Everyone (in the book trade) knows that a cast off is a calculation of how many book pages a manuscript will make. Benjamin Franklin tells us in 1784 “The compositors in your chapel* do not cast off their copy well”. To people who don’t work with books the meaning is unlikely to be quite so obvious.
The Oxford English Dictionary defines to cast off as “To count or reckon, so as to ascertain the sum of various numbers, orig. by means of counters, to the manipulation of which the word probably refers.” Think abacus — not a tool used today by most estimators. Originally the verb “cast” could stand on its own with this meaning: to cast a column of figures, or to cast accounts, but now it seems to need the support of the “off”. The cast off is one of these wonderful numbers that comes with its own guaranteed margin of error – the margin determined by the skill of the caster-offer. But it is a number that’s fundamental to all our activities, and severe retribution can result when it’s too far off target.
An analogous usage is to cast a horoscope – basically a calculation – also coming of course with its own built-in margin of error.
Cast and throw are almost synonyms. Thus we have in about 1400 “Palomydon cast off his clothis cantly & well.” And in 1697 Sir William Dampier refers to a quick acting tree “In a weeks time the Tree casts off her old Robes.” Milton appears to be the first to say “to cast off this yoke” in Paradise Lost, implying a revolutionary rejection of authority. Of closer personal impact perhaps we have the same rejection in the King James Version of the Bible (1611) in Psalm lxxi. 9 “Cast me not off in the time of old age.”
The worlds of hawking and hunting give us cast off meaning to unleash (dogs), to let fly (hawks). For example: “When a magpie is seen at a distance, a hawk is immediately to be cast off.”
Alchemy comes into the picture too, where cast off means to throw off, as for instance vapor, or to run off as melted metal.
Sailing and boating have two different usages of cast off: obviously setting off, untying the rope holding the boat to the dock (basically the same meaning as the hunting examples) and shortening sail as in “The gaskets cast off the fore topsail”.
In country dancing to cast off means to maneuver around other dancers to a position lower down the set. In knitting it implies completion of the job: “To cast off which is done by knitting two loops and pulling the first made loop over the last” 1880.
* chapel is an interesting usage in itself. Printing union ‘branches’ are still referred to as chapels, with the shop steward designated as the father of chapel.