It's odd alright.

The first thing google offers in search for μῆδος is the Medes, Medoi, an Iranian people, known for the Median empire, and mentioned in the bible. This would be a good rabbit whole for a sunday afternoon, but I was searching for ...

The second link that google offers, wiktionary, has an entry for μῆδος. It is the "hypothetical nominative/accusative/vocative singular of μήδεα (mḗdea) (both etymologies)" but "Both senses of μήδεα (mḗdea) occur only in the plural"; apparently that must be wrong, if you are correct.

There are two definitions, indeed (The third one for the proper name is on a separate page because it's capitalized). One is

> 1.

> (poetic, plural only) counsel, plan, art, prudence, cunning

> From Proto-Indo-European méh₁desh₂, from med- (“to measure, give advice”) …

the other

> 2.

> (Epic, plural only) male genitals

> Uncertain. Has been connected with etymology 1 (compare Old High German gimaht for a similar double meaning), Latin mentula, and with μαδάω (madáō) and Middle Irish mess. Could also be Pre-Greek.

This is also a rabbit whole, but one I feel comfortable burrowing myself in. Trying to connect that back to μειδάω I will only dig a whole for myself, but lets see.

The mentioned Gothic word exists in Old Saxon as well, and reflects in Modern High German as Gemächt. Wiktionary traces that to the root in "might", megh-. I'm not sure how that doubles the sense in the first, ie. "councel, ...", except for Wehr-Macht, mighty king and the like.

"might" may find an analogy in German Schwanz (tail, cock) if connecting a root swent- "vigor" (cf. sound, Ger. gesund "healthy"; potentialy connected to swenh2- "to sound, resound", Lat. sono; also consider Venus from wenH-, cf. win, wish). I like to think of (im)potence, in that sense, basically, but ...

"might" may further compare to "sexes", the two halfes, as in Russian pol "genitals", if from Proto-Slavic pol- "half" (but compare Rumanian pula, Engl. pillar, and many more p-l-, b-l- words accross European languages; if not Greek pelikos "same size", with h2el-), or Ger. Geschlecht "genus", idiomatic vom gleichen Schlag ("of the same cut [=flock, ilk, feather, ...]"). Also consider Schlager "folk song", Schwank "story" (cp. swing).

"might" further invites comparison to maiden, Ger. Magd, Mädchen, supposedly from a sense "virgin", PIE \magh-.

A sense "measure" coming close to "cut", "cut out", is imaginable: cp. OE mete, meet a decision; Ger. messen, bemessen* "to measure", beimessen "to judge"; seek out, to see, also sick (injured, cut?), and section, second (also hora).

"councel" may be seen in mayor, master (magister), if not clearly from mega-, \méǵh₂s-. Compare mass* 1, 2.

This could further allow interpreting the Medoi as "judges" (cf. book of judges). Vice versa, the Greek might have considered them dicks! This pun alone was worth the write-up ...

Ultimately, μειδάω "smile" can be compared to "meassure" if "to cut, split" has been gleaned in Beowulfs Grendel, interpreted as grin-divided--which is a bit weak. Perhaps a better approach is to be taken from Ger. mögen, ich mag as a cognate to may, which also translates I like. Proto-Germanic \laikana and \likana shouldn't be confused at that; A good PIE etymology for the latter seems to be missing. Incidently, we may well compare laughter, but it seems to be a Celtic-Germanic isogloss (i.e. influenced by exchange); instead, comparison to yell, call, loud Ger. gellen, schallen is in order. Schall "sound" may be of interest, if related to Schelte, further Schulz, Schultheiß, a kind of Germanic judge. This is more often compared to Schuld "debt, blame", perhaps further shall, should, Ger. Soll, Schuld; It comes to mind more likely because heißen "to call, name" incures images of a commander. On the other hand, cp. schmeicheln, schmiegen (to compliment, cuddle).

Oh boy, I could go on and on with loose associatiins and deceptive similarities (genus, gnosis; honour, Hohn; burden, Bürde, bard; …).

It does not help that I want to make Aphrodite basicly a lesbian (lover of virgins?).