Book V.

See N.

a 3. 5.

PROP. IV. THEOR.

F the first of four magnitudes has the fame ratio to the fecond which the third has to the fourth; then any equimultiples whatever of the firft and third fhall have the fame ratio to any equimultiples of the fecond and fourth, viz. the equimultiple of the firft shall have the fame ratio to that of the fecond, which the equimultiple of the third has to that of the fourth.'

Let A the firft, have to B the second, the fame ratio which the third C has to the fourth D; and of A and C let there be taken any equimultiples whatever

E, F; and of B and D any equi-
multiples whatever G, H: Then
E has the fame ratio to G, which
F has to H.

LF

Take of E and F any equimultiples whatever K, L, and of G, H, any equimultiples whatever M, N: Then, because E is the fame multiple of A, that F is of C; and of E and F have been taken equimultiples K, L; therefore K is the fame multiple of A, that L is of C: For the fame reafon, M KE is the fame multiple of B, that N is of D: And becaufe, as A is to b Hypoth. B, fo is C to Db, and of A and C have been taken certain equimultiples K, L; and of B and D have been taken certain equimultiples M, N; if therefore K be greater than M, L is greater than Ñ; and if equal, equal; if lefs, cs. def. 5. lefs. And K, L are any equimultiples whatever of E, F; and M, N any whatever of G, H: As therefore E is to G, fo is F. to H. Therefore, if the firft, &c. Q. E. D.

с

ABG M

C D H N

COR. Likewife, if the first has the fame ratio to the second, which the third has to the fourth, then alfo any equimulti

ples

ples whatever of the firft and third have the fame ratio to the fecond and fourth: And in like manner, the first and the third have the fame ratio to any equimultiples whatever of the fecond and fourth.

Let A the firft, have to B the fecond, the fame 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.

Book V.

Take of E, F any equimultiples whatever K, L, and of B, D any equimultiples whatever G, H; then it may be demonftrated, as before, that K is the fame multiple of A, that L is of C And becaufe A is to B, as C is te 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 lefs, lefs: s. def. 5. And, K, L are any equimultiples of E, F, and G, H any whatever of B, D; as therefore E is to B, fo is F to D: And in the fame way the other cafe is demonftrated.

IF

one magnitude be the fame multiple of another, See N. which a magnitude taken from the firft is of a magnitude taken from the other; the remainder fhall be the fame multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the fame multiple of CD, that AE taken from the firft, is of CF taken from the other; the remainder EB fhall be the fame multiple of the remainder FD, that the whole AB is of the whole CD.

G

A

Take AG the fame multiple of FD, that AE is of CF: Therefore AE is the fame multiple of CF, that EG is of CD: But AE, by the hypothefis, is the fame multiple of CF, that AB is of CD: Therefore EG is the fame mul- E tiple of CD that AB is of CD; wherefore EG is equal to AB. Take from them the common magnitude AE; the remainder AG is equal to the remainder EB. Wherefore, fince AE is the fame multiple of CF, that AG is of FD, and that AG is equal to EB; therefore AE is the fame multiple of CF, that EB is of FD; But AE is the fame multiple of CF,

a I. 5.

F

b I. Ax. Si

B D

that

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

IF

PROP. VI. THEOR.

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 fame E, F, the remainders GB, HD are either equal to E, F, or equimultiples of them.

A K
АК
C

Firft, Let GB be equal to E; HD is e-
qual to F: Make CK equal to F; and be
caufe AG is the fame multiple of E, that
CH is of F, and that GB is equal to E,
and CK to F; therefore AB is the fame
multiple of E, that KH is of F. But AB,
by the hypothefis, is the fame multiple of
E that CD is of F; therefore KH is the G
fame multiple of F, that CD is of F;

H

B

DEF

■ 1. Ax. 5. wherefore KH is equal to CD: Take a-
way 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.

B 2.5.

G

Ki

A

C

H

But let GB be a multiple of E; then HD is the fame multiple of F: Make CK the fame inultiple of F, that GB is of E: And becaufe AG is the fame multiple of E, that CH is of F; and GB the fame multiple of E, that CK is of F; therefore AB is the fame multiple of E, that KH is of Fb: But AB is the fame multiple of E, that CD is of F; therefore KH is the fame multiple of F, that CD is of it; wherefore KH is equal to CD): Take away CH from both; therefore the remainder KC is equal to the remainder B DEF HD: And becaufe GB is the fame multiple of E, that KC is of F, and that KC is equal to HD; therefore HD is the fame multiple of F, that GB is of E: If therefore two magnitudes, &c. Q. E. D.

PROP.

PROP. A.

THEOR.

Book V.

IF

F the first of four magnitudes has to the fecond, the see N. fame ratio which the third has to the fourth; then, if the first be greater than the fecond, the third is alfo greater than the fourth; and, if equal, equal; if lefs, lefs.

Take any equimultiples of each of them, as the doubles of each; then, by def. 5th of this book, if the double of the firft be greater than the double of the fecond, the double of the third is greater than the double of the fourth; but, if the first be greater than the fecond, the double of the firft is greater than the double of the fecond; 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 fecond, 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 propor- See N. tionals allo when taken inversely.

If the magnitude A be to B, as C is to D, then alfo 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 É 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 firft and third, G and H are equimultiples; and of B and D, the fecond and fourth, E and F are equimultiples; and that G is lefs than E, H is alfo lefs 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 lefs, lefs; and E, F are any equi multiples whatever of B and D, and G, H any whatever of A and C; therefore, as B

I

G A BE

a

HC D F 5. def. 5.

is

1

Book V. is to A, fo is D to C. If, then, four magnitudes, &c. Q. E. D.

See N.

a 3. 5.

PROP. C. THEOR.

IF the first be the fame multiple of the fecond, or the fame part of it, that the third is of the fourth; the firft is to the fecond, as the third is to the fourth.

Let the first A, be the fame multiple of B the fecond, 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 fame multiple of B that C is of D; and that E is the fame multiple of A, that Fis of C; E is the fame multiple of B, that Fis of D; therefore E and F are the fame 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 lefs; F is equal to H, or lefs than it. But E, F are equimultiples, any whatever, of A, C, and G, H any equimultiples whatever of B, 5. def. 5. D. Therefore A is to B, as C is to D.

c B. 5.

Next, Let the first A be the fame part of the fecond B, that the third C is of the fourth D: A is to B, as C is to D: For B is the fame multiple of A, that D is of C; wherefore, by the preceding cafe, B is to A, as D is to C; and inverfely A is to B, as C is to D. Therefore, if the first be the fame multiple, &c. Q. E. D.

A B C D
EGF H

A B C D

PROP.