17.

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 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 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.

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

A

F

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 given ratio to C; and because FB, GE have each of them a given ratio to C, they have a given ratio a to one another. But to FB, GE the given magnib 18. dat. tudes AF, DG are added; therefore the whole magnitudes AB, DE have either a

a 9. dat.

18.

D

G

given ratio to one another, or the excess of BCE one of them above a given magnitude has

a given ratio to the other.

PROP. XXVI.

If 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 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 HA is given: And because as a 2. dat. GD to AB, so is CG to HA, and so is CD to HB; the b 12. 5. ratio of CD to HB is given: Also because KD has a given ratio to EF, as KD to EF, so make CK to LE; therefore the ratio of CK to LE is given; and CK is given, wherefore LE is given: And because as KD to EF, so is CK to LE, and so 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 is given: And from HB, LF the given magnitudes 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.

Another Demonstration.

H

G

KE

c 9. dat.

BDF

Let AB, C, DE be three magnitudes, and let the excess 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.

F

G

D

d 19. dat.

Because the excess of C above a given magnitude has a given ratio to AB; therefore AB together with a given mag- a 14. dat. nitude has a 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 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

CE

b 9. dat.

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. c 19. dat.

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 magnitude 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, so make GH to CF; therefore the ratio of GH to CF is given, and CF is given, a 2. dat. wherefore a GH is given; and AG is given, wherefore the whole AH is given: And because as GB to CD, so is GH to CF, and so is the remainder HB to the remainder

b 19. 5.

FD; the ratio of HB to FD is given: And

A
G

C

HF

c 9. dat. the ratio of FD to E is given, wherefore c the ratio of HB to E is given: And AH is given; therefore HB the excess of AB above RDE a given magnitude AH has a given ratio

to E.

"Otherwise.

Let AB, C, D be three magnitudes, the excess EB of the first of which AB above the given magni- A

E

tude 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. Because EB has a given ratio to C, and F the excess of C above a given magnitude d 24. dat. has a given ratio to D; therefore the excess of EB above a given magnitude has a given ratio to D: Let this given magnitude be EF: therefore FB the excess of

B C D

EB above EF has a given ratio to D: And AF is given, be

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

If two lines given in position cut one another, the See N. point or points in which they cut one another are given.

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

If the extremities of a straight line be given in position; the straight line is given in position and magnitude.

26.

Because the extremities of the straight line are given, they can be found: Let these be the points, A, B, between which a 4. def. a straight line AB can be drawn b; this has an invariable position, because between two given points

there can be drawn but one straight

A

B

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.

2A2

b 1. Postulate.

If one of the extremities of a straight line given in position and magnitude be given; the other extremity shall also be given.

Let the point A be given, viz. one of the extremities of a straight 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 a 1. def. to it can be found: let this be the straight line D: From the greater straight line AC cut off AB equal to the less D: Therefore

A

the other extremity B of the straight
line AB is found: And the point B
has always the same situation; be-
cause any other point in AC, upon

B C

D

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: Thereb 4. def. fore the point B is given: And it is plain, another such point can be found in AC, produced upon the other side of the point

28.

a 31. 1.

A.

PROP. XXXI.

If a straight line be drawn through a given point parallel to a straight line given in position; that straight line is given in position.

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

a

Through A draw the straight
line DAE parallel to BC; the D
straight line DAE has always the
same position, because no other
straight line can be drawn through
A parallel to BC: Therefore the

B

b 4. def. straight line DAE which has been found is given in position.