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GH to CF: therefore the ratio of GH to CF A
is given; and CF is given, wherefore (2. dat.)
GH is given : and AG is given, wherefore the G
whole AH is given : and because as GB to
CD, so is GH to CF, and so is (19. 5.) the re-

F
mainder HB to the remainder FD; the ratio H
of HB to FD is given : and the ratio of FD to
E is given, wherefore (9. dat.) the ratio of HB

B G E to E is given: and AH is given; therefore HB, the excess of AB above a given magnitude AH, has a given ratio to E.

- Otherwise, Let AB, C, D, be three magnitudes, the excess EB of the first of which AB above the given magnitude AE has a given ratio to C, and the excess of C above a given magni

A tude has a given ratio to D: the excess of AB above a given magnitude has a given ratio

E to D. Because EB has a given ratio to C, and the

F excess of C above a given magnitude has a given ratio to D; therefore (24. dat.) the excess of EB above a given magnitude has a

B С D 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 giyen, because AE, EF are given : therefore FB, the excess of AB above a given magnitude AF, has a given ratio to D.”

PROP. XXVIII. If two lines given in position cut one another, the point or points in which they cut one another are given.*

Let two lines AB, GD given is position cut one another in the point E; the point E is given.

С Because the lines AB, CD are given in position, they have always

E the situation (4. def.), and

А.

B therefore the point, or points, in which they cut one another, have always the same situation : and because

D the lines AB, CD can be found (4.

E def.), the point, or points, in which

Athey cut one another, are likewise

B found; and therefore are given in position (4. def.)

С

D PROP. XXIX.

26. 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, they can

25.

same

* See Note.

be found (4. def.): let these be the points A, B, between which a straight line AB can be drawn (1. pos

A

-B tulate); this has an 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.

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If one of the extremities of a straight line given in position 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 (1. def.); let this be the straight line D: from the greater straight line AC cut off AB equal

A

B C to the lesser D: therefore the other ex

-1 tremity B of the straight line AB is found : and the point B has always the D 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 (4. def.): and it is plain another such point can be found in AC, produced upon the other side of the point A.

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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. Through A draw (31. 1.) the straight

D

A line DAE parallel to BC; the straight line DAE has always the same position, because no other straight line can be B

C drawn through A parallel to BC; there. fore the straight line DAE, which has been found, is given (4. def.) in position.

PROP. XXXII.

29. If 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 point

in it; the straight line drawn to C, a

F E which makes a given angle with CB, is given in position.

Because the angle is given, one equal to it can be found (1. def.); let this be the angle at D: at the given A

B point C, in the given straight line AB, make (23. 1.) 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 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.

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Ir 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.

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 position.

E
A

F Through the point A, draw (31. 1.) the straight line EAF parallel to BC; and because through the given point A, the straight line EAF is drawn parallel to BC,

B

D which is given in position, EAF is therefore given in position (31. dat.): and because the straight line AD meets the parallels, BC, EF, the angle EAD is equal (29. 1.) 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, AD is given (32. dat.) in position.

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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 equal to

* See Note.

it can be found (1. def.); let this be the straight line D: from the point A draw AE perpendicular to BC; and be

A cause 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 BC, cannot be less than AE. If therefore D be equal to AE, AE B E C is the straight line given in magnitude, drawn

Dfrom the given point A to BC: and it is evident that AE is given in position, (33. dat.), 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: because the circle GFH is given in position (6. def.), and the straight line BC is also given in position; therefore their intersection G is given (28. dat.);

A and the point A is given; wherefore AG is given in position (29. dat.), that is, the straight line AG

B (E

H given in magnitude, (for it is equal to D) and drawn from the given point A to the straight line BC

F given in position, is also given in

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

PROP. XXXV.

32. 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 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 (29. 1.) to the angle A

E H B 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 position, and makes a given angle HGD: the straight c

F G

D line HG is given in position (32. dat.): and AB is given in position: therefore the point H is given (28. dat.), and the point G is also given, wherefore GH is given in magnitude (29. dat.) and EF is equal to it, therefore EF is given in magnitude.

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Ir a straight line given in magnitude be drawn between 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 between the parallel straight lines AB, CD, which are A Ε Η Β 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 (1. def.): let this be G:in AB take a given point H, and from it draw

С (12. 1.) HK perpendicular to CD; therefore the

F K D straight line G, that is, EF, cannot be less than GHK: 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 absurd. Therefore the angle EFD is a right, and con. sequently 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 (6. def.) MLN, and the straight line CD, are given in position, the points M, N are (28. dat.) given: and the point H is given, wherefore the straight A E

B lines HM, HN, are given in position (29. dat.) and CD is given in position: therefore the angles

K HMN, HNM, are given in position (A. def.): of the straight lines C F 0 M L N D HM, HN, let HN be that which is

G not parallel to EF, for EF cannot be parallel to both of them; and draw EO parallel to HN: EO therefore is equal (34. 1.) 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, (29. 1.) 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 magnitude (1. def.); and consequently the angle EFC.

H

PROP. XXXVII.

Ir 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 straight line given in position, is given in position.*

* See Note.

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