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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 is to D, E-

G
b 11. 5. so bis E to F: again, be-
cause G, H are equimul- A-

C
tiples of C, D, as C is to
D, so is G to Ha; but B.

D-
as C is to D, so is E to
F. Wherefore, as Eis F-
to F, so 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 c 14. 5. fourth; and if equal, equal; if less, less s. 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 equimultiples whatever of A, B; and d 5. def. 5. G, H any whatever of C, D. Therefore, A is to C, as B to D4

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

H

PROP. XVII. THEOR,

See N.

IF magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately; 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 BE, 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 the same multiple of AE, that LM is of CF; wherefore GK is the same multiple of AB,

21.5.

that LM is of CF. Again, because LM is the same multiple of Book V. 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 multiple of CF, that GK is of AB; GK therefore is the same multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; and that KX is

х also the same multiple of EB, that NP is of FD; therefore HX is the same multiple of EB, that MP is of FD. And because

P

b 2. 5. 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 k equimultiples; if GK be greater than HX,

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

н: B

c5. def.5. greater than KX, by adding the common part HK to both, GK is greater than HX; wherefore also LN is greater than MP; E and by taking away MN from both, LM

F is greater than NP: therefore, if GH be greater than KX, LM is greater than NP. In like manner it may be demonstrated, G Å Ć L 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. Thereforec, as AE is to EB so is CF to FD. If, then, magnitudes, &c. Q. E. D.

DM

PROP. XVIII. THEOR.

IF magnitudes, taken separately, be proportionals, See N. 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 BE, DF take any whatever equi

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Book V. multiples KO, NP: and because KO, 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
of the same DF; and if KO be equal H
10 KH, NP shall be equal to NM; and
if less, less.
First, let KO not be greater than KH,

M therefore NP is not greater than NM:

pl and because GH, HK are equimultiples of AB, BE, and that AB is greater than K

N a 3. Ax.5. BE, therefore GH is greater a than HK;

but KO is not greater than KH, where-
fore GH is greater than KO. In like
manner it may be shown, that LM is
greater than NP. Therefore, if KO be

B
not greater than KH, then GH, the
multiple of AB, is always greater than

D

E
KO, the multiple of BE; and likewise
LM, the multiple of CD, greater than

G A

L
NP, the multiple of DF.

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

O b 5.5. the remainder AE that GH is of AB b: which is the same that LM is of CD.

H

M In like manner, because LM is the same multiple of CD, that MN is of

pl DF, the remainder LN is the same multiple of the remainder CF, that

K

N
the whole LM is of the whole CD b:
but it was shown that LM is the same
in ultiple of CD, that GK is of AE;

B
therefore GK is the same multiple of
'AE, that LN is of CF; that is, GK, El

D
LN are equimultiples of AE, CF:
and because Ko, NP are equimul-
tiples of BE, DF, if from KO, NP G А С L
there be taken KH, NM, which are likewise equimultiples

of BE, DF, the remainders HO, MP are either equal to BE, € 6.5. DF, or equimultiples of them c. First, Let HO, MP be

equal to BE, DF; and because AE is 10 EB, as ÇF to FD, and

F

143 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 be d Cor.4.5. greater than HO, LN is greater than MP; and if equal, equal; and if lesse, less.

e A.5. 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

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 H

P than Ko, taking KH from both, GK is greater than HO; wherefore also

M LN is greater than MP; and, consequently, adding NM to both, LM is

K greater than NP: therefore, if GH

N be greater than KO, LM is great

B er than NP. In like manner it

D may be shown, that if GH be equal

E to KO, LM is equal to NP; and if less, less. And in the case in which

G A с 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 BÉ, DF; therefore f, as AB is to BE so is CD to DF. If then magnitudes, &c. Q. E. D.

PROP. XIX. THEOR.

IF 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 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 reinainder 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.

Book V. BA is to AE as DC to CF: and because, if mag А

nitudes, taken jointly, be proportionals, they are 17.5. also proportionals b when taken separately; therefore, as BE is to DF so is EA to FC; and alter

с

E
nately, as BE is to EA, so is DF to FC: but, as
AE to CF, so by the hypothesis is AB to CD;

F
therefore also BE, the remainder, shall 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 mag-
nitude taken from the first is to a magnitude taken B
from the other; the remainder likewise is to the
remainder, as the magnitude taken from the first to that taken
from the other: the demonstration is contained in the preceding,

PROP. E. THEOR.

IF four magnitudes be proportionals, they are also proportionals by conversion, that is, the first is to its excess above the second, as the third to its excess above the fourth.

А

с

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 division a, AE is to EB as CF to FD; and by inver. sion b, BE is to EA as DF to FC. Wherefore, by composition c, BA is to AE, as DC is to CF. If, therefore, four, &c. Q. E. D.

a 17.5. b B. 5. c 18. 5.

E

F

B

PROP. XX. THEOR.

See N.

IF there be three magnitudes, and other three, which, taken two and two, have the same ratio; if the first be greater than the third, the fourth shall be greater than the sixth; and if equal, equal; and if less, less.

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