Old English steppa (Mercian), stæpe, stepe (West Saxon) "stair, act of stepping," from the source of step (v.). Compare Old Frisian, Middle Dutch, Dutch stap, Old High German stapfo, German Stapfe "footstep").

It is attested from late Old English as "degree on a scale." The figurative meaning "action which leads toward a result" is recorded from 1540s. In dancing, from 1670s. The meaning "type of military pace" is from 1798.

The warning phrase watch your step is attested from 1911 (Wycliffe, late 14c., has keep thy foot in essentially the same sense). Step by step indicating steady progression is from 1580s. To follow in (someone's) steps is from mid-13c.

early 14c., "a drawing out, delay;" late 14c., "trailing part of a skirt, gown, or cloak;" also "retinue, procession," from Old French train "tracks, path, trail (of a robe or gown); act of dragging," from trainer "to pull, drag, draw," from Vulgar Latin *traginare, extended from *tragere "to pull," back-formation from tractus, past participle of Latin trahere "to pull, draw" (see tract (n.1)).,

General sense of "series, progression, succession, continuous course" is from late 15c.; train of thought is attested from 1650s. The railroad sense "locomotive and the cars coupled to it" is recorded from 1820 (publication year, dated 1816), from the notion of a "trailing succession" of wagons or carriages pulled by a mechanical engine.

Proto-Indo-European root meaning "to walk, go." 

It forms all or part of: aggress; aggression; aggressive; centigrade; congress; degrade; degree; degression; digress; digression; egress; gradation; grade; gradual; graduate; grallatorial; gravigrade; ingredient; ingress; plantigrade; progress; progression; regress; regression; retrograde; retrogress; tardigrade; transgress; transgression.

It is the hypothetical source of/evidence for its existence is provided by: Latin gradus "a step, a pace, gait," figuratively "a step toward something, a degree of something rising by stages;" gradi "to walk, step, go;" Lithuanian gridiju, gridyti "to go, wander;" Old Church Slavonic gredo "to come;" Old Irish in-greinn "he pursues."  

"near, nearby, close together, adjacent," Middle English neigh, from Old English neah (West Saxon, Kentish), neh (Anglian), from Proto-Germanic *naehwa- (source also of Old Saxon nah, Old Frisian nei, Middle Dutch, Dutch na, Old High German nah, German nah, Gothic nehwa), of uncertain origin, with no cognates outside Germanic. The Old English progression was neah - near - niehsta, for "nigh - nigher - nighest." But the comparative near and the superlative nehst (see next) gradually evolved into separate words that were no longer felt as related to nigh. New comparative and superlative forms nigher, nighest developed 14c. as phonetic changes obscured the original relationships. As an adjective and preposition in Middle English.

"flat surface, simplest of all geometrical surfaces," c. 1600, from Latin planum "flat surface, plane, level, plain," noun use of neuter of adjective planus "flat, level, even, plain, clear," from PIE *pla-no- (source also of Lithuanian plonas "thin;" Celtic *lanon "plain;" perhaps also Greek pelanos "sacrificial cake, a mixture offered to the gods, offering (of meal, honey, and oil) poured or spread"), suffixed form of root *pele- (2) "flat; to spread."

Introduced (perhaps by influence of French plan in this sense) to differentiate the geometrical senses from plain, which in mid-16c. English also meant "geometric plane." The figurative sense, in reference to inanimate things, is attested from 1850.

"never-ending pattern," 1975, from French fractal, ultimately from Latin fractus "interrupted, irregular," literally "broken," past participle of frangere "to break" (from PIE root *bhreg- "to break"). Coined by French mathematician Benoit Mandelbrot (1924-2010) in "Les Objets Fractals."

Many important spatial patterns of Nature are either irregular or fragmented to such an extreme degree that ... classical geometry ... is hardly of any help in describing their form. ... I hope to show that it is possible in many cases to remedy this absence of geometric representation by using a family of shapes I propose to call fractals — or fractal sets. [Mandelbrot, "Fractals," 1977]

The term was suggested earlier in Mandelbrot's 1967 book, "How Long is the Coast of Britain -- Statistical Self-Similarity and Fractional Dimension."

1550s, "regular geometric body with six square faces," also "product obtained by multiplying the square of a quantity by the quantity itself," from French cube (13c.) and directly from Latin cubus, from Greek kybos "a six-sided die," used metaphorically of dice-like blocks of any sort, also "cake; piece of salted fish; vertebra," of uncertain origin. Beekes points out that "words for dice are often loans" and that "the Lydians claimed to have invented the game" of kybos.

The mathematical also was in the ancient Greek word: the Greeks threw with three dice; the highest possible roll was three sixes. The word was attested in English from late 14c. in Latin form. The 1960s slang sense of "extremely conventional person" (1959) is from the notion of a square squared. Cube-root is from 1550s (in Middle English this was simply a cubick).

mid-13c., "tool for measuring right angles, carpenter's square," from Old French esquire "a square, squareness," from Vulgar Latin *exquadra, back-formation from *exquadrare "to square," from Latin ex "out" (see ex-) + quadrare "make square, set in order, complete," from quadrus "a square" (from PIE root *kwetwer- "four").

Meaning "square shape or area" is recorded by late 14c. (Old English used feower-scyte). Geometric sense "four-sided rectilinear figure" is from 1550s; mathematical sense of "a number multiplied by itself" is first recorded 1550s. Sense of "open space in a town or park" is from 1680s; that of "area bounded by four streets in a city" is from c. 1700. As short for square meal, from 1882. Square one "the very beginning" (often what one must go back to) is from 1960, probably a figure from board games.

1580s, "church of a bishop," from phrase cathedral church (c. 1300) "principal church of a diocese," a phrase partially translating Late Latin ecclesia cathedralis "church of a bishop's seat," from a specific early Christian use of classical Latin cathedra "a teacher or professor's chair," commonly "an easy chair (principally used by ladies)," also metonymically, as in cathedrae molles "luxurious women," from Greek kathedra "chair, seat, bench," also "exalted seat occupied by men of eminent rank or influence," from kata "down" (see cata-) + hedra "seat, base, chair, face of a geometric solid" (from PIE root *sed- (1) "to sit"). The Greek word was used in Old Testament and New Testament translations.

The English word was born an adjective, and attempts to force further adjectivization onto it in 17c. yielded cathedraical (1670s), cathedratic (1660s), cathedratical (1660s), after which the effort seems to have been given up.

mid-15c., Latinized spelling of Middle English spere (c. 1300) "cosmos; space, conceived as a hollow globe about the world," from Anglo-French espiere, Old French espere (13c., Modern French sphère), from Latin sphaera "globe, ball, celestial sphere" (Medieval Latin spera), from Greek sphaira "globe, ball, playing ball, terrestrial globe," a word of unknown origin.

From late 14c. in reference to any of the supposed concentric, transparent, hollow, crystalline globes of the cosmos believed to revolve around the earth and contain the planets and the fixed stars; the supposed harmonious sound they made rubbing against one another was the music of the spheres (late 14c.). Also from late 14c. as "a globe; object of spherical form, a ball," and the geometric sense "solid figure with all points equidistant from the center." Meaning "range of something, place or scene of activity" is first recorded c. 1600 (as in sphere of influence, 1885, originally in reference to Anglo-German colonial rivalry in Africa).