multiples M, N; if therefore K be greater than M, L is greater than N; and if equal, equal; if less, less (5 Def. v.). And K, L are any equimultiples (Constr.) whatever of E, F; and M, N any whatever of G, H as therefore E is to G, so is (5 Def. v.) F to H. Therefore, if the first, etc. 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 have the same ratio to the second and fourth and in like manner, the first and the third 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 C has to the fourth D, and of A and C let E and F be any equimultiples whatever; then E is to B, as F to D.

Take of E, F, any 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 (Hyp.) A is to B, as C is to D, 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 (5 Def. v.).

And K, L are any (Constr.) equimultiples of E, F, and G, H any whatever of B, D; as therefore E is to B (5 Def. v.), so is F to D. And in the same way the other case is demonstrated.

PROPOSITION V.

THEOR. 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 (1. v.) the same multiple of CF, that EG is of CD:

G

C

-A

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 B AB is of CD; wherefore EG is equal to AB (1 Ax. v.):

E

Take from them the common magnitude AE, and the remainder AG s equal to the remainder EB.

Wherefore, since AE is the same multiple of CF, that AG is of FD, and that AG is equal to EB, therefore AE is the same multiple of CF, that EB is of FD: but AE is the same multiple of CF (Hyp.) that AB is of CD; therefore EB is the same multiple of FD, that AB is of CD. Therefore, if one magnitude, etc.

Q. E. D.

PROPOSITION VI.

THEOR. 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 be equimultiples of the two E, F, and AG, CH taken from the first two be equimultiples of the same E, F; the remainders GB, HD are either equal to E, F, or equimultiples

of them.

First, let GB be equal to E; HD is F qeut. to F:

Make CK equal

And because AG is the same multiple of E, that CH is of F, and that GB is equal to E, and CK to F; there- A fore 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 (1 Ax. v.):

B

K

D

Take away the common magnitude CH, then the remainder KC is equal to the remainder HD: but KC is equal (Constr.) to F; HD therefore is equal to F.

But let GB be a multiple of E; then HD is 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; and GB the same multiple of E, that CK is of F; JA therefore AB is the same multiple of E (Hyp.), that KH is of F (2. v.):

But AB is the same multiple of E (Hyp.), that CD is of F; therefore KH is the same multiple of F, that CD is of B it; wherefore KH is equal (1 Ax. v.) to CD:

-G

H

F

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, and that KC is equal to HD; therefore HD is the same multiple of F, that GB is of E. If, therefore, two magnitudes, etc. Q. E. D.

PROPOSITION A.

THEOR. If the first of four magnitudes has to the second the same ratio 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, etc. Q. E. D.

PROPOSITION B.

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

If the magnitude A be to B, as C is to D, then also inversely B is to A, as D to C.

Take of B and D, any equimultiples whatever E and F; and C, any equimultiples whatever G and H.

First, let E be greater than G, then G is less than E: and because (Hyp.) A is to B as C is to D; 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, H is also (5 Def. v.) less than F; that is, F is greater than H; if therefore E be greater than G, F is greater than H:

In like manner, if E be equal to G, F may be shown to be equal to H; and if less, less; and E, F are any equimultiples (Constr.) whatever of B and D, and G, H any whatever of A and C; therefore (5 Def. v.), as B is to A, so is D to C. If, then, four magnitudes, etc. Q. E. D.

PROPOSITION C.

H

and of A

ABE

C D F

THEOR. 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 B the second, that C the third is of the fourth D: A is to B as C is to 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 (Hyp.) multiple of B that C is of D; and that E is the same (Constr.) multiple of A that F is of C; E is the same multiple of B that F is of D (3. v.), therefore E and F are the same multiples of B and D:

But G and H are equimultiples (Constr.) of B and D ; therefore, if E be a greater multiple of B than G is, F is a greater multiple of D than H is of D; that is, if E be greater than G, F is greater than H:

In like manner, if E be equal to G, or less, F is equal to H, or less than it :

But E, F are any equimultiples (Constr.) whatever of A, C; and G, H any equimultiples whatever of B, D; therefore A is to B, as C is to D (5 Def. v.).

A

E

Next, let the first A be the same part of the second B, that the third C is of the fourth D: A is to B, as C is to D:

For 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 inversely (B. v.) A is to B, as C is to D. Therefore, if the first be the same multiple, etc. Q. E. D.

A B C

PROPOSITION D.

THEOR. 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 C is to D; and first, let A be a multiple of B; C is the same multiple of D.

Take E equal to A; and whatever multiple A or E is of B, make F the same multiple of D:

Then, because (Hyp.) A is to B, as C is to D; and of B the second, and D the fourth, equimultiples have been taken, E and F; A is to E, as C to F (Cor. 4. v.):

But A is equal (Constr.) to E, therefore C is equal to F (A. v.).

And Fis the same (Constr.) multiple of D, that A is of B. Wherefore C is the same multiple of D, that A is of B.

Next, let the first A be a part of the second B; C the third is the same part of the fourth D-(Sec figure at bottom of Proposition C.)

Because A is to B, as C is to D; then inversely, B is (B. v.) to A, as D to C: but A is a part of B, therefore

B is a multiple of A; and, by the preceding case, D is the same multiple of C; that is, C is the same part of D, that A is of B. Therefore, if the first, etc. Q. E. D.

PROPOSITION VII.

THEOR. Equal magnitudes have the same ratio to the same magnitude; and the same has the same ratio to equal magnitudes.

Let A and B be equal magnitudes, and C any other: A and B have each of them the same ratio to C, and C has the same ratio to each of the magnitudes A and B.

Take of A and B any equimultiples whatever D and E, and of C any multiple whatever F:

Then, because D is the same (Constr.) multiple of A,

that E is of B, and that A is equal (Hyp.) to B; D is (1 Ax. v.) equal to E;

Therefore, if D be greater than F, E is greater than F; and if equal, equal; if less, less;

And D, E are any equimultiples of A, B (Constr.), and F is any multiple of C. Therefore (5 Def. v.) as Á is to C, so is B to C.

Likewise C has the same ratio to A that it has to B.

For, having made the same construction, D may in like manner be shown equal to E: therefore, if F be greater than D, it is likewise greater than E; and if equal, equal; if less, less:

And F is any multiple whatever of C; and D, E are any equimultiples whatever of A, B; therefore, C is to A as C is to B (5 Def. v.). Therefore, equal magnitudes, etc. Q. E. D.

PROPOSITION VIII. 1

THEOR. Of unequal magnitudes the greater has a greater ratio to the same than the less has; and the same magnitude has a greater ratio to the less than it has to the greater.

Let AB, BC be unequal magnitudes, of which AB is the greater, and let D be any magnitude whatever: AB has a greater ratio to D than BC to D; and D has a greater ratio to BC than unto AB.

FIG./

If the magnitude which is not the greater of the two AC, CB, be not less than D, take EF, FG, the doubles of AC, CB, as in El fig. 1. But if that which is not the greater of the two AC, CB, be less than D (as in figs. 2 and 3), this magnitude can be multiplied, so as to become greater than D, whether it be AC or CB. Let it be multiplied until it become greater than D, and let the other be multiplied as often; and let EF be the multiple thus taken of AC, and FG the same multiple of CB; therefore EF and FG are each of them greater than D: and in every one of the cases, take H the double of D, K its triple, and so on, till the multiple of D be that which first becomes greater than FG let L be that multiple of D which is first greater than FG, and K the multiple of D which is next less than L.

го

H

Then, because L is the multiple of D, which is the first that becomes greater than FG, the next preceding multiple K is not greater than FG; that is, FG is not less than K:

And since EF is the same multiple of AC, that FG is of CB; FG is the same multiple of CB, that EG is of AB (1. v.); wherefore EG and FG are equimultiples of AB and CB:

And it was shown that FG was not less than K, and by the construction, EF is greater than D; therefore the whole EG is greater than K and DE together:

But K together with D is equal (Constr.) to L; therefore EG is greater than L: but FG is not greater than L; and EG, FG are equimultiples of AB, BC, and L is a (Constr.) multiple of D; therefore (7 Def. v.) AB has to D a greater ratio than BC has to D.

Also, D has to BC a greater ratio than it has to AB.

For, having made the same construction, it may be shown, in like manner, that L is

FIC.3

FIG. 2

A

F

A

C

C

B

C

L

KHD

K D

greater than FG, but that it is not greater than EG: and L is a multiple of D; and FG, EG are equimultiples of CB, AB; therefore D has to CB a greater ratio (7 Def. v.) than it has to AB. Wherefore, of unequal magnitudes, etc. Q. E. D.