where ad and bc denote (as in § 20), the areas of the rectangles contained by a and d and by b and c respectively.
This allows us to transform every proportion between four lines into an equation between two products.
It shows further that the operation of forming a product of two lines, and the operation of forming their ratio are each the inverse of the other.
If we now define a quotient a/b of two lines as the number which multiplied into b gives a, so that
| a | b = a, |
| b |
we see that from the equality of two quotients
| a | = | c |
| b | d |
follows, if we multiply both sides by bd,
| a | b·d = | c | d·b, |
| b | d |