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This pair of particles need to be considered together. While you may not always use specific words to translate them, they are extremely useful in highlighting the structure of larger sentences, and should never be ignored. An important subset is when they are used to indicate two different groups of nouns.
These particles are used to mark two elements in a sentence which are in some way contrasted with each other. Often initially learnt as meaning ‘on the one hand…, on the other hand…’, this is most often too heavy-handed in translation. Instead we can mark just one of the clauses (with ‘whereas’ or ‘while’ for example), leaving the other one unmarked. For example:
ἡμεῖς γὰρ ἐπεθυμήσαμεν, ὦ βουλή, Θεοδότου, Πλαταϊκοῦ μειρακίου, καὶ ἐγὼ μὲν εὖ ποιῶν αὐτὸν ἠξίουν εἶναί μοι φίλον, οὗτος δὲ ὑβρίζων καὶ παρανομῶν ᾤετο ἀναγκάσειν αὐτὸν ποιεῖν ὅ τι βούλοιτο.
Men of the council, we were both in love with Theodotus, a young boy from Plataea. I treated him well and considered him a friend, whereas he broke the law and abused him, and thought he would force him to do what he wanted.
Lysias Speeches 3.5
Alternatively, it is often possible to simply translate the δέ as ‘and’ or ‘but’ as usual: the contrast is then left implicit. For example:
καὶ τοὺς ἀκούοντας οὐ χαλεπῶς ἔπειθον· ἀποκτιννύναι μὲν γὰρ ἀνθρώπους περὶ οὐδενὸς ἡγοῦντο, λαμβάνειν δὲ χρήματα περὶ πολλοῦ ἐποιοῦντο.
They had no difficulty in persuading their audience. For they thought little of killing people, but put high store in making money.
Lysias Speeches 12.7
On some occasions an ‘on the other hand…’ translation is appropriate, particularly in analytical discursive passages. For example:
αἱ μὲν τοίνυν ὀλιγαρχίαι καὶ δημοκρατίαι τὰς ἰσότητας τοῖς μετέχουσι τῶν πολιτειῶν ζητοῦσι, καὶ τοῦτ’ εὐδοκιμεῖ παρ’ αὐταῖς, ἢν μηδὲν ἕτερος ἑτέρου δύνηται πλέον ἔχειν· ὃ τοῖς πονηροῖς συμφέρον ἐστίν· αἱ δὲ μοναρχίαι πλεῖστον μὲν νέμουσι τῷ βελτίστῳ, δεύτερον δὲ τῷ μετ’ ἐκεῖνον, τρίτον δὲ καὶ τέταρτον τοῖς ἄλλοις κατὰ τὸν αὐτὸν λόγον. καὶ ταῦτ’ εἰ μὴ πανταχοῦ καθέστηκεν, ἀλλὰ τό γε βούλημα τῆς πολιτείας τοιοῦτόν ἐστιν.
Oligarchies and democracies aim for equality for those who are part of them. They approve of the idea that no man should be able to have any more than another. But this is beneficial to the worst people! Monarchies, on the other hand, give the most to the best, the next most to the next best, and the same to the one third and fourth in line and so on. Even if this doesn't happen everywhere, this is the aim of monarchy, anyway.
Isocrates Letters 3 (Nicocles) 15
Two sub-groups
A significant number of μέν...δέ... sentences contrast two sub-groups of people or things, when a larger group has been previously been introduced or implied. For example:
καὶ τά τε εἰσφερόμενα ἀποδεκτέον καὶ ἃ μὲν ἂν αὐτῶν δέῃ δαπανᾶν σοὶ διανεμητέον, ἃ δ’ ἂν περιττεύειν δέῃ, προνοητέον καὶ φυλακτέον ὅπως μὴ ἡ εἰς τὸν ἐνιαυτὸν κειμένη δαπάνη εἰς τὸν μῆνα δαπανᾶται.
You must also take in the income and hand out what needs to be spent. You need to look after whatever's left over and you need to make sure that the amount to be spent in a year is not spent in a month.
Xenophon Economics 7.36
In these instances each group can be referred to by the article alone.
Ἠλέκτρα: σκοποὺς ἔπεμψε τούσδε τῶν ἐμῶν κακῶν.
Αὐτουργός: οὐκοῦν τὰ μὲν λεύσσουσι, τὰ δὲ σύ που λέγεις.
Electra: He has sent these men to find out about my difficulties.
Peasant: So they will see part of it and you will tell them the rest, I suppose.
Euripides Electra 355
Word order and focus
Given that, according to the usual rules of Greek word order, the first word of the clause is the element that the author wants to focus on, and that in μέν...δέ... sentences there are two contrasting elements being focussed on, the particle usually follows the word that is being contrasted. For example:
πρῶτον μὲν οὖν εὐσέβει τὰ πρὸς τοὺς θεούς, μὴ μόνον θύων, ἀλλὰ καὶ τοῖς ὅρκοις ἐμμένων· ἐκεῖνο μὲν γὰρ τῆς τῶν χρημάτων εὐπορίας σημεῖον, τοῦτο δὲ τῆς τῶν τρόπων καλοκαγαθίας τεκμήριον.
First then be respectful of religious rites, not only by sacrificing but also keeping your oaths. The former is a sign of great wealth, while the latter is proof of good character.
Isocrates On Demonicus 1.13
Here it is ‘the former’ and 'the latter' that are being contrasted.
The particles must come second in the clause, so in the case of an element which is referred to by more than one word, the particle will follow the first word. For example:
δυσκόλου δ’ ὄντος φύσει καὶ χαλεποῦ τοῦ βουλεύεσθαι, ἔτι πολλῷ χαλεπώτερον ὑμεῖς αὐτὸ πεποιήκατ’, ὦ ἄνδρες Ἀθηναῖοι· οἱ μὲν γὰρ ἄλλοι πάντες ἄνθρωποι πρὸ τῶν πραγμάτων εἰώθασι χρῆσθαι τῷ βουλεύεσθαι, ὑμεῖς δὲ μετὰ τὰ πράγματα
While decision-making is difficult by its very nature, you have made it even more difficult, Athenians. Whereas others are used to deliberate before the events, you deliberate afterwards.
Demosthenes On the Peace 5.2
However, μέν and δέ are not always preceded by the contrasting element. Another common pattern is to find the element that is common to each clause preceding them. For example:
ἐγὼ γὰρ ξυνῆν μὲν ἡλικιώταις ἡδόμενος ἡδομένοις ἐμοί, συνῆν δὲ ἐμαυτῷ, ὁπότε ἡσυχίας ἐπιθυμήσαιμι, διῆγον δ’ ἐν συμποσίοις πολλάκις μὲν μέχρι τοῦ ἐπιλαθέσθαι πάντων εἴ τι χαλεπὸν ἐν ἀνθρωπίνῳ βίῳ ἦν, πολλάκις δὲ μέχρι τοῦ ᾠδαῖς τε καὶ θαλίαις καὶ χοροῖς τὴν ψυχὴν συγκαταμιγνύναι, πολλάκις δὲ μέχρι κοίτης ἐπιθυμίας ἐμῆς τε καὶ τῶν παρόντων.
I spent time with my peers, enjoying them enjoying my company, and I spent time by myself when I wanted peace and quiet. I would go to parties often until I forgot everything difficult in human life, often until I lost my senses in song, dance and festivity, often until I and my companions longed for our bed.
Xenophon Hiero 6.2
Ellipsis
The two clauses often depend on each other for their interpretation. The δέ clause is often elliptical and information needs to be provided from the μέν clause. For example:
καλεῖ δὲ ὁ μὲν αὐτὴν δημοκρατίαν, ὁ δὲ ἄλλο, ᾧ ἂν χαίρῃ, ἔστι δὲ τῇ ἀληθείᾳ μετ’ εὐδοξίας πλήθους ἀριστοκρατία.
One man calls it democracy, another calls it whatever he likes, but it is in truth an aristocracy with the support of the people.
Plato Menexenus 238c
Distance
The two contrasted elements can be separated at some distance from each other. In these instances it is particularly important to remember to interpret them together. For example:
καὶ τούτου πότερα χρή, ὦ Σώκρατες, τὸν ἄνδρα αἰτιᾶσθαι ἢ τὴν γυναῖκα; πρόβατον μέν, ἔφη ὁ Σωκράτης, ὡς ἐπὶ τὸ πολὺ ἂν κακῶς ἔχῃ, τὸν νομέα αἰτιώμεθα, καὶ ἵππος ὡς ἐπὶ τὸ πολὺ ἂν κακουργῇ, τὸν ἱππέα κακίζομεν· τῆς δὲ γυναικός, εἰ μὲν διδασκομένη ὑπὸ τοῦ ἀνδρὸς τἀγαθὰ κακοποιεῖ, ἴσως δικαίως ἂν ἡ γυνὴ τὴν αἰτίαν ἔχοι· εἰ δὲ μὴ διδάσκων τὰ καλὰ κἀγαθὰ ἀνεπιστήμονι τούτων χρῷτο, ἆρ’ οὐ δικαίως ἂν ὁ ἀνὴρ τὴν αἰτίαν ἔχοι;
"And should we blame the woman or the man for this, Socrates?" "Well, in the case of a sheep, if it misbehaves, for the most part we blame the shepherd. And in the case of a horse, if it misbehaves, for the most part we blame the groom. When it comes to a woman, then, if she does wrong when her husband has taught her how to do right, perhaps it would be right for the woman to carry the blame. But if her husband keeps her ignorant of that and doesn't teach her what's right, surely the man should carry the blame?"
Xenophon Economics 3.11
Subtle contrast
Sometimes the ‘contrast’ between the two elements is very subtle indeed. However, there is always a reason to use the particles. For example:
οὑμὸς πατὴρ Κέφαλος ἐπείσθη μὲν ὑπὸ Περικλέους εἰς ταύτην τὴν γῆν ἀφικέσθαι, ἔτη δὲ τριάκοντα ᾤκησε, καὶ οὐδενὶ πώποτε οὔτε ἡμεῖς οὔτε ἐκεῖνος δίκην οὔτε ἐδικασάμεθα οὔτε ἐφύγομεν, ἀλλ’ οὕτως ᾠκοῦμεν δημοκρατούμενοι ὥστε μήτε εἰς τοὺς ἄλλους ἐξαμαρτάνειν μήτε ὑπὸ τῶν ἄλλων ἀδικεῖσθαι.
My father Cephalus was persuaded by Pericles to come to this country, and he lived here for thirty years. And during this time neither we nor he ever either brought a case or were called to defend a case. Instead we lived so democratically that we avoided doing wrong to our fellow-citizens as much as being done wrong to.
Lysias Speeches 12.4
In this example the ‘contrast’ is between the fact that, originally, Cephalus had to be persuaded to come live here, but that he lived here for 30 years in the end.
Some particular uses
A particular idiom may be noted with τοῦτο μεν... τοῦτο δε, meaning something like ‘firstly...secondly…’. Here the τοῦτο does not refer to an object within the sentence:
ἔπεμψαν ἡμέας Ἀθηναῖοι λέγοντες ὅτι ἡμῖν βασιλεὺς ὁ Μήδων τοῦτο μὲν τὴν χώρην ἀποδιδοῖ, τοῦτο δὲ συμμάχους ἐθέλει ἐπ’ ἴσῃ τε καὶ ὁμοίῃ ποιήσασθαι ἄνευ τε δόλου καὶ ἀπάτης, ἐθέλει δὲ καὶ ἄλλην χώρην πρὸς τῇ ἡμετέρῃ διδόναι, τὴν ἂν αὐτοὶ ἑλώμεθα.
The Athenians sent us the message that the Persian King first would give us back our land, and secondly that he wanted to become allies on an equal and fair footing without any deceit or trickery.
Herodotus 9.7δῆλον γὰρ ὅτι ἐγγύς που τοῦ λιμένος εἰκὸς ἦν τοῦτο γίγνεσθαι, τοῦτο μὲν μεθύοντος τοῦ ἀνδρός, τοῦτο δὲ νύκτωρ ἐκβάντος ἐκ τοῦ πλοίου·
For it is clear that this must have happened somewhere near the harbour, since the man was drunk and had left the ship at night.
Antiphon On the Murder of Herodes 5.27
There are some particularly tricky uses of this construction when elements are negated. Here, the position of the negative before the μέν...δέ...construction shows that the whole phrase needs to be negated (i.e. ‘it is not the case that he does A but does not do B’):
οὐ διαρρήδην εἰς τὰς ἐπιστολὰς γράφει “ἐμοὶ δ ̓ ἐστὶν εἰρήνη πρὸς τοὺς ἀκούειν ἐμοῦ βουλομένους”; καὶ οὐ γράφει μὲν ταῦτα, τοῖς δ ̓ ἔργοις οὐ ποιεῖ, ἀλλ ̓ ἐφ ̓ Ἑλλήσποντον οἴχεται...
Does he not clearly write in his letters that he is at peace with those who wish to listen to him? And it is not the case that he writes this but does not in fact act. Instead he has gone to Ambracia...
Demosthenes Third Phillipic 9.27
This use is seen in full glory in the famous passage below:
συνεπαινεσάντων δὲ πάντων καὶ οὐδενὸς εἰπόντος ἐναντίον οὐδέν, οὐκ εἶπον μὲν ταῦτα, οὐκ ἔγραψα δέ, οὐδ’ ἔγραψα μέν, οὐκ ἐπρέσβευσα δέ, οὐδ’ ἐπρέσβευσα μέν, οὐκ ἔπεισα δὲ Θηβαίους, ἀλλ’ ἀπὸ τῆς ἀρχῆς ἄχρι τῆς τελευτῆς διεξῆλθον, καὶ ἔδωκ’ ἐμαυτὸν ὑμῖν ἁπλῶς εἰς τοὺς περιεστηκότας τῇ πόλει κινδύνους.
Everyone applauded and no-one said anything in opposition. And it was not the case that I said this but did not call a vote, it was not the case that I called a vote but did not act as ambassador, it was not the case that I acted as ambassador but did not persuade the Thebans - instead I carried it out from beginning to end and completely surrendered myself to the dangers surrounding the city.
Demosthenes 18.179