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ratio of the remainder EB to the remainder FD, which is the b 19. 5. same b with the given ratio of AB to CD, shall be given.

But if the ratio of AB to CD be not the same with the ratio of AE to CF, either it is greater than the ratio of AE to CF, or, by inversion, the ratio of CD to AB is greater than the ratio of CF to AE. First, let the ratio of AB to CD be greater than the ratio of AE to CF, and as AB to CD, so make AG to CF; therefore the ratio of AG to

A
E G

B CF is given, and CF is given, c 2. dat. whereforec AG is given : and

с FD because the ratio of AB to CD,

that is, the ratio of AG to CF, d 10. 5. is greater than the ratio of AE to CF; AG is greaterd than

AĚ: and AG, AE are given, therefore the remainder EG is given; and as AB to CD, so is AG to CF, and so isb the re

nainder GB to the remainder FD; and the ratio of AB to CD is given : wherefore the ratio of GB to FD is given ; therefore GB, the excess of EB above a given magnitude EG, has a given ratio to FD. In the same manner the other case is demonstrated.

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IF to one of two magnitudes which have a given ratio to one another, a given magnitude be added, and from the other a given magnitude be taken; the excess of the sum above a given magnitude shall have a given ratio to the remainder.

Let the two magnitudes AB, CD have a given ratio to one another, and to AB let the given magnitude EA be added, and from CD let the given magnitude CF be taken ; the excess of the sum EB above a given magnitude has a given ratio the remainder FD.

Because the ratio of AB to CD is given, make as AB to

CD, so AG to CF: therefore the ratio of AG to CF is given, a 2. dat. and CF is given, wherefore a AG

E А

GB is given; and EA is given, there

fore the whole EG is given : and

С

F D because 'as AB to CD, so is AG b 19. 5.

1to CF, and so isb the remainder GB to the remainder FD; the ratio of GB to FD is given. And EG is given, therefore GB, the excess of the sum EB

above the given magnitude EG, has a given ratio to the remainder FD.

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IF two magnitudes have a given ratio to one ano. See N. ther, if a given magnitude be added to one of them, and the other be taken from a given magnitude; the sum, together with the magnitude to which the remainder has a given ratio, is given: and the remainder is given together with the magnitude to which the sum has a given ratio.

Let the two magnitudes AB, CD have a given ratio to one another; and to AB let the given magnitude BE be added, and let CD be taken from the given magnitude FD: the sum AE. is given, together with the magnitude to which the remainder FC has a given ratio.

Because the ratio of AB to CD is given, make as AB to CD, so (B to FD : therefore the ratio of GB to FD is given, and FD is given, wherefore GB is

G A в Е a 2. dat. given a; and BE is given, the whole GE is therefore given: and

I because as AB to CD, so is GB

F С

D to FD, and so is b GA to FC ; the

b 19. 5. ratio of GA to FC is given : and AE together with GA is given, because GE is given ; therefore the sum AE together with GA, to which the remainder FC has a given ratio, is given. The second part is manifest from prop. 15.

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IF two magnitudes have a given ratio to one ano-See N. ther, if from one of them a given magnitude be taken, and the other be taken from a given magnitude ; each of the remainders is given, together with the magnitude to which the other remainder has a given ratio.

Let the two magnitudes AB, CD have a given ratio to one another, and from AB let the given magnitude AE be taken,

and let CD be taken from the given magnitude CF: the remainder EB is given, together with the magnitude to which the other remainder DF has a given ratio.

Because the ratio of AB to CD is given, make as AB to CD,

so AG to CF: the ratio of AG to CF is therefore given, and a 2. dat. CF is given, wherefore a AG is

А
E B

G
given ; and AE is given, and
therefore the remainder EG is
given ; and because as AB to CD,

C

D so is AG to CF: and so is b the

F remainder BG to the remainder DF; the ratio of BG to DF is given: and EB together with BG is given, because EG is given: therefore the remainder EB together with BG, to which DF the other remainder has a given ratio, is given. The second part is plain from this and prop. 15.

b 19. 5.

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See N.

IF from two given magnitudes there be taken magnitudes which have a given ratio to one another, the remainders shall either have a given ratio to one another, or the excess of one of them above a given magnitude shall have a given ratio to the other.

Let AB, CD be two given magnitudes, and from them let the magnitudes AE, CF, which have a given ratio to one another, be taken ; the remainders EB, FD either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other. Because AB, CD are each of A

E B
them given, the ratio of AB to
CD is given : and if this ratio
be the same with the ratio of AE C

F D
to CF, then the remainder EB
has a the same given ratio to the
remainder FD.

But if the ratio of AB to CD be not the same with the ratio of AE to CF, it is either greater than it, or, by inversion, the ratio of CD to AB is greater than the ratio of CF to AE: first, let the ratio of AB to CD be greater than the ratio of AE to CF; and as AE to CF, so make AG to CD; therefore the ratio of AG to CD is given, because the ratio of AE to CF is given; and CD is given, wherefore b AG is

a 19. 5.

b 2. dat.

given ; and because the ratio of AB to CD is greater than the ratio of (AE to CF, that is, than

A

E GB the ratio of) AG to CD; AB is greater c than AG : and AB, AG

c 10.5. are given; therefore the remain

C F D der G is given : and because as AE to CF, so is AG to CD, and so is a EG to DF; the ratio of EG to FD is given : and GB is a 19. 5. given; therefore EG, the excess of EB above a given magnitude GB, has a given ratio to FD. The other case is shown in the same way.

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IF there be three magnitudes, the first of which See N. has a given ratio to the second, and the excess of the second above a given magnitude has a given ratio to the third; the excess of the first above a given magnitude shall also have a given ratio to the third.

Let AB, CD, E, be the three magnitudes of which AB has a given ratio to CD; and the excess of CD above a given magnitude has a given ratio to E: the excess of AB above a given magnitude has a given ratio to E.

Let CF be the given magnitude, the excess of CD above which, viz. FD has a given ratio to E: and because the ratio of AB to CD is given, as AB to CD, so make A AG to CF; therefore the ratio of AG to CF is given; and CF is given, wherefore a AG is

a 2. dat.

Ggiven : and because as AB to CD, so is AG to CF, and so is b GB to FD; the ratio of GB

b 19. 5. to FD is given.

F. And the ratio of FD to E is given, wherefore c the ratio of GB to E is

c 9. dat. given, and AG is given ; therefore GB the excess of AB above a given magnitude AG has a given ratio to E.

B

DI E Cor. 1. And if the first has a given ratio to the second, and the excess of the first above a given magnitude has a given ratio to the third; the excess of the second above a given magnitude shall have a given ratio to the third. For, if the second be called the first, and the first the second, this corollary will be the same with the proposition.

Cor. 2. Also, if the first has a given ratio to the second, and the excess of the third above a given magnitude has also a gira ratio to the second, the same excess shall have a given ratio m the first; as is evident from the 9th dat.

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IF there be three magnitudes, the excess of the first whereof above a given magnitude has a given ratio to the second ; and the excess of the third above a given magnitude has a given ratio to the same se cond: the first shall either have a given ratio to the third, or the excess of one of them above a given magnitude shall have a given ratio to the other.

Let AB, C, DE be three magnitudes, and let the excesses of each of the two AB, DE above given magnitudes have given ratios to C; AB, DE either have a given ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other.

Let FB the excess of AB above the given magnitude AF have a given ratio to C; and let GE the ex А cess of DE above the given magnitude DG have a given ratio to C; and because FB, GE

have each of them a given ratio to C, they F a 9. dat. have a given ratio a to one another. But to FB

D GE the given magnitudes AF, DG are addb 18. dat. ed; therefore b the whole magnitudes AB,

DE have either a given ratio to one another,
or the excess of one of them above a given
magnitude has a given ratio to the other. B с E

G

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IF there be three magnitudes, the excesses of one of which above given magnitudes have given ratios to the other two magnitudes; these two shall either have a given ratio to one another, or the excess of one of them above a given magnitude shall have a given ratio to the other.

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