This is actually an extremely interesting and expansive topic. And one interesting thing is that historical linguists don’t even need written texts to reconstruct languages that haven’t been spoken for thousands of years – just a big sample of descendant languages.
Anyway, there’s a couple answers to your question. The first thing is that many ancient authors *wrote* about how their own language sounded. For something like ancient Latin, there are surviving texts that describe how to properly pronounce things and common mistakes that uneducated speakers made. Things like poetry, songs, and other linguistic works will inform us of the pronunciations and stress locations in words. In other instances we can find common spelling mistakes that tell us some things about how a language was pronounced. For example, in a thousand years historical linguists will be able to tell that in Modern English the words there/their/they’re are pronounced exactly the same since using the wrong one is a common spelling mistake. Ancient writers were not immune to these types of mistakes as well – especially at times when spellings weren’t standardized and people just kinda wrote things how they sounded.
The other aspect of this is that languages have a tendency to change in specific ways over time. Look at the pronunciation of ‘Caesar’ – in classical Latin a ‘C’ was always a hard sound in that it was pronounced similar to the modern English ‘K’ sound. So it was pronounced more like ‘Kai-sar.’ However, in the modern languages descended from Latin this is not apparent as the beginning consonant is pronounced softly – in French and Spanish it is pronounced with an ‘S’ sound while in Italian it is pronounced with a ‘ch’ sound. It turns out this is a common sound shift where the pronunciation of the consonant moves from the back of the throat (‘K’) to the front of the mouth (‘S’) known as assibilation. And the interesting thing is you can apply this shift to other words in descendant languages and reverse the process to see what the original pronunciation likely was. If you have a bunch of different descendant languages, you will find doing this process will end up giving you a language that is pretty much what was spoken by the tens of millions of inhabitants of Roman provinces – ‘Vulgar’ or Common Latin. This process of reconstructing a parent language by comparing pronunciations, grammar, and other aspects is known as the field of Comparative Linguistics.
And this process can be taken back even further in time. As it turns out, most languages in Europe (including English, Latin, Celtic, and Slavic languages) and many in the Middle east (Persia), and Northern India (Hindu and related languages) are actually related to each other – they all came from an original mother language that was spoken probably about 5-6 thousand years ago. Fittingly, this language is known as ‘Proto-Indo-European’ Since its modern descendants are spoken from India to Europe. By comparing modern and ancient Indo-European languages, linguists have been able to roughly reconstruct what this language sounded like. [Here’s one example of a famous story in reconstructed Proto-Indo-European](https://youtu.be/pEcABb8MnxI). One should not that this is an approximation of what it sounded like and there is still considerable debate about exact details.
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