Etymology
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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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Maxim 

single-barreled water-cooled machine gun, 1885 (Maxim gun), named for inventor, U.S.-born British engineer Sir Hiram S. Maxim (1840-1916).

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maximist (n.)

"one who has a fondness for quoting or using maxims," by 1827, from maxim + -ist.

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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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*meg- 
Proto-Indo-European root meaning "great."

It forms all or part of: acromegaly; Almagest; Charlemagne; maestro; magisterial; magistral; magistrate; Magna Carta; magnate; magnitude; magnum; magnanimity; magnanimous; magni-; Magnificat; magnificence; magnificent; magnify; magniloquence; magniloquent; Magnus; maharajah; maharishi; mahatma; Mahayana; Maia; majesty; major; major-domo; majority; majuscule; master; maxim; maximum; may (v.2) "to take part in May Day festivities;" May; mayor; mega-; megalo-; mickle; Mister; mistral; mistress; much; omega.

It is the hypothetical source of/evidence for its existence is provided by: Armenian mets "great;" Sanskrit mahat- "great, mazah- "greatness;" Avestan mazant- "great;" Hittite mekkish "great, large;" Greek megas "great, large;" Latin magnus "great, large, much, abundant," major "greater," maximus "greatest;" Middle Irish mag, maignech "great, large;" Middle Welsh meith "long, great."
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gnome (n.2)

"short, pithy statement of general truth," 1570s, from Greek gnōmē "judgment, opinion; maxim, the opinion of wise men," from PIE root *gno- "to know."

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pom-pom (n.)
"Maxim automatic gun," 1899, of imitative origin, soldiers' slang from the Boer War. For the ornamental tuft, see pompom.
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vox populi (n.)
1540s, Latin, literally "voice of the people." The full maxim (first attested in Medieval Latin) is vox populi, vox Dei "the voice of the people is the voice of God." Short form vox pop attested by 1964.
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dicte (n.)

also dict, "a saying, maxim, statement," late 14c., from Latin dictum "a thing said" (see dictum). A word long obsolete, but it figures in what is probably the first book printed in English, Caxton's issue of Anthony Woodville's "Dictes and Sayings of the Philosophers."

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dictate (n.)
Origin and meaning of dictate

1590s, "positive order or command;" 1610s "authoritative rule, maxim, or precept," from Latin dictatum "a thing said, something dictated," noun use of neuter past participle of dictare "say often, prescribe," frequentative of dicere "to say, speak" (from PIE root *deik- "to show," also "pronounce solemnly").

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