Etymology
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mis- (1)

prefix of Germanic origin affixed to nouns and verbs and meaning "bad, wrong," from Old English mis-, from Proto-Germanic *missa- "divergent, astray" (source also of Old Frisian and Old Saxon mis-, Middle Dutch misse-, Old High German missa-, German miß-, Old Norse mis-, Gothic missa-), perhaps literally "in a changed manner," and with a root sense of "difference, change" (compare Gothic misso "mutually"), and thus possibly from PIE *mit-to-, from root *mei- (1) "to change."

Productive as word-forming element in Old English (as in mislæran "to give bad advice, teach amiss"). In 14c.-16c. in a few verbs its sense began to be felt as "unfavorably," and it came to be used as an intensive prefix with words already expressing negative feeling (as in misdoubt). Practically a separate word in Old and early Middle English (and often written as such). Old English also had an adjective (mislic "diverse, unlike, various") and an adverb (mislice "in various directions, wrongly, astray") derived from it, corresponding to German misslich (adj.). It has become confused with mis- (2).

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mis- (2)

word-forming element of Latin origin (in mischief, miscreant, misadventure, misnomer, etc.), from Old French mes- "bad, badly, wrong, wrongly," from Vulgar Latin *minus-, from Latin minus "less" (from suffixed form of PIE root *mei- (2) "small"), which was not used as a prefix in Latin but in the Romanic languages was affixed to words as a depreciative or negative element. The form in French perhaps was influenced in Old French by *miss-, the Frankish (Germanic) form of mis- (1).

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step (n.)
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"). From late Old English as "degree on a scale." Figurative meaning "action which leads toward a result" is recorded from 1540s. In dancing, from 1670s. Meaning "type of military pace" is from 1798. Warning phrase watch your step is attested from 1911 (Wyclif (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.
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step (v.)
Old English steppan (Anglian), stæppan (West Saxon) "take a step," from West Germanic *stap- "tread" (source also of Old Frisian stapa, Middle Dutch, Dutch stappen, Old High German stapfon, German stapfen "step"), from PIE root *stebh- "post, stem; to support, place firmly on" (see staff (n.); source also of Old Church Slavonic stopa "step, pace," stepeni "step, degree"). The notion is perhaps "a treading firmly on; a foothold."

Transitive sense (as in step foot in) attested from 1530s. Related: Stepped; stepping. Originally strong (past tense stop, past participle bestapen); weak forms emerged 13c., universal from 16c. To step out "leave for a short time" is from 1530s; meaning "to go out in public in style" is from 1907. Step on it "hurry up" is 1923, from notion of gas pedal.
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step- 
Old English steop-, with connotations of "loss," in combinations like steopcild "orphan," related to astiepan, bestiepan "to bereave, to deprive of parents or children," from Proto-Germanic *steupa- "bereft" (source also of Old Frisian stiap-, Old Norse stjup-, Swedish styv-, Middle Low German stef-, Dutch stief-, Old High German stiof-, German stief-), literally "pushed out," from PIE *steup-, from root *(s)teu- (1) "to push, stick, knock," with derivatives referring to fragments (see steep (adj.)). Barnhart suggests the forms in -f- are by assimilation of the first sound in following words for "father."

Etymologically, a stepfather or stepmother is one who becomes father or mother to an orphan, but the notion of orphanage faded in 20c. and came to denote simply relation through marriage. For sense evolution, compare Latin privignus "stepson," related to privus "deprived." Compare orphan (n.).
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step-ladder (n.)
also stepladder, one with flat steps instead of rungs, 1728, from step (n.) + ladder.
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step-daughter (n.)
Old English stepdohtor; see step- + daughter (n.). Similar formation in German Stieftochter.
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lock-step (n.)

1802, in military writing, to describe a very tight style of mass marching, from lock (n.1) + step (n.).

Lock-step. A mode of marching by a body of men going one after another as closely as possible, in which the leg of each moves at the same time with and closely follows the corresponding leg of the person directly before him. [Thomas Wilhelm, "Military Dictionary and Gazetteer," Philadelphia, 1881]

Figurative use by 1836.

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goose-step (n.)
1806, originally a military drill to teach balance; "to stand on each leg alternately and swing the other back and forth." This, presumably, reminded someone of a goose's way of walking. In reference to "marching without bending the knees" (as in Nazi military reviews) it apparently is first recorded 1916. As a verb by 1854.
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quick-step (n.)

1802, "step used in marching in quick-time" (110 steps per minute); by 1811 as "a march in quick-time," from quick (adj.) + step (n.). Also of music adapted to such a march. By 1880 as "a fast dance." From 1906 as a verb. Related: quick-stepped; quick-stepping.

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