I have been arguing about this with quite a number of people and it seems we all cannot find the answer (med students, duh😅)
My question is: Where should the stress on words like mastoideus and styloideus be? on Ideus or idEus.
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As far as I can tell, there are no classical precedents for the specific form of the ending -oideus. It ultimately comes from Ancient Greek -οειδής, an ending found mostly on third-declension adjectives with the meaning "-shaped" or "-like". These adjectives are compounds containing the linking vowel -ο- followed by the element εἶδος "form, shape". The direct Latinization of this Greek ending is -oides. How this was altered to -oideus is difficult for me to explain. I couldn't think of any other similar examples of Greek -ης corresponding to Latin -eus, but when browsing this site, I found a previous question where Cairnarvon's answer states that eugenēus/eugenīus is a Latinization of Greek εὐγενής.
Looking through all the Greek words containing -οειδ- in the LSJ Dictionary, I only found a few ending in anything other than -ο-ειδής, and none with an ending comparable to -oideus. So I think the extension of the Greek ending with -eus must have been created in Latin. But there are also no words ending in "oideus" in Lewis and Short, so "-oideus" seems to be a post-Classical formation, possibly occurring exclusively in "anatomical Latin".
Latin stress, which is what would be most relevant here (since the ending -us is Latinate or Latinized), is (almost always) predictable if you know the length of the vowel in the second-to-last syllable of a word.
If the e is short, the Latin pronunciation would stress the i (which represents the Greek diphthong ει, and which did not form a diphthong in Latin with the preceding o, despite the usual English pronunciation of -oi- as a diphthong in the ending -oid). So the stress would be -o-í-de-us.
If the e is long, the Latin pronunciation would stress the e. So the stress would be o-i-dé-us.
Unfortunately, it's quite unclear to me what length this e "should" be.
Latin has both words ending in -ĕus with short e and words ending in -ēus with long e.
The ending -ĕus is a common Latin adjective-forming suffix.
The ending -ēus is less common. It violates the general rule that a vowel before another vowel is short, but can be found nevertheless in a fair number of words, such as Bacchēus. I believe Bacchēus is derived from Greek Βάκχειος/Βακχεῖος, meaning what we see here is an example of Greek ει before a vowel being Latinized as long ē. I think most other words ending in -ēus have the same origin in Greek -ειος. Another example: spondēus
(Some words can end in -e͡us, with a diphthong eu, but I'm pretty sure that's only used for words where -eus is from Greek -ευς, and so it would be implausible in -oideus words.)
So which do we find in words ending in -oideus?
Based on what I've read, I would lean towards the pronunciation with a short e and stress on the i. I have found an article, "Medical Language" by H. Zimmerer in the Medical Brief (Vol. 36, 1908), that argues for this interpretation of the form, while also showing that variability in the accentuation of the ending has existed for more than a century:
By far the most frequent of all adjectival suffixes in medical onomatology is -ideus -ο- ειδής. [...] Only in the seventeenth century they were used as Latin epithets in anatomy, mostly through J. Riolanus; many of them are still in use today. The corresponding Greek adjectives end in -ειδής. In Latin it is permissible to contract the diphthong ει to ι and to change the end syllable ης to -eus, so that, for instance, αδεν-ο-ειδής may be expressed by aden-o-ideus. In this word the i is long, because it stands for ει, while the e is short, as is the case in all adjectives in -eus (aureus, argenteus, ferreus, plumbeus, etc.). The only correct pronunciation, therefore, is adenoideus, and this also applies to the similar words adenoid, alcaloid, sarcoid, myxoid, etc., which have been formed in imitation of the Greek precept, and which ought to have the long accent on the i. It has become an objectionable practice among anatomists to reverse the accentuation, and the desire that such custom should no longer obtain, is voiced by all linguists.
I found the similarly formed word theoideus marked theoídeus in the dictionary Botanisch-Gärtnerisches Taschenwörterbuch, by Reinhold Metzner.
However, I don't know of a definite argument that the e cannot be long as Bacchēus. In fact, the previous question linked above about eugenēus seems to show an analogous example where a long vowel apparently was used (however, I'm not sure how the dictionary-makers actually know the vowel was long in that word).
I'm not familiar enough with the relevant rules for Greek word formation to be able to say whether it would theoretically be possible to derive forms in -οειδειος from -οειδής in Greek. In purely phonological terms, it looks possible to me, as adjectives like -οειδής have a stem ending in /es/, and the Greek sound change rules of deleting intervocalic /s/ and contracting adjacent vowels should turn -es-ios into -ειος. For comparison, it looks like there are a few abstract nouns ending in -οείδεια, which would be the outcome of -o-eides-ia. And the name Ἀριστοφάνης, which I believe is etymologically a third-declension /es/ stem* like -οειδής adjectives, has the derived adjective Ἀριστοφάνειος. However, I don't know if it makes any morphological sense in Greek to attach the adjective-forming suffix -ios to a word that is already an adjective.
*Apparently third-declension masculine ης nouns may have eventually shifted to the first declension, but I don't think that's relevant here