From the given point A let the straight line AED be drawn to the two parallel ftraight lines FG, BC, and let the ratio of the fegments AE, AD be given; if one of the parallels BC be given in pofition, the other FG is alfo given in pofition.

From the point A draw AH perpendicular to BC, and let it meet FG in K. and because AH is drawn from the given point A to the ftraight line BC given in position, and makes a given angle AHD;

c. 19. Dat. tude. and, becaufe FG, BC are paral

B

keis, as AE to AD, fo is AK to AH; and F E K
the ratio of AE to AD is given, where-

G

fore the ratio of AK to AH is given; but AH is given in magni

d. z. Dat. tude, therefore AK is given in magnitude; and it is alfo given in e. 30. Dat.position, and the point A is given; wherefore the point K is given. and because the ftraight line FG is drawn thro' the given point K f. 31. Dat. parallel to BC which is given in pofition, therefore f FG is given in pofition.

37.38.

See N.

IF the ratio of the fegments of a straight line into which it is cut by three parallel ftraight lines, be given; If two of the parallels are given in pofition, the third alfo is given in pofition.

Let AB, CD, HK be three parallel ftraight lines, of which AB, CD are given in pofition; and let the ratio of the fegments GE, GF

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

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 pofition. and CD is given in pofition, wherefore the a. 33. Dat. point M is given ; and the point L is given, LM is therefore given b. 28. Da“, in magnitude. and because the ratio of GE to GF is given, and c. 29. Dat.

Cor.G:

or

e 2 Dat.

as GE to GF, fo is NL to NM; the ratio of NL to NM is given;
and therefore the ratio of ML to LN is given. but LM is givend.
in magnitude, wherefore LN is given in magnitude; and it is alfo 7.Dat.
given in pofition, and the point L is given; wherefore f the point
N is given. and because the straight line HK is drawn thro' the given
point N parallel to CD which is given in pofition, therefore HK is
given in pofition &

IF

PROP. XLI.

f. 30 Dat

2. 31. Dae

F.

a ftraight line meets three parallel ftraight lines See N which are given in pofition; the fegments into which they cut it, have a given ratio.

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

In AB take a given point L, and draw

LM perpendicular to CD, meeting EF in A

N; therefore LM is given in pofition;

and CD, EF are given in pofition, where- CHM

fore the points M, N are given. and the

GL

B

a. 33. Daf

D

b. 29. Daf

N F

given,

c. 1. Dat given. but as LM to MN, fo is GH to HK; wherefore the ratio of GH to HK is given.

39.

See N.

2. 22. I.

b. 8. 1.

PRO P. XLII.

IF each of the fides of a triangle be given in magnitude; the triangle is given in fpecies.

Let each of the fides of the triangle ABC be given in magnitude; the triangle ABC is given in species.

a

A.

D

Make a triangle DEF the fides of which are equal, each to each, to the given straight lines AB, BC, CA; which can be done, becaufe 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 becaufe the two fides ED, DF are equal to the two BA, AC, each to each, and the bafe EF equal to the bafe BC; the angle EDF is

B

F

equal to the angle BAC. therefore becaufe the angle EDF, which e. 1. Def. is equal to the angle BAC, has been found, the angle BAC is given, in like manner the angles at B, C are given. and because the fides d. 1. Dat. AB, BC, CA are given, their ratios to one another are given, theree. 3. Def. fore the triangle ABC is given 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 magni

tude; the triangle ABC is given in fpecies.

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

each

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 pofition, making the given angle EDF; therefore DF is given in pofition b. in like manner EF alfo is given in pofition; wherefore the b. 32. Dat. point F is given. and the points D, E are given; therefore each of

the ftraight lines DE, EF, FD is given in magnitude. wherefore c. 29. Dat. the triangle DEF is given in fpecies; and it is fimilar to the tri- d 41. Dat. angle ABC; which therefore is given in fpecies.

IF

PRO P. XLIV.

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

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

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 pofition, making the given angle EDF; therefore FD is given in pofition. and because the ratio of BA to AC is given, make the ratio of ED to DF the fame with it, and join EF. and becaufe the ratio of ED to DF is given, and ED is gi

ven, therefore b DF is given in magni

D

AA

CE F

tude; and it is given alfo in position, and the point D is given,

e.

4.6.

1. Def.

(6.

41,

a. 32. Dat.

b. 2. Dat.

wherefore the point F is given . and the points D, E are given, c. 30. Dat. wherefore DE, EF, FD are given in magnitude; and the triangle d. 19. Dat. DEF is therefore given in fpecies. and because the triangles ABC, c. 41. Dat. DEF have one angle BAC equal to one angle EDF, and the fides about thefe angles proportionals; the triangles are f fimilar. but f. 6. 6. the triangle DEF is given in fpecies, and therefore alfo the triangle ABC.

42.

See N.

PROP. XLV.

IF the fides of a triangle have to one another given ratios; the triangle is given in fpecies.

Let the fides of the triangle ABC have given ratios to one another. the triangle ABC is given in fpecies.

a

Take a ftraight line D given in magnitude; and because the ratio of AB to BC is given, make the ratio of D to E the fame a. 2. Dat. with it; and D is given, therefore a E is given. and because the ratio of BC to CA is given, to this make the ratio of E to F the fame; and E is given, and therefore F. and because as AB to BC, fo is D to E, by compofition A B and B C together are to BC, as D and E to E; but as EC to

whofe fides are equal to D, E, F, fo 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; f. 42. Dat. therefore the triangle GHK is given f in fpecies. but as AB to EC, fo is (D to E, that is) GH to HK; and as BC to CA, fo is (E to F, that is) HK to KG; therefore, ex aequali, as AB to AC, so is GH to GK. wherefore the triangle ABC is equiangular and fimilar to the triangle GHK. and the triangle GHK is given in fpecies; therefore alfo the triangle ABC is given in fpecies.

8.5.6.

COR. If a triangle is required to be made the fides of which fhall have the fame ratios which three given ftraight lines D, E, F have to one another; it is neceffary that every two of them be greater than the third.

PROP.