Book V. of A and C, M and G are equimultiples: And of B and D, N and K are equimultiples; if M be greater than N, G is greater b s. def. s. than K; and if equal, equal; and if less, less; but G is greater than K, therefore M is greater than N: But His not greater than L; and M, H are equimultiples of A, E; and N, L equimultiples of B, F: Therefore A has a greater ratio

a 7. def. 5.0 B, than E has to F2. Wherefore, if the first, &c. QE. D. COR. And if the first has a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the fixth; it may be demonstrated in like manner that the first has a greater ratio to the second than the fifth has to the fixth.

Sọc N.

8.5

PROP. XIV. THEOR.

IF the first has to the second the fame ratio, which the third has to the fourth; then, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; and if less, less.

Let the first A have to the second B the same ratio, which the third C has to the fourth D; if A be greater than C, B is greater than D.

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio than C to B2; But as A is to B, so

b 13.5.

is C to D; therefore also C has to D a greater ratio than C has to Bb: But of two magnitudes, that to which the same has the

C 10. 5.

greater ratio is the lesser: Wherefore D is less than B; that is, B is greater than D.

d 9. 5.

Secondly, If A be equal to C, B is equal to D: For A is to B, as C, that is A, to D; B therefore is equal to Dd.

Thirdly, If A be less than C, B shall be less than D: For C is greater than A, and because C is to D, as A is to B, D is greater than B by the first cafe; wherefore B is less than D.

Therefore, if the first, &c. Q. E. D.

PROP,

M

PROP. XV. THEOR.

AGNITUDES

have the fame ratio to one another

which their equimultiples have.

Let AB be the fame multiple of C, that DE is of F: Cis to F, as AB to DE.

Because AB is the fame multiple of C that DE is of F; there

Book V.

are as many magnitudes in AB equal to C,
as there are in DE equal to F: Let AB be
divided into magnitudes, each equal to C,
viz. AG, GH, HB; and DE into magni-
tudes, each equal to F, viz DK, KL, LE: G
Then the number of the first AG, GH, HB,
fhall be equal to the number of the last DK,
KL, LE: And because AG, GH, HB are H
all equal, and that DK, KL, LE are also
equal to one another; therefore AG is to
DK, as GH to KL, and as HB to LE;
And as one of the antecedents to its confe-

A

D

K

L

BCEF

quent; so are all the antecedents together to

all the confequents together; wherefore, as AG is to DK, so b 12. S. is AB to DE: But AG is equal to C, and DK to F: There-. fore, as C is to F, so is AB to DE. Therefore magnitudes, &c. Q. E. D.

PROP. XVI. THEOR.

IF four magnitudes of the fame kind be proportionals, they shall alfo be proportionals when taken alternate

ly.

Let the four magnitudes A, B, C, D be proportionals, viz. 25 A to B, fo C to D: They shall al'o be proportionals when taken alternately, that is, A is to C, as B to D.

Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H: And

:

Book V. because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another which their equimultiples have; therefore A is to B, as E is to F; But as A is to B, so is C to

a 15. 5.

D: Wherefore, as CE

G

is to D, so is G to FH

H2; but as C is to

C 14. 5.

D, so is E to F. Wherefore, as E is to F, so is G to Hb. But, when four magnitudes are proportionals, if the first be greater than the third, the second shall be greater than the fourth; and if equal, equal; if less, less. Wherefore, if E be greater than G, F likewife is greater than H; and if equal, equal; if less, less: and E, F are any equimultiples whatever of A, B; and, G, H #s. def. 5. any whatever of C, D. Therefore A is to C, as B to Dd. If then four magnitudes, &c. QE. D.

PROP. XVII, THEOR.

IF magnitudes taken jointly be proportionals, they shall
alfo be proportionals when taken separately; that
is, if two magnitudes together have to one of them, the
fame ratio which two others have to one of these, the
remaining one of the first two shall have to the other, the
fame ratio which the remaining one of the last two has
to the other of these.

Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, fo is CD to DF; they shall also be proportionals taken separately, viz. as AE to EB, fo CF to FD.

Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM, MN; and again, of EB, FD, take any equimultiples whatever KX, NP: And because GH is the fame multiple of AE that HK is of EB, therefore GH is the fame multiple of AE, that GK is of AB: But GH is the fame multiple of AE, that LM is of CF; wherefore GK is the same multiple of AB,

1

:

that

=

Again, because LM is the fame multiple of Book V.

Pb2.5.

that LM is of CF. CF that MN is of FD; therefore LM is the same multiple of CF, that LN is of CD: But LM was shewn to be the fame a I. 5. multiple of CF, that GK is of AB; GK therefore is the fame multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of AB, CD. Next, because HK is the fame multiple of EB, that MN is of FD; and that KX is also the fame multiple of EB, that NP is of FD; therefore HX is the fame multiple b of EB, that MP is of FD. And because AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD, HX and MP are K equimultiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if less, less: But if GH be H greater than KX, by adding the commen part HK to both, GK is greater than HX; wherefore also LN is greater than MP; and by taking away MN from both, LM is greater than NP: Therefore, if GH be greater than KX, LM is greater than NP. In like manner it may be demonstrated, that if GH be equal to KX, LM likewise is equal to NP; and if less, less: And GH, LM are any equimultiples whatever of AE, CF, and KX, NP are any whatever of EB, FD. Therefore, as AE is to EB, so is CF to FD. If then magnitudes, &c. Q. E. D.

N

c 5. def. S

B

DM

F

E

GACL

PROP. XVIII. THEOR.

IF magnitudes taken feparately be proportionals, they see N
shall alfo be proportionals when taken jointly, that is,
if the first be to the second, as the third to the fourth,
the first and fecond together shall be to the second, as
the third and fourth together to the fourth.

Let AE, EB, CF, FD be proportionals; that is, as AE to EB, fo is CF to FD; they shall alfo be proportionals when taken jointly; that is, as AB to BE, so CD to DF.

Take of AB, BE, CD, DF any equimultiples whatever GH, HK, LM, MN; and again, of BE, DF, take any whatever equimultiples KO, NP: And because KO, NP are equimultiples

of

Book V. of BE, DF; and that KH, NM are equimultiples likewife of BE, DF, if KO the multiple of BE be greater than KH, which is a multiple of the fame BE, NP likewise the multiple of DF shall be greater than NM the multiple f of the fame DF; and if KO be equal

and if less, less.

to KH, NP shall be equal to NM; Of

First, Let KO not be greater than KH, therefore NP is not greater than NM: And because GH, HK are equi- K multiples of AB, BE, and that AB is greater than BE, therefore GH is

M

1

P

N

a 3. Ax.

greater than HK; but KO is not

greater than KH, wherefore GH is

B

greater than KO. In like manner it

may be shewn, that LM is greater than
NP. Therefore, if KO be not great-
er than KH, then GH the multiple of
AB is always greater than KO the GA

D

E

F

OL

multiple of BE; and likewife LM the

multiple of CD greater than NP the multiple of DF.

Next, Let KO be greater than KH; therefore, as has been shewn, NP is greater than NM: And because the whole GH is the fame multiple of the whole AB, that HK is of BE, the remainder GK is the fame multiple of the remainder AE that GH is of AB, O

bs. s.

which is the fame that LM is of CD.

In like manner, because LM is the H

P

fame multiple of CD, that MN is of
DF, the remainder LN is the fame
multiple of the remainder CF, that

M

the whole LM is of the whole CDb: K

But it was shewn that LM is the fame

multiple of CD that GK is of AE;
therefore GK is the fame multiple of
AE that LN is of CF; that is, GK,
LN are equimultiples of AE, CF:

And because KO, NP are equimul-
tiples of BE, DF, if from KO, NP

there be taken KH, NM, which are likewise equimultiples of BE, DF, the remainders HO, MP are either equal to BE, DF, or equimultiples of them. First, Let HO, MP be equal to BE, DF; and because AE is to EB, as CF to FD, and