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Book V.

M

PROP. XV. THEOR.

AGNITUDES have the fame ratio to one another
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.

A

G

H

D

K

L

a 7.5.

BCEF

Becaufe 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, fhall be equal to the number of the last DK, KL, LE: And becaufe 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 DK, fo b 12. 5. 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.

four magnitudes of the fame kind be proportionals,

nately.

Let the four magnitudes A, B, C, D be proportionals, viz. as A to B, fo C to D: They fhall alfo be proportionals when taken alternately; that is, A is to C, as B to D.

Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H: And

because

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Book V. 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

a 15. 5.

b II. 5.

€ 14. 5.

d

D: Wherefore, as CE
is to D, fob is E to A-
F: Again, because
G, H are equimul- B-
tiples of C, D, as C
is to D, fo is G to F
H; but as C is to

G

C

D

H

D, fo is E to F. Wherefore, as E is to F, fo is G to Hb. But, when four magnitudes are proportionals, if the first be greater than the third, the fecond thall 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, H any whatever of C, D. Therefore A is to C, as B ta 5. def. 5. Dd. If then four magnitudes, &c. Q. E. D.

Sec N.

PROP. XVII. THEOR.

F magnitudes, taken jointly, be proportionals, they shall 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 these, the remaining 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

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b 2. 5.

that LM is of CF. Again, because LM is the fame multiple of Book V. CF, that MN is of FD; therefore LM is the fame multiple of CF, that LN is of CD: But LM was fhown to be the fame a 1. 5. multiple 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, becaufe 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 EB and FD, HX and MP are K 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 Hgreater 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, fo is CF to FD. If then magnitudes, &c. Q. E. D.

E

N

c 5. def. s.

B

DM

F

GACL

PROP. XVIII THEOR.

IF magnitudes, taken feparately, be proportionals, they see N. fhall alfo be proportionals when taken jointly, that is, if the first be to the fecond, as the third to the fourth, the first and fecond 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 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 because KO, NP are equimultiples

of

Book V. of BE, DF; and 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 equal to KH, NP fhall be equal to NM; and if lefs, lefs.

2. 3. Ax.

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
greater than BE, therefore GH is
greater a
a than HK; but KO is not
greater than KH, wherefore GH is
In like manner it
greater than KO.

may be shown, that LM is greater than
NP. Therefore, if KO be not great.

er than KH, then GH, the multiple of

Н

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L

b 5. 5.

c 6.5.

AB, is always greater than KO, the GA
multiple of BE; and likewife LM, the

multiple of CD, greater than NP, the multiple of DF.

Next, Let KO be greater than KH; therefore, as has been fhown, NP is greater than NM: And becaufe the whole GH is the fame multiple of the whole AB, that HK is of BE, the remainder GK is the fame multiple of

P

M

B

DN

F

CL

the remainder AE, that GH is of ABb:
which is the fame that LM is of CD.
In like manner, becaufe LM is the H-
fame multiple of CD, that MN is of
DF, the remainder LN is the fame
multiple of the remainder CF, that
the whole LM is of the whole CD: K
But it was fhown that LM is the fame
multiple of CD, that GK is of AE;
therefore GK is the fame multiple of
AE, that LN is of CF; that is, GK,
LN are equimultiples of AE, CF:
And becaule KO, NP are equimul-
tiples 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 e-
qual to BE, DF; and because AE is to EB, as CF to FD, and

E

GIA

that

d Cor. 4. 5.

that GK, LN are equimultiples of AE, CF; GK fhall be to Book V. EB, as LN to FDd: But HO is equal to EB, and MP to FD; wherefore GK 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.

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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 fhown in the preceding cafe. If therefore GH be greater than KO, H 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 K than NP: Therefore, if GH be greater than KO, LM is greater than NP. In like manner it may be shown, 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 great- G er than KH, it has been fhown that

M

N

D

E

B

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 Ax. 5.

f 5. def. 5.

IF.

PROP. XIX. THEOR.

Fa whole magnitude be to a whole, as a magnitude see N. taken from the firft, is to a magnitude taken from the other; the remainder hall 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, fhall 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. 5. BA

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