Yes, you are right, and the whole story of the laryngeal theory is quite exciting. – Saussureʼs initial evidence was a set of Sanskrit forms; and his argumentation went like:
- There is e.g. a root meaning ‘carry’ having the full grade “bʰar” and a corresponding zero grade “bʰr̩” (within the regular ablaut system of Sanskrit).
- Then there is e.g. a root meaning ‘clean’ having the full grade “pavi” and a corresponding zero grade “pū”.
- So we have “bʰar” : “pavi” = “bʰr̩” : “pū” or, re-grouped, “bʰar” : “bʰr̩” = “pavi” : “pū”.
- We already know (since the times of the great Indian grammarians) that, in ablaut, “v” corresponds to “u” (samprasāraṇa). All synchronic observations, by the way, about these different kinds of roots were already made by the Indian grammarians in the first millennium BC, too; and they called the roots à la “bʰar” “aniṭ” ‘without i’ and the roots à la “pavi” “sēṭ” ‘with i’.
- “bʰar” : “bʰr̩” is the regular ablaut pattern understandable as: ‘the full grade has the short vowel (“a” in Sanskrit, “e” in PIE), the zero grade lacks it.’
- Saussures brilliant and simple idea was to trace back “pavi” : “pū” to this very same basic pattern.
- To make this work he assumed a sound in the ‘clean’-root that became “i” between consonants but vanished with compensatory lengthening after sonantic “u”. (Saussure denoted this sound here with the cover symbol “A” in small caps.) So the older, regular pattern can be reconstructed as (in more modern notation):
- “bʰar” : “bʰr̩” = “pavH̩” : “puH”, which yields:
- “bʰar” : “bʰr̩” = “pavi” : “pū” as attested.
This argumentation is fairly compelling IMHO because it complies with Occamʼs razor by assuming that a second complicated, seemingly irregular morphological pattern leads back to the simpler, regular pattern we have to assume anyways, and that this simpler pattern was complicated by sound change – which is the normal way of linguistic change.
But it got even better.
- Saussure then drew attention to the Sanskrit verb formations of the seventh and ninth class (again, already classified and extensively described by the Indian grammarians), which both had an infix, i.e. a morphological element inserted into the root (not prepended like a prefix or appended like a suffix):
- 7th, e.g. “yunakti” ‘yokes (up)’ (the English word is a cognate), built like “yu·na·k-ti” with zero grade “yug” (and “g” → “k” before voiceless “t”), “na”-infix and personal ending “ti”.
- 9th, e.g. “punāti” ‘cleans’ – our pavi/pū-root again. But now with a short “u”? And with an infix “nā” instead of “na” as in “yunakti”? So … “u” instead of “ū” and “nā” instead of “na” … and both is already explained by the coefficient, because then we have to reconstruct:
- “pu·na·H-ti”, because the na-infix had to be inserted before the last consonant of the root. And this formation “pu·na·H-ti” is, again, exactly the same pattern as:
- “yu·na·k-ti”, just with “H” : “k”.
So far this argument justifies to assume one “coefficient”, in the case of the “pavi”-root denoted as “A” (in small caps) by Saussure and “h₂” nowadays. Saussure assumed two – denoting the other as “O̬”, nowadays “h₃” – because he also already noticed the coloring effect you explained in your comment below: The compensatorily lengthened Sanskrit “ā” sometimes corresponds to e.g. Greek and Latin “ā”, sometimes to Greek and Latin “ō”; for the latter Saussure introduced the “O̬”. His argumentation here is more difficult and partly outdated, because he wrote his mémoire (published 1879) in a time when another major discovery was not yet fully taken into account: So far, Indo-Europeanists had assumed that Sanskrit “a” originated from Proto-Indo-European “a”. When Saussure wrote his mémoire, it had become clear that it was necessary to assume at least two diffent vowels here, which both became Indian “a”.