# “With respect to” in mathematics

The expression "with respect to" is common in mathematics. Consider these example sentences:

1. The derivative of x^2y with respect to y is x^2.
2. Let us reflect the point A with respect to the line L and call the new point B.

(This SE site doesn't support MathJax, so I'm not formatting the formulas at all. Using dollars would probably confuse users here.)

I would like to be able to translate these sentences like these in Latin. I believe such things can be found in scientific texts written in Latin, but I am not familiar with this literature. Is there an attested Latin idiom for the mathematical "with respect to"?

By best guess is respectu with genitive. Using my examples:

1. Derivativus expressionis x^2y respectu (variabilis) y est x^2.
2. Punctum A reflectimus respectu lineae L et punctum novum B appellamus.

My feeling is that using a mere preposition or case is not enough. (Although in a sense the whole "with respect to" is a preposition.) As a mathematician myself, I find it beneficial to have a somewhat heavier structure to make it more easily recognizable.

• How close to 'in respect to' is the legal expression in rem +genitive? – Hugh Sep 6 '16 at 3:00
• @Hugh, I'm not familiar with that legal expression, so I can't tell. Quick searches suggest that not very close. – Joonas Ilmavirta Sep 6 '16 at 3:23
• quoad is my first-draw reaction, but I can't back that up with actual Latin mathematical texts. – brianpck Sep 6 '16 at 14:50
• @brianpck, how would you use that in a sentence? For example, what would be "reflection with respect to a line"? – Joonas Ilmavirta Sep 6 '16 at 14:58
• A quick Google search seems to suggest I'm right: Principia calculi differentialis or just search for "quoad x." I'll try to turn that into an answer when I have time. – brianpck Sep 6 '16 at 15:03

There is one word that seems to fit the bill: quoad. Although this word has a temporal ("as long as") and spatial ("as far as") meaning, Lewis and Short also gives the following meaning:

B.3: With respect to, as to

It also suggests that it derives from quod attinet ad. An interesting note is that all three classical examples given are dubious readings, so we cannot say that there is a rock-solid classical testimony that this word can be used like a normal PREP + ACC.

That said, here are several usages in later mathematical works that use quoad in the same context as the OP's example sentences. (I don't know how to do do mathematical notation: feel free to edit.)

Principia calculi differentialis et integralis, Andrea Caraffa

denotabunt itaque fx'(x,y,z,...), fx"(x,y,z,...), ... derivatas primi, secundi, ... ordinis ex f(x,y,z,...) quoad x, item fy'(x,y,z,...), fy"(x,y,z,...), ... derivatas primi, secundi, ... ordinis ex f(x,y,z,...) quoad y; etc. (pg. 28)

Another quote from the same page uses the same word, but also uses respectu ipsius to mean something else. (My multi-variable differential calculus is rusty, so perhaps a comment could clarify what the distinction should be):

Ex functione μ = f(x, y, z...) exsistunt etiam partiales variorum ordinum derivatae per vices quoad binas, ternas, ... variabiles v. gr. quoad y, x, quoad z, y, x, etc.; sic axa-1cos(y) est partialis derivata secundi ordinis ex functione xasin(y) quoad y, x, nimirum derivata quoad x ex derivata quoad y respectu ipsius xasin(y). (pg. 28)

Another,example of this usage comes from the Tentamen of Wolfgangus Bolyai de Bolya (pg. 221), talking about differentials. This is significant because he is explicitly instructing on proper terminology:

tum vz(dot) dicitur differentiale quoad z et uẋ quoad x acceptum, atque v dicitur derivata ipsius U quoad z et u derivata quoad x accepta...

Searching for "quoad x" with quotes reveals numerous other examples.

• Good answer! I assume the mathematical "with respect to" in the modern sense as an idea did not exist in antiquity, so attested classical examples are too much to ask for. If I had a declinable word instead of a variable name, would quoad take a nominative or an accusative? – Joonas Ilmavirta Sep 8 '16 at 14:31
• The examples given in L&S use the accusative. I also remembered why I remembered the word so quickly: it's very common in Scholastic Latin. Just try searching for quoad in the Corpus Thomisticum – brianpck Sep 8 '16 at 15:10