The usual story I've heard is that Numbers were held in great esteem by the Pythagoreans, that the discovery of numbers that were not ratios were held to be contra reason and thus irrational, as suggested for example by wikipedia which explains that this discovery was made by (apocraphally) Hippasus and:
[who], however, was not lauded for his efforts: according to one legend, he made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans “…for having produced an element in the universe which denied the…doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios. Another legend states that Hippasus was merely exiled for this revelation.
An alternative reading is that irrational means simply not a ratio. This is a clear, definitative and unambiguous description of what such a number is.
(It also appears a signifier of a break between geometry and number, with geometry taking the lead as it could deal with magnitudes that were not rational).
The question is, if this is true, when did irrational actually denote something that is against reason? And was that read back into the discovery of irrational magnitudes?