a + b : a ∷ c + d : c
a : a - b ∷ c : c - d
a + c : a - c ∷ b + d : b - d.

In these and all others it must be observed, that when such expressions as a-b and c-d occur, it is supposed that a is greater than b, and c greater than d.

186. If four numbers be proportional, and any two dissimilar terms be both multiplied, or both divided by the same quantity, the results are proportional. Thus, if a: b ∷ c: d, and m and n be any two numbers, we have also the following:

ma : b ∷ mc : d
a : mb ∷ c : md
a : mb ∷ c : md
nn
ma : nb ∷ mc : nd
a : b ∷ c : d
mmmm
a : b ∷ c : d
mmnn

and various others. To prove any one of these, recollect that nothing more is necessary to make four numbers proportional except that the product of the extremes should be equal to that of the means. Take the third of those just given; the product of its extremes is

a × md, or mbc ,
nn

while that of the means is

mb × c , or mad .
nn

But since a : b ∷ c : d, by (181) ad = bc,

whence, by (180), mad = mbc, and mad = mbc .
nn