But since a divided by b gives x, x multiplied by b will give a, or a = bx. For a similar reason, c = dx. Put bx and dx instead of a and c in the four expressions just given, recollecting that when quantities are multiplied together, the result is the same in whatever order the multiplications are made; that, for example, bxbxbx is the same as bbbxxx.
Hence, 2aaa + 3aab = 2bxbxbx + 3bxbxb
= 2bbbxxx + 3bbbxx
which is bbb multiplied by 2xxx + 3xx
or bbb (2xxx + 3xx)[29]
Similarly, 2ccc + 3ccd = ddd (2xxx + 3xx)
Also, bbb + abb = bbb + bxbb
= bbb multiplied by 1 + x
or bbb(1 + x)
Similarly, ddd + cdd = ddd (1 + x)
Now, bbb : bbb ∷ ddd : ddd