are given: Therefore FB the excess of AB above a given maghitude AF has a given ratio to D."

PROP. XXVIII.

IF two lines given in pofition cut one another, the point See N. or 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 gi

ven.

C

A

Because the lines AB, CD are given in pofition, they have always the fame fituation, and therefore the point, or points in which they cut one another have always the fame fituation : And because the lines AB, CD can be found", the point, or points, in which they cut one another, are likewife found; and therefore are given in position *.

A

25.

F

IF

the extremities of a ftraight line be given in pofition; the ftraight line is given in pofition and magnitude.

Because the extremities of the straight line are given, they

can be found: Let thefe be the points A, B, between which a 4 def.

a ftraight line AB can be drawn ;

this has an invariable pofition, be- A

caufe between two given points there

B

can be drawn but one straight line: And when the ftraight line AB is drawn, its magnitude is at the fame time exhibited, or given: Therefore the ftraight line AB is given in pofition and magnitude.

Bb 2

PROP.

b J, Pofiulate.

27.

a I. def.

PROP. XXX.

IF one of the extremities of a ftraight line given in pofition and magnitude be given; the other extremity fhall 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 alfo given. Because the ftraight line is given in magnitude, one equal to it can be found; let this be the ftraight line D: From the greater ftraight line AC cut off AB A BC equal to the leffer D: "Therefore the other extremity B of the ftraight line AB is found: And the point B has al- D ways the fame fituation ; because any

other point in AC, upon the fame fide of A, cuts off between it and the point A a greater or lefs ftraight line than AB, that is, b 4, def. than D: Therefore the point B is given: And it is plain another fuch point can be found in AC produced upon the other fide of the point A.

18.

a 31. 1.

PROP. XXXI.

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

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

a

Through A draw the straight line.

DAE parallel to BC; the ftraight D A

line DAE has always the fame pofi

tion, because no other ftraight line B

can be drawn through A parallel to

E

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

4. def. given in pofition.

PROP.

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 pofition.

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

Because the angle is given, one equal to it can be found"; let this be the angle at D, at the given point C, in the given ftraight line AB, make the angle ECB equal to the angle at D: Therefore the ftraight line EC has always the fame fituation, because any other ftraight line FC, drawn to the

G

D

29.

FE

F

a 1. def.

B

b 23. 1.

point C, makes with CB a greater or lefs angle than the angle. ECB, or the angle at D: Therefore the ftraight line EC, which has been found, is given in pofition.

It is to be obferved, 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.

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

fition

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

B

A

D

30.

F

a 31. I.

given in pofition, EAF is therefore given in pofition: And b 31. dat. because the ftraight line AD meets the parallels BC, EF, the

Bb3

angle

C 29. T.

с

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 ftraight line EF given in pofition, and makes with it a given angle EAD, d 3. dat. AD is given in pofition.

3. I.

PROP. XXXIV.

See N.. IF from a given point to a ftraight 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 straight line given in position, a ftraight line given in magnitude drawn from the point A to BC is given in pofition.

A

E C

Because the flraight line is given in magnitude, one equal to a 1. def. it can be found; let this be the ftraight line D: From the point A draw AE perpendicular to BC; and because AE is the fhorteft of all the ftraight 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 BC, cannot be less than AE. If therefore D D be equal to AE, AE is the ftraight line given in magnitude drawn from the given point A to BC: And it is evident that b. 33. dat. 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.

€ 6. def.

d 28. dat.

e 29. dat.

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 diftance AF, defcribe the circle GFH, and join AG, AH: Because the circle GFH is given in pofition, and the ftraight line BC is alfo given in polition; therefore their interfection G is given; and the point A is given; wherefore AG is given in pofition ", that is, the ftraight Ine AG given in magnitude, (for it is equal to D) and drawn from the given point A 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

B G E HC

D

F

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

IF

PROP. XXXV.

F a ftraight line be drawn between two parallel ftraight lines given in pofition, and makes given angles with them, the straight 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

32.

GH a 31. 1.
EF,

EH

В

Bb 29 1.

In CD take the given point G, and through G draw parallel to EF: And because CD meets the parallels GH, the angle EFD is equal to the angle HGD: And EFD is a given angle; 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; C the ftraight line HG is given in pofition: And AB is given in pofition; therefore the point H is c 32. dat. given ; 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.

IF

PROP. XXXVI.

FG D

33

a ftraight line given in magnitude be drawn between See N. two parallel straight lines given in pofition, it fhall make given angles with the parallels.

Ε Η Β

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

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

Bb4

A

a 1. def,

FK D

b 12. 1.

G

that