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Book V.

PROP. XV. THEOR.

MAGNITUDES have the same ratio to one another which their equimultiples have.

Let AB be the same multiple of C, that DE is of F: C is 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

А 10 C, as there are in DE equal to F: let AB be. divided into magnitudes, each

D
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,

KI
GH, HB, shall be equal to the number of
the last DK, KL,LE: and because AG, H-
GH, HB are all equal, and that DK, KL,

L 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 B C E F to its consequent, so are all the antecedents together to all a 7.5. the consequents together b; wherefore, as AG is to DK, so is AB to DE: but AG is equal to C, and DK to F: therefore, b. 12. 5. as C is to F, so is AB to DE. Therefore' magnitudes, &c. Q. E. D.

PROP. XVI. THEOR.

IF four magnitudes of the same kind be proportionals, 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 shall also be probortionals when taken alternately; that is, A is to C, as B to D.

Take of A and B any equimultiples wnatever E and E; 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 a 15. 5. equimultiples have a; therefore A is to B, as E is to F: but as

A is to B, so is C to
D:. wherefore as C E

Gb 11. 5. is to D, sob is. E to

F: again, because G, A
H are equimultiples
of C, D, as C is to D, B-

D
so is G to H; but as
C is to D, so is E to F.

H
F. Wherefore, as E
is to F, so is G to Hb. But, when four magnitudes are propor-

tionals, if the first be greater than the third, the second shall c 14. 5. be greater than the fourth ; and if equal, equal; if less, lesse,

Wherefore, if E be greater than G, F likewise is greater than

H; and 'if equal, equal; if less, less; and E, F are any equi

| multiples whatever of A, B ; and G, H any whatever of C, D. a 5. def. 5. Therefore A is to C, as B to Dd. If then four magnitudes,

&c. Q. E. D.

PROP. XVII. THEOR.

IF magnitudes, taken jointly, be proportionals, See Note. they shall also be proportionals when taken sepurately;

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; tbnt is, as AB to BF, so is CD to DF; they shall 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 same multiple of AE, that HK is of EB, wherefore GH is the same multiple a 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, because LM is the same multiple Book v. of CF, that MN is of FD; therefore LM is the same multiple a, of CF, that LN is of CD: but LM was shown to be the same a 1. 5. inultiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of €D; that is, GK, LN are equi. multiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; and that KX is

1.3.

х also the same multiple of EB, that NP is of FD; therefore HX is the same multiple

P b. of EB, that MP is of FD. And because

b 2. 5. AB is to BE, as CD is to DF, and that of AB and CD, GK and LN are equimulti

K ples, and of EB and FD, HX and MP are

N equimultiples; if GK be greater than HX,

N then LN is greater than MP; and if equal, equal; and if less, lesse: but if GH be

HE greater than KX, by adding the common

MI c 5. def. 5.. part HK to both, GK is greater than HX;

B D 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, G A C L that if GH be equal to KX, LN 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. Thereforec, as AE is to EB, so is CF to FD. If then magnitudes, &c. Q. E. D.

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PROP. XVIII. THEOR.

IF magnitudes, taken separately, be proportionals, See Note. they 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, so 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 BÉ, DF, take any whatever equi

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Book v. multiples KO, NP: and because KỘ, NP are equimultiples

of BE, DF; and 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 same BE, NP, likewise the multiple of DF,
shall be greater than NM, the multiple

H
of the same DF; and if KO be equal
to KH, NP shall be equal to NM; and
if less, less.

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

K

N
greater than BE, therefore, GH is
23. Ax. 5. greatera than HK; but KO is not

greater than KH, wherefore GH is
greater than Ko. In like manner it
may be shown, that LM is greater than B

D
NP. Therefore if KO be not greater
than KH, then GH, the multiple of E FI
AB, is always greater than KO, the
multiple of BE; and likewise LM, the
multiple of CD, greater than NP, the G

A

CL multiple of DF.

b 5,5.

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N

Next, Let KO be greater than KH : therefore, as has been shown, NP is greater than NM: and because the whole GH is the same multiple of the whole AB, that HK is of BE, the re. mainder GK is the same multiple of H the remainder AE that GH is of ABb: which is the same that LM is of CD. 01

M In like manner, because LM is the

! same multiple of CD, that MN is of

P-
DF, the remainder LN is 'the same
multiple of the remainder CF, that

K
the whole LM is of the whole CDb:
but it was shown that LM is the same
multiple of CD, that GK is of AE; B
therefore GK is the same multiple of
AE, that LN is of CF; that is, GK, E E

D
LN' are equimultiples of AE, CF:

F1
and because KO; NP are equimul-

1 tiples of BE, DF, if from KO, NP G А) CL 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 that GK, LN are equimultiples of AE, CF; GK shall be to Book V, EB, as LN to FD d: but HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP. If therefore GK bed Cor. 4. 5. greater than HO, LN is greater than MP; and if equal, equal; and if lesse, less.

c 6. 5.

*

e Ax. 5. But let HO, MP be equimultiples of EB, FD; and because AE is to EB; as CF to FD, and that of AE, CFare taken equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LN

O is greater than MP; and if equal, equal; and if less, less f': 'which was

f 5. def. 5. likewise shown in the preceding case. If therefore GH be greater than KO, H

P. taking KH from both, GK is greater than H0; wherefore also LN is greater

MU than MP; and, consequently, adding NM to both, LM is greater than ŅP: therefore, if GH be greater than KO, K в

N. LM'is greater than NP. In like man

D it

may be shown, that if GH be equal to KOLM is equal to NP; and E if less, less. And in the case in which

G A

L KO is not greater than KH, it has been shown 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; thereforel, as AB is to BE, so is CD to DF. If then magnitudes, &c. Q. E. D.

ner

ACT

bit L

PROP. XIX. THEOR,

IF a whole magnitude be to a wbole, as a magni. See Note. 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 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; likewise, alternately a, a 16. 5.

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