27

PROP. XXX.

IF one of the extremities of a straight line given in pofition and magnitude be given; the other extremity thall alfo 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 straight line AC given in pofition; the other extremity is also given.

Becaufe the ftraight line is given in magnitude, one equal a 1. def. to it can be found a; let this be the ftraight line D: From the greater ftraight line AC cut off AB equal to the leffer D: Therefore the A other extremity B of the ftraight line AB is found: And the point B has al- D

ways the fame fituation; because any

B C

other point in AC, upon the fame fide 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 another fuch point can be found in AC produced upon the other fide of the point A.

IF

PROP. XXXI.

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

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

Through A draw a the ftraight line DAE parallel to BC; the ftraight line DAL has always the fame pofi. tion, becaufe no other ftraight line B can be drawn through A parallel to

AE
C

BC. Therefore the ftraight line DAE which has been found is

b 4. def. given in pofition.

PROP

PROP. XXXII.

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

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

G

29.

F

E

F

a 1. def.

C

B

b 23. I

ftraight line EC has always the

D

Because the angle is given, one equal to it can be found a; let this be the angle at D, at the given point C in the given ftraight line A AB make b the angle ECB equal to the angle at D: Therefore the

fame fituation, becaufe any other ftraight line FC drawn to the

point C makes with CB a greater or lefs 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 ftraight lines EC, GC upon one fide of AB that make equal angles with it, and which make equal angles with it when produced to the other fide.

PROP. XXXIII.

IF a ftraight line be drawn from a given point, to a ftraight line given in pofition, and makes a given angle with it; that ftraight line is given in pofition.

From the given point A, let the ftraight line AD be drawn to the ftraight line BC given in pofition, and make with it a given angle ADC; AD is given in po- E

Lition.

B

A

F

30%

a 31.

Thro' the point A, draw a the straight line EAF parallel to BC; and because thro' the given point A the ftraight line EAF is drawn parallel to BC which is given in pofition, EAF is therefore given in pofition b: And b 31. datu because the ftraight line AD meets the parallels BC, EF, the

Bb 3

D C

angla

29. angle EAD is equal to the angle ADC; and ADC is given, wherefore alfo the angle EAD is given: Therefore, because the ftraight line DA is drawn to the given point A in the straight line EF given in pofition, and makes with it a given angle EAD, d 31. dat. AD is given d in pofition..

3 1.

See N.

PROP. XXXIV.

IF from a given point to a straight line given in pofition, a ftraight line be drawn which is given in magnitude; the fame is alfo given in pofition.

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

A

Because the ftraight line is given in magnitude, one equal to a. def. it can be found a; let this be the ftraight line D: From the point A draw AE perpendicular to BC; and becaufe AE is the fhorteft of all the ftraight lines which can be drawn from the point A to BC, the ftraight line D, to which one equal is to be drawn from the point A to B BC, cannot be icfs than AE. If therefore D be equal to AE, AE is the ftraight line given in magnitude drawn from the given point A to BC: And 33. dat. it is evident that AE is given in pofition b, because it is drawn from the given point A to BC which is given in pofition, and makes with BC the given angle AEC.

D

E C

But if the ftraight 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, defcribe the circle GFH, and join AG, AH: Becaufe the circle GFH is given in pofic 6. def. tion, and the ftraight line BC is alfo given in pofition; there28. dat. fore their interfection G is given d;

and the point A is given; where

29. dat. fore AG is given in pofitione, that B GE
is, the ftraight line AG given in
magnitude (for it is equal to D)

and drawn from the given point A

HC

F

D

to the ftraight line BC given in pofition, is alfo given in pofition: And in like manner AH is given in pofition: Therefore in this cafe there are two ftraight lines AG, AH of the fame

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

IF

PROP. XXXV.

a ftraight line be drawn between two parallel ftraight lines given in pofition, and makes given angles with them; the ftraight line is given in magnitude.

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

A

EH B

32.

b 29. I.

In CD take the given point G, and through G draw a GHa 31. 1. parallel to EF: And becaufe CD meets the parallels GH, EF, the angle EFD is equal b to the angle HGD: And EFD is a given angle; wherefore the angle HGD is given: And becaufe HG is drawn to the given point G in the ftraight line CD given in pofition, and makes a given angle HGD ; C F G the ftraight line HG is given in pofi

D

tion c And AB is given in pofition; therefore the point H isc 32. đặt. given d; and the point G is alfo given, wherefore GH is given d 28. dat. in magnitude : And EF is equal to it; therefore EF is given e 29. dat. in magnitude.

PROP. XXXVI.

33°

IF a ftraight line given in magnitude be drawn be- See N. tween two parallel ftraight lines given in pofition;

it fhall make given angles with the parallels.

Let the ftraight line EF given in magnitude be drawn between the parallel ftraight lines AB, CD which are given in pofition; the angles AEF, EFC fhall be given.

A

Because EF is given in magnitude, a ftraight line equal to it can be found a ; let this be G: In AB take a given point C H, and from it draw b HK perpendicu lar to CD: Therefore the ftraight line G,

Bb4

G

EHB

e 6. def.

that is, EF, cannot be lefs than HK: And if G be equal to HK, EF alfo is equal to it; wherefore EF is at right angles to CD; for if it be not, EF would be greater than HK, which is abfurd: Therefore the angle EFD is a right, and confequently a given, angle.

But if the straight line G be not equal 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, defcribe the circle MLN, and join HM, HN: And because the circle c MLN, and the ftraight line CD are given in pofition, the points M, d 28. dat. Nared given; and the point H is given; wherefore the ftraight lines HM, HN are

A

H B

E

F

JOMEND

K

G

e 29. dat, given in pofitione; And CD is given in pofition; therefore the angles HMN, HNM are C f A. def. given in pofition f: Of the ftraight lines HM, HN, let HN be that which is not parallel to EF, for EF cannot be parallel to both of them; and draw EO parallel to HN: EO therefore is equal g to HN, that is, to G; and EF is equal to G, wherefore EO is equal to EF, and the angle EFO 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. def. magnitude k; and confequently the angle EFC.

g 34. I.

h 29. 1.

E.

PROP. XXXVII.

Fee N. IF a ftraight line given in magnitude be drawn from a point to a straight line given in pofition, in a given angle; the ftraight line drawn through that point parallel to the ftraight line given in pofition, is given in pofition.

AHF

Let the ftraight line AD given in magnitude be drawn from the point A to the ftraight line BC given in pofition, in the given angle ADC; the E ftraight line EAF drawn through A parallel to BC is given in position.

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

D G C