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Why does Proto-Balto-Slavic have form *aśís (with one s in the stem)? despite of the fact that Old Prussian has word assis (compare PIE *h₂eḱs-i-s)?

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The most probable reason for the double <S> in the Old Prussian word assis (“axis”) is that it was recorded by a German who used the German orthography. In German, a single <S> between two vowel letters is invariably pronounced as [z], a voiced sibilant:

Germ. Hase [ˈhaːzə] — Eng. “hare”
Germ. lesen [ˈleːzən] — Eng. “read”
Germ. Basis [ˈbaːzis] — Eng. “basis, base”
Germ. Kasus [ˈkaːzʊs] — Eng. “case (linguistics)”

To write the voiceless [s] between two vowels, the most common way out is to double the <S>:

Germ. Wasser [ˈvasɐ] — Eng. “water”
Germ. müssen [ˈmʏsən] — Eng. “must”

In foreign names the double <S> for the intervocalic [s] is used often. For example, the name of Vasilisa the Beautiful (Василиса [vɐsʲɪˈlʲisə]), a recurring character of Russian fairy tales, is spelled as “Die schöne Wassilissa” in German.

Since in all the rest of the Balto-Slavic languages the consonant in the stem of the word for “axis” is single (Baltic: Lith. ašis, Latv ass (“-s” being the Nominative Singular suffix, Gen. sg. asi ), Slavic: os, osa, oś [ɔɕ], ось [osʲ], вісь [visʲ]), it is most probable that it was single in Old Prussian, too. It was just spelled as a double <S> to assure its reading as the voiceless sibilant [s]. Don't forget that German (and English too) has no long/gemminated consonant sounds, the double consonant letters do not mean the consonant sounds they stand for are in any way long. In German orthography, double <S> is always pronounced as [s], never as [sː].

The reason why the PIE *ḱs gave the PBS *ś is explained like this (Wiki):

It appears that palatovelars yielded fricatives in Proto-Balto-Slavic before the effect of RUKI law, so that *ḱs appears simply as *ś. Compare:

  • Slavic *desnъ "right (i.e. opposite to left)" (OCS desnъ, Russian désnyj, Serbo-Croatian dèsnī), Lithuanian dẽšinas < Proto-Balto-Slavic *deśinas < PIE *déḱsinos (Latin dexter, Sanskrit dákṣiṇa)
  • Slavic *osь "axle, axis" (OCS osь, Russ. os’, SCr. ȏs), Lithuanian ašis < Proto-Balto-Slavic *aśís < PIE *h₂éḱsis (Latin axis, Sanskrit ákṣas).
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