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A

E

GB

C 10. 5.

F

D

given; and becaufe the ratio of AB to CD is greater than the
ratio of (AE to CF, that is, than
the ratio of) AG to CD; AB is
greater c than AG: And AB,
AG are given; therefore the re-
mainder BG is given: And be-

cause as AE to CF, fo is AG to CD, and fo is a EG to FD; a 19. §.
the ratio of EG to FD is given: And GB is given; therefore
EG, the excefs of EB above a given magnitude GB, has a gi-
ven ratio to FD. The other cafe is fhewn in the fame way.

IF

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there be three magnitudes, the first of which has See N. a given ratio to the fecond, and the excefs of the fecond above a given magnitude has a given ratio to the third; the excefs of the first above a given magnitude fhall alfo 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 magritude 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 excefs 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; therefore the ratio of AG to CF is given; and CF is given, wherefore a AG is given And becaufe as AB to CD, fo is AG to CF, and fo is b GB to`FD: the ratio of GB to FD is given. And the ratio of FD to E is given, wherefore the ratio of GB to E is given, and AG is given; therefore GB the excefs of AB above a given magnitude AG has a given ratio to E.

G

a z. dat.

F

b 19. s.

c 9. dat.

BDE COR. 1. And if the firft has a given ratio to the fecond, and the excefs of the firft above a given magnitude has a given ratio to the third; the excefs of the fecond above a given magnitude shall have a given ratio to the third. For, if the second be called the firft, and the first the fecond, this corollary will be the fame with the propofition.

Cor.

17.

COR. 2. Alfo 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 fecond, the fame excefs fhall have a given ratio to the first; as is evident from the 9th dat.

PROP. XXV.

IF there be three magnitudes, the excess of the first whereof above a given magnitude has a given ratio to the fecond; and the excefs of the third above a given magnitude has a given ratio to the fame fecond: The first fhall either have a given ratio to the third, or the excess of one of them above a given magnitude fhall have a given ratio to the other.

Let AB, C, DE be three magnitudes, and let the exceffes 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 excefs of AB above the given magnitude AF have a given ratio to C; and let GE the ex A cefs of DE above the given magnitude DG have a given ratio to C; and because FB, GEF. have each of them a given ratio to C, they 2. 9. dat. have a given ratio a to one another. But to FB GE the given magnitudes AF, DG are add

6. 18. dat. ed; therefore b the whole magnitudes AB, DE

28.

D

G+

have either a given ratio to one another, o B CE
the excess of one of them above a given mag-
nitude has a given ratio to the other.

PROP. XXVI.

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

Let

Let AB, CD, EF be three magnitudes, and let GD the excefs of one of them CD above the given magnitude CG have a given ratio to AB; and alfo let KD the excefs 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 exceís 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, fo 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 a 2 dat. GD to AB, fo is CG to HA, and fo is b CD to HB; the ra- b 12.5. tio of CD to HB is given: Alfo becaufe KD has a given ratio to EF, as KD to EF, fo make CK to LE; H therefore the ratio of CK to LE is given; and CK is given, wherefore LEa is given: And becaufe as KD to EF, fo is CK to LE. and A fo b is CD to LF; the ratio of CD to LF is given But the ratio of CD to HB is given, wherefore e the ratio of HB to LF is given: and from HB, LF the given magnitudes HA, LE being taken, the remainders AB, EF fhall

E

c9. dat.

D F

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 demonftration.

Let AB, C, DE be three magnitudes, and let the exceffes 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 excefs of one of them above a given magnitude has a given ratio to the other.

G

D

Because the excefs of C above a given magnitude has a given ratio to AB; therefore a AB together with a given mag- a 14. dat! nitude has a given ratio to C: Let this given F magnitude be AF, wherefore FB has a given ratio to C: Alfo, because the excefs of C above A a given magnitude 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 has a given ratio to C, therefore b the ratio b of FB to GE is given: And from FB GE the given magnitudes AF DG being taken, the remainders AB. DE either have a given rio to one another, or the excefs of one of them above a given magnitude has a given ratio to the other.”

Bb

B

PROP.

9.

dat.

c 19. dat.

19.

PROP. XXVII.

IF there be three magnitudes: the excess of the first of which above a given magnitude has a given ra tio to the fecond; and the excefs of the second above a given magnitude has alfo a given ratio to the third: The excefs of the first above a given magnitude fhall have a given ratio to the third.

Let AB, CD, E be three magnitudes, the excess of the firft of which AB above the given magnitude AG, viz. GB, has a given ratio to CD; and FD the excefs of CD above the given magnitude CF, has a given ratio to E: The excefs of AB above a given magnitude has a given ratio to E.

Becaufe the ratio of GB to CD is given, as GB to CD, fo make GH to CF; therefore the ratio of GHA a 2. dat. to CF is given; and CF is given, wherefore a

H

F

GH is given; and AG is given, wherefore G the whole AH is given: And because as GB b 19.5. to CD, fo is GH to CF, and fo is b the remainder HB to the remainder FD; the ratio of HB to FD is given: And the ratio of FD to E is given, wherefore e the ratio of HB to E is given: And AH is given; therefore HB the excefs of AB above a given magnitude AH has a given ra tio to E.

c9 dat.

" Otherwife.

B DE

Let AB, C, D be three magnitudes, the excefs EB of the firft of which AB above the given magnitude AE has a given rato to C, and the excefs of C above a given

magnitude has a given ratio to D: The excels Al
of AB above a given magnitude has a given ra-
tic to D.

E

Becaufe EB has a given ratio to C, and the excets of C above a given magnitude has a gi-F d 24. dat. ven ratio to D; therefored the excefs of EB above a given nagnitude has a given ratio to

D: Let this given magnitude be EF; therefore
FB the excefs of EB above EF has a given ra- B C D
tio to D And AF is given, becaufe AE, EF

are

:

are given: Therefore FB the excess of AB above a given magnitude AF has a given ratio to D."

PROP. XXVIII.

25.

two lines given in pofition cut one another, the see N. point or points in which they cut one another are

given.

Let two lines AB, CD given in pofition cut one another in the point E; the point E is gi

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IF the extremities of a ftraight line be given in pofition; the straight line is given in pofition and mag

nitude.

Because the extremities of the ftraight line are given, they can be found a : let these be the points A, B, between which a 4. des. a ftraight line AB can be drawn b;

this has an invariable position, becaufe between two given points there

A

b. 1. Pofta

B

late.

can be drawn but one ftraight line: And when the ftraight line AB is drawn, its magnitude is at the fame time exhibited, or given: Therefore the ftraight line AB is given in pofition and magnitude.

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