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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 gi-G ven, wherefore the whole AH is given : And because as GB to CD, so is GH to 19. 5. CF, and so is the remainder HB to the HF remainder FD; the ratio of HB to FD is

given: And the ratio of FD to E is

C

given, BDE

• 9 Dat. wherefore the ratio of HB to E is given : And AH is given; therefore HB the excess of AB above a given magnitude AH has a given ratio to E.

d24

" 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 A " ratio to D: The excess of AB above a F " given magnitude has a given ratio to D.

F

"Because EB has a given ratio to C, " and the excess of C above a given mag" nitude has a given ratio to D; therefored the excess of EB above " magnitude has a given ratio to D: Let "this given magnitude be EF; therefore FB the excess

Dat."

a

given

B

CD

" of EB above EF has a given ratio to D: and AF is

" given, because AE, EF, are given: Therefore FB the

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

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Ir 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 ano

ther in the point E; the point E is given.

Because the lines AB, CD,

С.

are given in position; they have

always the same situation, and therefore the point, or points, in A.

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which they cut one another have always the same situation: And because the lines AB, CD, can be founda, the point, or points, in which they cut one another, are likewise found; and therefore are given in positiona.

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PROP. XXIX.

Ir 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 founda: Let these be the points A, B, be- a 4 Def.

tween which a straight line AB

can be drawn; this has an in

A

variable position, because be

-Bb 1 Postu

late.

tween two given points there can be drawn but one straight line: And when the straight line AB is drawn, its magni

tude is at the same time exhibited, or given : Therefore the

straight line AB is given in position and magnitude.

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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, to wit, 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 • 1 Def. equal to it can be founda: let this be the straight line D:

From the greater straight line AC A

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cut off AB equal to the lesser D:

Therefore the other extremity B of

the straight line AB is found: And D

the point B has always the same situation; 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 4 Def. AB, that is, than D: Therefore the 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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Ir 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 31. 1.

Through A drawa the straight D

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line DAE parallel to BC; the

straight line DAE has always the

same position, because no other B

C

straight line can be drawn through A parallel to BC:

Therefore the straight line DAE which has been found is

4 Def. given in position.

PROP. XXXII.

IF a straight line be drawn to a given point in a straight line given in position, and makes a given angle with it; that straight line is given in position.

29.

Let AB be a straight line given in position, and Ca

given point in it, the straight line
drawn to C, which makes a given G

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angle with CB, is given in posi

F

tion.

Because the angle is given, one equal to it can be founda; A let this be the angle at D. At the given point C, in the given straight line AB, make the an

C

B1 Def.

b 23.1.

gle ECB equal to the angle at

D

D: Therefore the straight line EC has always the same situation, because any other straight line FC, drawn to the point C, makes with CB a greater or less angle than the angle ECB, or the angle at D: Therefore the straight line EC, which has been found, is given in position.

It is to be observed, that there are two straight lines EC, GC, upon one side of AB that make equal angles with it, and which make equal angles with it when produced to the other side.

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If a straight line be drawn from a given point to a straight line given in position, and make a given

angle with it, that straight line is given in position.

From the given point A let the straight line AD he drawn to the straight line BC given in position, and make

with it a given angle ADC: AD E

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is given in position.

Through the point A draw a

31.1.

the straight line EAF parallel to

BC; and because through the B

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given point A the straight line EAF is drawn parallel to

BC, which is given in position, EAF is therefore given in

position. And because the straight line ADmeets the paral- 31 Dat.

e

€ 29. 1. lels BC, EF, the angle EAD is equal to the angle ADC; and ADC is given, wherefore also the angle EAD is given : Therefore, because the straight line DA is drawn to the given point A in the straight line EF given in position, d 32 Dat. and makes with it a given angle EAD, AD is given d in position.

PROP. XXXIV.

See N. Ir from a given point to a straight line given in position, a straight line be drawn which is given in magnitude; the same is also given in position.

Let A be a given point, and BC a straight line given in position, a straight line given in magnitude, drawn from the point A to BC, is given in position.

Because the straight line is given in magnitude, one *1 Def. equal to it can be founda; let this be the straight line D: From the point A draw AE perpendicular to BC: and because AE is the shortest of all the

A

BEC

straight lines which can be drawn from
the point A to BC, the straight line
D, to which one equal is to be drawn
from the point A to BC, cannot be less
than AE. If therefore D be equal to
AE, AE is the straight line given in magnitude drawn
from the given point A to BC: And it is evident that AE

D

b 33 Dat. is given in position, because it is drawn from the given point A to BC, which is given in position, and makes with BC the given angle AEC.

But if the straight line D be not equal to AE, it must be greater than it: Produce AE, and make AF equal to D; and from the centre A, at the distance AF, describe the circle GFH, and join AG, AH: Because the circle 6. Def. GFH is given in positions, and the straight line BC is also d 28 Dat. given in position; therefore their intersection G is givend;

and the point A is given;

wherefore AG is given in

* 29 Dat. positione; that is, the straight

A

line AG given in magnitude B GE

HC

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A to the straight line BC given in position, is also given in

(for it is equal to D), and

drawn from the given point

position; And in like manner AH is given in position:

Therefore in this case there are two straight lines AG,

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