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Sanskrit has been studied by Western scholars since the late 18th century. In the 19th century, the study of Sanskrit played a crucial role in the development of the field of comparative linguistics of the Indo-European languages.
During the British Raj (1857-1947), Western scholars edited many Sanskrit texts which had survived in manuscript form.
The discovery of Indo-European first started with a British judge named William Jones who was stationed in India in 1780. Jones, a bright fellow with classical training in Greek and Latin, had determined to master the ancient Sanskrit tongue. He wanted to brush up on native Indian law codes--many of which were written in Sanskrit script--before administering British law in the region.
Jones was shocked to discover a regular pattern of similarities between ancient Sanskrit words and ancient words in classical Western languages.
Other Sanskrit words were similar to Greek terms. For instance, the Greek word trias ("three") is close to trayas and tres in the chart above. The Greek word pente ("five") is close to Sanskrit panca ("five"), and so on. Jones began systematically charting the similarities, finding literally thousands of such parallels between Sanskrit, Greek, and Latin. He presented his findings on February 2nd, 1786, to the "Asiatick Society in Calcutta." He declared boldly that Sanskrit had a stronger affinity than could possibly have been produced by accident; so strong, indeed, that no philologer could examine them all three without believing them to have sprung from some common source, which perhaps no longer exists.
What Jones had uncovered, without realizing it initially, was the existence of a lost mother tongue, what scholars call proto-Indo-European --a single, ancient, prehistoric language that led to the development of many languages in Europe, India, Russia, and the Middle East. It required nearly ninety years of comparative linguistics to fill in all the gaps.
Some Hindu ultra-nationalists will argue that Sanskrit originated within India, but Sanskrit is markedly Indo-European, and bears no resemblance to languages such as Tamil.
In any event, Sanskrit, Latin, and Greek are all Indo-European languages, meaning that they shared a common linguistic ancestor, known as Proto-Indo-European. Around 2000 BC, the Indo-Europeans began to migrate away from their homeland in all directions.
Before Jones, earlier scholars had long ago noted that many languages shared such similarities. It was no news, for instance, that Romance languages shared cognates with each other. Spanish caballo (horse) was a cognate for Portuguese cabalo (horse), Italian caballo (horse), Provençal caval (horse), French cheval (horse), and English cavalry (horse-riding troops). Scholars had long known that all these words ultimately came from the vulgar Latin term caballus (horse), and that French and Spanish and other Romance languages had developed from Roman provincial speech--with some voiced /v/'s changing to unvoiced /b/'s, or some hard velar stops (/k/ sounds) changing to aspirated 's. Likewise, Germanic languages like Low and High German, Frisian, Dutch, Swedish, and Norse shared many cognates with each other in much the same way, tracing their origins back to a proto-Germanic tongue in prehistoric times.
What astonished linguists was that Sanskrit had cognates to more than just Latin and Greek words. Philologists found that Dutch, German, Old Norse, Gothic, Old Slavic, and Old Irish had similar patterns of words with Sanskrit. These cognates had a related meaning and they also sounded similar to each other either in terms of vowels or consonants (or both!). (Sanskrit is a fascinating and complex language, but it contains many features that bear a striking resemblance to Ancient Greek, Latin, Avestan, Tocharian, and many other ancient Indo-European languages.)
If you study Sanskrit, or any modern Indo-*** language (e.g. Hindi, Bengali, Punjabi, Gujarati), you will note a wealth of cognates (for example, confer Hindi "do" (two) with English "two", Spanish "dos", Portuguese "dois", etc) and analogous structures.
All that remained for scholars to do was (1) to trace what rules governed these changes linguistically--a task taken up by Jakob Grimm and later Karl Verner, and (2) to reconstruct as far as possible what this original language must have sounded like and how it functioned. This is tricky, given that proto-Indo-European is a prehistoric language existing before the written word, but not impossible given the wealth of linguistic information we can garner from surviving languages today.
It's hard to escape the conclusion that these words must have come from a common source--especially if you chart the words out on a map of where each language is spoken. In the case of the words for father, a linguist can almost visually see the unvoiced /t/ sounds changing to voiced /d/ sounds as people migrated westward across the map, and then these letters changing to as they moved north through Europe along the Germanic branch. In the case of the words for brother, the same sort of linguistic change is occurring with unvoiced /t/ and voiced /d/ sounds, but another pattern is happening simultaneously with voiced /b/ and unvoiced /p/ sounds. Multiply the examples above for a few thousand other words, and the evidence looks fairly air-tight.