PROP. XV. THEOR.

MAGNITUDES 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 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 magnitudes, each equal to F, viz. DK, KL, LE. then the number of the first AG, GH, HB

A

fhall be equal to the number of the last DK, G

D

K

BCEF

KL, LE. and because AG, GH, HB are all e-
qual, and that DK, KL, LE are alfo equal to H
one another; therefore AG is to DK, as GH to'
KL, and as HB to LE. and as one of the an-
tecedents to its confequent, fo are all the ante-
cedents together to all the confequents toge-
ther; wherefore as AG is to DK, fo is AB to DE. but AG is
èqual to C, and DK to F. therefore as C is to F, fo is AB to DE.
Therefore magnitudes, &c. Q. E. D.

IF

PROP. XVI. THEOR.

F four magnitudes of the fame kind be proportionals, they fhall also be proportionals when taken alternately.

Let the four magnitudes A, B, C, D be proportionals, viz. as A

to B, fo C to D. they

fhall alfo be proporti- E

onals when taken alternately; that is, A

A

is to C, as B to D.

B

G

C

H

b. 12.

ver E and F; and of C and D take any equimultiples whatever &

Book V. and H. and because E is the fame multiple of A, that F is of B, and that magnitudes have the fame ratio to one another which their a. 15. 5 equimultiples have; therefore A is to B, as E is to F. but as A b. 11. 5. is to B, fo is C to D. wherefore as C is to D, fo bis E to F. again, because G, Hare equi

G

C

D

H

four magnitudes are proportionals, if the first be greater than the third, the fecond fhall be greater than the fourth; and if equal, c. 14. 5. equal; if lefs, lefs. Wherefore if E be greater than G, F likewise is greater than H; and if equal, equal; if lefs, lefs. and E, F are any equimultiples whatever of A, B; and G, H any whatever of C, d. 5. Def. 5.D. Therefore A is to C, as B to Dd. If then four magnitudes, &c. Q.E. D.

Sde N.

a. 1. 5.

F magnitudes taken jointly be proportionals, they fhall alfo be proportionals when taken feparately, that is, if two magnitudes together have to one of them, the fame -ratio which two others have to one of thefe, the remain

ing one of the first two fhall 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 fhall also be proportionals taken feparately, 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 fame multiple of AB, that LM is of CF. Again, be

a

cause

a

caufe 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 shewn to be the fame multiple of CF, that GK is of AB; GK there- a. s. 5. fore is the fame multiple of AB, that LN is of CD; that is, GK,

Next, because HK is the fame

K

P

b. 2. S.

N

B

1

DM c. s. Def. s.

E

F

LN are equimultiples of AB, CD.
multiple of EB, that MN is of FD; and that
KX is also the fame multiple of EB, that NP X
is of FD; therefore HX is the fame mul-
tiple 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 equi-
multiples; if GK be greater than HX, then
LN is greater than MP; and if equal, equal; H+
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 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 likewife is equal to NP; and if lefs, 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, fo is
CF to FD. If then magnitudes, &c. Q. E. D.

GACL

PROP. XVIII. THEOR.

If paken when taken jointly, that is,

magnitudes taken feparately be proportionals, they see N.

if the first be to the fecond, as the third to the fourth, the first and second together shall be to the fecond, 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 becaufe 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 likewife the multiple of DF fhall be greater than NM the multiple of the fame DF; and if KO be fl equal to KH, NP fhall be equal to NM; and if lefs, lefs.

C

L

tiple of AB is always greater than KO GA
the 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 muitiple of the remainder AE

b. 5. 5. that CH is of ABb, which is the fame

that LM is of CD. In like manner, be-
caufe LM is the fame multiple of CD, that
MN is of DF, the remainder LN is the

O
H

P

fame multiple of the remainder CF, that

M

DF, if from KO, NP there be taken KH, NM, which are likewife equimultiples of BE, DF, the remainders HO, MP are either equal c. 6. s. 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

to FD4. but HO is equal to EB, and MP to FD; wherefore GK Book V.
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.

P

d. Cor. 4. 5.

f. 5. Def. s.

But let HO, MP be equimultiples of EB, FD; and becaufe AE c. A. s. 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 lefs, lefs f; which was likewife shewn in the preceeding cafe. If therefore GH be greater than KO, taking KH from H 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

K

M

B

N

D

E

F

fhewn that if GH be equal to KO, LM

is equal to NP; and if lefs, lefs. And GA

in the cafe in which KO is not greater

C L

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, so is CD to DF. If then magnitudes, &c, Q. E. D.

PROP. XIX. THEOR.

IF a a as a

[F a whole magnitude be to a whole, as a magnitude see N.

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 shall be to the remainder FD, as the whole AB, to the whole CD.

Because AB is to CD, as AE to CF; likewife, alternately, a. 16. s.
BA

I 4