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

A

GB the ratio of) AG to CD; AB is

C 10. S. greater c than AG: And AB,

С F D AG are given ; therefore the remainder BG is given : And because as AE O CF, fo is AG to CD, and so is a EG to FD; a 19. s. the ratio of EG to FD is given : And GB is given ; therefore EG, the excess of EB above a given magnitude GB, has a given ratio to FD. The other cale is shewn in the same way.

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IF there be three magnitudes, the first of which has see N.

a given ratio to the fecond, 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. S.

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 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

Al AG to CF; therefore the ratio of AG to CF is given ; and CF is given, wherefore a AG is given : And because as AB to CD, so is AG

с to CF, and to is b GB to FD: the ratio of GB to FD is given. And the ratio of FD to E is F given, wherefore c the ratio of GB to ,E is given, and AG is given; therefore GB the ex: cess of AB above a given magnitude AG has a given ratio to E.

B D 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 mag. nitude 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 fame with the propofition.

Cor.

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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 fame excefs fhall have a given ratio to the first; as is evident from the gth dat.

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IF
F 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 thall 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.

1

Let AB, C, DE be three magnitudes, and let the exceffes of cach 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

A cess of DE above the given magnitude DC

D 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,

GH GE the given magnitudes AF, DG are add 6. 18. dat. ed; therefore b the whole magnitudes AB, DE

have either a given ratio to one another, o. B'C' E
the excess of one of them above a given mag-
nicude has a given ratio to the other.

PRO P. 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 magnitudie ihall have a giyen ratio to the other.

Let AB, CD, EF be three magnitudes, and let GD the ex. cess of one of them CD above the given magnitude CG have a given ratio to AB; and also let KD the excess of the same CD above the given magnitude CK have a given ratio to EF, either AB has a given ratio to EF, or the excels 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 10 AB, so make CG to HA; therefore the ratio of CG 10 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 so is b CD 10 HB; the ra. b 12. s. tio of CD to HB is given : Also because KD has a given ratio to EF, as KD to EF, so makę CK LE ; H therefore the ratio of CK to LE is given ; and CK is given, wherefore LE a is given : And C because 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, K wherefore c the ratio of HB to LF is given : and from HB, LF the given magnitudes HA, LE being taken, the remainders AB, EF fhall

BI DI 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.

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" Another demonstration. Let AB, C, DE be three magnitudes, and let the excefies 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.

Becaufe the excess 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: Also, because the excefs of Carovi 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

B

C ratio io C: And FB has a given :atio to C, therefore by the ratio 1 9. dar. of FB O GE is given : And from IB GE the given magnitudes AF DG being taken, the remainders AB. DE erher have a given crio to one another, or the excess of one of them above a given magnitude has a given ratio to the other."

Bb

PROP.

C 19. date

19.

PRO P. XXVII.

IF there be three magnitudes : the excess of the first

of which above a given magnitude has a given ra. tio to the second; and the excess of the second above a given magnitude has also a given ratio to the third : The excess of the firít above a given magnitude shall have a given ratio to the third.

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

Because the ratio of GB to CD is given, as GB to CD, fo

makc GH to CF; therefore the ratio of GHA a 2. dat. to CF is given ; and CF is given, wherefore a

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 re-H+ F

mainder HB to the remainder FD; the ratio

of HB to FD is given: And the ratio of FD
cg dat. 1o E is given, wherefore c the ratio of HB 10 B D E

E is given : And AH is given ; therefore HB
the excess of AB above a given magnitude AH has a given ra-
tio to E.

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• Otherwise.

Let AB, C, D be three magnitudes, the excess EB of the first of which AB above the given magnitude AE has a givea raco to C, and the excess of C above a given magnitude has a given ratio to D: The excess of AB above a given magniiude has a given ratic to D.

Because 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 excess of EB

above a given nagnitude has a given ratio to
D: Let this given magnitude be LF; therefore
FB the excets of EB above EF has a given ra. B C D
tio to Di And AF is given, becaufe AE, ET

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

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IF two

two lines given in position cut one another, the See N.

point or points in which they cut one another are given.

-B

a 4. def.

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

Because the lines AB, CD are given in position, they have A always the same situation a, and therefore the point, or points in which they cut one another

D have always the same situation :

E And because the lines AB, CD A

-B can be found a, the point, or points, in which they cut one C another, are likewise found; and therefore are given in position 4.

PROP. XXIX.

IF the extremities of a straight line be given in posi

tion; the straight line is given in position and may

nitude.

Because the extremities of the straight line are given, they can be found a : let these be the points A, B, between which a 4

def. a straight line AB can be drawn b;

b 1. Pofta this has an invariable position, be

A

B late. cause between two given points there can be drawn but one straight line: And when the straight line AB is drawn, its magnitude is at the fame time exhibited, or given : Therefore the straight line AB is given in pofition and magnitude. B b 2

PROP.

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