# A phrase of L. Euler on functions

I'm trying to understand the following sentence from Leonhard Euler's Institutionum calculi integralis Vol. III Chap. 2, bottom of p.40:

Huiusmodi functiones arbitrarias, prouti hic feci, eiusmodi signandi modo f:y indicabo, unde cauendum erit ne littera f pro quantitate habeatur, quocirca ipsi colon suffigere visum est.

A translation can be found here, but I'm unable to make sense of the second half:

Arbitrary functions of this kind, as I have made here, I will indicate by being marked in this way f:y , from which there will be a caution, without the letter f for the quantity that may be considered, concerning which a colon is considered sufficient.

I'd appreciate your help rephrasing this.

(Some context: The passage is interesting since Euler is largely considered responsible for introducing and popularising the f(x) notation in mathematics, which he at that time wrote as f:x. This is one of the few instances where he explains something about it.)

• Welcome to the site! Nice question! Commented May 29, 2018 at 13:11

If I get you right, the part you have trouble with is the one starting with unde, right?

The translation seems clumsy to me. Euler is explaining the notation he will use for functions.

• unde: hence/from where
• cauendum erit: it is to be taken care of
• ne: that not/so that not/not to
• littera f pro quantitate habitur: letter f be taken as a quantity (i.e., constant)
• quocirca: for which reason
• ipsi: to itself
• colon: colon (:)
• suffigere: suffix (as a verb)
• visum est: has been seen.

A possible, idiomatic(-ish) translation, could be,

hence, care is to be taken not to consider letter f as a constant, that is why [it is seen that] a colon was suffixed

I'm not a native English speaker, but I hope at least the idea is clear.

To complement Rafael's excellent answer, here's my own translation.

The original, as copied from above:

Huiusmodi functiones arbitrarias, prouti hic feci, eiusmodi signandi modo f:y indicabo, unde cauendum erit ne littera f pro quantitate habitur, quocirca ipsi colon suffigere visum est.

Somewhat literally:

Arbitrary functions of this sort, just as I have made here, I will indicate, as they must be marked in this way: f:y. Care must be taken for the letter f not to be used (read?) as a quantity (number?), for which reason a colon is seen to have been appended.

Much more idiomatically, taking some liberties with the text:

I will indicate functions like these by writing them in the form f:y. Be careful not to take f as a simple number; to keep them separate, function symbols have a colon after their name.

One particular point: the Latin text you've cited has suffigere visum est "is seen appended to [the letter]" in the last clause. But the fact that the translation says "is considered sufficient" makes me wonder: that would be a better translation of sufficere visum est, with a c instead of a g. If I saw this in an ancient text, I'd attribute it to different manuscripts; in this case, it seems like a straight-up error.

• I find the most natural translation of quantitas to be simply "number", as opposed to "function". Some numbers are regarded as variables, some as constants, some as parameters, and some as something else. The division is somewhat arbitrary, but many mathematicians will understand the difference in nuance. I think Euler means numbers in general, not any specific type, judging by this passage alone. Commented May 29, 2018 at 23:10
• At the time of Euler and way until the 20 century what was officially called the function was f(x) and not f. Euler called f the "character of the function f(x)". Commented May 30, 2018 at 16:45

videtur usually means "it seems," but it can also mean "it seems good," which appears to be the case here. senatui videtur means "the Senate decides" (literally, "it seems good to the Senate"). The translation should be as follows:

..., for which reason it has seemed good (to me) to append a colon to it (i.e., to the letter f).

or

..., for which reason I have decided to append a colon to it.

Another thing: Euler correctly wrote "habeatur." habitur doesn't exist.