and because A is the same multiple of B, that Cis of D,
and that EK is equal to A, and GL equal to C:

therefore EK is the same multiple of B, that GL is of D: for the same reason, KF is the same multiple of B, that LH is of D: and so, if there be more parts in EF, GH, equal to A, C: therefore, because the first EK is the same multiple of the second B, which the third GL is of the fourth D,

and that the fifth KF is the same multiple of the second B, which the sixth LH is of the fourth D;

EF the first, together with the fifth, is the same multiple of the second * B, (v. 2.)

which GH the third, together with the sixth, is of the fourth D.

If, therefore, the first, &c. Q. E.D.

PROPOSITION IV. THEOREM.

If the first of four magnitudes has the same ratio to the second which the third has to the fourth; then any equimultiples whatever of the first and third shall have the same ratio to any equimultiples of the second and fourth, viz, 'the equimultiple of the first shall have the same ratio to that of the second, which the equimultiple of the third has to that of the fourth.'

Let A the first have to B the second, the same ratio which the third C has to the fourth D;

and of A and Clet there be taken any equimultiples whatever E, F; and of B and D any equimultiples whatever G, H.

Then E shall have the same ratio to G, which F has to H.

Take of E and F any equimultiples whatever K, L, and of G, H any equimultiples whatever M, N: then because E is the same multiple of A, that Fis of C; and of E and F have been taken equimultiples K, L; therefore K is the same multiple of A, that L is of C: (v. 3.) for the same reason, M is the same multiple of B, that N is of D. And because, as A is to B, so is C to D, (hyp.)

and of A and C have been taken certain equimultiples K, L,
and of B and D have been taken certain equimultiples M, N;
therefore if K be greater than M, L is greater than N;
and if equal, equal; if less, less: (v. def. 5.)

but K, L are any equimultiples whatever of E, F, (constr.)
and M, N any whatever of G, H;

therefore as E is to G, so is F to H. (v. def. 5.)
Therefore, if the first, &c. Q. E. D.

COR. Likewise, if the first has the same ratio to the second, which the third has to the fourth, then also any equimultiples whatever of

the first and third shall have the same ratio to the second and fourth; and in like manner, the first and the third shall have the same ratio to any equimultiples whatever of the second and fourth.

Let A the first have to B the second the same ratio which the third Chas to the fourth D.

and of A and Clet E and F be any equimultiples whatever.
Then E shall be to B as F to D.

Take of E, Fany equimultiples whatever, K, L,
and of B, D any equimultiples whatever G, H:

then it may

be demonstrated, as before, that K is the same multiple of A, that L is of C:

and because A is to B, as Cis to D, (hyp.)

and of A and C certain equimultiples have been taken viz., K and L; and of B and D certain equimultiples G, H;

therefore, if K be greater than G, L is greater than H;
and if equal, equal; if less, less: (v. def. 5.)

but K, L are any equimultiples whatever of E, F, (constr.)
and G, H any whatever of B, D;

therefore as E is to B, so is F to D. (v. def. 5.)
And in the same way the other case is demonstrated.

PROPOSITION V. THEOREM.

If one magnitude be the same multiple of another, which 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 the magnitude 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 AE is of CF: therefore AE is the same multiple of CF, that EG is of CD: (v. 1.) but AE, by the hypothesis, is the same multiple of CF, that AB is of CD; therefore EG is the same multiple of CD that AB is of CD; wherefore EG is equal to AB: (v. ax. 1.)

take from each of them the common magnitude AE; and the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF, that AG is of FD, (constr.)

and that AG has been proved equal to EB;

therefore AE is the same multiple of CF, that EB is of FD:
but AE is the same multiple of CF that AB is of CD: (hyp.)
therefore EB is the same multiple of FD, that AB is of CD.
Therefore, if one magnitude, &c. Q.E.D.

PROPOSITION VI. THEOREM.

If two magnitudes be equimultiples of two others, and if equimultiples of these be taken from the first two; the remainders are either equal to these others, or equimultiples of them.

Let the two magnitudes AB, CD he equimultiples of the two E, F, and let AG, CH taken from the first two be equimultiples of the same E, F.

Then the remainders GB, HD shall be either equal to E, F, or equimultiples of them.

First, let GB be equal to E:
HD shall be equal to F.

Make CK equal to F:

and because AG is the same multiple of E, that CH is of F: (hyp.) and that GB is equal to E, and CK to F;

therefore AB is the same multiple of E, that KH is of F: but AB, by the hypothesis, is the same multiple of E, that CD is of F; therefore KH is the same multiple of F, that CD is of F: wherefore KH is equal to CD: (v. ax. 1.)

take away the common magnitude CH,

then the remainder KC is equal to the remainder HD:
but KC is equal to F: (constr.)
therefore HD is equal to F.

Next let GB be a multiple of E.

Then HD shall be the same multiple of F.

Make CK the same multiple of F, that GB is of E: and because AG is the same multiple of E, that CH is of F:

(hyp.)

and GB the same multiple of E, that CK is of F; therefore AB is the same multiple of E, that KH is of F: (v. 2.) but AB is the same multiple of E, that CD is of F; (hyp.) therefore KH is the same multiple of F, that CD is of F; wherefore KH is equal to CD: (v. ax. 1.) take away CH from both;

therefore the remainder KC is equal to the remainder HD: and because GB is the same multiple of E, that KC is of F, (constr.) and that KC is equal to HD;

therefore HD is the same multiple of F, that GB is of E.
If, therefore, two magnitudes, &c. Q. E.D.

PROPOSITION A. THEOREM.

If the first of four magnitudes has the same ratio to the second, which the third has to the fourth; then, if the first be greater than the second, the third is also greater than the fourth; and if equal, equal; if less, less.

Take any equimultiples of each of them, as the doubles of each: then, by def. 5th of this book, if the double of the first be greater than the double of the second, the double of the third is greater than the double of the fourth:

but if the first be greater than the second,

the double of the first is greater than the double of the second; wherefore also the double of the third is greater than the double of the fourth;

therefore the third is greater than the fourth:

in like manner if the first be equal to the second, or less than it, the third can be proved to be equal to the fourth, or less than it. Therefore, if the first, &c. Q. E.D.

PROPOSITION B. THEOREM.

If four magnitudes are proportionals, they are proportionals also when taken inversely.

A

G

Let A be to B, as Cis to D.

Then also inversely, B shall be to A, as D to C.

Take of B and D any equimultiples whatever E and F;
and of A and C any equimultiples whatever G and H.
First, let E be greater than G, then G is less than E:
and because A is to B, as C is to D, (hyp.)

and of A and C, the first and third, G and H are equimultiples; and of B and D, the second and fourth, E and F are equimultiples; and that G is less than E, therefore H is less than F; (v. def. 5.) that is, Fis greater than H;

if, therefore, E be greater than G,
Fis greater than H;

in like manner, if E be equal to G,
F may be shewn to be equal to H;
and if less, less;

but E, F, are any equimultiples whatever of B and D, (constr.)
and G, H any whatever of A and C;

therefore, as B is to A, so is D to C. (v. def. 5.)
Therefore, if four magnitudes, &c, Q. E. D.

PROPOSITION C. THEOREM.

If the first be the same multiple of the second, or the same part of it, that the third is of the fourth; the first is to the second, as the third is to the fourth.

Let the first A be the same multiple of the second B,

that the third C is of the fourth D.

Take of A and C any equimultiples whatever E and F; and of B and D any equimultiples whatever G and H. Then, because A is the same multiple of B that Cis of D; (hyp.) and that E is the same multiple of A, that F is of C; (constr.) therefore E is the same multiple of B, that Fis of D; (v. 3.) that is, E and F are equimultiples of B and D: but G and H are equimultiples of B and D; (constr.) therefore, if E be a greater multiple of B than G is of B, Fis a greater multiple of D than H is of D; that is, if E be greater than G,

Fis greater than H:

in like manner, if E be equal to G, or less than it,
F may be shewn to be equal to H, or less than it,
but E, F are equimultiples, any whatever, of A, C; (constr.)
and G, H any equimultiples whatever of B, D;

therefore A is to B, as Cis to D. (v. def. 5.) Next, let the first A be the same part of the second B, that the third Cis of the fourth D.

For since A is the same part of B that C is of D, therefore B is the same multiple of A, that D is of C: wherefore, by the preceding case, B is to A, as D is to C; and therefore inversely, A is to B, as Cis to D. (V. B.) Therefore, if the first be the same multiple, &c. Q. E.D.

If the first be to the second as the third to the fourth, and if the first be a multiple, or a part of the second; the third is the same multiple, or the same part of the fourth.

Let A be to B as Cis to D:
and first, let A be a multiple of B.
Then C shall be the same multiple of D.

B-
E

Take E equal to 4,

D

F

and whatever multiple A or E is of B, make F the same multiple

of D:

then, because A is to B, as C is to D; (hyp.)

and of B the second, and D the fourth, equimultiples have been taken, E and F;

therefore A is to E, as C to F: (v. 4. Cor.)
but A is equal to E, (constr.)

therefore C is equal to F: (V. A.)