b

ง

X

a.

P b. 2. 5.

cause LM is the fame multiple of CF that MN is of FD; therefore Book V. LM is the fame multiple of CF, that LN is of CD. but LM was fhewn to be the fame multiple of CF, that GK is of AB; GK 1. 5. 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 same multiple 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 K and LN are equimultiples, and of EB and FD, HX and MP are equimultiples; if GK be greater than HX, then LN is H greater than MP; and if equal, equal; and if lefs, lefs. but if GH be greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore alfo 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.

IF

N

B

D M

E

c. 5. Def. 5.

F

GACL

PROP. XVIII, THEOR.

magnitudes taken feparately be proportionals, they see N. shall also be proportionals when taken jointly, that is, if the first be to the fecond, as the third to the fourth, the first and second together fhall 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 fhall alfo be proportionals when taken jointly; that is, as AB to BE, fo 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 BE, DF; and

Book V. 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 of the fame DF; and if KO be equal to KH, NP shall be equal to NM; and if lefs, lefs.

H

o!

First, Let KO not be greater than KH, therefore NP is not greater than NM. and because GH, HK are equimultiples of AB, BE, and that AB K is greater than BE, therefore GH is a. 3. Ax. 5. greater a than HK; but KO is not greater than KH, wherefore GH is greater than KO. In like manner it may be shewn that LM is greater than NP. Therefore if KO be not greater than KH, then GH the multiple of AB is always greater than KO the

b. 5. 5.

E

JBA

GA

M

P

N

multiple of BE; and likewise LM the multiple of CD greater than NP the multiple of DF.

Next, Let KO be greater than KH; therefore, as has been fhewn, 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", which is the O

fame that LM is of CD. In like
manner, because LM is the fame H
multiple of CD, that MN is of DF,
the remainder LN is the fame mul-
tiple of the remainder CF, that the K
whole LM is of the whole CD b.

P

B

D N

c. 6. 5.

and because KO, NP are equimultiples of BE, DF, if from KO, NP there be taken KH, NM, which are likewife 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 that GK, LN are equimultiples of AE, CF; GK fhall be to EB, as LN to FD4. but HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP. If therefore GK be greater than HO, LN is greater than MP; and if equal, equal; and if lefs, lefs.

But let HO, MP be equimultiples of EB, FD; and because AE is to EB, as CF to FD, and that of AE, CF are taken equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LN is greater than

MP; and if equal, equal; and if less, lefs f; which was likewife fhewn in

K

Book V.

d. Cor. 4. 5.

c. A. 5.

f. 5. Def. 5.

P

M

B

D N

E

the preceding cafe. If therefore H.
GH be greater than KO, taking KH
from both, GK is greater than HO;
wherefore alfo LN is greater than
MP; and confequently, adding NM
to both, LM is greater than NP.
therefore if GH be greater than KO,
LM is greater than NP. In like
manner it may be fhewn that if GH
be equal to KO, LM is equal to
NP; and if lefs, lefs. And in the cafe in which KO is not greater
than KH, it has been fhewn that GH is always greater than KO,
and likewife LM than NP. but GH, LM are any equimultiples of
AB, CD, and KO, NP are any whatever of BE, DF; therefore fas
AB is to BE, fo is CD to DF. If then magnitudes, &c. Q. E. D.

IF

GA

PROP. XIX. THEOR.

F a whole magnitude be to a whole, as a magnitude See N. taken from the first is to a magnitude taken from the other; the remainder fhall be to the remainder as the whole to the whole.

Let the whole AB be to the whole CD, as AE a magnitude taken from AB to CF a magnitude taken from CD; the remainder EB fhall be to the remainder FD, as the whole AB, to the whole CD.

Book V.

2. 16. 5.

R. 75

17. 5. q. B.5. f. 18. 5.

Because AB is to CD, as AE to CF; like wife, alternately a BA is to AE, as DC to CF. and because if magnitudes taken jointly be proportionals, they are alfo proportionals b when taken feparately; therefore as BE is to EA, fo is DF to FC; and alternately, as BE is to DF, fo is EA to FC. but as AE to CF, fo, by the Hypothefis, is AB to CD; therefore alfo BE the remainder fhall be to the remainder DF, as the whole AB to the whole CD. Wherefore if the whole, &c. Q. E. D.

COR. If the whole be to the whole, as a magnitude taken from: the first is to a magnitude taken from the other; the remainder likewife is to the remainder, as the magnitude taken from the first to that taken from the other. the Demonftration is contained inthe preceding.

I'

PROP. E. THEOR,

F four magnitudes be proportionals, they are also proportionals by converfion, that is, the first is to its excefs above the fecond, as the third to its excess above the fourth,

Let AB be to BE, as CD to DF; then BA
is to AE, as DC to CF.

Because AB is to BE, as CD to DF, by E
divifion 2, AE is to EB, as CF to FD; and
by inverfion, BE is to EA, as DF to FC.
Wherefore, by Compofition, BA is to AE,
as DC is to CF. If therefore four, &c.
Q. E. D.

A

C

F

B D

See N.

PROP. XX. THEOR.

E there be three magnitudes, and other three, which taken two and two have the fame ratio; if the first be greater than the third, the fourth fhall be greater than the fixth; and if equal, equal; and if lefs, less,

Let A, B, C be three magnitudes, and D, E, F other three, Book V. which taken two and two have the fame ratio, viz. as A is to B,

fo is D to E; and as B to C, fo is E to F. If A be greater than C, D fhall be greater than F; and if equal, equal; and if less, less.

Because A is greater than C, and B is any other magnitude, and that the greater has to the fame magnitude a greater ratio than the lefs has to it;

therefore A has to B a greater ratio than C has A B C
to B. but as D is to E, fo is A to B; therefore b
D has to E a greater ratio than C to B. and be- DEF
caufe B is to C, as E to F, by inverfion, C is to
B, as F is to E; and D was fhewn to have to Ea
greater ratio than C to B; therefore D has to Ea
greater ratio than F to E. but the magnitude
which has a greater ratio than another to the fame magnitude, is
the greater of the two ", D is therefore greater than F.
Secondly, Let A be equal to C; D fhall be equal to F, because

A and C are equal to one another, A
is to B, as C is to B. but A is to B,
as D to E; and C is to B, as F to
E; wherefore D is to E, as F to Ef;
and therefore D is equal to F 8.

Next, Let A be lefs than C; D

b. 13.5.

c.Cor.13.5.

d. 10. 5

f. II. 5.

A B C

g. 9. 5.

A B C

DEF

shall be less than F. for C is great- DEF

er than A, and, as was fhewn in the first case, C is to B, as F to E, and in like manner B is to A, as E to D; therefore F is greater than D, by the first cafe; and therefore

D is lefs than F. Therefore if there be three, &c. Q. E, D,

,

there be three magnitudes, and other three, which have the fame ratio taken two and two, but in a crofs order; if the firft magnitude be greater than the third, the fourth fhall be greater than the fixth; and if equal, equal; and if less, less,