the means and extremes

When a proportion is written in the colon form a : b = c : d, the four numbers have traditional names borrowed from how they sit on the page. The two on the outside, a and d, are the extremes (the ends); the two in the middle, b and c, are the means. So in 2 : 3 = 8 : 12, the extremes are 2 and 12, and the means are 3 and 8.

The key fact, the means-extremes property, is that in any true proportion the product of the means equals the product of the extremes: b·c = a·d. This is exactly the cross-multiplication rule wearing classical clothes. In 2 : 3 = 8 : 12 the means give 3·8 = 24 and the extremes give 2·12 = 24 — equal, so it is a genuine proportion. Running it backwards, if you ever find that the product of the means equals the product of the extremes, the four numbers must form a proportion.

These names earn their keep in a special case. When the two means are the same number — a : x = x : d — that repeated middle term x is called the mean proportional, or geometric mean, of a and d, and the property tells us x^2 = a·d. This single idea drives the altitude-on-hypotenuse relations in a right triangle and reappears throughout geometry whenever one length is 'in proportion between' two others.