The editors of Recovering Biblical Manhood and Womanhood provide a long, drawn out expository discussion of the Greek word kephale which means, “head” and argue very convincingly that it is best defined as “ruling or leading.” For them, this also means having an “authority over” something. However I personally try not to confuse the word “authority” with “leading” because there is another Greek word for authority, exousia.
Egalitarian circles have construed the meaning of kephale as “source” yet this is only a rare and more abstract meaning which forsakes the overwhelmingly obvious meaning as dictated by most historical texts, Greek scholars, and all the Greek lexicons. It is best understood as leading or ruling and the Greek translation (LXX) of the Old Testament uses the word in this verse:
So the LORD cut off from Israel head and tail… (Isa. 9:14)
καὶ ἀφεῖλεν κύριος ἀπὸ Ισραηλ κεφαλὴν καὶ οὐράν… (Isa. 9:14 LXX)
The Hebrew word for head here is ro’sh which according to Strong’s Hebrew Dictionary means “chief, head, top of a mountain.” The Greek translation of the Old Testament was translated by seventy-two Jewish scholars in the third century B.C. This basic meaning of the word kephale is at least 2,200 years old. Egalitarians insisting that the word means “source” are forced to dig through an overwhelming amount material to find just one variance on the definition. This is clearly not good practice.
We read in the very next verse,
the elder and honored man is the head,
and the prophet who teaches lies is the tail (Isa. 9:14)
πρεσβύτην καὶ τοὺς τὰ πρόσωπα θαυμάζοντας αὕτη ἡ ἀρχή
καὶ προφήτην διδάσκοντα ἄνομα οὗτος ἡ οὐρά… (Isa. 9:14)
Interestingly, the Jewish scholars don’t translate head (Hebrew ro’sh) as kephale in the next verse. Instead they choose the Greek word, archē which means “beginning, principality.”
The meaning should be clear just from these few examples. However it should be noted that it is not quite the same as exousia which means “authority.”
 Piper, et al. Recovering Biblical Manhood and Womanhood. Edited by John Piper and Wayne Grudem. Wheaton: Crossway Books, 1991. 32. 467.
 Strong’s Hebrew Dictionary: רֹאשׁ rôʼsh, roshe; from an unused root apparently meaning to shake; the head (as most easily shaken), whether literal or figurative (in many applications, of place, time, rank, etc.):—band, beginning, captain, chapiter, chief(-est place, man, things), company, end, × every (man), excellent, first, forefront, (be-)head, height, (on) high(-est part, (priest)), × lead, × poor, principal, ruler, sum, top.
 ἀρχή archḗ, ar-khay’; from G756; (properly abstract) a commencement, or (concretely) chief (in various applications of order, time, place, or rank):—beginning, corner, (at the, the) first (estate), magistrate, power, principality, principle, rule.