Several verses are related as indicated by the occurrence of words based on the root δουλ-.
Λέγω δέ ἐφ᾽ ὅσον χρόνον ὁ κληρονόμος νήπιός ἐστιν οὐδὲν διαφέρει δούλου κύριος πάντων ὤν
Now I say, as long as the heir is a child, he is no different than a slave, although master of all.
In this verse, there is the Greek word δούλου (doulou), the genitive declension of the noun δοῦλος (doulos), meaning "servant," "slave."
οὕτως καὶ ἡμεῖς ὅτε ἦμεν νήπιοι ὑπὸ τὰ στοιχεῖα τοῦ κόσμου ἦμεν δεδουλωμένοι
Likewise, when we were children, we were enslaved under the elements of the world.
In this verse, there is the perfect passive participle δεδουλωμένοι, meaning "enslaved." It is conjugated from the root verb δουλόω (douloō), meaning "to serve (as a slave)," which shares the same stem (δουλ-) as the noun δοῦλος.
Ἀλλὰ τότε μὲν οὐκ εἰδότες θεὸν ἐδουλεύσατε τοῖς μὴ φύσει οὖσιν θεοῖς
But then, when you did not know God, you served those who are by nature not God.
In this verse, there is the verb ἐδουλεύσατε (edouleusate), which is conjugated from the root verb δουλεύω (douleuō), meaning "to be a slave."
νῦν δὲ γνόντες θεόν μᾶλλον δὲ γνωσθέντες ὑπὸ θεοῦ πῶς ἐπιστρέφετε πάλιν ἐπὶ τὰ ἀσθενῆ καὶ πτωχὰ στοιχεῖα οἷς πάλιν ἄνωθεν δουλεύειν θέλετε
But now, after you knew God (but rather, after you are known by God), how do you return to the weak and beggarly elements, to which you wish to be a slave again?
In this verse, there is the infinitive δουλεύειν (douleuein), conjugated from the root verb δουλεύω (douleuō), meaning "to be a slave."
Now, I believe most agree that the Galatians were Gentiles, although there were possibly some Jews in the congregation. That being acknowledged, notice that Paulos states that the Galatians were "returning" (ἐπιστρέφετε πάλιν) to "the weak and beggarly elements" (τὰ ἀσθενῆ καὶ πτωχὰ στοιχεῖα). Could these "elements," then, be related to the Torah (the Law of Moshe)? Fact of the matter is, these Gentile Galatians never knew anything about the Torah before, nor practiced it before. So, they could not be "returning" to its practice. They could only be "returning" to something they had practiced before. What did Gentiles, in general, practice before coming to Christ? Pagan idolatry.
Paganism adopted many practices; perhaps one could write tomes about the subject. But, could we actually determine the Galatians' actual practice from the context? I believe we can.
It's commonly believed that Gal. 4:10 refers to the Galatians keeping Jewish holy days and appointed times.
ἡμέρας παρατηρεῖσθε καὶ μῆνας καὶ καιροὺς καὶ ἐνιαυτούς
You observe days and months and times and years.
But, again, the Galatians never practiced these before and could not be "returning" to this practice. Furthermore, appointed times such as the Jubilee, in which the land was to be given rest, typically only applied to the land of Canaan (Israel), not anywhere in the diaspora.
In any case, I wish to bring your attention to Gen. 1:14, in which it is written,
εἶπεν ὁ θεός γενηθήτωσαν φωστῆρες ἐν τῷ στερεώματι τοῦ οὐρανοῦ εἰς φαῦσιν τῆς γῆς τοῦ διαχωρίζειν ἀνὰ μέσον τῆς ἡμέρας καὶ ἀνὰ μέσον τῆς νυκτὸς καὶ ἔστωσαν εἰς σημεῖα καὶ εἰς καιροὺς καὶ εἰς ἡμέρας καὶ εἰς ἐνιαυτοὺς (LXX)
And God said, "Let there be lights in the firmament of the heaven to divide the day from the night, and let them be for signs, and for seasons, and for days, and years. (Translation of Masoretic)
These "lights" in the firmament of heaven were the sun, moon, and stars (Gen. 1:16-18). Notice that they were to be for "signs, seasons, days, and years." The LXX translates this as «εἰς σημεῖα καὶ εἰς καιροὺς καὶ εἰς ἡμέρας καὶ εἰς ἐνιαυτοὺς». Coincidentally, three of the four nouns in the LXX match those in Gal. 4:10:
Gen. 1:14 > Gal. 4:10
- καιροὺς > καιροὺς ("times")
- ἡμέρας > ἡμέρας ("days")
- ἐνιαυτοὺς > ἐνιαυτούς ("years")
As Daniel Cordova suggested, elsewhere στοιχεῖα can be used in the sense of heavenly bodies (i.e., planets, stars, moon, etc.).
For exampe, in his Dialogue,1 Justin Martyr asked Trypho the Jew, «ὁρᾶτε ὅτι τὰ στοιχεῖα οὐκ ἀργεῖ οὐδὲ σαββατίζει», that is, "Dont you see that the elements (στοιχεῖα) neither rest nor keep the Sabbath?" In his second apology,2 he wrote that God «τὰ οὐράνια στοιχεῖα εἰς αὔξησιν καρπῶν καὶ ὡρῶν μεταβολὰς
κοσμήσας», that is, "arranged the heavenly elements (στοιχεῖα) for the increase of fruits and change of seasons."
While describing God, Theophilus of Antioch3 wrote that «ἥλιος καὶ σελήνη καὶ ἀστέρες στοιχεῖα αὐτοῦ εἰσιν, εἰς σημεῖα καὶ εἰς καιροὺς καὶ εἰς ἡμέρας καὶ εἰς ἐνιαυτοὺς γεγονότα, πρὸς ὑπηρεσίαν καὶ δουλείαν ἀνθρώπων», that is, "The sun, moon, and stars are His elements, made for signs, and seasons, and days, and years, for the service and servitude of men." Theophilus is alluding to Gen. 1:14. However, note what Theophilus states at the end. God made the elements "for the service and servitude of men." Meaning, the elements were made to serve men, being for "signs, seasons, days, and years." This is remarkably similar to the apostle's comment in Gal. 4:10. But, instead of the elements serving men, the apostle wrote that men served the elements, i.e. idolatry!
It is interesting to note that Philo, a Jewish philosopher who lived in the 1st century A.D., used the exact same phrase, «τὰ στοιχεῖα τοῦ κόσμου»,4 in reference to the «τέτταρα στοιχεῖα, ἐξ ὧν τόδε τὸ πᾶν ἐδημιουργήθη», that is, "the four elements, from which this universe was made."5 These four elements were «γῆ καὶ ὕδωρ καὶ ἀὴρ καὶ πῦρ»6, that is, "earth, and water, and air, and fire." Sometimes Philo used a slightly different phrase, «τὰ στοιχεῖα τοῦ παντὸς»,7 which had the same meaning.
In summary, I believe the Galatians were committing idolatry by worshipping, and thus becoming enslaved to, both the primitive elements (fire, earth, air, water) and those things that were composed of the elements, such as the moon, stars, and sun (remember, the ancients believed the cosmic bodies were composed of these four elements; however, science has since enlightened in that matter). Even the Israelites had succumbed to this practice on occasion (cp. Deut. 17:3), perhaps learning it from the Gentile nations who surrounded them.
1 Justin Martyr. Dialogue to Trypho the Jew (πρὸς Τρύφωνα Ἰουδαῖον Διάλογος), Ch. XIII. | English |
2 Justin Martyr. Apology for Christians to the Roman Senate (Ἀπολογία ὑπὲρ Χριστιανῶν πρὸς τὴν Ῥωμαίων Σύγκλητον), Ch. V. | English |
3 Theophilus of Antioch. To Autolykos (Προς Αυτολυκον), Book I, Ch. IV. | English |
4 Philo. On the Incorruptibility of the World (Περι αφθαρσιας κοσμου/ De Incorruptibilitate Mundi), §109. | English |
5 Philo. On the Creation of Moses (Περι της Μωυσεως κοσμοποιιας/ De Opificio Mundi), §52. | English |
6 Philo. On the Life of Moses (Περι βιου Μωσεως/ De Vita Mosis), Book I, §96. | English |
7 Philo. On the Three Virtues (Περι τριων αρετων/ De Virtutibus), §73. | English |
(I would also recommend reading Rambam's Hilkhot Avodat Kokhavim in his Mishneh Torah.)