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any equimult 'ples of A, C, and G, H, any other equimultiples of B, D: I say, E will be to G, as F is to H.

For let K, L be equimultiples of E, F, and m, nequimultiples; of G, H.

Then because e is the same multiple of A, as F is of c, and K, L are equimultiples of E, F; [by 3. 5.] K will be the same multiple of A, as L is of c. By

the same reason M will be the I same multiple of B, as n is of d. K E AB M And because as A is to B, so is

ċ to D, and K, L, are taken equimultiples of A, C, and M, N other equimultiples of B, D: If K exceeds M [by 5. def. 5.1 will exceed 'n ; if k be equal to M, I will be equal to n; if k be

less than M, L will be less than n; I I

and K, L are equimultiples of LT CD II N E, F: But m, n any other equiF

F multiples of G, H: Therefore [by 5. def. 5.)'as E is to G, so will be to H.

Wherefore if the first of four magnitudes has the same ratio to the second, as the third has to the fourth; then will any equimu'tiples of the first and third have the same ratio to any equimultiples of the second and fourth, when compared to one another. Which was to be demonstrated.

Corollary. Therefore because it has been demonftratcd if k exceeds M, I will also exceed N; if the one be equal to the one, the other will be equal to the other ; if less, less : it is manifest if M exceeds K, N will exceed L; if M be equal to K, N will be equal to L; if less, less : Therefore as G is to E, so is H to F. Wherefore from hence it is manifest, that if four magnitudes be proportionals, they will be proportionals inversely.

PROP.

PROP. V. THEOR. If one magnitude be the same multiple of another, as

a part taken away from the one is of a part taken away from the other ; then all the part remaining of the one be the same multiple of the part remaining of the other, as one of the whole magnitudes is of the other u.

For let the magnitude A B be the same multiple of the magnitude c D, as the part A E taken away of the one is of the part of taken away of the other: Í fay, the part E B remaining of the one, will be the same multiple of the part F remaining of the other, as the whole magnitude A B is of the whole magnitude c D.

For make EB the same multiple of cG, as A e is of cf.

Then because [by 1. 5.) Ae is the same multiple of cf as A B is of GF, and Ą.E

A is the same multiple of C F, as A B is of

G CD; AB is the same multiple of GF, CD: Wherefore GF is equal to cd.

E Take away C F, which is common, from both: Then the remainder g cis equal to the remainder DF. And because A E

F is the fame multiple of CF, as E B is of GC, and c B is equal to DF; A E will

B be the same multiple of C F, as e B is of

D FD. But a E is put the fame multiple of c F, as A B is of CD: Therefore E B is the same multiple of FD, as AB is of of cD: Wherefore the remainder E B will be the same multiple of the remainder FD, as the whole A B is of the whole c D.

Therefore if one magnitude be the same multiple of another, as a part taken away from the one is of a part taken away from the other ; then shall the part remaining of the one be the same multiple of the part remaining of the other, as one of the whole magnitudes is of the other. Which was to be demonstrated.

u This proposition, proposed here only in multiple propor. tion, is universally demonstrated at prop. 19. and the seventh and eighth propositions of the feventh book answer to it in numbers,

PROP.

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PRO P. VI. THEOR.
If two magnitudes be equimulriples of two other mag.

. nitudes, and some parts taken away from them be equimultiples of the same magnitudes ; then the remainders will either be equal to the same magnitudes, or equimultiples of them w.

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For let two magnitudes A B, CD be equimultiples of two other magnitudes E, F, and the parts AG, CH taken away from them be fome equimultiples of these other magnitudes E, F: I say, the remainders GB, HD are either equal to E, F, or equiinultiples of them.

For first let G B be e-
A
K
AT IK

qual to E: I say, HD is

equal to F; for put ck C

equal to F.

Then because AG is

the same multiple of E, G

H as ch is of F, and GB

is equal to E, and c.K to TF; A B (by 2.5.] will be

į the same multiple .f E, as B DEF B D E F KH is of F. But A B is put the same multiple of E, as cd is of F; therefore KH is the same multiple of F, as cd is of F: Wherefore because K H, ce are each an equimultiple of F; KH will be equal to co Take

away
the common part c H,

and then the remainder K c is equal to the remainder HD.

But kc is equal to F: Therefore h D is equal to F. If therefore G B be equal to E, HD will be equal to F.

In like manner we demonftrate, whatever multiple GB is of E (as in the second figure) the same will HD be of F.

Therefore if two magni udes be equimultiples of two other magnitudes, and some parts taken away from them be equimultiples of the same magnitudes ; then the re

; mainders will be either equal to the same magnitudes or equimultiples of them. Which was to be demonstrated,

* This proposition being only in multiple proportion, is universally demonstrated at prop. 24. of all kinds of proportion.

PROP.

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PROP. VII, THEOR. Equal magnitudes bave the same ratio to the same

magnitude, and the same magnitude has the same ratio to equal magnitudes *.

Let A, B be equal magnitudes, and c any magnitude whatsoever : I say, each of thele magnitudes A, B have the same ratio to c: and also c has the fame ratio to A; or B

For take D, E equimultiples of A and B, and take any other multiple F of c.

Then becaule D is the same multiple of A, as E is of B, and a is equal to B ; D will be equal to E. But F is any other multiple of c: Therefore if d exceeds F, E will exceed F too; and if it be equal to F, E will be equal to F; if lefs, less. D A But D, E are equimultiples of A, B, E and F is any multiple whatsoever of c: Therefore [by 5. def. 5.) A will

. be to, c, as B is to c.

I say moreover that c has the same ratio to A or B.

For the construction remaining the same, we demonstrate in like manner that d is equal to E. other magnitude. Therefore if F exceeds D, it will also exceed E, if F be equal to D, it will be equal to É; if

E lefs, less. But F is a multiple of c, and D, E are any equimultiples of A, B: Therefore [by 5. def. 5.) as c is to A, so will c be to B.

Therefore equal magnitudes have the same ratio to the same magnitude, and the same magnitude has the same ratio to equal magnitudes. Which was to be demonstrated.

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But F is any

* This seventh, with the eighth, ninth, tenth, eleventh, and twelfth propofitions following, are taken by some to be mere axioms requiring no demonstration at all. They are indeed yery evident in numbers. But since they are applicable to all magnitudes in general, lines, planes, solids, commenfurable, and incommensurable, Euclid could not but demonitrate them.

PROP. greater than c.

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PROP. VIII, THEOR. The greater of (two] unequal magnitudes has a greater

ratio to the same magnitude than the lefser ; and the same magnitude to the leffer of two) unequal magnitudes bas a greater ratio than it has to the greater (of these two) magnitudes. Let AB, c be two unequal magnitudes, and let A B be

Also let d be any other magnitude whatsoever : I say, A B has a greater ratio to D, than c has to

D: and D has a greater ratio to FT

c, than it has to A B.

For because A B is greater than TE

C, make [by 3. 1.] Be equal to C; GH

then the lifler of thesetwomagni

tudes A E, E B being inultiplied, B

will at length exceed D [by 4. def. 5.] First let A E be less than EB, and multiply A E so often till it exceeds D : Ler FG be this multiple of AE, which is greater than

D: Also make GH the same mulKHC D L M N is of A E.

tiple of E B, and K of c, as F G

And take L double to D, M triple to it, and so forwards greater by one, until the magnitude taken be a multiple of D, and [in the first place] greater that K. Let n be this magnitude, being four times the magnitude D, and (in the first place] greater than K.

Then because K [in the first place] is less than N, K will not be lesser than M. And since F G is the same multiple of A E as HG is of e B, [by 1. 5.) F G will be the same multiple of a E as Fh is of AB. But F G is the same multiple of A E as K is of c: Therefore FH is the faine multiple of AB as K is of c. And fo FH, K are equimultiples of A B and c. Again, because G H is the same multiple of E B as k is of c, and EB is equal to c; GH will be equal to K. But K is not less than M. Therefore G H is not less tha: M. But [by conftr.] F G is greater than D: Therefore the whole F H will be greater than

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