In what sense did "average" ever come to mean a statistical quantity? I was researching the etymology of "average" to enlighten the debate as to whether "average" can in fact mean a "median" (or if the presenter was merely trying to sweep up their tracks so to speak). I was surprised by the etymology: unknown possible from Arabic meaning losing an eye, and via Italian/French meaning damage to shipping goods according to wiktionary.org
The first use for a statistical sense was 1735 (I am guessing it was De Moivre measures of central tendency). Was there any justification for the use of the word in this sense? The median does in fact have a sampling distribution which tends to normal in most cases, but my suspicion was that this early work focused on arithmetic quantities.
My questions are:


*

*What is the document attributed to the 1735 usage for the statistical of an "average"?

*Is the definition "(statistics) Any measure of central tendency, especially any mean, the median, or the mode. [from 1735]" actually reflective of what the 1735 treatise reflected?

*If not, at what point (if ever) did average start to reflect other statistical quantities than the arithmetic mean?


1: https://en.wiktionary.org/wiki/average
 A: Consolidation of my comments to an answer.
This doesn't address the etymology, but the underlying concern of the OP as to whether "average" can mean median.
There are times when "average" is used to refer to medians. It's common to say "The average American household earns 50,000 per year," and intend that the median household income is 50,000. 
Compare, for example, the headline and data presented here.  The headline:

Here’s how much the average American earns at every age

and the data presented:

Below, check out the median earnings for Americans at every age
  bracket, according to data from the Bureau of Labor Statistics for the
  second quarter of 2017.

(Also later in the article, use of "average man".)
As a different example from a statistical source, Grissom and Kim, 2012, Effect Sizes for Research, 2nd ed. The first sentence of chapter 2:

Recall that the location of a population is a parameter that is
  usually defined as a measure of its "average," commonly its mean or
  median.

To be clear, I think that "average" usually indicates the mean. But to the original question, it's possible the speaker was using average in a way that implies a median value. 
As a final note, I wonder if when we say, "man of average height" if we are thinking of the median and not the mean. Probably neither per se. But I could see how speaking this way, I might use the median as the criterion.
A: Average had for at least two centuries since its induction in European languages implied damage to shipping goods, Words with similar roots are noted among shipping peoples (English, Dutch) and other traders in the Southern Mediterranean (Italian, French). The root awar- is from Arabic meaning damaged. 
King Louis the XIV reissued and enforced a set of pan-European maritime laws after nearly a millenia of lawlessness. This was in 1681. The Maritime Law of General Average states that damage to shipping goods should result in a proportionate share of losses to all parties: receiver and shipper and among the shipper all the crew.
It seems like no coincidence that the usage of average to imply a mean in a general sense is attributed only 50 years later to another French writer, I guess it would be De Moivre. 
Since the Maritime Law refers to a sum being divided with equal weights, if there is any bearing on the conversation, I think we can probably say it is the arithmetic mean.
A: The word average came into English from Middle French avarie, a derivative of an Arabic word meaning “damaged merchandise.” Avarie originally meant damage sustained by a ship or its cargo, but came to mean the expenses of such damage.
