This is a great question. Not that it matters all that much, but it's always good to periodically revisit the directions one's discipline has taken.

First, the problem is that there's linguistics and then there's study of aspects of language for a particular purpose.

Lexicography, pedagogic grammar, philosophy of language - they all have a long tradition in the West. You could almost say that applied linguistics came first.

One of the earliest significant works of linguistic theory was the Port Royal grammar of 1660 but it can in no way be seen as even a pre-cursor of modern linguistics. Its importance was retrospectively recognised by Chomsky but it had almost no direct influence on the development of the discipline.

The true starting point of linguistics as a separate discipline is generally identified in the work of William Jones (more than 100 years after the Port Royal grammar). Jones recognized commonalities between several Indo-European languages and thus started what is still recognizable as modern philology. From him, we can trace the developments of the following century almost in a direct line.

It would be another 100 years before we could point to something that looks like truly modern linguistics. By this I mean, an empirical research paradigm aimed at discovering the principles behind the workings of individual languages, their building blocks and language as a general phenomenon. Some names that stand out along the way are Wilhelm von Humboldt and Hermann Paul but it's not until the work of people like Otto Jespersen, Ferdinand de Saussure, Jan Baudouin de Courtenay and Vilém Mathesius at the start of the 20th century that we get output that we can read and still find linguistic affinities with (note: philologists can go all the way to Jones). de Saussure is by far the most famous but mostly through the efforts of his students. The competence/performance dichotomy can be directly traced to him (Chomsky's inventing history by pointing to the Port Royal grammar as his true antecedent. It was really the structuralists.) And, of course, it is only slightly later that Edward Sapir and Leonard Bloomfield contributed their significant syntheses that echo in the work of linguists to this day.

We should also not neglect the developments in affiliated disciplines which have been developing along side (if often slightly behind) general linguistics. Phonetics and phonology, psycholinguistics, and even philosophy of language each have their own interesting histories and intertwining but separate interests from those of linguistics. Then there are those of subdisciplines like sociolinguistics, contact linguistics, discourse analysis, corpus linguistics, etc. which each also have trajectories that are worth pursuing most of them not really starting until the 1950s. In many people's mind, linguistics is identified with generative linguistics but that is only one of the many subdisciplines of the field whose importance was artificially inflated at least in part due to US defense funding of AI research (see Frederick Newmeyer's histories on this).

Sadly Pāṇini's is always only mentioned as a footnote. Yet, his influence on all the Sanskrit scholars must have been significant. When you compare his meticulous treatment of Sanskrit grammar (including phonology) from at least 400 BCE with the meager output of European grammarians since the days of Plato, you cannot be in awe. Arguably the work of Indian grammarians provided models of best empirical practice for European students of Indo European languages but it is hard to estimate exactly what impact it's had on linguistics as we know it today. But it is without question the greatest work of empirical and theoretical linguistic inquiry prior to about the mid 1800s.

Finally, let's talk about the question of "scientific" study of language. Chomsky and his followers often cover up their embarrassing ignorance of the vast field of linguistics by dismissing anybody not in their formalist tradition as somehow not scientific enough. Whatever you think about Chomsky's own theory (and I think it is an impressive achievement if not really that much about the empirical phenomenon most people would describe as language), this is just pure and unadulterated nonsense. It's a rhetorical rather than an empirical device that is unfortunately all too common in academic discourse. But it is no less disreputable by its ubiquity.