I thank prof. David Crystal for
his book The Stories of English.
 AM: O.E. eom "to remain," (Mercian eam,
Northumbrian am), from PIE *esmi- (cf. O.N. emi, Goth. im,
Hittite esmi, O.C.S. jesmi, Lith. esmi), from base *es-, *s-,
the S-ROOT, which also yielded Gk. esti-, L. est, Skt. as-, and Ger.
ist. In O.E. it existed only in present tense, all other forms being
expressed in the W-BASE (see were, was). This cooperative verb is
sometimes referred to by linguists as *es-*wes-. Until the distinction
broke down 13c., *es-*wes- tended to express "existence," with beon
meaning something closer to "come to be" . O.E. am had two plural
forms: sind/sindon, sie and earon/aron (Mercian). The s- form (also
used in the subjunctive) fell from use in the early 13c. (though it
continues in Ger. sind, the 3rd person plural of "to be") and was
replaced by forms of be, but aron (aren, arn, are, from Pgm. *ar-,
probably a variant of PIE base *es-) continued and encroached on some
uses that had previously belonged to be as the two verbs merged. By the
early 1500s it had established its place in standard Eng. Art became
archaic in the 1800s.