27

PROP. XXX.

IF one of the extremities of a straight line given in position 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 ftraight 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. dcf. to it can be found a; let this be the straight line D: From the

greater straight line AC cut off AB
equal to the leffer D: Therefore the A

other extremity B of the straight line
AB is found: And the point B has al- D

ways the fame fituation; because any

BC

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

28.

PROP. XXXI.

IF a straight line be drawn through a given point parallel to a ftraight line given in position; that ftraight 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.

can be drawn through A parallel to

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

b 4. def. given b in pofition.

PROP. PROP. ΧΧΧΙΙ.

IF a straight line be drawn to a given point in a straight line given in pofition, 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

29.

poin in it, the straight line drawn
to C which makes a given angle
with CB, is given in pofition.
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 straight line A

G

AB make b the angle ECB equal to the angle at D: Therefore the straight line EC has always the same situation, 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 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 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 pofition.

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

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

A

F

DC

a 31.

3

given in pofition, EAF is therefore given in position b: And b 31. dat

because the straight line AD meets the parallels BC; EF, the

Bb3

C 29.

angle EAD is equal to the angle ADC; and ADC is given, wherefore alfo 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 d in position.

3.

Sce N.

PROP. XXXIV.

IF from a given point to a straight line given in position, a straight line he drawn which is given in magnitude; the fame is also given in position.

Let A be a given point, and BC a fstraight 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 equal to as. 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
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

D

A

EC

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

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 position, and makes with BC the given angle AEC.

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, describe the circle GFH, and join AG, AH: Because the circle GFH is given in pofitions, and the straight line BC is also given in position; there

c 6. def.

d 28. dat. fore their interfection G is given d;

A

and the point A is given; where

29. dat. fore AG is given in pofitione, that BG/E

HC

is, the ftraight line AG given in

magnitude (for it is equal to D)

F

and drawn from the given point A

D

to the straight line BC given in position, is also given in pofi.

tion: And in like manner AH is given in position: Therefore

in this cafe there are two straight lines AG, AH of the fame

given

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

PROP. XXXV.

IF 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 mag-

nitude.

Let the straight 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.

34.

In CD take the given point G, and through G draw a GHa 3t. 1. parallel to EF: And because CD meets the parallels GH, EF, the angle EFD is equal b to the angle HGD:

ЕН В

b 29. 1.

And EFD is a given angle; A

wherefore the angle HGD is given: And
because HG is drawn to the given point
G in the straight line CD given in pofi-
tion, and makes a given angle HGDC

FGD

the straight line HG is given in pofi-
tion c: And AB is given in position; therefore the point Hisc 32. dat.
given d; and the point G is also given, wherefore GH is givend 28. dat.
in magnitude: And EF is equal to it; therefore EF is givene 29. dat.
in magnitude.

PROP. XXXVI.

33:

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

tween the parallel straight lines AB, CD
which are given in position; the angles
AEF, EFC shall be given.

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

Bb4

A

C

a 1. def.

FKD

b12.

that

G

c 6. def.

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 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, describe the circle MLN, and join HM, HN: And because the circle - MLN, and the straight line CD are given in position, the points M,

d 28. dat. Nared given; and the point

E

H B

His given; wherefore the A straight lines HM, HN are e 29. dat. given in pofitione: And CD is given in position; therefore the angles HMN, HNM are CF

G

f A. def. given in pofition f: Of the straight lines HM, HN, let

g 34. 1.

h 29. 1.

K

OMIND

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

kr. def. magnitude k; and confequently the angle EFC.

E.

See N.

PROP. XXXVII.

IF a ftraight 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 straight line given in position, is given in pofition.

Let the ftraight line AD given in magnitude be drawn from

the point A to the straight line BC given in

position, in the given angle ADC; the E

AHF

straight 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 DGC

to a given point G in the straight line BC gi

ven