ments GE, GF into which the straight line GEF is cut by the three parallels, be given; the third parallel HK is given in position.

In AB take a given point L, and draw LM perpendicular to CD, meeting HK in N; because LM is drawn from the given point L to CD which is given in position and makes a given angle LMD; LM is given in positiona; and CD is given in a 33. dat. position, wherefore the point M is givenb; and the point L is b 28. dat. given, LM is therefore given in magnitudec; and because the c 29. dat. ratio of GE to GF is given, and as GE to GF, so is NL to

Cor.

6. or

e 2. dat.

NM; the ratio of NL to NM is given; and therefored the ratio of ML to LN is given; but LM is given in magnituded, where-d foree LN is given in magnitude; and it is also given in position, 7. dat. and the point L is given; wherefore f the point N is given, f 30. dat. and because the straight line HK is drawn through the given point N parallel to CD which is given in position, therefore HK is given in positiong.

g 31. dat.

IF a straight line meets three parallel straight lines See Note. which are given in position, the segments into which they cut it have a given ratio.

Let the parallel straight lines AB, CD, EF given in position, be cut by the straight line GHK; the ratio of GH to HK is given.

In AB take a given point L, and A draw LM perpendicular to CD, meeting EF in N; therefore a LM is given in position; and ED, CF are given in position, wherefore the points M, Nare given; and the point L is given; therefore b the straight lines LM,MN are given in magnitude; and the ratio E

G L

B

H

M

a 33. dat.

D

b 29. dat.

K

N

F

c 1. dat.

39.

See Note.

a 22. 1.

of LM to MN is therefore givenc: but as LM to MN, so is GH to HK ; wherefore the ratio of GH to HK is given.

PROP. XLII.

IF each of the sides of a triangle be given in magnitude, the triangle is given in species.

Let each of the sides of the triangle ABC be given in magnitude, the triangle ABC is given in species.

A

D

Make a triangle a DEF, the sides of which are equal, each to each, to the given straight lines AB, BC, CA, which can be done; because any two of them must be greater than the third; and let DE be equal to AB, EF to BC, and FD to CA; and because the two sides ED, DF are equal to the two BA, AC, each to each,

b 8. 1.

c 1. def.

d 1. def.

e 3. def.

EDF, is equalb to the angle BAC; therefore, because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is givenc, in like manner the angles at B, C are given. And because the sides AB, BC, CA are given, their ratios to one another are givend, therefore the triangle ABC is givene in species.

IF each of the angles of a triangle be given in magnitude, the triangle is given in species.

Let each of the angles of the triangle ABC be given in magnitude, the triangle ABC is given

in species.

Take a straight line DE given in position and magnitude, and at the 23. 1. points D, E make a the angle EDF equal to the angle EAC and the angle DEF equal to ABC; therefore the other angles EFD, BCA are

B

A

D

CE F

equal, and each of the angles at the points A, B, C, is given,

wherefore each of those at the points D, E, F is given and because the straight line FD is drawn to the given point D in DE which is given in position, making the given angle EDF; therefore DF is given in position b. In like manner EF also is b 32. dat. given in position; wherefore the point F is given: and the points D, E are given; therefore each of the straight lines DE, EF, FD is givenc in magnitude; wherefore the triangle DEF is given in species d: and it is similare to the triangle ABC: d 42. dat. which is therefore given in species.

c 29. dat.

4. 6.

e 1 def.

6.

PROP. XLIV.

IF one of the angles of a triangle be given, and if the sides about it have a given ratio to one another; the triangle is given in species.

Let the triangle ABC have one of its angles BAC given, and let the sides BA, AC about it have a given ratio to one another; the triangle ABC is given in species.

Take a straight line DE given in position and magnitude, and at the point D in the given straight line DE, make the angle EDF equal to the given angle BAC; wherefore the angle EDF is given; and because the straight line FD is drawn to the given point D in ED which is given in position, making the given angle EDF; therefore FD

is given in position a. the ratio of BA to make the ratio of

And because

AC is given,

ED to Df the

same with it, and join EF; and because the ratio of ED to DF is given, B

Α

D

C E F

41.

a 32. dat.

and ED is given, thereforeb DF is given in magnitude: and it b 2. dat. is given also in position, and the point D is given, wherefore the point F is givenc; and the points D, E are given, where-c 30. dat. fore DE, EF, FD are givend in magnitude: and the triangled 29. dat. DEF is therefore given in species; and because the triangles e 42. dat. ABC DEF have one angle BAC equal to one angle EDF, and the sides about these angles proportionals; the triangles are f similar; but the triangle DEF is given in species, and there-f 6. 6. fore also the triangle ABC.

42.

See Note.

PROP. XLV.

IF the sides of a triangle have to one another given ratios; the triangle is given in species.

Let the sides of the triangle ABC have given ratios to one another, the triangle ABC is given in species.

Take a straight line D given in magnitude; and because the ratio of AB to BC is given, make the ratio of D to E the same a 2. dat. with it; and D is given, therefore a E is given. And because the ratio of BC to ČA is given, to this make the ratio of E to F the same; and E is given, and therefore a F; and because as AB to BC, so is D to E; by composition AB and BC together are to BC, as D and E to F;

f 42. dat.

*g 5. 6.

are equal to D, E, F, so that GH be equal to D, HK to E, and KG to F; and because D, E, F are each of them given, therefore GH, HK, KG are each of them given in magnitude; there fore the triangle GHK is givenf in species: but as AB to BC, so is (D to E, that is) GH to HK; and as BC to CA, so is (E to F, that is,) HK to KG; therefore, ex æquali, as AB to AC, so is GH to GK. Wherefore the triangle ABC is equiangular and similar to the triangle GHK; and the triangle GHK is given in species; therefore also the triangle ABC is given in species.

COR. If a triangle is required to be made, the sides of which shall have the same ratios which three given straight lines D, E, F have to one another; it is necessary that every two of them be greater than the third.

PROP. XLVI.

43.

IF the sides of a right-angled triangle about one of the acute angles have a given ratio to one another; the triangle is given in species.

Let the sides AB, BC about the acute angle ABC of the triangle ABC, which has a right angle at A, have a given ratio to one another; the triangle ABC is given in species.

A

D

F

C

E

G

d 6. def.

Take a straight line DE given in position and magnitude; and because the ratio of AB to BC is given, make as AB to BC, so DE to EF; and because DE has a given ratio to EF, and DE is given, therefore a EF is given; and because as AB a 2 dat. to BC, so is DE to EF; and AB is less than BC, therefore b 19. 1. DE is less than EF. From the point D draw DG at right an- c A. 5. gles to DE, and from the centre E at the distance EF, describe a circle which shall meet DG in two points; let G be either of them, and join EG; therefore the circumference of the circle is givend in position; and the straight line DG is givene in position, because it is drawn toe 32. dat. the given point D in DE given in position, in a given angle, therefore f the point G is given; and the points D, E are given, f 28. dat. wherefore DE, EG, GD are giveng in magnitude, and the tri-g 29. dat. angle DEG in speciesh. And because the triangles ABC, DEG h 42. dat. have the angle BAC equal to the angle EDG, and the sides about the angles ABC, DEG proportionals, and each of the other angles BCA, EGD less than a right angle; the triangle ABC is equiangulari and similar to the triangle DEG: buti 7. 6. DEG is given in species; therefore the triangle ABC is given in species and, in the same manner, the triangle made by drawing a straight line from E to the other point in which the circle meets DG is given in species.