Classical learning in the west primarily consists (or consisted if you want to take a pessimistic view of the state of classical learning) of the study of the major ancient languages – Latin and Greek. However, at the end of the 18th century another classical language entered the tradition: Sanskrit. Having always existed at the periphery of the Classical World (from a Western perspective, of course), Sanskrit the other great ancient descendant of PIE or Proto-Indo-European.
This is something that British scholars quickly realized during the process of colonization – the similarities between Sanskrit and the languages that they had studied in University were too striking. It couldn’t be coincidence. And returning to Europe with the staggering volume of writing that they had found, Sanskrit had something of a renaissance in the West – or rather, a naissance. It was quickly introduced into English and German universities where the similarities between it and the Greco-roman tradition ensued. During the 19th century this cross study of the three major languages led to the development of a whole new field of study: comparative linguistics. As well as the theory of a much, much older origin language, PIE.
Sanskrit verbs conjugation was very similar to that of Greek, though slightly less complicated. What was extremely complicated, and hence very interesting to this generation of European scholars (these were the people that eventually deciphered cuneiform and hieroglyphs) was the complicated rules regarding sound agreement, called Sandhi. These sound changes where significantly more complicated than those found in Greek (sound agreement and elision in Latin is hardly worth mentioning as it is almost entirely relegated to poetry and in the form of small contractions in the active perfect 3rd plural and a few other cases). This, combined with the fact that there was an enormous amount of material to be read led to a huge explosion of academic interest. The results of which can still be seen today as the major centers of Sanskrit studies are in the USA, England, and Germany not India.
However, this interest was not sustained and the study largely died off after the comparative aspects came to be understood. The demise of the classical education (at least as a requirement), began after the first Word War and as fewer men took up the study of their own tradition, even less took an interest in foreign traditions. Western classicist now fear a similar decline of Latin and Greek that they have seen with Sanskrit: that is, it is now relegated to a very small number of extremely specialized academics. And last time Latin and Greek retreated into the monasteries it took centuries for it to return in the form if the Renaissance.
If you have any interest in this study, University programs still exist at schools that still have serious classics programs. There is also something of a Sanskrit renaissance going on in India as the country rediscovers its roots and it’s own tradition resurfaces (it never stopped being used in a religious context). One of the best things about Sanskrit is the fact that the best literature (or what is generally regarded as the best), that of Kalidasa, is much easier to read than the majority of the Vedic works. His writing was mostly drama and truly excellent poetry.
Despite its complicated Sandhi system, the ancient Indians were extraordinarily serious about the study of grammar. Their intense study resulted in the language not changing nearly as much as one would expect over such a long period of time. This work all started with one of the first true grammarians: Panini.
I studied Sanskrit in school for a few semesters and it was extraordinarily rewarding. However I could not maintain three classical languages no matter how related they were!