MA

PROP. XV. THEOR.

AGNITUDES have the fame ratio to one ano-
ther which their equimultiples have.

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

G

Book V.

A

D

K

H

L

В С E F

a. 7. 5.

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 firft AG, GH, HB fhall be equal to the number of the last DK, KL, LE. and because AG, GH, HB are all equal, and that DK, KL, LE are alfo 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 confequent, fo are all the antecedents together to all the confequents together; wherefore as AG is to b. 12. 5, DK, fo is AB to DE. but AG is equal to C, and DK to F. therefore as C is to F, fo is AB to DE. Therefore magnitudes, &c. Q. E.D.

I'

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 propor- E

tionals when taken A alternately; that is,

A is to C, as B to D. B

Take of A and F

B any equimultiples

G

C

D

H

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

a. 15. 5. b. 11. 5.

Book V. G and H. and becaufe E is the fame multiple of A, that F is of ✔ B, and that magnitudes have the fame ratio to one another which their equimultiples have ; therefore A is to B, as E is to F. but as A is to B, fo is C to D. wherefore as C is to D, fob is E to F. again, becaufe G, H are equimultiples of EC, D, as C is to D, Afo is G to H; but

C. 14. 5.

A

as C is to D, fo is E B

to F. Wherefore as F

E is to F, fo is G to

G

C

D

H

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 lefs, lefs . Wherefore if E be greater
than G, F likewife is greater than H; and if equal, equal; if lefs,
lefs. and E, F are any equimultiples whatever of A, B; and G,
Therefore A is to C, as B to D. If

d. 5. Def. 5. H any whatever of C, D.
then four magnitudes, &c. Q. E. D.

IF

PROP. XVII. THEO R.

F magnitudes taken jointly be proportionals, they fhall also 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 these, the remaining one of the first two fhall have to the other, the fame ratio which the remaining one of the laft 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 alfo 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

X

133

a. 1. 5.

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 fhewn to be the fame muitiple 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 alfo the fame multiple of EB, that NP is of FD; therefore HX is the fame multiple of EB that MP is of FD. And becaufe AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EBand FD, HX and MP are equimultiples; if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if lefs, lefs. but if GH be greater

K

H

P

b. 2. 5.

N

B

D M

E

c. 5. Def..

F

GACL

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 demonftrated, that if
GH be equal to KX, LM likewife is equal to NP; and if lefs, lefs.
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

PROP. XVIII. THEOR.

c

F magnitudes taken feparately be proportionals, they See fhall also be proportionals when taken jointly, that is, if the first be to the second, as the third to the fourth, the first and second together fhall 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 shall also 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 likewife the multiple of DF fhall be greater than NM the multiple of the fame DF; and if

H
KO be equal to KH, NP fhall be e-
qual to NM; and if lefs, lefs.

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

2. 3.Ax. 5. greater than HK; but KO is not

b. s. s.

c. 6. 5.

B

E

N

greater than KH, wherefore GH is
greater than KO. In like manner it
may be fhewn 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
multiple of BE; and likewife LM the multiple of CD greater than
NP the multiple of DF.

GA

C

L

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 ABb, which is the O

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 Book V. 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 d. Cor. 4. 5. LN to MP. If therefore GK be greater than HO, LN is greater than MP; and if equal, equal; and if lefs, less *.

H

K

B

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 lefs, lefs f; which was likewife fhewn in the preceding cafe. If therefore 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 f as AB is to BE, fo is CD to DF. If then magnitudes, &c. Q. E. D,

E

GA

P

N

F

CL

e. A. 5.

f. 5. Def. s.

PROP. XIX. THEOR.

IF a whole magnitude be to a whole, as a magni- See N,

tude 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 remain. der EB fhall be to the remainder FD, as the whole AB, to the whole CD,