# What is the historical origin for the naming of the word 'function' in its mathematical context?

I tried Wikipedia but it doesn't explain why functions are called functions.

The reason why I'm asking is because this word just doesn't make any sense for what it does.

If we were to reinvent all the words that don't make sense, and they become normal in a given culture, communication would be far more efficient and clear.

• math.stackexchange.com/questions/926110/… Commented Jun 18, 2017 at 11:43
• Please do not cross-post on multiple SE sites. Commented Jun 18, 2017 at 12:07
• "Function" makes perfect sense to me. It's an input-output machine that serves a particular function (in the ordinary sense). For instance, f(x)=x^2 is the input-output machine that serves the function of squaring a number. Commented Jun 18, 2017 at 15:07
• why was it named 'function'? -- it was first used to designate a geometric object associated with a curve, it doesn't make any sense today. im sure that ppl that study the origins of words and the linguists has plenty of examples (which they've already given examples for) of the structure of words just not making any sense @LeeMosher
– ambw
Commented Jun 18, 2017 at 20:27
• @StoneyB I did some digging up of old MSE posts, and Shog9 ♦ says, "there's value in having similar questions posted to multiple sites, so long as they're actually on-topic for the sites they're posted to." Example 1 and 2
– NVZ
Commented Jun 19, 2017 at 9:58

This is what OED has to say about the origin:

This use of the Latin functio is due to Leibniz and his associates. A paper in the Acta Eruditorum for 1692, pp. 169–170, signed ‘O.V.E.’, but probably written by Leibniz, uses functiones in a sense hardly different from its ordinary untechnical sense, to denote the various ‘offices’ which a straight line may fulfil in relation to a curve, viz. its tangent, normal, etc. In the same journal for 1694, p. 316, Leibniz defines functio as ‘a part of a straight line which is cut off by straight lines drawn solely by means of a fixed point, and of a point in the curve which is given together with its degree of curvature’; the examples given being the ordinate, abscissa, tangent, normal, etc. As the functiones (in Leibniz' sense) of a curve are variable quantities having a fixed mutual relation, this use of the word easily developed into the modern sense, which occurs in the writings of the Bernoullis early in the 18th cent. A somewhat peculiar use occurs about 1713, in Leibniz' Hist. et Origo Calc. Diff. (Math. Schriften ed. Gerhardt V. 408), where he says that just as constant quantities have their ‘functions’, viz. powers and roots, so variables have also ‘functions’ of a third kind, viz. differentials.

Acta Eruditorum can be found here, but it's in Latin.

A quote taken from the website Earliest Known Uses of Some of the Words of Mathematics indicates that there is an earlier use by Leibniz, in 1673:

The word FUNCTION first appears in a Latin manuscript "Methodus tangen tium inversa, seu de fuctionibus" written by Gottfried Wilhelm Leibniz (1646-1716) in 1673. Leibniz used the word in the non-analytical sense, as a magnitude which performs a special duty. He considered a function in terms of "mathematical job"- the "employee" being just a curve. He apparently conceived of a line doing "something" in a given figura... From the beginning of his manuscript, however, Leibniz demonstrated that he already possessed the idea of function, a term he denominates relatio.

• You could link to the Earliest Known Uses... website directly: jeff560.tripod.com/f.html (you have to scroll down, there's no way to link specifically to the FUNCTION entry).
– user1635
Commented Jun 18, 2017 at 16:49
• @Rahul TY, fixed. I didn't realize it still existed. The site must have moved, since that wasn't the link the paper cited.
– Laurel
Commented Jun 18, 2017 at 17:06
• I took the liberty of updating the current name of the website.
– user1635
Commented Jun 18, 2017 at 17:14
• please link what oed is, and spell out what it is, ppl on the Web who come across this would not understand what you're saying, also please include a 1-line summary of the answer, maybe like 'first used to designate a geometric object associated with a curve' but a few years later was used for 'general case of quantities dependent on other quantities' -- en.wikipedia.org/wiki/History_of_the_function_concept
– ambw
Commented Jun 18, 2017 at 20:36

Suppose you carry peaches in a bag, and each peach weighs 150 grams. What's the weight of the peaches you carry?

Well, you can't answer until you know how many there are. Because, the weight of the peaches is a function of their number.

No need to invoke Leibniz or Archimedes. Just refer to people who spoke English.

Denote P the total weight, and n the number of peaches. Then

P = n x 150

This is a simple function, a linear one.

Sometimes, P is denoted P(n), to stress that it depends on n.

You may even add that P is the dependent variable, and n is the independent variable.

• The question is about the "historical origin" of the word usage. I would think Leibniz and Bernoulli are relevant to that. Commented Jun 21, 2017 at 12:47