## Proposition 4

 If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. Let a first magnitude A have to a second B the same ratio as a third C to a fourth D, and let equimultiples E and F be taken of A and C, and G and H other, arbitrary, equimultiples of B and D. I say that E is to G as F is to H. Take equimultiples K and L of E and F, and other, arbitrary, equimultiples M and N of G and H. Since E is the same multiple of A that F is of C, and equimultiples K and L of E and F have been taken, therefore K is the same multiple of A that L is of C. For the same reason M is the same multiple of B that N is of D. V.3 And, since A is to B as C is to D, and equimultiples K and L have been taken of A and C, and other, arbitrary, equimultiples M and N of B and D, therefore, if K is in excess of M, then L is in excess of N; if it is equal, equal; and if less, less. V.Def.5 And K and L are equimultiples of E and F, and M and N are other, arbitrary, equimultiples of G and H, therefore E is to G as F is to H. V.Def.5 Therefore, if a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order. Q.E.D.

Note how Euclid uses the definition to prove that the two ratios pa:qb and pc:qd are the same. (Here, a and b are magnitudes of one kind, and c and d are magnitudes of another kind, but p and q are numbers.) We are given a:b = c:d. That means for any numbers m and n that

if ma >=< nb, then mc >=< nd.

We have to prove that pa:qb = pc:qd for any numbers p and q. That means, we have to prove that for any m and n,

if mpa >=< nqb, then mpc >=< nqd.

But that's just a special case of the given relation

if ma >=< nb, then mc >=< nd.

#### Use of this proposition

Proposition V.4 is used in the proof of one other proposition, namely, V.22.

Next proposition: V.5

Previous: V.3

 Select from Book V Book V intro V.Def.1-2 V.Def.3 V.Def.4 V.Def.5-6 V.Def.7 V.Def.8-10 V.Def.11-13 V.Def.14-16 V.Def.17-18 V.1 V.2 V.3 V.4 V.5 V.6 V.7 V.8 V.9 V.10 V.11 V.12 V.13 V.14 V.15 V.16 V.17 V.18 V.19 V.20 V.21 V.22 V.23 V.24 V.25 Select book Book I Book II Book III Book IV Book V Book VI Book VII Book VIII Book IX Book X Book XI Book XII Book XIII Select topic Introduction Table of Contents Geometry applet About the text Euclid Web references A quick trip