Etymology
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filly (n.)
c. 1400, filli, fyly, "a young mare, female colt or foal," possibly from Old Norse fylja, fem. of foli "foal," from Proto-Germanic *fuljo, from PIE *pulo- "young of an animal," suffixed form of root *pau- (1) "few, little." Slang sense of "lively young girl" is from 1610s.
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pony (n.)

1650s, powny, "a very small horse" (less than 13 hands in height), from Scottish, apparently from obsolete French poulenet "little foal" (mid-15c.), diminutive of Old French poulain "foal," from Late Latin pullanus "young of an animal," from Latin pullus "young of a horse, fowl, etc." (from PIE root *pau- (1) "few, little") [Skeat's suggestion, still accepted]. Compare, from the same source, foal, filly, Sanskrit potah "a young animal," Greek pōlos "foal," secondarily also of other young animals; Latin pullus "young animal," Lithuanian putytis "young animal, young bird."

A small horse, especially one of a small breed, as opposed to a colt or filly, words which indicate merely young horses. German, sensibly, indicates this animal by attaching a diminutive suffix to its word for "horse," which might yield Modern English *horslet. Modern French poney is a 19c. borrowing from English.

The Shetland breed of ponies are stoutly built, active and hardy, with very full mane and tail, and of gentle, docile disposition. In western parts of the United States all the small hardy horses (mustangs or broncos) used by the Indians are called ponies. [Century Dictionary, 1897] 

Meaning "crib of a text as a cheating aid," especially a translation of a Greek or Latin author used unfairly in the preparation of lessons (1827) and "small liquor glass" (1849) both are from notion of "smallness" (the former also "something one rides," a translation being something that enables a student to "get along fast").

As the name of a popular dance, it dates from 1963. The U.S. Pony Express began 1860 (and operated about 18 months before being superseded by the transcontinental telegraph). The figurative one-trick pony is 1897, American English, in reference to circus acts.

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*pau- (1)

Proto-Indo-European root meaning "few, little."

It forms all or part of: catchpoll; encyclopedia; filly; foal; few; hypnopedia; impoverish; orthopedic; Paedophryne; paraffin; parvi-; parvovirus; paucity; Paul; pauper; pedagogue; pederasty; pedo-; pedophilia; poco; poltroon; pony; pool (n.2) "game similar to billiards;" poor; poulterer; poultry; poverty; puericulture; puerile; puerility; puerperal; pullet; pullulate; Punch; Punchinello; pupa; pupil (n.1) "student;" pupil (n.2) "center of the eye;" puppet; pusillanimous; putti.

It is the hypothetical source of/evidence for its existence is provided by: Sanskrit potah "a young animal," putrah "son;" Avestan puthra- "son, child;" Greek pauros "few, little," pais (genitive paidos) "child," pōlos "foal;" Latin paucus "few, little," paullus "little," parvus "little, small," pauper "poor," puer "child, boy," pullus "young animal;" Oscan puklu "child;" Old English feawe "not many, a small number," fola "young horse;" Old Norse fylja "young female horse;" Old Church Slavonic puta "bird;" Lithuanian putytis "young animal, young bird;" Albanian pele "mare."

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colt (n.)

Old English colt "a young horse," also "young ass," in Biblical translations also used for "young camel," perhaps from Proto-Germanic *kultaz (source also of Swedish dialectal kult "young boar, piglet; boy," Danish kuld "offspring, brood") and akin to child. Commonly and distinctively applied to the male, the young female being a filly. Applied to young or inexperienced persons from early 13c.

COLT'S TOOTH An old fellow who marries, or keeps a young girl, is ſaid to have a colt's tooth in his head. ["Classical Dictionary of the Vulgar Tongue," 1796]

The image is in Chaucer. Colts shed their first set of teeth beginning at about three years.

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