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COR. 2. Also if the first has a given ratio to the fecond, and the excess of the third above a given magnitude has also a given ratio to the second, the fame excess shall have a given ratio to the first; as is evident from the oth Dat.

PRO P. XXV.

I

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 giren magnitude has a given ratio to the same second. the first thall either have a given ratio to the third, or the exces 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 o her.

Lct FB the excess of AB above the given magnitude AF have a given ratio to C; and let CE the excess of DE above the given magnitude DG have a gi- A ven ratio to C; and because FB, GE have each F.

D of them a given ratio to C, they have a gi

Gt a. 9. Dat. von ratio a to one another. but to FB, GE the

given magnitudes AF, DG are added; thereB. 18. Dat. fore b the whole magnitudes AB, DE have either

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

18.

PRO P. XXVI.
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.

Let

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 same 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 b CD to HB ; the ratio of CD to HB is given. b. 12. 5. allo because KD has a given ratio to EF, ?s KD to EF, fo make CK to LE; therefore the ratio of CK to LE is given; and CK is given, where C fore. LE is given. and because as KD to EF, A fo is CK to LE, and so bis CD to LF; the ratio

C of CD to LF is given. but the ratio of CD to IIB is given, wherefore the ratio of HB to LF is given, and from HB, LF ihe given magnitudes B! DI HA, LE being taken, the remainders AB, EF fhall either have a given ratio to one another, or the excess of one of thein 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 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 giren magnitude has a given ratio to the other.

Because the excets of C above a given magnitude has a given ratio to AB, therefore * 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, F because the excess of C above a given magnitude has a given ratio to DE, therefore * DE together

A 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 B! C to C, therefore b the ratio 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 ratio to one another, or the excess of one of them above a given magnitude has a given ratio to the other."

PROP. c. 19. Dat,

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19.

XXVII.

PRO P.
If there be three magnitudes the excess of the first of

which above a given magnitude has a given ratio to the second; and the excess of the second above a given mag. nitude has also a given ratio to the third. the excess of the first 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 make

GH to CF; therefore the ratio of GH to CF is a. . Dat. given; and CF is given, wherefore - GH is gi- A

ven ; and AG is given, wherefore the whole G+

AH is given. and because as G B to CD, so is, b. 19. 5. GH to CF, and so is o the remainder HB to the H

F
remainder FB; the ratio of HB to FD is given.
c. 9. Dat. and the ratio of FD to E is given, wherefore C

the ratio of HB to E is given. and AH is given; B D E
therefore HB the excess of AB abore the given
magnitude AH has a given ratio to E.

BD

• Otherwise,
Let AB, C, D be three magnitudes, the excess EB of the first
of which AB above the given magnitude AE has a given ratio to C,
and the excess of C above a given magnitude has a given ratio to D.
the excess of AB above a given magnitude has a
given ratio to D.

A
Because EB has a given ratio to C, and the

E
excess of C above a given magnitude has a gi-
d. 24. Dat. ven ratio to D; therefore d the excess of EB a-

FH
bove a given magnitude has a given ratio to D.
let this given magnitude be EF, therefore FB
the excess of EB above EF has a given ratio to
D. and AF is given, because AE, EF are given.

B C D

there

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

PROP.

XXVIII.

25.

IF two lines given in polition cut one another, the point See N.

or points in which they cut one another are given.

a. 4. Dcf.

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

С given in position, they have al

E ways the same situation, and A

B therefore the point, or points, in which they cut one another have

D always the same situation, and be

E cause the lines AB, CD can be A

-B found, the point, or points, in in which they cut one another,

C

D are likewise found; and therefore are given in position '.

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IF the extremities of a straight line be given in position ;

the straight line is given in position and magnitude.

late.

Because the extremities of the straight line are given, they can be found ; let these be the points A, B, between which a straight a. 4. Def. line AB can be drawn b; this has an

b. I. Poftu. invariable position, because between A

-В two given points there can be drawn but one straight line. and when the straight line AB is drawn, its magnitude is at the same time exhibited, or given, therefore the Straight line AB is given in position and magnitude.

PROP.

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IF

one of the extremities of a straight live given in pofi.

tion and magnitude be given; the other extremity shall also be given.

Let the point A be given, to wit ane of the extremities of a Nraight line given in magnitude, and which lies in the straight line AC given in position; the other extremity is also given.

Because the straight line is given in magnitude, one equal to it a. 1. Def. can be found ; let this be the straight line D. from the greater

straight line AC cut off AB equal to the
leffer D. therefore the other extremity A B С
B of the straight line AB is found, and
the point B has always the fame fitua. D
tion, because any other point in AC,
upon the same side of A, cuts off between it and the point A a

greater or less straight line than AB, that is than D. therefore the 1.4. Def. point B is given b. and it is plain another such point can be found

in AC produced upon the other side of the point A.

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TF a straight livre be drawn thro' a given point parallel

to a straight line given in position; that straight line is given in position.

2. 31. I.

Let A be a given point, and BC a straight line given in position; the straight line drawn thro' A parallel to BC is given in pofition.

Thro' A drawa the straight line DAE
parallel to BC; the straight line DAE D A

E
has always the same position, because
no other straight line can be drawn B

thro' A parallel to BC. therefore the b. 4. Def. straight line DAE which has been found is given b in position.

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PROP. ]

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