One might assume that the loan process between Sanskrit and, say, Korean went something like this:

Sanskrit > standard Middle Chinese > Korean

However, it may have been more like this:

Sanskrit > Middle Indic languages > Middle Chinese dialects > Korean

After multiple filters, it is no wonder that Skt Tuṣita- 'name of a Buddhist heaven' ended up becoming K Tosol (instead of *Tusit) or Vietnamese Đâu Suất (instead of *Tu Sít).

An expert on Middle Indic (not me!) might find much to mine in the variants of Buddhist terminology in Chinese: e.g., these transcriptions of Skt Tuṣita- from Soothill's A Dictionary of Chinese Buddhist Terms:

Sanskrit t u

t a
Middle Chinese 都史多 *t *o *t *a
Middle Chinese 覩史多
Middle Chinese 鬭瑟多 *əw *h *i
Middle Chinese 兜率(哆)
*w *t(t *a)
Middle Chinese 兜術 *t

Discrepancies from Sanskrit:

1. 都 *to 'capital' (hence the "urban" in the title) and 覩 *toʔ 'to see', like 兜 *təw 'helmet' and its near-homophone 鬭 *təwh 'to fight', were the closest available approximations of Skt tu because Middle Chinese had no syllable *tu (because Old Chinese *tu had become Middle Chinese *tɕu).

2. The glottal stops may correspond to Sanskrit short vowels. I suspect that Middle Chinese open syllables had long vowels.

3. I cannot explain the *-h of 鬭 *təwh 'to fight'. Preaspiration of the following sibilant (cf. the hs < *ch- of some transcriptions of Burmese)? Doubtful.

4. The voiced of 術 *ʑwit 'art' may reflect Middle Indic intervocalic voicing.

5. I still have no idea why there is a *-w- in the last two transcriptions. Was there a Middle Chinese dialect in which 率 and 術 had no *-w-?

6. The of 史 *ʂɨʔ 'history' may correspond to a zero vowel between *ṣ and *ṭ in an Indic variant Tuṣṭa-.

7. The Chinese sequence *-t-t- may imply a preceding short vowel in the original Indic: cf. the nonetymological geminates in Thai after Indic short vowels in words such as witthayaa < Skt vid 'knowledge' (cognate to Eng wit).

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