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V.

Book V. A part of a greater magnitude is greater than the same part of m

a less.

PROP. I. THEOR.

B

TF any number of magnitudes contain as many

others the same number of times; all the first magnitudes taken together shall contain all the others that same number of times : and if each of the first contain its other exactly, the whole shall contain the whole exactly.

Let any number of magnitudes AB, CD contain as many E, F the same number of times; as many times as AB contains E, so many times shall AB and CD together contain E and F together.

If AB contain E but once, it is less a than double of E: and a Def. A. Så CD is less than double of F; therefore AB, CD together are less than double of E, F together; and therefore they contain E, F but once.

If AB contain E more than once, from it cut off AG, GH equal each of them to E, and from CD cut off CK, KL equal to F, so that the remainders BH, DL contain E, F but once; therefore AB contains E as many times as there are magnitudes AG, GH, HB; and CD contains F as many

times as

there are magnitudes CK, KL, LD: but AB, CD con- H

L tain E, F equally; therefore the number of

K the magnitudes AG, GH, HB is equal to the

I number of the others CK, KL, LD: and be

A E C F cause HB contains E once, and LD contains F once; therefore BH and DL together contain E and Fonce : and because GH is equal to E, and KL to F; therefore GH, and KL together are equal to E, and F together.

to E, and F together. For the b2.Ax. 1. same reason, AG and CK together are equal to E and F together. Wherefore as many times as E is contained in AB, so

many tiines are E, F together contained in AB, CD together.

Likewise, if BH be equal to E, and DL to F; BH, DL together are equal to E, F together: and GH, KL are equal to E, F; as also are AG, CK; therefore AB, CD together contain E, F exactly ; that is, AB together with CD is the same multiple of E together with F, that AB is of E. The same demonstras tion holds if there be more magnitudes. Wherefore, &c. Q.E.D.

P

PROP.

D

Book V.

PROP. II. THEOR.

IF

2

F to two magnitudes which contain two others the

same number of times, equimultiples of these others be added; the wholes shall contain these others the fame number of times : and if the first two be equimultiples of the others, the wholes shall also be equimultiples of them.

Let AB contain C the same number of times that DE contains F; and let BG be the same multiple of C that EH is of F: Then shall the whole AG contain C the same number of times that the whole DH contains F.

Because BG is the same multiple of C, that EH is of F, as a 2. Def. 5. many magnitudes as are in BG equal to C, fo

many are there. in EH equal to F. Divide BG into BK, KG, each of them equal to C, and EH into EL, HL, each equal to F: therefore the number of the magnitudes BK, KG is equal to the

B

E number of the others EL, LF: and because

F AB contains C the same number of times

L

K
that DE contains F, and that BK is equal to
C, and EL to F; therefore AK contains C

EC
the same number of times that DL contains
F: and KG is equal to C, and LH to F; therefore AG con-
tains the same number of times that DH contains F. Like-
wise, if AB contain C exactly, AK and AG contain C exactly ;
and if DE contain F exactly, DL and DH contain F exactly ;
therefore, if AB, DE be equimultiples of C, F; AG, DH are
also equimultiples of them. Wherefore, &c. Q. E. D.

PROP. III. THEOR.

F two magnitudes be equimultiples of two others;

any equimultiples of the first two are also equimultiples of the other two.

If

Let AB be the same multiple of C that DE is of F, and let AG, DH be equimultiples of AB, DE; then AG is the same

multiple of C that DH is of F. a 2. Def. 5. Let the magnitudes equal a to AB, in AG, be AB, BK, KG;

and those equal to DE, in DH, be DE, EL, LH: and because

AG, DH are equimultiples of AB, DE, the number of the bDef. B. 5. magnitudes AB, BK, KG is equal to the number of the others

DE,

DE, EL, LH: And because AB is the same multiple of C Book V.
that DE is of F, and that BK is equal to AB, and EL to ED;
therefore BK is the same multiple of C that
EL is of F: For the same reason, KG is the

H
fame multiple of C that LH is of F: And be-
cause to AB, DE, which are equimultiples of K.

L.
C, F, there are added BK, EL, which are also
equimultiples of C, F; therefore the whole

B AK is the same multiple of C that the whole DL is of F. For the same reason, AG

1 is the same multiple of C that DH is of F.

1

A C
Wherefore, &c. Q. E. D.

C 2. S.

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PROP. IV. THEOR.
F the first of four magnitudes has the same ratio

to the second, which the third has to the fourth,
and any equimultiples be taken of the first and third,
and also any equimultiples of the second and fourth :
the multiple of the first shall have to that of the se-
cond, the same ratio which the multiple of the third
has to that of the fourth.

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Let A the first, have to B the second, the same ratio which the
third C has to the fourth D; and of A and C,
let E and F be any equimultiples; and of B
and D, let GH and KL be any equimultiples :
then shall E have the same ratio to GH that

1
F has to KL.

A B C D
Take ME any multiple of E greater than

F K
GH, and MF the same multiple of F: also
take NG, the least multiple of GH, that is
greater than ME, and NK the same multiple
of KL: Therefore NH, NL are either equal

L

HT
to GH, KL, or equimultiples of them : But
GH, KL are themselves equimultiples of B,
D; therefore NH, NL are equimultiples a of

a 3. 5.
B, D: For the same reason, NG, NK are
equimultiples of B, D; and ME, MF, of A,
C: and because, as A to B, so is C to D,
and ME, MF are equimultiples of A, C;

M
ME contains B the same number of times

b 5. Def.5. that MF contains D: NG also and NK con

CA. Def.se
tain B and D the same number of times ,
because they are equimultiples of them; but
ME does not contain B so many times as NG

N
P 2

contains

M

N

Book V. contains B, for ME is less than NG; therefore MF does not

contain D so many times as NK does : MF is therefore less d d 2. Ax. 5. than KN. In like manner, because ME is not less than NH,

for NG is the last multiple of GH greater than ME, it may be proved, that MF is not less than NL: and because ME is not

less than NH, but less than NG the next greater multiple of e def. A.5. GH; ME and NH contain GH the same number of times .

For the same reason, MF and NL contain KL the same number of times : but NH, NL contain GH, KL equally, for they are either equal to GH, KL, or equimultiples of them; therefore ME, MÊ contain GH, KL equally: and ME, MF are aby equimultiples whatever of E, F; as many times therefore as

any multiple of E contains GH, so many times does the same 6 5. Def. 5. multiple of F contain KL: Wherefore, as E is to GH, fo is Foto

KL. Therefore, if the firft, &c. Q. E. D.

Cok. Likewise, if the first be to the second, as the third to the fourth, then also any multiple of the first is to the second, as the same multiple of the third to the fourth : and in like manner, the first is to any multiple of the second, as the third to the same multiple of the fourth.

For, it was proved in the proposition, that as many times as ME any multiple of E, contains B, so many times does MF the same multiple of F, contain B; therefore, as E is to B, so iso F to D.

And if ME, MF be any equimultiples of A, C, of which ME is greater than GH; and NG, NK be taken as in the propofition: it may be demonstrated, as before, that ME contains GH the same number of times that MF contains KL: there. fore, as A to GH, so is b C to KL.

PROP. V. THEOR.

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Fone magnitude be the same multiple of another,

that a maguitude taken from the firit is of one taken from the other; the remainder shall be the same multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the same multiple of CD, that AE taken from the first is of CF taken from the other; the remainder EB fall be the same multiple of the remainder FD, that the whole AB is of the whole CD.

Take AG the fame multiple of FD, that AE is of CF: therefore AE is the fame multiple a of CF, that EG is of CD:

But

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But AE, by the hypothefis, is the same multiple of CF that Book V. AB is of CD: Therefore EG is the same multiple of CD that AB is of CD; wherefore EG is equal to AB: Take from them the common magnitude

b 4. Ax. s. AE, and the remainder AG is equal to the remainder EB. Wherefore, since AE is the same multiple of CF that AG is of FD, and that AG A is equal to EB; therefore AE is the same multiple of CF that EB is of FD: But AE is the same multiple of CF, that AB is of CD; therefore EB is the same multiple of FD that AB is of CD. Therefore, &c. Q. E. D.

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IF

F from two magnitudes, which contain two others

the same number of times, there be taken equimultiples of these others; the remainders, if they be not less than the other magnitudes, shall contain them the same number of times.

Let the magnitude AB contain E the same number of times that CD contains F; and let AG, CH, equimultiples of E, F, be taken from AB, CD, so that the remainder GB be not less than E: GB, HD shall contain E, F the same number of times.

Because AG is the same multiple of Ethat CH is of F, there are as many magnitudes in AG equal to E, as there are in CH equal to Fa: Divide AG into AK, KG,

a Def.B.5. each equal to E, and CH into CL, LH, each equal to F; therefore the number of the first AK, KG is equal to the number of the last

С CL, LH: and because AB contains E the K same number of times that CD contains F; and AK, CL are equal to E, F; the remain

G der BK contains E the same number of times

Η that the remainder DL contains F. In like manner, if from BK, DL, which contain E, F equally, there be taken KG, LH, which are

BE

DT equal to E, F; the remainder BG contains E the same number of times that the remainder DH contains F. Wherefore, &c. Q. E. D.

PROP.

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