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Book V. that EB is of FD: But AE is the same multiple of CF that

AB is of CD; therefore EB is the same multiple of FD, that
AB is of CD. Therefore, if one 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 will be either equal to these others or equimultiples of them.

Let the two magnitudes AB, CD be equimultiples of the
two E, F, and let AG, CH, equimultiples of E, F, be taken
from AB, CD; the remainders GB, HD will be either equal
to E, F, or equimultiples of them.
First, Let GB be equal to E, then will

K
HD be equal to F: Make CK equal to A
F; and because AG is the same multiple
of E, that CH is of F, and GB is equal

C
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 multi- H
ple of E that CD is of F; therefore, KH
is the same multiple of F, that CD is of F;

B D E F a l. Ax. 5. wherefore, KH is equal to CD a : 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.

Next, Let GB be a multiple of E; then will HD be the same multiple of F: Make CK the same multiple of F, that GB is of E: And K because AG is the same multiple of E, that CH is of F; and GB the same mul- A tiple of E, that CK is of F; therefore, AB is the same multiple of E, that KH is

C
of Fb; but AB is the same multiple of
b 2. 5.

E, that CD is of F; therefore, KH is G
the same multiple of F that CD is of it; H
wherefore, KH is equal to CDa. Take
away CH from both; therefore, the re-
mainder KC is equal to the remainder
HD: And because GB is the same mul-

B D E F tiple of E, that KC is of F, and KC is equal to HD; therefore, HD is the same multiple of F, that GB is of Ed: “ If, therefore, two magnitudes," &c.

Q. E. D.

PROP. A. THEOR.

Book v.

If the first of four magnitudes has to the second the Sec N. 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. 5. 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.

If four magnitudes are proportionals, they are see N. 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 of A and Cany 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 equimulti- G A B E ples; and because G is less than E, H is also CDF i less than F; that is, F is greater than H;

a 5, Def. 5. if, therefore, E be greater that 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; therefore, as B

Book v. is to A, so is D to C.

J Q. E. D.

If, then, four magnitudes,&c.

PROP. C. THEOR.

See N.

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If the first of four magnitudes 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; then A is to B as C is to D.

For, of A and C take any equimultiples

whatever E and F; and of B and D any equia Hyp. multiples whatever G and H: Then, because *

A is the same multiple of B that C is of D; b Conste and

b E is the same multiple of A, that Fis A B C D

of C; E is also the same multiple of B, that E G F II c 3. 5. F is of DC; therefore E and F are the same

multiples of B and D: But G and H are equi-
multiples of B and D: therefore, if E be a
greater multiple of B than G is, F is a great-
er 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
than G, F is equal to H, or less than H. But
E, F are any equimultiples whatever of A, C,

and G, H any equimultiples whatever of B, D. d 5. def. 5. Therefore, A is to B as C is to D d.

Next, Let the first A be the same part of the second B, that the third C is of the fourth D; then 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, e B. 5. B is to A, as D is to C; and inversely

A is to B as C is to D. Therefore, if
the first be the same multiple,&c. Q. E. D. A B C D

e

Book V.

PROP. D. THEOR.

If the first be to the second as the third to the fourth, See N. 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: First, Let A be a multiple of B; then will be 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, the equimultiples E and F have been taken; A is to E, as C to Fa: But A is

a cor. 4. 5. equal to E, therefore C is equal to Fb: And

b A. 5. F is the same multiple of D, that A is of B. Wherefore, C is the same multiple of D, A B C D that A is of B.

E Next, Let the first A be a part of the

F See the

figure at second B; then will С the third be the same

the foot of part of the fourth D.

the precedBecause A is to B, as C is to D; then, inversely, B is c to A, as D to C: But A is

c B. 5. 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,” &c. Q. E. D.

ing page.

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 magnitude. 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

Book V. of C any multiple whatever F: Then, because D is the same

multiple of A, that E is of B, and A is equal a l. Ax. 5. to B; D is also a equal to E: Therefore, if

D be greater than F, E is also greater than
F; and if equal, equal; if less, less : And
D, E are any equimultiples whatever of A,

B, and F is any multiple whatever of C. b 5. def. 5. Therefore, as A is to C, so is B to C.

D A
Likewise, C has the same ratio to A, that
it has to B: For, having made the same con- E B
struction, D may, in like manner, be shown

CF
to be equal to E: Therefore, if F be greater
than D, it is also 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
Cis to A, as C is to Bb. Therefore, “equal
magnitudes,&c. Q. E. D.

PROP. VIII. 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.

Let AB, BC be unequal magnitudes, of which AB is the
greater, and let D be any magnitude what-
ever; then AB has a greater ratio to D Fig. 1.
than BC to D: And D has a greater ratio E
to BC than to AB.

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

с
is not the greater of the two AC, CB,
be less than D, (as in Fig. 2. and 3,) this

GB
magnitude can be multiplied, so as to
become greater than D, whether it be L K HD
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

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