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many will there be in CD equal to F: Divide A B into parts equal to E, viz. AG, GB, and c d into parts equal to F, viz. C H, HD: Then the multitude

A of the parts c H, HD of the one is equal to the multitude of the parts AG, GB of

G

E the other. And because A G is equal to E, and c H to F; AG, CH will be equal to E, F. By the same reason G B is equal to B E, and HD to F: Therefore GB, HD are equal to E, F: Wherefore there are as

с many parts in A B equal to E, as there are in A B, C D equal to E, F: Therefore A B, CD will be the same multiple of E, F, as H. F AB is of E.

Therefore if there be how many soever magnitudes equimultiples of as many other magnitudes, each of each; the same multiple one magnitude is of one, all shall be of all. Which was to be de monftrated,

• The fifth and fixth propofitions of the seventh book of Euclid answer to this in numbers. And what is here pronounced of multiple ratio only, is afterwards demonstrated universally at prop. 12. of any proportionals rational, or irrational whatsoever. And this proposition was first put down to prove others, which were necessary towards the demonstration of the twelfth proposition,

PROP. II. THEOR,

If the first of four magnitudes be the same multiple of

the second, as the third is of the fourth, and there be a fi; ib the same multiple of the second, as a sixth magnitride is of th fourth; then the first and fifth taken together will be the same multiple of the second, as the third and fixth are of the fourth

For let the first of four magnitudes A B be the same multiple of the second c, as the third D E is of the fourth F. And let a fifth magnitude B G be the same multiple of the second 4, as a fixth eh is of the fourth r: I say, A G the first

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and

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D and fifth taken together will be

the same multiple of the fe

cond c, as D H the third and А

fixth taken together is of the
fourth F.

For because AB is the same
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multiple of c, as D E is of F, B

F there will be as many magni,

tudes in AB equal to c, as there
are in D E equal to F. And by
the same reason as many as are
in B G equal to ç, so many
there are in E H equal to F.
Therefore as many as there are

in the whole A G, equal to c,
H

so many there will be in the whole D H equal to F: Therefore D H is the same multiple of F, as A G is of c: Wherefore a G the first and fifth taken together, is the same multiple of the second c, as Dh the third and fixth, is of the fourth F.

Wherefore, if the first of four magnitudes be the same multiple of the second, as the third is of the fourth, and a fifth magnitude be the same multiple of the second, as a fixth is of the fourth, then the first and fifth taken together will be the same multiple of the second, as the third and fixth is of the fourth.

* This propofition fuprofes the fifth and fixth magnitudes to be equimultiples of the second and fourth. But the proposition would hold the same, if the fifth were only equal to the fecond, and the fixth to the fourth.

PROP. III. THEOR. .
If the first of four magnitudes be the same multiple of

the second as the third is of the fourib, and any equi-
multiples of the first and third be taken ; iben, by
equ zlity, esch of the assumed equimultiples will be
an equimultiple of tach, the one of the Second, and
the other of the fourth magnitude ť

For let the first of four magnitudes A be the same multiple of the second B, as the third c is of the fourth D; and

take

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take E F, G H equimultiples of a, c; I say, E F is the same multiple of B, as G H is of D.

For because E F is the same multiple of A, as Gh is of Ç; there will be as many magnitudes in Gh equal to co as there are in E F equal to A. Divide E F into the magnitudes E K, K F equal to A, and G H into the magnitudes GL, LH equal to c; then will the multitude of EK, KF bc equal to the multitude of

L GL, LH. And because a is the

K same multiple of B, as c is of D, and Ek is equal to A, and GL equal to C; Ek will be the same

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II multiple of B, as G L is of D. By the same reason KF is the same

A B GCD multiple of B, as L H is of D. Then because the first ek is the same multiple of the second B, as the third G L is of the fourth D; and the fifth KF is the same multiple of the second B, as the sixth L H is of the fourth D: the magnitude E F made up of the first and fifth will [by 2. 5.) be the same multiple of the second B, as G H the third and fixth is of the fourth D.

If therefore the first of four magnitudes be the same multiple of the second, as the third is of the fourth, and any equimultiples of the first and third be taken; then, by equality, each of the affumed magnitudes will be an equimultiple of each, the one of the second, and the other of the fourth magnitude. Which was to be demonstrated,

* This is demonstrated nniversally at prop. 22. it being here only proposed in multiple ratio; and that for the sake of demonftrating the fourth proposition.

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

second, as the third bus to the fourth; then will any equimultiples of the first and third bave obe same ratio to any equimultiples of the second and fourthy when compared to one anotker.

For let the first magnitude a have the same ratio to the second B, as the third c has to the fourth D. Taķe E, r

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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, N equimultiples of G, H.

Then because E is the same multiple of A, as F is of c, and K, I 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

same multiple of B, as n is of D. KE A BGM And because as A is to B, so is

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

M, L will be equal to n; if k be II

less than M, L will be less than n;

and K, L are equimultiples of L TC D II N E, F: But M, N any other equin multiples of G, H: Therefore [by 5. def. 5.]'as E is to G, so will F 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 demonstrated if k exceeds M, L 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.

PRO P.

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PRO P. V. THEOR. If one magnitude be the same multiple of anotber, 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 wbolo magnitudes is of the other u.

For let the magnitude A B be the same multiple of the magnitude CD, as the part A E taken away of the one is of the part C F taken away of the other: I fay, the part E B remaining of the one, will be the same multiple of the part FD 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 Ę is of cf.

Then because [by 1.5.]ae is the same multiple of c F as A B is of GF, and Ą.E

A А is the same multiple of c F, as A B is of CD; AB is the same multiple of GF, CD: Wherefore GF is equal to c D. 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 same multiple of C F, 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 ek is of

D FD. But A E is put the same multiple of cf, as A B is of CD: Therefore E B is the same multiple of F D, as A B is of of cD: Wherefore the remainder EB will be the same multiple of the remainder F D, 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 propofition, 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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