Encyclopaedia Britannica, 11th Edition, "Geodesy" to "Geometry" / Volume 11, Slice 6

by Various

c : a < c : b.

The proof depends on Definition 7.

Prop. 9 (converse to Prop. 7). If

a : c :: b : c,

or if

c : a :: c : b, then a = b.

Prop. 10 (converse to Prop. 8). If

a : c > b : c, then a > b,

and if

c : a < c : b, then a < b.