Etymology
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axiom (n.)

"statement of self-evident truth," late 15c., from French axiome, from Latin axioma, from Greek axioma "authority," literally "that which is thought worthy or fit," from axioun "to think worthy," from axios "worthy, worth, of like value, weighing as much" (from PIE adjective *ag-ty-o- "weighty," from root *ag- "to drive, draw out or forth, move").

Axioms in philosophy are not axioms until they are proved upon our pulses. [Keats, letter, May 3, 1818]
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axiomatic (adj.)
"of the nature of a self-evident truth," 1797, from Greek axiomatikos, from axioma (genitive axiomatos); see axiom. Form axiomatical is attested from 1580s.
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aphorism (n.)

1520s, "concise statement of a principle" (especially in reference to the "Aphorisms of Hippocrates"), from French aphorisme (corrected from Old French aufforisme, 14c.), from Late Latin aphorismus, from Greek aphorismos "definition; short, pithy sentence," from aphorizein "to mark off, divide," from apo "from" (see apo-) + horizein "to bound" (see horizon).

General sense of "short, pithy statement containing a truth of general import" (e.g. "life is short, and art is long") is from 1580s in English. Distinguished from an axiom, which is a statement of self-evident truth; an epigram is like an aphorism, but lacking in general import. Maxim and saying can be used as synonyms for aphorism, but maxims tend to be practical and sayings tend to be more commonplace and have an author's name attached.

[F]or aphorisms, except they should be ridiculous, cannot be made but of the pith and heart of sciences ; for discourse of illustration is cut off ; recitals of examples are cut off ; discourse of connexion and order is cut off ; descriptions of practice are cut off. So there remaineth nothing to fill the aphorisms but some good quantity of observation : and therefore no man can suffice, nor in reason will attempt, to write aphorisms, but he that is sound and grounded. [Francis Bacon, "The Advancement of Learning," 1605] 
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*ag- 

Proto-Indo-European root meaning "to drive, draw out or forth, move."

It forms all or part of: act; action; active; actor; actual; actuary; actuate; agency; agenda; agent; agile; agitation; agony; ambagious; ambassador; ambiguous; anagogical; antagonize; apagoge; assay; Auriga; auto-da-fe; axiom; cache; castigate; coagulate; cogent; cogitation; counteract; demagogue; embassy; epact; essay; exact; exacta; examine; exigency; exiguous; fumigation; glucagon; hypnagogic; interact; intransigent; isagoge; litigate; litigation; mitigate; mystagogue; navigate; objurgate; pedagogue; plutogogue; prodigal; protagonist; purge; react; redact; retroactive; squat; strategy; synagogue; transact; transaction; variegate.

It is the hypothetical source of/evidence for its existence is provided by: Greek agein "to lead, guide, drive, carry off," agon "assembly, contest in the games," agōgos "leader," axios "worth, worthy, weighing as much;" Sanskrit ajati "drives," ajirah "moving, active;" Latin actus "a doing; a driving, impulse, a setting in motion; a part in a play;" agere "to set in motion, drive, drive forward," hence "to do, perform," agilis "nimble, quick;" Old Norse aka "to drive;" Middle Irish ag "battle."

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maxim (n.)

early 15c., maxime, "an axiom, statement of a self-evident truth," from Old French maxime, from Late Latin maxima, shortened from phrases such as maxima propositio (Boethius), maxima sententarium "axiom," literally "greatest or chief premise, greatest among propositions" (one which is general and absolute), from fem. of maximus "greatest," from PIE *mag-samo-, superlative form of root *meg- "great."

The modern meaning "summary statement of an established or accepted proposition serving as a rule or guide, a proposition ostensibly expressing some general truth" is from 1590s.

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principle (n.)

late 14c., "origin, source, beginning" (a sense now obsolete), also "rule of conduct; axiom, basic assumption; elemental aspect of a craft or discipline," from Anglo-French principle, Old French principe "origin, cause, principle," from Latin principium (plural principia) "a beginning, commencement, origin, first part," in plural "foundation, elements," from princeps  (genitive principis) "first man, chief leader; ruler, sovereign," noun use of adjective meaning "that takes first," from primus "first" (see prime (adj.)) + root of capere "to take" (from PIE root *kap- "to grasp").

 The English -l- apparently is by analogy of participle, manciple, etc., also principal. From the notion of "one of the fundamental tenets or doctrines of a system, a law or truth on which others are founded" comes the sense of "a right rule of conduct" (1530s).

It is often easier to fight for principles than to live up to them. [Adlai Stevenson, speech, New York City, Aug. 27, 1952]

Scientific sense of "general law of nature," by virtue of which a machine or instrument operates, is recorded from 1802.

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