COR. 2. Alfo if the first has a given ratio to the second, and the excefs of the third above a given magnitude has also a given ratio to the fecond, the fame excefs hall have a given ratio to the firft; as is evident from the 9th Dat.

IF

PROP. XXV.

F there be three magnitudes, the excefs of the first wher of 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 firft 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 excefs of one of them above a given magnitude has a given ratio to the other,

A

D

Let FB the excess of AB above the given magnitude AF have a given ratio to C; and let GE the exccfs of DE above the given magnitude DG have a gi ven ratio to C; and becaufe FB, GE have each of them a given ratio to C, they have a gia. 9. Dat. ven ratio to one another. but to FB, GE the given magnitudes AF, DG are added; there

b. 18. Dat. fore the whole magnitudes AB, DE have either

a given ratio to one another, or the excefs of BCE
one of them above a given magnitude has a given ratio to the other.

IF

F there be three magnitudes the exceffes of one of which above given magnitudes have given ratios to the other two magnitudes; thefe two fhall either have a given ratio to one another, or the excefs of one of them above a given magnitude fhall have a given ratio to the other.

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 alfo 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 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 to AB, fo make CG to HA; therefore the ratio of CG to HA is given; and CG is gi

a

C

ven, wherefore HA is given. and becaufe as GD to AB, fo is CG a. 2. Dat, to HA, and fo is CD to HB; the ratio of CD to HB is given. b. 12. 5. alfo because KD has a given ratio to EF, as KD H to EF, fo make CK to LE; therefore the ratio. of CK to LE is given; and CK is given, wherefore LE is given. and becanfe as KD to EF, fo is CK to LE, and fo is CD to LF; the ratio of CD to LF is given. but the ratio of CD to HB is given, wherefore the ratio of HB to LF

c

KE

G

is given. and from HB, LF the given magnitudes BD F HA, LE being taken, the remainders AB, EF fhall 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 4.

"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 cxccfs of one of them above a given magnitude has a given ratio to the other.

A

G

D

c. 9. Dat.

d. 19. Dat.

Because the excefs of C above a given maguitude 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. alfo, K becaufe the excefs of C above a given magnitude has a given ratio to DE, therefore* 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 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 excefs of one of them above a given magaitude has a given ratio to the other."

B CE

b. 9. Dat.

C. 19. Dat.

19.

b. 19. 5.

PRO P. XXVII.

IF there be three magnitudes the excefs of the first of which above a given magnitude has a given ratio to the fecond; and the excefs of the fecond above a given magnitude has alfo a given ratio to the third. the exccfs of the firft above a given magnitude fhall have a given ratio to the third.

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

Because the ratio of GB to CD is given, as GB to CD, so mat GH to CF; therefore the ratio of GH to CF is

H+

a. 2. Dat. given; and CF is given, wherefore CH is gi-A
ven; and AG is given, wherefore the whole
AH is given. and becaufe as GB to CD, fo is
GH to CF, and fo is the remainder HB to the
remainder FB; the ratio of HB to FD is given.
9. Dat. and the ratio of FD to E is given, wherefore
the ratio of IIB to Eis given. and AH is given;
therefore HB the excefs of AB above the given
magnitude AH has a given ratio to E.

"Otherwife.

BDE

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

given ratio to D. the excess of AB above a gi-A,

ven magnitude has a given ratio to D.

Becaufe EB has a given ratio to C, and the E-
excefs of C above a given magnitude has a given.

d. 24. Dat. ratio to D; therefore the excess of EB above a F
given magnitude has a given ratio to D. let this
given magnitude be EF, therefore FB the ex-
cefs of EB above EF has a given ratio to D.
and AF is given, becaufe AE, EF are given.

B C D

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

PROP. XXVIII.

25.

IF

F 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 given.

Because the lines AB, CD are given in pofition, they have always the fame fituation, and therefore A the point, or points, in which they cut one another have always the fame fituation. and because the lines AB, CD can be found', the point, or points, in which they cut one another, are likewife found; and therefore are given in pofition".

A

B

a. 4. Def.

D

IB

D

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

nitude.

Because the extremities of the ftraight line are given, they can

be found; let these be the points A, B, between which a itraight a. 4. Def.

line AB can be drawn ; this has an

invariable pofition, becaufe between A

two given points there can be drawn

B

but one ftraight line. and when the ftraight line AB is drawn, its magnitude is at the fame time exhibited, or given. therefore the

b. 1. Poftu

late.

ftraight line AB is given in pofition and magnitude.

27.

2. 1. Def.

IF

PROP. XXX.

F one of the extremities of a ftraight line given in pofition and magnitude be given; the other extremity fhall also be given.

Let the point A be given, to wit one of the extremities of a ftraight line given in magnitude, and which lies in the ftrag line AC given in pofition; the other extremity is also given.

Because the ftraight line is given in magnitude, one equal to can be found; let this be the ftraight line D. from the gre ftraight line AC cut off AB equal to the A

leffer D. therefore the other extremity

B of the ftraight line AB is found. and-
the point B has always the fame fitua
tion, becaufe any other point in AC,

D

B C

upon the fame fide of A, cuts off between it and the point A

greater or lefs ftraight line than AB, that is than D. therefore the b. 4. Def. point B is given. and it is plain another fuch point can be found in AC produced upon the other fide of the point A.

28.

a. 31. 1.

PROP. XXXI.

IF a ftraight line be drawn through a given point parallel to a ftraight line given in pofition; that ftraight line is given in pofition.

Let A be a given point, and BC a straight line given in pofition; the ftraight line drawn thro' A parallel to BC is given in poftica. Thro' A draw the straight line DAE parallel to BC; the ftraight line DAEhas always the fame pofition, becaufeB no other ftraight line can be drawn through A parallel to BC. therefore the

A E

C

b. 4. Def. ftraight line DAE which has been found is given in pofition,