Book V. AB is of CD; therefore EB is the same multiple of FD, that AB is of CD. Therefore, if any magnitude, &c. Q. E. D.

PROP. VI. THEOR.

See N. 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.

K

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 equal to F: Make CK equal to F; and A because AG is the same multiple of E, that CH is of F, 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 G multiple of E that CD is of F; therefore KH is the same multiple of F, that CD * 1 Ax. 5. is of F; wherefore KH is equal to CD a:

7

C

H

BDE F

Take away the common magnitude CH, then the remainder KC is equal to the remainder HD: But KC is equal 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 mul-
tiple of E, that CH is of F; and GB the
same multiple of E, that CK is of F;
therefore AB is the same multiple of E,
2. 5. that KH is of Fb: But AB is the same

multiple of E, that CD is of F; there-
fore KH is the same multiple of F, that
CD is of it; Wherefore KH is equal to

A

H

K

CDa: Take away CH from both; there- B DE F fore 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, &c. Q. E. D.

BOOK V.

PROP. A. THEOR.

Ir the first of four magnitudes has to the second See N. 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, &c. Q. E. D.

PROP. B. THEOR.

Ir four magnitudes are proportionals, they are pro- see N. portionals 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 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, 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 equimul- GAB tiples; and that G is less than E, H is

E

also a less than F; that is, F is greater H CDF5 Def. 5. 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 whatever of B and D, and G, H any whatever of A and C :

BOOK V. therefore, as B is to A, so is D to C. If then four magnitudes, &c. Q. E. D.

PROP. C. THEOR.

See N. 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 multiple of B that C

is of D; and that E is the same multiple A B C D of A, that F is of C: E is the same multi- E G FH

3. 5. ple of B, that F is of Da; therefore E and

F are the same multiples of B and D: But G and H are equimultiples 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 equimultiples, any whatever, of A, C, and G, H, any equimultiples whatever of B, D. 5 Def. 5. Therefore A is to B, as C is to D.

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 inB. 5. versely A is to B, as C is to D: There

с

fore, if the first be the same multiple, A B C D

&c. Q. E. D.

BOOK V.

PROP. D. THEOR.

Ir the first be to the second as the third to the Sce N. fourth, and if the first be a multiple or 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 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 Fa: But A is equal to E, therefore C is equal

to Fb: and F is the same multiple of D, A B C
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.

Because A is to B, as C is to D; then inversely, B is 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

a Cor. 4. 5.

DA. 5.

E

F

See the fi

same multiple of C; that is, C is the same part of D, that A is of B: Therefore, if the first, &c. Q. E. D.

gure at the
foot of the
preceding
page.
c B. 5.

PROP. 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 Cany multiple whatever F: Then because D is the

BOOK V. same multiple of A, that E is of B, and that A is equal to B; D is equal to E: 1 Ax. 5. 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, and F is any multiple of C. 5 Def. 5. Therefore, as A is to C, so is B to C.

E B

D
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 Bb. There-
fore, equal magnitudes, &c. Q. E. D.

PROP. VIH. THEOR.

See N. 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.

Fig. 1.

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.

E

If the magnitude which is not the F
greater of the two AC, CB, be not less
than D, take EF, FG, the doubles of
AC, CB, as in Fig. 1. But if that which
is not the greater of the two AC, CB,
be less than D (as in Fig. 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: There-

G

L

C

A.

B

KHD