Etymology
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Words related to ordinal

order (n.)

c. 1200, "body of persons living under a religious discipline," from Old French ordre "position, estate; rule, regulation; religious order" (11c.), from earlier ordene, from Latin ordinem (nominative ordo) "row, line, rank; series, pattern, arrangement, routine," originally "a row of threads in a loom," from Proto-Italic *ordn- "row, order" (source also of ordiri "to begin to weave;" compare primordial), which is of uncertain origin. Watkins suggests it is a variant of PIE root *ar- "to fit together," and De Vaan finds this "semantically attractive."

The original English word reflects a medieval notion: "a system of parts subject to certain uniform, established ranks or proportions," and was used of everything from architecture to angels. Old English expressed many of the same ideas with endebyrdnes. From the notion of "formal disposition or array, methodical or harmonious arrangement" comes the meaning "fit or consistent collocation of parts" (late 14c.).

Meaning "a rank in the (secular) community" is first recorded c. 1300. Sense of "a regular sequence or succession" is from late 14c. The meaning "command, directive" is first recorded 1540s, from the notion of "that which keep things in order." Military and honorary orders grew out of the fraternities of Crusader knights.

The business and commerce sense of "a written direction to pay money or deliver property" is attested by 1837; as "a request for food or drink in a restaurant" from 1836. In natural history, as a classification of living things next below class and next above family, it is recorded from 1760. Meaning "condition of a community which is under the rule of law" is from late 15c.

In order "in proper sequence or arrangement" is from c. 1400; out of order "not in proper sequence or orderly arrangement" is from 1540s; since 20c. principally mechanical, but not originally so ("and so home, and there find my wife mightily out of order, and reproaching of Mrs. Pierce and Knipp as wenches, and I know not what," - Pepys, diary, Aug. 6, 1666).

Phrase in order to "for the purpose of" (1650s) preserves etymological notion of "sequence." In short order "without delay" is from 1834, American English; order of battle "arrangement and disposition of an army or fleet for the purposes of engagement" is from 1769. The scientific/mathematical order of magnitude is attested from 1723.

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*ar- 
also arə-, Proto-Indo-European root meaning "to fit together."

It forms all or part of: adorn; alarm; aristarchy; aristo-; aristocracy; arm (n.1) "upper limb of the body;" arm (n.2) "weapon;" armada; armadillo; armament; armature; armilla; armistice; armoire; armor; armory; army; art (n.) "skill as a result of learning or practice;" arthralgia; arthritis; arthro-; arthropod; arthroscopy; article; articulate; artifact; artifice; artisan; artist; coordination; disarm; gendarme; harmony; inert; inertia; inordinate; ordain; order; ordinal; ordinance; ordinary; ordinate; ordnance; ornament; ornate; primordial; subordinate; suborn.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit irmah "arm," rtih "manner, mode;" Armenian arnam "make," armukn "elbow;" Greek arti "just," artios "complete, suitable," artizein "to prepare," arthron "a joint;" Latin ars (stem art-) "art, skill, craft," armus "shoulder," artus "joint," arma "weapons;" Old Prussian irmo "arm;" German art "manner, mode."
cardinal (adj.)
Origin and meaning of cardinal

"chief, pivotal," early 14c., from Latin cardinalis "principal, chief, essential," a figurative use, literally "pertaining to a hinge," from cardo (genitive cardinis) "that on which something turns or depends; pole of the sky," originally "door hinge," which is of unknown origin. Related: Cardinally.

The cardinal numbers (1590s) are "one, two, three," etc. as opposed to ordinal numbers "first, second, third," etc.; they are so called because they are the principal numbers and the ordinals depend on them.

The cardinal points (1540s) are north, south, east, west. The cardinal sins (c. 1600) are too well known to require rehearsal. The cardinal virtues (c. 1300) were divided into natural (justice, prudence, temperance, fortitude) and theological (faith, hope, charity). The natural ones were the original classical ones, which were amended by Christians. But typically in Middle English only the first four were counted as the cardinal virtues:

Of þe uour uirtues cardinales spekeþ moche þe yealde philosofes. ["Ayenbite of Inwyt," c. 1340]

By analogy of this, and cardinal winds (late 14c.), cardinal signs (four zodiacal signs marking the equinoxes and the solstices, late 14c.), etc., the adjective in Middle English acquired an association with the number four.

cardinal number (n.)
1590s, "one, two, three," etc. as opposed to ordinal numbers "first, second, third," etc.; so called because they are the principal numbers and the ordinals depend on them (see cardinal (adj.)).
coordinal (adj.)

also co-ordinal, 1849, in mathematics and geometry, "having (a certain number) of coordinates;" see co- + ordinal.