third shall have the same ratio to any equimultiples whatever of the second and fourth.
Let A the first have the same ratio to B the second, that C the third has to D the fourth ; and of A and C let there be taken any equimultiples whatever E and F: E shall be to B as F is to D.
Take of E and F any equimultiples whatever K and L, and of B and D any equimultiples whatever G and H.
Then it may be shewn, as before, that K is the same multiple of A that L is of C.
And because A is to B as C is to D, [Hypothesis.
and of A and C have been taken certain equimultiples K and L, and of B and D have been taken certain equimultiples G and H;
therefore if K be greater than G, L is greater than H; and if equal, equal ; and if less, less. [V. Definition 5.
But K and L are any equimultiples whatever of E and F. and G and H are any equimultiples whatever of B and D,
therefore E is to B as F is to D. [V. Definition 5.
In the same way the other case may be demonstrated.
PROPOSITION 5. THEOREM.
If one magnitude he the same multiple of another that a magnitude taken from the first is of a magnitude taken from the other, the remainder shall be the same multiple of the remainder that the whole is of the whole.
Let AB be the same multiple of CD, that AE taken from the first, is of CF taken from the other : the remainder EB shall be the same multiple of the remainder FD, that the whole AB is of the whole CD.
Take AG the same multiple of FD, that AEis of CF;
therefore AE is the same multiple of CF that EG is of CD. [V. 1.
But AE is the same multiple of CF that AB is of CD;
therefore EG is the same multiple of CD that AB is of CD;
therefore EG is equal to AB. [V. Axiom 1.
