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Any Latinists here please? 1. Why did Gauss choose modulus? How does it relate to $n >1$ where $a - b = kn$ for some integer $k$?

  1. And why the Ablative Case?

ag.algebraic geometry - The Origin(s) of Modular and Moduli - MathOverflow. See also AYun's answer.

The word modulus (moduli in plural, cf. radius and radii, focus and foci, locus and loci) comes from Latin as a word meaning "small measure" or "unit of measure". This is why the absolute value of a complex number z is sometimes called the modulus of z and why the word is used in physics for Young's modulus. In 1800 Gauss introduced the congruence relation $a \equiv b \bmod m $ with m being called the modulus because this equivalence relation on integers a and b was being "measured" according to the integer m.

modulo, prep. and adj. : Oxford English Dictionary

Etymology: < classical Latin modulō, ablative of modulus modulus n. Compare earlier mod prep.

The preposition was first used in this sense in a Latin context by Gauss in 1801 (see note s.v. mod prep.).

mod, prep. : Oxford English Dictionary

Etymology: Shortened < classical Latin modulō modulo prep. and adj. (originally as a graphic abbreviation).

The notation bc (mod. a) (‘b is congruent to c modulo a’) was introduced by the German mathematician Carl Friedrich Gauss (compare gauss n.) in Disquisitiones Arithmeticae (1801) 2. Compare earlier modulus n. 2b.

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Gauss uses accusative modulum, not ablative modulo, when introducing the term in the very first sentence of Disquisitiones Arithmeticae:

"Si numerus $a$ numerorum $b$, $c$ differentiam metitur, $b$ et $c$ secundum a congrui dicuntur, sin minus, incongrui; ipsum $a$ modulum appelamus." (If a number $a$ measures the difference between two numbers $b$ and $c$, $b$ and $c$ are said to be congruent, if not, incongruent; $a$ itself is called modulus).

Modulo is used later in the text when grammatically appropriate. The structure of the sentence is similar to Euclid's in the Elements, book X, definition 1:

"Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure."

Under Gauss's definition, $b$ and $c$ can be reached from the same starting point (any residue) in increments of $a$, so it is a common measure. Modulus as a translation of Greek μέτρα (measure) has been used by mathematicians before, so Gauss's choice of words is not particularly surprising. Cotes used it in 1722 and De Moivre in 1738 in different contexts, see Earliest Known Uses of Some of the Words of Mathematics.

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    $\begingroup$ Good answer. Some more notes: the last clause I would translate: "'a' itself we may call the modulus." (It seems awkward to leave out the definite article.) Also "b modulo a" (modulus in the ablative) means "'b' with modulus 'a'", which might be one way the phrasing came about; it could also be coined from adding -o to a stem, to give a sense of role or purpose, when forming an adjective. But this is all my guessing $\endgroup$ Nov 18, 2021 at 16:24
  • $\begingroup$ I too lean towards the ablative of means, or instrumental, as in "a is congruent to b with/when using modulus m" $\endgroup$ Sep 6, 2023 at 22:38

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