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Under- and over- seem to be prefixes with contrary meanings. But with the example of undertake and overtake, I don't see contrary meanings between them. Am I missing something?

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If you're a reckless overtaker, you might find yourself giving business to the undertaker. – Barrie England Oct 10 '11 at 7:24
Thanks! Would you explain what you mean? – Tim Oct 10 '11 at 7:37
An undertaker deals with funeral arrangements. Although the name is derived from the fact that they "undertake" the arrangements. – Urbycoz Oct 10 '11 at 8:09
As Urbycoz said. Perhaps not found in AmEng. – Barrie England Oct 10 '11 at 8:53
up vote 12 down vote accepted

By themselves, under and over are antonyms, but as prefixes the same rule doesn't always apply.

The prefix under has several meanings:

beneath, below : underarm, undercarriage

lower in rank : undersecretary

not enough : underdeveloped

The prefix over also has several meanings:

excessively, completely : overconfident, overburdened, overjoyed

upper, outer, over, above : overcoat, overcast

Over and under are sometimes contrary prefixes, but only when they both refer to the same aspect of something. For example, one can overapply or underapply sunscreen depending on how much they use. In this case, over refers to its meaning of "excessively" while under refers to its meaning of "not enough".

Now, in your specific examples, these rules do not apply. Undertake is formed partially from German loan words, rather than from putting together two English words to form a compound. In this case,

"The under in this word may be the same one that also may form the first element of understand."

Because of this, overtake and undertake won't have contradictory meanings.

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I wonder why Etymonline says that it is "a partial loan-translation" from German. The OED doesn't say anything about this. – Colin Fine Jul 27 '11 at 10:40
There's also the question of what under and over are in relation to in each word. When undertaking a task, you're on top of it (the task). When overtaking someone, you're above him/her. It can be seen as a matter of perspective. – onomatomaniak Oct 10 '11 at 8:14

The common meaning of overtake is to catch up with, or surpass.

Most common meanings of undertake involve taking on a task or obligation

The closest I can get to 'opposite meaning' is that undertake can be used in a motoring context, to mean "pass by another vehicle on the inside (nearest the kerb)" (normally one passes on the outside, nearest the centre of the road). I think that usage occurs in the British Highway Code, where it's specifically presented as bad practice (even on motorways where several lanes may actually be moving at much the same speed).

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Incidentally, that meaning of "on the inside" is bizarre to Americans. While what we actually say is "passing on the right" [left hand drive imperialists that we are], e.g. "outside lanes" means those furthest from the centerline. – Random832 Jul 27 '11 at 5:13
not only is it "bad practice" its actually illegal in the UK to pass someone on the nearside UNLESS their queue of traffic is moving slower than the one you are in (i.e. a filter lane on the left allowing the nearside queue to move whilst the offside queue remains still). – Mauro Jul 27 '11 at 7:18
Are you sure it's used in the Highway Code? I can't find it there, and I would always regard such usage as humorous. – Colin Fine Jul 27 '11 at 10:52
@Colin Fine: I haven't read the Highway Code since passing my test over 40 years ago. I recall thinking the word "undertake" was an unusual usage at the time, but I really couldn't say if I read it or if it was said by my father who taught me to drive. All I know for sure is I first met the usage while I was learning to drive, and at the time it wasn't being used humorously. – FumbleFingers Jul 27 '11 at 17:27
...anyway, Wikipedia doesn't think it's a joke... en.wikipedia.org/wiki/Undertaking_(driving) – FumbleFingers Jul 27 '11 at 17:29

As the other answers indicate, 'overtake' and 'undertake' are not really antonyms.

Undertaking has a meaning 'person who manages burials (of dead people)'. I don't think I've ever heard anyone use 'overtaking' to refer to the opposite processes (whether that's managing a birth or grave-robbing).

In the context of driving, undertaking and overtaking are similar operations (moving past a slower-moving vehicle), and the oppositeness comes from which side of the vehicle you go past (and that, in turn, depends on whether your country drives on the left or the right of the road). There are grim jokes about undertaking (on a highway) leading to the undertakers.

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