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Chris, quite rightly you point out that the Scoti crossed from Ireland to Scotland, however the Scoti were also Celts.
Infact they spoke Goidelic, which is the old name for Gaelic, and the Irish also spoke Gaelic.
Irish is in fact a Gaelic language, although the name Gaelic is used nowadays to define the Celtic language spoken in the Hebrides and some parts of the Highlands.
Some historians claim that the Scoti, who were at home in Argyl and parts of the Hebrides, may actually not have crossed from Ireland, but simply spoke the same version of Celtic as the Irsish.
However, they did live in common with the Picts in some areas of Scotland. The Picts spoke Brythonic Celtic, and the two groups eventually merged into the Scottish People, forming the Kingdom of Alba. In Gaelic, Scotland is Alba.
There is also historic evidence that a tribe of Picts lived in Antrim, in north eastern Ireland.
The Irish, before moving to Ireland, lived in Britain, and it is even plausible that they originally crossed to Ireland from Scotland, from an area where the coast of Ireland is visible from Scotland.