Book v.

PROP. XV. THEOR.

AGNITUDES have the same ratio to one ano

ther which their equimultiples have.

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

Because AB is the same 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

A
into magnitudes, each equal to F, viz. DK,

D
KL, LE. then the number of the first AG,

G
GH, HB shall be equal to the number of
the last DK, KL, LE. and because AG,

K
GH, HB are all equal, and that DK, KL,
LE are also equal to one another; there-

H

L
fore AG is to DK, as GH to KL, and as
HB to LE, and as one of the antecedents

ВС
to its consequent, fo are all the antecedents

a. 7.5

EF together to all the consequents together b; wherefore as AG is to b. 11. 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.

PROP. XVI. THEOR.

IF

tionals, they shall also be proportionals when taken alternately.

Let the four magnitudes A, B, C, D be proportionals, viz. as A
to B, so C to D. they
Thall also be propor- E

G
tionals when taken A

C
alternately; that is,

D
A is to C, as B to D.

B-
Take of A and F

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

b. II.s.

Book v. G and H. and because E is the same multiple of A, that F is of mB, and that magnitudes have the same ratio to one another which

their equimultiples have à; 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, so b is E to F.
again, because G, H
are equimultiples of E

-G
C, D, as C is to D,
A-

C
fo is G to H; but
as C is to D, fo is E
B.

D.
to F. Wherefore as F-

H E is to F, fo 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, lefse Wherefore if I be greater than G, F likewife is greater than H ; and if equal, equal; it lets,

less. and E, F are any equimultiples whatever of A, B; and G, d. 5. Def. 5. H any whatever of C, D. Therefore A is to C, as B to DJ. If

then four magnitudes, &c. Q. E. D.

t. 14. 5.

PROP. XVII. THEO R.

Ste N.

IF
F magnitudes taken jointly be proportionals, they

shall also be proportionals when taken feparately, that is, if two magnitudes together have to one of them, the same ratio which two others have to one of these, the remaining one of the first two shall have to the other, the same 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 BF, so is CD to DF;
Mall also be proportionals taken separately, viz. as AE to EB, so
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 same multiple of AE, that LM is of CF; wherefore GK is the same multiple of AB, that LM is of CF. Again, be

1

a. I. 5.

b. 2. s.

DM
F

caufe LM is the same multiple of CF that MN is of FD; therefore

Book v. LM is the same multiple of CF, that LN is of CD. but LM was Thewn to be the same multiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of CD; that is, GK, I.N are equimultiples of AB, CD. Next, becaule HK is the fame multiple of EB, that MN is of FD; and that KX is also the fame multiple of X 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

N

H GK be greater than HX, then LN is

B greater than MP; and if equal, equal ; and if lefs, less'. but if GH be greater

E

c. 5. Def.s. thao KX, by adding the common part HK to both, GK is greater than HX; wherefore allo LN is greater than MP;

G A Ċ L and by taking away MN from both, LM

greater than NP, therefore if GH bę 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 19 EB, so is CF to FD. If then magnitudes, &c. Q. E. D,

PROP. XVIII. THEOR. F

be shall 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 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 also 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 BE, DF; and

If
magnitudes taken feparately be proportionals

, they see W;

greater than

Book V. that KH, NM are equimultiples likewise 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 e-

H
qual to NM; and if less, less.

O
First, Let KO not be greater than
KH, therefore NP is not greater than

M
NM. and because GH, HK are equi-
multiples of AB, BE, and that AB K

P is greater than BE, therefore GH is 4. 3. Ax. 5. greater « than HK; but KO is not

N greater than KH, wherefore CH is

B
greater than Ko. In like manner it

D
may be shewn that LM is greater than E
NP. Therefore if KO be not greater

F
than KH, then GH the multiple of
AB is always greater than Ko the G | A CIL
multiple of BE; and likewise LM the multiple of CD
NP the multiple of DF.

Next, Let KO be greater than KH; therefore, as has been shewn, No is greater than NM. and because the whole GH is the same multiple of the whole AB, that HK is of BE, the remainder GK is

the same multiple of the remainder b. s. s.

AE that GH is of ABb, which is the O
fame that LM is of CD. In like

H

P
manner, because LM is the same
multiple of CD, that MN is of DF,
the remainder LN is the same mul-
tiple of the remainder CF, that the K

B
whole L M is of the whole CD b.
but it was thewn that LMis the fame

E
multiple of CD that GK is of AE;
therefore GK is the same multiple of
AE that LN is of CF; that is, GK,

G A

L
LN are equimultiples of AE, CF.
and because KO, NP are equimultiples 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, c. 6. s. equimultiples of them. First, Let HO, MP be equal to BE, DF;

M/

D

N

F!

or

and because AE is to EB, as CF to FD, and that GK, LN are Book V. equimultiples of AE, CF; GK Mall be to EB, as LN to FD4. but w 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, lesse.

e. A. s. 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,

f. s. Def. s. lessf; which was likewise shewn in

H

P the preceding cafe. If therefore GH be greater than KO, taking KH from both, GK is greater than HO;

M wherefore alfo LN is greater than K MP; and consequently, adding NM В

N to both, LM is greater than NP. therefore if GH be greater than KO, E

F LM is greater than NP. In like manner it may be shewn that if GH

G A be equal to KO, LM is equal to

С L NP; and if lefs, less. And in the case in which KO is not greater than KH, it has been shewn that GH is always greater than KO, and likewise 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, so is CD to DF. If then magnitudes, &c. Q. E. D,

MI

PROP. XIX. THE O R.

IF
F 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 shall 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 Mall be to the remainder FD, as the whole AB, to the whole CD