Conversely, an equation between two fractions can be put into a proportion. By means of these simple principles all the various properties of proportion can be proved in the most direct and easy manner.
3. If we take the proportion a : b :: c : d, and multiply the first and third terms, each by m, and second and fourth, each by n, we get the four multiples, ma, nb, mc, nd; and we want to prove that if ma is greater than nb, mc is greater than nd; if equal, equal; and if less, less.
| Dem.—Since | a : b | :: c : d, | |||||||||
| we have |  | =  . | |||||||||
Hence, multiplying each by  we get | |||||||||||
 | =  . | ||||||||||
Now, it is evident that if
is greater than unity,
is greater than unity; but if
is greater than unity, ma is greater than nb; and if