hence

a/b = a/b × e/f
c/dc/d × e/f

which corresponds to the second formula[17] in (124). In a similar manner it may be shewn, that the other formulæ of the same article are true when the letters there used either represent fractions, or are removed and fractions introduced in their place. All formulæ established throughout this work are equally true when fractions are substituted for whole numbers. For example (54), (m + n)a = ma + na. Let m, n, and a be respectively the fractions

p , r , and b
q s c

Then m + n is

p + r, or ps + qr
q s qs

and (m + n)a is

ps + qr × b, or (ps + qr)b
qs c qsc
or psb + qrb .
qsc
But this (112) is psb + qrb, which is pb + rb ,
qscqscqcsc
since psb = pb , andqrb = rb (103).
qscqcqscsc