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IF
F one of the extremities of a straight line given in

polition and inagnitude be given; the other extremity thall 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 pofition; the other extremity is also given.

Because the straight line is given in magnitude, one equal a 1. def. to it can be found a ; let this be obe straight line D: From the

greater straight line AC cut off AB
egal to the leffer D): Therefore the A B С
other extremity B of the straight line
AB is found : And the point B has al-D
ways the same timation; 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, b 4. def. than D): Therefore the point B is given b: And it is plain an

other such point can be found in AC produced upon the other fide of ihe point A.

28.

PRO P. XXXI.

JF a straight line be drawn through a given point pa

rallel to a straight line given in position; that itraight 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 pofition.

Through A draw a the straight line

31. 1.

Α Ε

C

line DAC has always the same poli-
tion, because no other straight line B
can be drawn through A parallel to

BC. Therefore the straight line DAE which has been found is b 4. def. given bin position, .

PROP. .

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F E

a 1. def.

IF
F 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.

Let AB be a straight line given in position, and C a given poin in it, the straight line drawn to C which makes a given angle with CB, is given in pofition.

G Because the angle is given, one

F equal to it can be found a; let this be the angle at D, at the given point C in the given straight line

B AB make b the angle ECB equal to the angle at D: Therefore the straight line EC has always the fame firuation, because any other

D 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 pofition.

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

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PROP. XXXIII.

30

IF a straight line be drawn from a given point, to

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

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From the given point A, let the straight line AD be drawn to the straight line BC given in position, and make with it a given angle ADC; AD is given in po. E A F fition.

Thro' the point A, draw a the straight line EAF parallel to BC; and because thro' the given point A the straight line EAF is drawn parallel to BC which is

B given in pofition, EAF is therefore given in position b: And b) 31. datu because the straight line AD meets the parallels BC; EF, the

angle

Bb 3

6 29: 1. angle LAD is equal c 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 angle EAD, d 32. dat. AD is given a in position.',

3

PRO P. XXXIV.

Sec N.

IF from a given point to a straight line given in pofi

tior, a straight line be drawn which is given in mag. nitude; the fame is also given in position.

Let A be a given point, and BC a straight line given in pofition, 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 equal to a 1. def. it can be found a ; 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

Α,
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

B E C
BC, cannot be less than AE. If therefore

D
D be equal to AE, AE is the straight line

given in magniiude drawn from the given point A to BC: And be 33. dat. it is evident that AE is given in position b, because it is drawn

from the given point A to BC which is given in position, and makes with BC ihe given angle AEC.

But if the straight line D bę not equal to AE, it must be greater thary 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 GFH is given in polic 6. slef. tione, and the straight line BC is also given in polition; there. – 23. dat. fore their intersection G is given d.;

A А and the point A is given ; where† 29. dat. fore AG is given in pofition, that

B G/E H C is, the straight line AG given in magnitude (for it is equal to D) and drawn from the given point A 10 the straight line BC given in pofition, is also given in pofi

. tion : And in like manner AH is given in position: Therefore in this case there are two straight lines AG, AH of tbe fame

D

given magnitude which can be drawn from a given point A to a straight line BC given in position.

PRO P. XXXV.

32.

II
F a straight line be drawn between two parallel

straight lines given in position, and makes 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 a GH2 31. 1. Parallel to EF : And because CD meets the parallels GH, EF, the angle EFD is equal b to the angle

b 29.1. HGD : And EFD is a given angle ;

A EH B wherefore the angle HGD is given : And because HG is drawn to the given point G in the straight line CD given in pofition, and makes a given angle HGD

;

с F G D the straight line HG is given in pofi. tion c: And AB is given in position ; therefore the point H is c 32. dat. given d; and the point G is also given, wherefore GH is given d 28. dat. in magnitude e : And EF is equal to it; therefore EF is given e 29. data in magnitude.

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PRO P. XXXVI.

33:

IF a straight line given in magnitude be drawn be. See N.

tween two parallel Itraight lines given in position; it shall make given angles with the parallels.

Let the straight line EF given in magnitude be drawn between the parallel straight lines AB, CD which are given in position; the angles

A E HE AEF, EFC shall be given.

Because EF is given in magnitude, a straight line equal to it can be found a ;

a 1. dets let this be G: In AB cake a given point C F KD H, and from it draw b HK perpendicular to CD: Therefore the straight line G,

that

b 12. im

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that is, EF, cannot be less than HK: And if G be equal to HK, EF also is equal to it; wherefore EF is at right angles to CD; for if it be nor, EF would be greater than HK, which is absurd: Therefore the angle EFD is a right, and consequently a given, angle.

But if the straight line G be not cqual to HK, it must be greater than it : Produce HK, and take HL, equal to G; and

from the centre H, at the distance HL, describe the circle c 6. def. MLN, and join HM, HN: And because the circle c MLN,

and the straight line CD are given in position, the points M, d 28. dat. Nared given ; and the point

H is given ; 'wherefore the A E H B

straight lines HM, HN are € 29. dat. given in pofitione; And CD is given in position ; therefore

R the angies HMN, HNM are

с

OM Ľ ND { A. dcf. given in position f: Of the

straight lines HM, HN, let G-
HN be that which is not parallel to EF, for EF cannot be paral-

lel io both of ihem; and draw EO parallel to HN: EO there8 34. I.

fore is equal to HN, that is, to G; and EF is equal to G, wherefore EO is equal to EF, and the angle E FO to the angle EOF, that is h, to the given angle HNM, and because the angle HNM which is equal to the angle EFO or EFD has been found,

therefore the angle EFD, that is, the angle AEF, is given in k 1. def. magnitude k; and confequently the angle EFC.

F

h 20. I.

PRO P. XXXVII,

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IF a straight line given in magnitude be drawn from

a point to a straight line given in position, in a given angle ; the straight line drawn through that point parallel to the straicht line given in position, is given in position.

Let the firaight line AD given in magnitude be drawn from the point A to ihe straight line BC given in poficion, in the given angle ADC; the E A HF traight line EAF drawn through A parallel to BC is given in pofition.

In BC take a given point G, and draw GH parallel to AD: And because HG is drawn B D G C to a given point G in the straight line BC gi

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