Old English sceal, Northumbrian scule "I owe/he owes, will have to, ought to, must" (infinitive sculan, past tense sceolde), from *skulanan, a common Germanic preterite-present verb (along with can, may, will), from Proto-Germanic *skul- (source also of Old Saxon sculan, Old Frisian skil, Old Norse and Swedish skola, Middle Dutch sullen, Old High German solan, German sollen, Gothic skulan "to owe, be under obligation"). This is said to be related, via a past tense form, to Old English scyld "guilt," German Schuld "guilt, debt;" also Old Norse Skuld, name of one of the Norns.
These Germanic words are reconstructed (Watkins, Pokorny) to be from a PIE root *skel-(2) "to be under an obligation." The basic sense of the Germanic word probably was "I owe," hence "I ought." Cognates outside Germanic include Lithuanian skelėti "to be guilty," skilti "to get into debt;" Old Prussian skallisnan "duty," skellants "guilty." But Boutkan gives the group no PIE etymology and writes that the alleged root, limited as it is to Germanic and Balto-Slavic, "is likely to represent an innovation on the basis of North European substrate material."
Shall survives as an auxiliary. The original senses are obsolete; the meaning shifted in Middle English from obligation to include futurity. It has no participles, no imperative, and no infinitive. Its past-tense form has become should (q.v.) and has acquired special senses of its own.
"if stars are infinitely and uniformly distributed through the sky, their number should counterbalance their faintness and the night sky should be as bright as the day;" named for German astronomer H.W.M. Olbers (1758-1840), who propounded it in 1826.
Old English ahte "owned, possessed," past tense of agan "to own, possess; owe" (see owe). As a past tense of owe, it shared in that word's evolution and meant at times in Middle English "possessed" and "under obligation to pay." It has been detached from owe since 17c., though he aught me ten pounds is recorded as active in East Anglian dialect c. 1825. As an auxiliary verb expressing duty or moral obligation (the main modern use, attested from late 12c.), it represents the past subjunctive.
Ought, Should. Ought is the stronger, expressing especially obligations of duty, with some weaker use in expressing interest or necessity: as, you ought to know, if any one does. Should sometimes expresses duty: as, we should be careful of others' feelings; but generally expresses propriety, expediency, etc.: as, we should dot our i's and cross our t's. [Century Dictionary, 1895]