If one takes the French rumin- or better yet Latin (because ultimately we don't want to give Monsieur Grenouille any credit for this interesting little factoid), we get throat or gullet, which is part way there. As ruminant animals (like cows) do, to 'ruminate' is to 'chew something over' (we can also see this in the metaphor 'to chew the cud' - cud from Old English cwidu: 'what is chewed'). The first compartment of a cow's stomach (cows only have one, not multiple stomachs - but the stomach does have four compartments) is the rumen, so to ruminate over something implies the same process as a cow bringing up grass for a second chew! Food for thought, eh?
Cattle, generally, are interesting. Not in a more-than-a-pet-and-less-than-a-lover kind of way (bona et catalla, anyone?), but in that they have been a fair constant across human existence and civilisation: which can be seen in language and will no doubt crop up again in future writings. The word 'cattle' itself is worth a lot for this discussion, and has only fairly recently referred specifically to walking-stakes-to-be. 'Cattle', and its Anglo-Norman (i.e. post-1066 Old-English/French language cocktail) precursor chattel has about it the meaning, not of ruminates but 'wealth' generally (as does a cognate in a few other languages including Anglo-Saxon) and links to that increasingly important word 'capital'. 'Capital' itself is from the latin caput meaning 'head' (think 'baseball/skullcap', the capital - top - on a stone column, capital cities and the phrase: per capita, 'per head'). Chattel as a word comes more to represent property and more specifically moveable property in Anglo-Norman, the most common of which was the moo-vable, cow. Carlos Linnaeus - a Swedish bloke from the 1700's aka Mr. Binomial Nomenclature (he liked to name things) - gave European (and African) cattle the general label, Bos taurus - which became Bos primigenius taurus a bit later on. The other two main groups are Bos primigenius indicus (the zebu) and Bos primigenius primigenius (the Aurochs- extinct 1627). In the former of the three names you'll probably already have spotted the 'Taurus' of star-sign fame. The other greek bit, 'Bos' (the 'cow' bit) crops up in the Balkans with the 'Istanbul Strait', or Bosphorus / Bosporus in what was called 'Rumelia' (translating to 'land of the romans' though I can't help but see that cowish rum-, again) by the Turkish. Understanding the other Greek bit of Bosphorus roughly to mean 'a water crossing', one finds in the name another place where wealth per capita is measured predominantly by the ability to ruminate: Oxford!
De Factoid the Second: Tracing the Sauce - coming soon