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Tacitus, Annales 4.3:

pellit domo Seianus uxorem Apicatam, ex qua tres liberos genuerat, ne paelici suspectaretur.

The translation on Perseus (Church and Brodribb) gives:

Sejanus, to avert his mistress's jealousy, divorced his wife Apicata, by whom he had had three children.

How does ne paelici suspectaretur mean to avert his mistress's jealousy?

Ne suspectaretur is easy: "lest he should be suspected". Paelici is a dative of paelex "mistress". But I'm not seeing what a dative could be doing in this clause.

Apparently, too, there's an alternate reading of this phrase: L&S under suspecto quote this as ne pellici suspectaretur, with a passive infinitive of pellicio "allure, entice". It would then mean "lest he be suspected of being enticed". This at least works grammatically, but I'm not sure what it's supposed to mean.

What's going on with this phrase?

This can be read as a dativus auctoris. It should then be translated thus:

lest he should be suspected by his mistress / be suspect to his mistress

Common in gerundive constructions (hostis nobis vincendus est), the dativus auctoris is also occasionally used with other passive verbs; it is then most common with past participles (mihi cognitum est: "it is known by me" or "it is known to me"). It indicates the person by whom something is done. Especially with verbs expressing a judgement, it is somewhat similar to the ordinary dative of videtur mihi "it seems to me" etc. Lewis & Short say the verb in this particular quotation is passive, i.e. not the deponent verb suspector.

According to Kühner–Stegmann¹, this dativus auctoris was only used in the perfect and with a personal pronoun in older Latin (pace their own Accius example, see below); but Cicero and various later authors used it with other passive forms and substantive nouns, too. K.–S. also mention Tacitus specifically.

In poetry, it is (especially) used by the Augustan poets, even non metri causa. They say the construction is originally Latin, but was extended under Greek influence (after all, the agent in a passive construction is normally expressed by a dative in Greek).

Here follow a few examples taken from K.–S. and Gildersleeve:

Accius, Tragoediae 284:

DIOMEDES:

Ergo me Argos referam, nam hic sum gnobilis

ne cui cognoscar noto.

"Then I will go back to Argos, for here I am notable, lest I be recognised by someone I know."

Cicero, Tusculanae Disputationes 5.68.4:

"Sumatur...nobis quidam praestans vir..."

"Let us assume an excellent man..."

Tacitus, Annales 2.50.13:

... ut exemplo maiorum propinquis suis ultra ducentesimum lapidem removeretur suasit.

" ...he suggested that, according to ancient custom, she should be removed by her next of kin beyond the two-hundredth milestone."

Tacitus, Annales 12.1.1:

Aelia Paetina...Narcisso fovebatur.

"Aelia Paetina was favoured by Narcissus."

The past participle or adjective suspectus is often used with a dative, which may also have influenced Tacitus to use the dativus auctoris with a passive imperfect (as in the quotation in question). Two examples with suspectus:

Sallustius, Bellum Iugurthinum 70.1.1:

Bomilcar...suspectus regi et ipse eum suspiciens novas res cupere...

"Bomilcar, suspected by the King [or: suspect to the King], and himself suspecting the King, desired a change of power..."

Tacitus, Annales 14.44.10:

Suspecta maioribus nostris fuerunt ingenia servorum...

"To our ancestors, the temper of their slaves was suspect..."

¹) Satzlehre vol. I (1912), §76 8d, p. 324: