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Otherwise, 66 Let AB, C, D, be three magnitudes, the excess “ EB of the first of which AB above the given magni“ tude AE has a given ratio to C, and " the excess of C above a given mag AJ “ nitude has a given ratio to D: the excess of AB above a given magni

다. “ tude has a given ratio to D.

6 Because EB has a given ratio to “ C, and the excess of C above a given

magnitude has a given ratio to D; " therefore * the excess of EB above a

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

ci D * 24 Dat.

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

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 given.

Because the lines AB, CD, are given in position; they

C have always the same situation, and therefore the point,

# 4 Def. or points, in which they cut

A

-В one another have always the same situation: and because

D the lines AB, CD, can be found *, the point, or points, E

A

4 Def. in which they cut one another, are likewise found; and there

D fore are given in position *.

• 4 Def.

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E B

С

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

the straight line is given in position and magnitude. Because the extremities of the straight line are given,

* 4 Def.

1 Postulate.

they can be found *: let these be the points A, B, be-
tween which a straight line AB
can be drawn *; this has an

A

-B invariable position, because 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 same time exhibited, or given: therefore the straight line AB is given in position and magnitude.

27.

PROP. XXX.

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If one of the extremities of a straight line given in posi

tion 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 equal to it can be found*: let this be the straight line D: from the greater straight line AC cut off AB equal to the

А

B lesser D: therefore the other extremity B of the straight line AB is found : and the point B D 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 AB, that is, than D: therefore 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 A.

* 4 Def.

28.

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.

* 31. 1.

Through A draw * the straight line DAE parallel to

D

A

E BC; the straight line DAE has always the same position, because no other straight line

B

с can be drawn through A parallel to BC: therefore the straight line DAE which has been found is given * in position.

* 4 Def.

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* 1 Def.

If a straight line be drawn to a given point in a straight

line given in position, and make a given angle with it ; that straight line is given in position.

Let AB be a straight line given in position, and C a given point in it, the straight line drawn to C, which

E makes a given angle with

G CB, is given in position.

Because the angle is given, one equal to it can be found *; let this be the angle at D. At A

B the given pointc, in the given straight line AB make * the angle ECB equal to the angle at D: therefore

D the straight line EC has always the sanje situation, because any other straight line FČ, drawn to the point C, makes with CB a greater or less angle than the angle EBC, 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.

23. 1.

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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 be drawn to the straight line BC given in position, and

* 31. 1.

make with ita given angleADC:
AD is given in position.

EA F
Through the point A draw*
the straight line EAF parallel
to BC; and because through
the given point A the straight

B

D C line EAF is drawn parallel to BC, which is given in position, EAF is therefore given in position * And because the straight line AD meets the parallels 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, and makes with it a given * 32 Dat. angle EAD, AD is given * in position.

* 31 Dat.

* 29. 1.

31.

PROP. XXXIV.

See N.

If 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 found * ; 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 straight lines which can be drawn

A,
from the point A to BC, the straight
line D, to which one equal is to be
drawn from the point A to BC, can-
not be less than AE. If therefore

BE
D be equal to AE, AE is the straight D
line given in magnitude drawn from

the given point A to BC: and it is evident that AE is * 33 Dat. 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: be6 Def. cause the circle GFH is given in position *, and the

straight line BC is also given in position; therefore their intersection G is

• 28 Dat. given *; and the point A is

А, given ; wherefore AG is given in position *; that is,

* 29 Dat.

B G EI the straight line AG given

H C in magnitude (for it is equal to D), and drawn from the D.

F given point A to the straight Jine BC given in position, is also given in position ; and in like manner AH is given in position: therefore in this case there are two straight lines AG, AH, of the same given magnitude which can be drawn from a given point A to a straight line BC given in position.

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*

* 29. 1.

If a straight line be drawn between two parallel straight

lines given in position, and make given angles with them; the straight line is given in magnitude.

Let the straight line EF be drawn between the parallels AB, CD, which are given in position, and make the given angles BEF, EFD: EF is given in magnitude.

In CD take the given point G, and through G draw * * 31. 1. GH parallel to EF: and because CD meets the parallels GH, EF, the angle EFD is equal to the angle HGD: and EFD is a given angle; wherefore the angle A E H B HGD is given : and because HG is drawn to the given point G, in the straight line CD, given in position, and makes a

С F G D given angle HGD; the straight ine HG is given in position *: and AB is given in posi- * 32 Dat. tion: therefore the point H is given*: and the point G is * 28 Dat. also given, wherefore GH is given in magnitude *: and • 29 Dat. EF is equal to it, therefore EF is given in magnitude.

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If a straight line given in magnitude be drawn between See N.

two parallel straight lines given in position, it shall
make given angles with the parallels.
Let the straight line EF given in magnitude be drawn

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