονιμον αυτων, τας δε ψυχας αθανατους αει διαμενειν, και συμπλεκεσθαι μεν εκ του λεπτοτατου φοιτωσας αιθερος ωσπερ ειρκταις τοις σωμασιν ιυγγι τινι φυσικη κατασπωμενας.

For their doctrine is this, that bodies are corruptible, and that the matter they are made of is not permanent, but that the souls are immortal and continue for ever; and that they come out of the most subtle air and are united to their bodies as to prisons, into which they are drawn by a certain natural enticement.

Επειδαν δε ανεθωσι των κατα σαρκα δεσμων, οια δη μακρας δουλειας απηλλαγμενας τοτε χαιρειν και μετεωρους φερεσθαι. και ταις μεν αγαθαις ομοδοξουντες παισιν Ελληνων αποφαινονται την υπερ ωκεανον διαιταν αποκεισθαι και χωρον ουτε ομβροις ουτε νιφετοις ουτε καυμασι βαρυνομενον, αλλ ον εξ ωκεανου πραυς αει ζεφυρος επιπνεων αναψυχει ταις δε φαυλαις ζοφωδη και χειμεριον αφοριζονται μυχον γεμοντα τιμωριων αδιαλειπτων.

But, when they are set free from the bonds according to the flesh, they then, as released from a long bondage, rejoice and mount upward. And this is like the opinions of the Greeks, that good souls have their habitations beyond the ocean, in a region that is oppressed neither with storms of rain or snow nor with intense heat, but that this place is such as is refreshed by the gentle breathing of a west wind that is perpetually blowing from the ocean, while they allot to bad souls a dark and tempestuous den, full of never-ceasing punishments.

Romans 1.3-4:

...περι του υιου αυτου του γενομενου εκ σπερματος Δαυιδ κατα σαρκα, του ορισθεντος υιου θεου εν δυναμει κατα πνευμα αγιωσυνης εξ αναστασεως νεκρων, Ιησου Χριστου του κυριου ημων.

...concerning his son who was from the seed of David according to the flesh, who was declared the son of God in power according to the spirit of holiness by the resurrection of the dead, Jesus Christ our Lord.

Romans 9.3-5:

Ηυχομην γαρ αναθεμα ειναι αυτος εγω απο του Χριστου υπερ των αδελφων μου των συγγενων μου κατα σαρκα, οιτινες εισιν Ισραηλιται, ων η υιοθεσια και η δοξα και αι διαθηκαι και η νομοθεσια και η λατρεια και αι επαγγελιαι, ων οι πατερες, και εξ ων ο Χριστος το κατα σαρκα, ο ων επι παντων, θεος ευλογητος εις τους αιωνας, αμην.

For I could wish that I myself were cursed away from Christ on behalf of my brethren, my kinsmen according to the flesh, who are Israelites, whose are the adoption and the glory and the testament and the lawgiving and the worship and the promises, whose are the fathers, and from whom is the Christ according to the flesh, the one who is over all things, God blessed unto the ages, amen.

1 Corinthians 10.18:

Βλεπετε τον Ισραηλ κατα σαρκα· ουχ οι εσθιοντες τας θυσιας κοινωνοι του θυσιαστηριου εισιν;

Look at Israel according to the flesh. Are not those who eat the sacrifices sharers of the sacrificial altar?

2 Corinthians 5.16:

Ωστε ημεις απο του νυν ουδενα οιδαμεν κατα σαρκα· ει και εγνωκαμεν κατα σαρκα Χριστον, αλλα νυν ουκετι γινωσκομεν.

Thus from now on we are not aware of anyone according to the flesh; if we also have known Christ according to the flesh, yet now rather we no longer know him thus.

Refer also to Romans 4.1; 8.4-5, 12-13; 1 Corinthians 1.26; 2 Corinthians 1.17; 10.2-3; 11.18; Galatians 4.23, 29; Ephesians 6.5; Colossians 3.22.

1 Clement 32.2:

Εξ αυτου γαρ ιερεις και Λευιται παντες οι λειτουργουντες τω θυσιατηριω του θεου· εξ αυτου ο κυριος Ιησους το κατα σαρκα· εξ αυτου βασιλεις και αρχοντες και ηγουμενοι κατα τον Ιουδαν· τα δε λοιπα σκηπτρα αυτου ουκ εν μικρα δοξη υπαρχουσιν, ως επαγγειλαμενου του θεου οτι, Εσται το σπερμα σου ως οι αστερες του ουρανου.

For from [Jacob] are the priests and Levites, all who minister unto the altar of God; from him is the Lord Jesus according to the flesh; of him are kings and rulers and leaders according to Judah; and the rest of his scepters possess no small glory, since God promised them: Your seed shall be as the stars of heaven.

Ignatius to the Ephesians 8.2:

Οι σαρκικοι τα πνευματικα πρασσειν ου δυνανται, ουδε οι πνευματικοι τα σαρκικα, ωσπερ ουδε η πιστις τα της απιστιας, ουδε η απιστια τα της πιστεως. α δε και κατα σαρκα πρασσετε, ταυτα πνευματικα εστιν· εν Ιησου γαρ Χριστω παντα πρασσετε.

The fleshly cannot practice spiritual things, nor the spiritual fleshly things, just as faith cannot practice the things of unfaithfulness, nor unfaithfulness the things of faith. But the things which you practice according to the flesh, even these things are spiritual; for you practice all things in Jesus Christ.

Ignatius to the Magnesians 6.2a:

Παντες ουν ομοηθειαν θεου λαβοντες εντρεπεσθε αλληλους, και μηδεις κατα σαρκα βλεπετω τον πλησιον, αλλ εν Ιησου Χριστω αλληλους δια παντος αγαπατε. μηδεν εστω εν υμιν ο δυνησεται υμας μερισαι, αλλ ενωθητε τω επισκοπω και τοις προκαθημενοις εις τυπον και διδαχην αφθαρσιας.

Do this, all of you, imitating the same divine conduct, pay respect to one another, and let no one look upon his neighbor according to the flesh, but through everything love each other in Jesus Christ.

Ignatius to the Romans 8.3:

Αιτησασθε περι εμου, ινα επιτυχω. ου κατα σαρκα υμιν εγραψα, αλλα κατα γνωμην θεου. εαν παθω, ηθελησατε· εαν αποδοκιμασθω, εμισησατε.

Ask concerning me, that I may attain it. I have written to you, not according to the flesh, but rather according to the thought of God. If I suffer, you have wished me well; but if I am rejected, you have hated me.

Ignatius to the Philadelphians 7.1a:

Ει γαρ και κατα σαρκα με τινες ηθελησαν πλανησαι, αλλα το πνευμα ου πλαναται, απο θεου ον.

For if some wished to deceived me according to the flesh, yet rather the spirit, being from God, is not deceived.

Ignatius to the Smyrnaeans 1.1-2:

...πεπληροφορημενους εις τον κυριον ημων αληθως οντα εκ γενους Δαυιδ κατα σαρκα, υιον θεου κατα θελημα και δυναμιν, γεγεννημενον αληθως εκ παρθενου, βεβαπτισμενον υπο Ιωαννου ινα πληρωθη πασα δικαιοσυνη υπ αυτου, αληθως επι Ποντιου Πιλατου και Ηρωδου τετραρχου καθηλωμενον υπερ ημων εν σαρκι.

...being fully persuaded with respect to our Lord that he was truly of the seed of David according to the flesh, and was the son of God according to the will and power of God, that he was truly born of a virgin, was baptized by John in order that all righteousness might be fulfilled by him, and was truly, under Pontius Pilate and Herod the tetrarch, nailed for us in his flesh.

Aristides, Apology 15.6, writing of the Christians:

Ξενον εαν ιδωσιν, υπο στεγην εισαγουσι και χαιρουσιν επ αυτω ως επι αδελφω αληθινω· ου γαρ κατα σαρκα αδελφους εαυτους καλουσιν, αλλα κατα ψυχην.

And if they should see a stranger, they lead him under their roof and rejoice over him as over a true brother; for they call them brethren, not according to the flesh, but rather according to the soul.

The epistle to Diognetus 5.8, writing of the Christians:

Εν σαρκι τυγχανουσιν, αλλ ου κατα σαρκα ζωσιν.

They are found in the flesh, but they do not live according to the flesh.

The Marcionite version of the epistle to the Romans probably lacked Romans 1.3, or at least the part that derives the ancestry of Christ from David according to the flesh, since Tertullian, in his discussion of the Marcionite version of this epistle in Against Marcion 5.13-14, fails to discuss the matter, which would have been quite an egregious oversight had the docetic Marcion included a mention of Christ according to the flesh.

Irenaeus, Against Heresies 3.22.1:

Et apostolus autem Paulus in epistola quae est ad Galatas, manifeste ait: Misit deus filium suum, factum de muliere. et rursus in ea quae est ad Romanos: De filio autem, inquit, eius, qui factus est ex semine David secundum carnem, qui praedestinatus est filius dei in virtute, secundum spiritum sanctificationis, ex resurrectione mortuorum, Iesu Christi domini nostri.

The apostle Paul, moreover, in the epistle to the Galatians manifestly says: God sent his son, made of a woman. And again in that to the Romans he says: Concerning his son, who was made of the seed of David according to the flesh, who was predestinated as the son of God with power, according to the spirit of holiness, by the resurrection from the dead, Jesus Christ our Lord.

Tertullian, On the Flesh of Christ 22.3:

Sed et Paulus, utpote eiusdem evangelii et discipulus et magister et testis quia eiusdem apostolus Christi, confirmat Christum ex semine David secundum carnem, utique ipsius. ergo ex semine David caro Christi. sed secundum Mariae carnem ex semine David, ergo ex Mariae carne est dum ex semine est David.

But Paul also, being a disciple and teacher and witness of the same gospel because he is an apostle of the same Christ, confirms that Christ is from the seed of David according to the flesh, evidently his own flesh. Therefore the flesh of Christ is from the seed of David. But it is from the seed of David in according to the flesh of Mary, and therefore from the flesh of Mary, since it is from the seed of David.

Origen, On Matthew 11.17:

Και γαρ χρησιμος, οιμαι, εσται σοι η τουτων συναγωγη προς το ιδειν την διαφοραν των προσιοντων, τινες μεν οιονει τω γενομενω εκ σπερματος Δαυιδ κατα σαρκα προσιασι, τινες δε τω ορισθεντι υιω θεου εν δυναμει κατα πνευμα αγιωσυνης.

For the gathering of these passages will be of more use to you, I suppose, in seeing the difference of those who proceed to him; some indeed proceed as to him who was born from the seed of David according to the flesh, but others come to him who was declared to be the son of God in power according to the spirit of holiness.

Refer also to Origen, On John 10.4.

Eusebius, History of the Church 3.11.1-2:

Ιωμεν δ επι τα εξης. μετα την Ιακωβου μαρτυριαν και την αυτικα γενομενην αλωσιν της Ιερουσαλημ λογος κατεχει των αποστολων και των του κυριου μαθητων τους εις ετι τω βιω λειπομενους επι ταυτον πανταχοθεν συνελθειν αμα τοις προς γενους κατα σαρκα του κυριου· πλειους γαρ και τουτων περιησαν εις ετι τοτε τω βιω· βουλην τε ομου τους παντας περι του τινα χρη της Ιακωβου διαδοχης επικριναι αξιον ποιησασθαι.

But let us go on with the rest. After the martyrdom of James and the conquest of Jerusalem which immediately followed, it is said that those of the apostles and disciples of the Lord that were still living came together from all directions with those that were related to the Lord according to the flesh, for the majority of them also were still alive, to take counsel as to who was worthy to succeed James.

Eusebius, History of the Church 3.20.1:

Ετι δε περιησαν οι απο γενους του κυριου υιωνοι Ιουδα του κατα σαρκα λεγομενου αυτου αδελφου ους εδηλατορευσαν ως εκ γενους οντας Δαυιδ. τουτους ο ηουοκατος ηγαγεν προς Δομετιανον Καισαρα. εφοβειτο γαρ την παρουσιαν του Χριστου ως και Ηρωδης.

And there still survived those from the lineage of the Lord, the grandsons of Jude, who was called his brother according to the flesh. These they informed against as being from the lineage of David. The evocatus led these to Domitian Caesar. For he feared the advent of Christ, as also Herod did.

In History of the Church 1.7.11, Eusebius quotes Julius Africanus as attributing a set of traditions to the kinsmen of the savior according to the flesh (του... σωτηρος οι κατα σαρκα συγγενεις).

Richard Bauckham, Jude and the Relatives of Jesus in the Early Church, page 128:

In these phrases 'according to the flesh' designates the realm of merely physical relationships, by contrast with relationships 'according to the Spirit' (cf. Rom 1:3-4; Gal 3:23, 29; Philem 16).

Apostolic Constitutions 8.35.1:

Καγω, Ιακωβος, αδελφος μεν κατα σαρκα του Χριστου, δουλος δε ως θεου μονογενους υιου, επισκοπος δε υπ αυτου του Χριστου και των αποστολων Ιεροσολυμων χειροτονηθεις, ταδε φημι.

And I, James, brother of Christ according to the flesh, and servant of the only begotten son as of God, and bishop handpicked by Christ himself and by the apostles of Jerusalem, say these things.

Refer also to Didascalia 24; the History of Joseph the Carpenter 2, 17, 28, 30 (century IV).

Porphyry, On Abstinence from Animal Food 3.7:

And, indeed, so far as pertains to sense and the remaining organization according to the sensoria and the flesh [αισθησεως της τε αλλης οργανωσεως της τε κατα τα αισθητηρια και της κατα σαρκα], nearly everyone will grant that these are similarly disposed in us as they are in brutes.

Ibidem, 4.13:

But, when they are no longer detained by the bonds of the flesh [ανεθωσι των κατα σαρκα δεσμων], then, as if liberated from a long slavery, they will rejoice and ascend to the celestial regions.

From Lactantius, Divine Institutes 4.13:

On which account the Milesian Apollo, being asked whether he was God or man, replied in this manner: He was a mortal according to the flesh, wise in miraculous works; but, being arrested by an armed force by command of the Chaldean judges [θνητος εην κατα σαρκα, σοφος τερατωδεσιν εργοις, αλλ υπο Χαλδαιοισι δικασπολιαισιν αλωκως], with nails and the cross he endured a bitter end.

From Athenaeus, Banquet 7.11:

For, on account of his stomach and of the rest of his pleasures according to the flesh [και της αλλης της κατα σαρκα ηδονης], the man was always flattering Idomeneus and Metrodorus.

G. A. Wells, Who Was Jesus?, page 73:

The reply has often been made that, to the Jewish way of thinking, legal paternity completely took the place of natural paternity, so that Jesus can be regarded as Son of David even though Joseph was only his legal father.... [This position] certainly fails to account for the insistence of Paul—the earliest extant Christian writer, who of course knows nothing of the virgin birth—that Jesus was "born of the seed of David according to the flesh" (Romans 1:3).