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I read here:

Une règle générale qui ne souffre que très peu d’exceptions : quand les consonnes sont doublées, et que ce n’est pas par raison d’étymologie, c’est presque toujours parce que les syllabes qu’elles forment sont brèves.

But in relation to that rule r is not discussed.

Considering this rule:

On peut encore établir une règle générale pour le doublement des consonnes, c’est que toutes les fois qu’un mot commence par les voyelles a ou o, et qu’elles y sont employées comme prépositions inséparables, les consonnes qui les suivent se doublent. On connaît que ces voyelles sont employées comme prépositions inséparables dans un mot lorsqu’en les retranchant de ce mot celui qui est un mot français qui entrait dans la composition du premier. Ainsi, en retranchant la voyelle a du mot apprendre, il reste prendre qui est un autre mot français. La voyelle a y était donc employée comme préposition inséparable. Par conséquent, apprendre est un mot composé, dont le simple est prendre.

That doesn't seem to apply to arrière.

As for the etymology of arrière, it is inherited from Old French ariere, from Vulgar Latin ad retrō, from Latin ad + retrō.

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The etymology seems to be more exactly that given below.

  • (Wiktionnary) Du latin populaire ad retro, équivalent à l’ancien français a riere. (Vers 980) aredre (Passion du Christ).

Using the link "a riere", the word « riere » is seen to have the meaning "derrière" in Ancient French. It follows that the second rule does apply, but in Ancient French only, as the word "riere" is not a part of the modern vocabulary.

It seems to me that, whereas the usual doubling is issued from pure formation by prefixing latin "ad" (a in French) to a French word, the doubling in "arrière" results from bringing the spelling into conformity with the spelling that results from the habitual process of doubling (when occurs the contraction of "a riere as "ariere" or, more likely, some time later).

Both spellings are found in the 13th century according to a dictionary from 1874, Dictionnaire de la langue française - Volume 1 - Page 202 Emile Littré · 1874.

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The spelling "arriere" is found in the 12th century; as well, the spellings "arrier" and "erriere", both rejected, are found side by side. In the 11th century, the spelling was still "arere".

It is to be noted that if a two-word form is not found for "derrière", which is as well constructed from the latin "retro" (TLFi, étymologie), but comes directly from the "bas latin" deretro, a similar evolution from a one r to a two r spelling is also characteristic of this word (Littré) ; the change is also recorded in the 12th century.

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