Etymology
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cardinal (n.)
Origin and meaning of cardinal
early 12c., "one of the ecclesiastical princes who constitute the sacred college," from Medieval Latin cardinalis, originally as a noun "one of the presbyters of the chief (cardinal) churches of Rome," short for cardinalis ecclesiae Romanae or episcopus cardinalis, from Latin cardinalis (adj.) "principal, chief, essential" (see cardinal (adj.)).

The North American songbird (Cardinalis virginianus) is attested from 1670s, so named for its fine red color, resembling the cardinals in their red robes.
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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.

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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.)).
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cardinality (n.)
1520s, "condition of being a cardinal," from cardinal (n.) + -ity. Mathematical sense is from 1935 (see cardinal (adj.)).
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carding (n.)
"wool-dressing," late 15c., verbal noun from card (v.2).
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cardio- 
before vowels cardi-, word-forming element meaning "pertaining to the heart," from Latinized form of Greek kardia "heart," from PIE root *kerd- "heart."
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cardiogram (n.)
"a tracing of the beating of the heart made with a cardiograph," 1876, from cardio- + -gram.
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cardiograph (n.)

"apparatus for recording by tracing the beating of the heart," 1867, from cardio- + -graph "something written."

Although the work does not treat of the recent means of diagnosis—the thermometer, laryngoscope, cardiograph, etc.,—still it is complete as far as it goes. [book review in Medical Investigator, May 1867, p.94]
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cardiology (n.)
"the study of the heart," 1847, from cardio- + -logy. Cardiologist attested from 1885.
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cardiopulmonary (adj.)
also cardio-pulmonary, "pertaining to both the heart and the lungs," 1879, from cardio- + pulmonary.
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