Means and Extremes of a Proportion

Date: 09/16/2004 at 17:27:43
From: Kathy
Subject: means and extremes

In the proportion a/b = c/d, why are the a and the d called the
extremes and the b and c called the means of the proportion?

Date: 09/16/2004 at 18:09:31
From: Doctor Schwa
Subject: Re: means and extremes

Hi Kathy -

In the old days, instead of writing 1/2 = 3/6, people would write 
1:2 = 3:6 and conventionally they would be written in that order so
that the "middle" numbers, the means, are actually in the middle, and 
the "extreme" numbers are the ones way out on the ends.

At least, that's as much as I know about it!

Then I tried the Oxford English Dictionary, and found:

  1570 BILLINGSLEY Euclid 153b, A right line is sayd to be deuided by
  an extreme and meane proportion, when the whole is to the greater
  part, as the greater part is to the lesser.

That is, originally "extremes" were the whole thing and the small 
piece, while the "means" were two EQUAL numbers in the middle of the
proportion, like 9:6 = 6:4.  Now it makes sense: in fact, 6 is called
the "geometric mean" in that case, and 9 and 4 are reasonable
"extremes" around the "mean" of 6.

I guess it was later that the same terms were used when the middle 
parts were unequal!

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/