√.

A part of a greater magnitude is greater than the fame part of a lefs.

IF

PROP. I. THEOR.

any number of magnitudes contain as many others the fame number of times; all the first magnitudes taken together fhall contain all the others that fame number of times: and if each of the first contain its other exactly, the whole fhall contain the whole exactly.

Let any number of magnitudes AB, CD contain as many E, F the fame number of times; as many times as AB contains E, fo many times fhall AB and CD together contain E and F together.

BOOK V.

If AB contain E but once, it is lefs than double of E: and a Def. A. 5« CD is less than double of F; therefore AB, CD together are

lefs than double of E, F together; and therefore they contain E,

F but once.

II

L

D

K

I I

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, fo 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 B CD contains F as many times as there are magnitudes CK, KL, LD: but AB, CD con- H tain E, F equally; therefore the number of G the magnitudes AG, GH, HB is equal to the number of the others CK, KL, LD: and becaufe HB contains E once, and LD contains F once; therefore BH and DL together contain E and F once : and becaufe GH is equal to E, and KL to F; therefore GH, and KL together are equal to E, and F together. For the b2.Ax. 1. fame reafon, AG and CK together are equal to E and F together. Wherefore as many times as E is contained in AB, so many times are E, F together contained in AB, CD together.

b

A E C F

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 alfo are AG, CK; therefore AB, CD together contain E, F exactly; that is, AB together with CD is the fame multiple of E together with F, that AB is of E. The fame demonftration holds if there be more magnitudes. Wherefore, &c. Q. E. D.

Book V.

יז

PROP. II. THEOR.

to two magnitudes which contain two others the fame number of times, equimultiples of these others be added; the wholes fhall 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 fame number of times that DE contains F; and let BG be the fame multiple of C that EH is of F: Then fhall the whole AG contain C the fame number of times that the whole DH contains F.

2

many are there.

K

E

F

Because BG is the fame multiple of C, that EH is of F, as 22.Def. 5. many magnitudes as are in BG equal to C, fo 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 number of the others EL, LF: and because AB contains C the fame number of times that DE contains F, and that BK is equal to C, and EL to F; therefore AK contains C the fame number of times that DL contains F: and KG is equal to C, and LH to F; therefore AG contains C the fame number of times that DH contains F. Likewife, 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.

a 2. Def. 5.

PROP. III. THEOR.

G

H

F two magnitudes be equimultiples of two others; any equimultiples of the first two are alfo equimultiples of the other two.

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

a

Let the magnitudes equal to AB, in AG, be AB, BK, KG; and thofe equal to DE, in DH, be DE, EL, LH: and becaufe 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

b

DE,

G

H

~

DE, EL, LH: And because AB is the fame 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 fame multiple of C that EL is of F: For the fame reason, KG is the fame multiple of C that LH is of F: And because to AB, DE, which are equimultiples of K C, F, there are added BK, EL, which are also equimultiples of C, F; therefore the whole AK is the fame multiple of C that the whole DL is of F. For the fame reason, AG is the fame multiple of C that DH is of F. Wherefore, &c. Q. E. D.

IF

PROP. IV. THEOR.

B

L

E

€ 2. 5.

I

A C

DF

F the first of four magnitudes has the fame ratio to the fecond, which the third has to the fourth, and any equimultiples be taken of the firft and third, and alfo any equimultiples of the fecond and fourth: the multiple of the first fhall have to that of the second, the fame ratio which the multiple of the third has to that of the fourth.

Let A the first, have to B the fecond, the fame 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 fhall E have the fame ratio to GH that F has to KL.

a of

A B

Take ME any multiple of E greater than GH, and MF the fame multiple of F: alfo E G take NG, the leaft multiple of GH, that is greater than ME, and NK the fame multiple of KL: Therefore NH, NL are either equal to GH, KL, or equimultiples of them: But GH, KL are themfelves equimultiples of B, D; therefore NH, NL are equimultiples B, D: For the fame reafon, NG, NK are equimultiples of B, D; and ME, MF, of A, C: and because, as A to B, fo is C to D, and ME, MF are equimultiples of A, C; ME contains B the fame number of times that MF contains D: NG alfo and NK con- M tain B and D the fame number of times c because they are equimultiples of them; but ME does not contain B fo many times as NG contains

P 2

I

بر

BOOK V. contains B, for ME is lefs than NG; therefore MF does not contain D fo many times as NK does: MF is therefore less & In like manner, because ME is not lefs than NH, for NG is the leaft multiple of GH greater than ME, it may be

d 2. Ax. 5. than KN.

proved, that MF is not lefs than NL: and because ME is not lefs than NH, but lefs than NG the next greater multiple of e def. A. 5. GH; ME and NH contain GH the fame number of times ©. For the fame reason, MF and NL contain KL the fame number of times but NH, NL contain GH, KL equally, for they are either equal to GH, KL, or equimultiples of them; therefore ME, MF contain GH, KL equally and ME, MF are any equimultiples whatever of E, F; as many times therefore as any multiple of E contains GH, fo many times does the fame b5. Def. 5. multiple of F contain KL: Wherefore, as E is to GH, so is Fb to KL. Therefore, if the firft, &c. Q. E. D.

:

COR. Likewife, if the firft be to the fecond, as the third to the fourth, then alfo any multiple of the first is to the second, as the fame multiple of the third to the fourth and in like manner, the first is to any multiple of the fecond, as the third to the fame multiple of the fourth.

For, it was proved in the propofition, that as many times as
ME any multiple of E, contains B, fo many times does MF the
fame multiple of F, contain B; therefore, as E is to B, fo is
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 demonftrated, as before, that ME contains GH the fame number of times that MF contains KL: there. fore, as A to GH, fo is C to KL..

IF

PROP. V. THEOR.

F one magnitude be the fame multiple of another, that a magnitude taken from the firit is of one taken from the other; the remainder fhall be the fame multiple of the remainder, that the whole is of the whole.

Let the magnitude AB be the fame multiple of CD, that AE taken from the firft is of CF taken from the other; the remainder EB fhall be the fame 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: 21. 5. therefore AE is the fame multiple 2 of CF, that EG is of CD : But

a

G

b 4. Ax. s.

But AE, by the hypothefis, is the fame multiple of CF that Book V. AB is of CD: Therefore EG is the fame multiple of CD that AB is of CD; wherefore EG is equal to AB: Take from them the common magnitude AE, and the remainder AG is equal to the remainder EB. Wherefore, fince AE is the fame multiple of CF that AG is of FD, and that AG A is equal to EB; therefore AE is the fame multiple of CF that EB is of FD: But AE is the fame multiple of CF, that AB is of CD; therefore EB is the fame multiple of FD that AB is of CD. Therefore, &c. Q. E. D.

E

IF

PROP. VI. THEOR.

from two magnitudes, which contain two others the fame number of times, there be taken equimultiples of these others; the remainders, if they be not less than the other magnitudes, fhall contain them the fame number of times.

C

Let the magnitude AB contain E the fame number of times that CD contains F; and let AG, CH, equimultiples of E, F, be taken from AB, CD, fo that the remainder GB be not lefs than E: GB, HD fhall contain E, F the fame number of times. Because AG is the fame multiple of E that CH is of F, there are as many magnitudes in AG equal to E, as there are in CH equal to F 2: Divide AG into AK, KG, 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 CL, LH: and because AB contains E the K fame number of times that CD contains F; and AK, CL are equal to E, F ; the remainder BK contains E the fame number of times 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 equal to E, F, the remainder BG contains E the fame number of times that the remainder DH contains F. Wherefore, &c. Q. E. D.

G

BE

L

H

D F

PROP.

a Def. B.5.