Proposition 8

Let the number AB be the same parts of the number CD that AE subtracted is of CF subtracted.

I say that the remainder EB is also the same parts of the remainder FD that the whole AB is of the whole CD.

Make GH equal to AB.

Therefore AE is the same parts of CF that GH is of CD.

Divide GH into the parts of CD, namely GK and KH, and divide AE into the parts of CF, namely AL and LE. Then the multitude of GK and KH equals the multitude of AL and LE.

Now since AL is the same part of CF that GK is of CD, and CD is greater than CF, therefore GK is also greater than AL.

Make GM equal to AL.

Then GK is the same part of CD that GM is of CF. Therefore the remainder MK is the same part of the remainder FD that the whole GK is of the whole CD.

Again, since EL is the same part of CF that KH is of CD, and CD is greater than CF, therefore HK is also greater than EL.

Make KN equal to EL.

Therefore KN is the same part of CF that KH is of CD. Therefore the remainder NH is the same part of the remainder FD that the whole KH is of the whole CD.

But the remainder MK was proved to be the same part of the remainder FD that the whole GK is of the whole CD, therefore the sum of MK and NH is the same parts of DF that the whole HG is of the whole CD.

But the sum of MK and NH equals EB, and HG equals BA, therefore the remainder EB is the same parts of the remainder FD that the whole AB is of the whole CD.

Therefore, if a number is the same parts of a number that a subtracted number is of a subtracted number, then the remainder is also the same parts of the remainder that the whole is of the whole.