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Gradus and passus both have multiple meanings. Passus, as in its meaning related to feet, is also described as a unit of measurement, of five Roman feet. Gradus also has distinct meanings, including that on one which steps while ascending (as a stair), things resembling stairs, a march, a stage, a gait, a position, stand firm, of rank (in a hierarchy including of age), a degree of a circle, and even as a braid of hair.

However, both gradus and step are described, as a step, or a pace. What are the differences, if any, between the two, in describing steps or paces. It's possible that including differences as units of measurement may help in answering the question.

Dictionaries used were Smith's smaller Latin English Dictionary, Lewis and Short Dictionary, and Le Gaffiot (French). The reason for raising this question is as followup to this Stack Exchange Question about steps in mille passus. If no else includes them I will add as an answer the differences in Le Gaffiot to the definitions given in the English dictionaries.

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    Passsus is really a measure of length, between open arms finger tip to tip, Cord, fathom 1.5 m, from patere , to stand open. As a standard geographical measure its a double step with milia equal to 1.5 km. Gradus is a single step, often uphills, on stairs or on a ladder. So its use as non-metrical, often intensity measure (fr. degree, engl. degree, it. grado, ger. Grad), for angles, temperature, academic ladders on a scale (gr. σκαλα stairs) as a measurement device.
    – Roland F
    Commented Jun 24 at 18:10

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First of all, passus is normally the perfect active participle of patior, so any other use of this word is much less common, almost obscure.

Secondly, passus as a measurement does not mean a step, except figuratively, it means pacing out. It's derivation is from pando, meaning to spread, so inherently it means spread out, so I guess the idea is that when a man strides forward he "spreads out". So the primary sense of passus is a stride, not a step, and it can only mean step to that extent that you consider a man's stride to be a step.

gradus, on the other hand, is a more common word and specifically means a step and its verb, gradior means to walk in steps or to march. Hence, one can talk of stepping up stairs or stepping down a hillside.

Just because a word can be translated to English in a certain way in a certain context, it does not necessarily follow that the word "means" the same thing as the English word. For example, angustia means a narrow place. Yes, angustiae can sometimes be translated as "difficulties", but that is only what it suggests, not what it means.

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