# When was “co-” first used to mean duality in mathematics?

As one can see on Wiktionary, the meaning of the prefix is "co-" as used in mathematics is different from its meaning as used in the rest of the English language, and does not seem to be a natural development from the other meanings.

This leads to my question: when and why did the prefix "co-" first begin to be used in mathematics? Why does is its meaning different in this field than in other contexts?

My guess is as follows: the only place I know "co-" being used in mathematics as in its typical usage elsewhere is in the term "covariant" as in "covariant tensor". This also has a natural interpretation as being the dual object of another important object, leading to other mathematical terms describing dual objects via analogy with covariant tensors, even though the "co-" in the original term with which the analogy was being drawn did not have that meaning.

If it wasn't "covariant tensor", then my other guess would be "cosine". This then would have lead to the name for the "cotangent function", which then would have lead to the name for "cotangent space", which is obviously a dual object and which would have started the trend for naming dual objects "co-". Perhaps the cosine function was called that because people did not feel motivated to come up with an original name for it after translating the Arabic term for "sine".

• Some insights may perhaps be gleaned from "C" page the "Earliest Known Uses of Some of the Words of Mathematics" collection. – Blue Sep 10 '16 at 6:51
• Actually "covariant" is using exactly the usual meaning of "co": A covariant tensor transforms (varies) the same way (co) as the basis. The opposite is a contravariant tensor, which transforms the opposite (contra) way. – celtschk Sep 10 '16 at 6:55
• Also looking at the Definition of "counion" at Wiktionary (Wikipedia doesn't have that term), it doesn't really fit the definition of "co-" at the page you lined to (it's definitely not the opposite of an union), even though it is linked there as example. – celtschk Sep 10 '16 at 7:06
• Looking again at the Wiktionary page, I notice the meaning "partner". That IMHO automatically applies to any case where the "co-" could be interpreted as "counterparts" (i.e. what the Wiktionary page lists as "mathematical meaning"): A counterpart of something is always a natural partner of it. – celtschk Sep 10 '16 at 7:56
• I suspect the coset is another candidature for being the first –  V. Rogov Jul 20 '17 at 23:08