Etymology
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Words related to intercalate

inter- 
word-forming element used freely in English, "between, among, during," from Latin inter (prep., adv.) "among, between, betwixt, in the midst of" (also used extensively as a prefix), from PIE *enter "between, among" (source also of Sanskrit antar, Old Persian antar "among, between," Greek entera (plural) "intestines," Old Irish eter, Old Welsh ithr "among, between," Gothic undar, Old English under "under"), a comparative of root *en "in."

A living prefix in English from 15c. and used with Germanic as well as Latinate words. Spelled entre- in French; most words borrowed into English in that form were re-spelled 16c. to conform with Latin except entertain, enterprise. In Latin, spelling shifted to intel- before -l-, hence intelligence, etc.
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calendar (n.)
c. 1200, "the year as divided systematically into days and months;" mid-14c. as "table showing divisions of the year;" from Old French calendier "list, register," from Latin calendarium "account book," from calendae/kalendae "the calends" the first day of the Roman month, when debts fell due and accounts were reckoned.

This is from calare "to announce solemnly, call out," as the priests did in proclaiming the new moon that marked the calends, from PIE root *kele- (2) "to shout." In Rome, new moons were not calculated mathematically but rather observed by the priests from the Capitol; when they saw it, they would "declare" the number of days till the nones (five or seven, depending on the month). The word was taken by the early Church for its register list of saints and their feast days. Meaning "list of documents arranged chronologically" is from late 15c.

The -ar spelling in English is 17c. to differentiate it from the now-obscure calender "cloth-presser." Related: Calendarial; calendary.
intercalary (adj.)
"inserted into the calendar," 1610s, from Latin intercalarius "intercalary, of an intercalary month," from intercalare "proclaim an intercalary day" (see intercalate). General sense of "interpolated" is attested from 1798.
intercalation (n.)
1570s, from Latin intercalationem (nominative intercalatio) "insertion of an intercalary day," noun of action from past participle stem of intercalare "proclaim an intercalary day" (see intercalate). The general sense "insertion of any addition into an existing series" is from 1640s.
*kele- (2)
*kelə-, Proto-Indo-European root meaning "to shout." Perhaps imitative.

It forms all or part of: acclaim; acclamation; Aufklarung; calendar; chiaroscuro; claim; Claire; clairvoyance; clairvoyant; clamor; Clara; claret; clarify; clarinet; clarion; clarity; class; clear; cledonism; conciliate; conciliation; council; declaim; declare; disclaim; ecclesiastic; eclair; exclaim; glair; hale (v.); halyard; intercalate; haul; keelhaul; low (v.); nomenclature; paraclete; proclaim; reclaim; reconcile.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit usakala "cock," literally "dawn-calling;" Latin calare "to announce solemnly, call out," clamare "to cry out, shout, proclaim;" Middle Irish cailech "cock;" Greek kalein "to call," kelados "noise," kledon "report, fame;" Old High German halan "to call;" Old English hlowan "to low, make a noise like a cow;" Lithuanian kalba "language."