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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 another, if See N.

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.

a. 2. Dat.

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 GB to FD. therefore the ratio of GB to FD is given, and FD is given, wherefore GB is given : G A BE and BE is given, the whole GE is therefore given. and because as AB to CD, so is GB to FD, and so is

E C

D O GA to FC; the 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 manifeft from Prop. 15.

b. 19. 5

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IF two magnitudes have a given ratio to one another, Sce N.

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 le:

CD be taken from the given magnitude CF; the remainder EB iş 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, fo

AG to CF, the ratio of AG to CF is therefore given, and CF is 4. 2. Dat. given, wherefore a AG is given;

and AE is given, and therefore the A E B G
remainder EG is given. and be-
cause as AB to CD, so is AG to

C D T 0. 19. I.

CF, and so is b the remainder BG
to the remainder DF; the ratio of BG to DF is given and EB to-
gether with BG is given, because EG is given. therefore the re-
mainder EB together with BG to which DF the other remainder
has a given ratio is given. the second part is plain from Prop. 15,

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See N.

IF
F from two given magnitudes there be taken magni.

tudes 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.

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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

B them given, the ratio of AB to CD is given. and if this ratio be C F D

the same with the ratio of AE to 4. 19. 5. 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 CF to AE. first, let the ratio of AB to CD be greater than the ratio of AE to CF; and as AE ta 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, where

c. 10. 5.

fore 6 AG is given; and because the ratio of AB to CD is greater b. 2. Dat. than the ratio of (AE to CF, that A E is, than the ratio of) AG to CD; AB is greater than AG. and AB, C AG are given, therefore the remainder BG is given. and because as AE to CF, so is AG to CD, and fo is a EG to FD; the ratio a. 19. 5. of EG to FD is given. and GB is given, therefore EG the excess of EB above the given magnitude GB, has a given ratio to FD. the other case is thewn in the same way.

F D

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PROP. XXIV.

13. IF F there be three magnitudes, the first of which has a See N.

given ratio to the second, and the excess of the fecond 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.

a. 2. Dat.

b. 19. 50

Let AB, CD, E be 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 AG to CF; there- A
fore the ratio of AG to CF is given; and CF is
given, wherefore a AG is given. and because as

C
AB to CD, so is AG to CF, and so is 6 GB to G
FD; the ratio of GB to FD is given. and the

TH ratio of FD to E is given, wherefore the ratio

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

BD BD 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 fecond 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.

Аа 4 4

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 given ratio to the second, the same excess shall have a given ratio to the first; as is evident from the oth 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 second. 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.

A

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 excess of DE above the given magnitude DG have a gi

D ven ratio to C; and because FB, GE have each of them a given ratio to C, they have a given

G ratio a to one another. but to FB, GE the given a. Ja Dat. magnitudes AF, DG are added; therefore b the Þ. 18. Dat. whole magnitudes AB, DE have either a given

ratio to one another, or the excess of one of B' C'E them above a given magnitude has a given ratio to the other,

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IF
F 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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b. 12. 5.

Let AB, CD, EF be three magnitudes, and let GD the excess of one of them CD above the given magnitude CG have a given ratio to AB; and also let KD the excess of the fame CD above the given magnitude CK have a given ratio to EF. either AB has a given ratio to EF, or the excess of one of them above a given magnitude has a given ratio to the other.

Because GD has a given ratio to AB, as GD to AB, so make CG to HA; therefore the ratio of CG to HA is given; and CG is given, wherefore a HA is given. and because as GD to AB, so is CG a. 2. Dat. to HA, and so is 6 CD to HB; the ratio of CD to HB is given. also because KD has a given ratio to EF, as KD

H to EF, so make CK to LE; therefore the ratio of CK to LE is given; and CK is given, wherefore a LE is given. and because as KD to EF, At fo is CK to LE, and so b is CD to LF; the ratio

G of CD to LF is given. but the ratio of CD to

KE . HB is given, wherefore the ratio of HB to LF

C. g. Dat. is given. and from HB, LF the given magnitudes B'DIT HA, LE being taken, the remainders AB, EF shall 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 d.

d. 19. Dat. " Another Demonstration. Let AB, C, DE be three magnitudes, and let the excesses of one of them C above given magnitudes have given ratios to AB and DE. either AB, DE 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 the excess of C above a given magnitude has a given ratio to AB, therefore a AB together with a given magnitude has a a. 14. Dat. given ratio to C. let this given magnitude be AF, wherefore FB has a given ratio to C. also, because the excess of C above a given magni- At D tude has a given ratio to DE, therefore a DE together with a given magnitude has a given ratio to C. let this given magnitude be DG, wherefore GE has a given ratio to C. and FB B' C'E has a given ratio to C, therefore the ratio of

b. 9. Dat. FB to GE is given. and from FB, GE the given magnitudes AF, DG being taken, the remainders 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,"

C. 19. Dat.

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