Pronounce the c?

In classical Latin a C is always hard, so Principia would not sound as "principle" does in English.

Whether this means that this is "correct" is debatable. Cs in Latin have been universally soft for several hundred years so using the classical pronunciation is anachronistic.

Perhaps appropriate for a English book with a Latin title. Pronounce it however you fancy, really.

There are two schools of thought: pronunciation as per the Catholic church, where the sound has evolved (to "ch") over hundreds of years of Latin being used as a working language, and pronunciation as per the Romans, for whom "c" was pronounced "k" (and "v" was pronounced "w").

Neither is really right or wrong, certainly neither is 'anachronistic'. Latin can be regarded as a classical, dead language, or as an ecclesiastical, priestly language, or as something else.

In practice, different pronunciations are used by people with different educational, cultural and religious backgrounds. In many cases the 'accepted' pronunciation is used to signify 'belonging' to a particular tribe (eg academic archeologists vs theologians educated at seminaries). In Scotland it was a sectarian marker (one of about a million obscure indicators of community allegiance) back in the days when Latin was widely taught in grammar schools.

FWIW, I have only ever heard mathematicians refer to the Russell book with a hard C.

Indeed, I doubt that Newton would have pronounced Principia with a hard 'c'; Allen observes that in England, schoolmasters only started to observe meter and the hard 'c' and 'g' starting from the mid-nineteenth, and only in parts.

I wanted the commenter to elaborate, since it seems to me that the most likely reason they remarked on its pronunciation would be that they discovered the reconstructed pronunciation of it and thought that it was correct to the point of being knowledge of how it ought to be pronounced, rather than as one of several ways of pronouncing it to communicate it.

Ranieri contends that many casual speakers end up observing only partly observing the hard 'c' and 'g' and little of the other differences and end up with a hybrid "not even worthy of importing". https://www.youtube.com/watch?v=IjcX3MVSdyA

>Cs in Latin have been universally soft for several hundred years so using the classical pronunciation is anachronistic.

Well, latin are anachronistic themselves - so the classical pronunciation would still matter.

Aren't Latin C's mostly K's? As in, Julius Kaesar? I never quite looked into it in depth, with all the gradual evolution, Church latin etc.

Yes, as an "s" or "k". The "correct" way to pronounce it is prinkipia.

> "s" or "k"

More common for post latin spoken languages would be ʧ ("ch·eese"). (You missed the foremost contender.)