of LM to MN is therefore given e: But as LM to MN, fo is GH to HK; wherefore the ratio of GH to HK is given.

39.

See N.

a 22. I.

PROP. XLII.

Feach of the fides of a triangle be given in magni

Itude, 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.

Make a triangle a DEF the fides 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 e

qual to AB, EF to BC,

A.

D

F

b 8. 1.

c I. def.

EDF, is equl b to the angle BAC; therefore, because the angle EDF, which is equal to the angle BAC, has been found, the angle BAC is given c, in like manner the angles at B, C are given. And because the fides AB, BC, CA are given, their ratios to one another are given d, therefore the triangle e 3. def. ABC is given e in species.

d I dat.

[F each of the angles of a triangle be given in magInfach, the tangle of a

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

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

ven,

1

ven, 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 pofition, making the given angle EDF; therefore DF is given in pofition b. In like manner b 32. dat. EF alfo is given in pofition; wherefore the point F is given : And the points D, E are given; therefore each of the straight lines DE, EF, FD is given in magnitude; wherefore the c 29. dat. triangle DEF is given in species d: and it is fimilar e to the d 42 dat. triangle ABC: which therefore is given in fpecies.

IF

PROP. XLIV.

F 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 species.

Take a straight line DE given in pofition and magnitude, and at the point D in the given ftraight 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 a.

And

CE

F

e

41.

4.

6.

1.def.

6.

a 32. dat.

is given, and ED is given, therefore b DF is given in mag-b 2. dat. nitude and it is given alfo in pofition, and the point D is

given, wherefore the point F is given : and the points D, c 30. dat. E are given, wherefore DE, EF, FD are givend in magni- d 29. dat. tude and the triangle DEF is therefore given e in fpecies; e 42. dat. and because the triangles ABC, DEF have one angle BAC equal to one angle EDF, and the fides about these angles proportionals; the triangles are f fimilar; but the triangle DEF f 6. 6. is given in fpecies, and therefore alfo the triangle ABC.

PROP.

42.

See N.

a 2. dat.

b 22. 5.

C 20. I.

d A. 5.

e 22. I.

I

PROP. XLV.

F 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 species.

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 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 a F. And because as AB to BC, fo is D to E; by compofition AB and BC together are to BC, as D and E to E; but as BC to CA, fo is E to F; therefore, ex æqualib, as AB and BC are to CA, fo are D and E to F, and AB and BC are greater c than CA; therefore D and E are greater d than F. In the fame manner any two of the three D, E, F are greater than the third. Makee the triangle GHK whose

B

A

C

G

DEF

H

Ак

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 magf 42. dat. nitude; therefore the triangle GHK is given fin fpecies: But as AB to BC, 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 æquali, as AB to AC, fo is GH to GK. Wherefore g 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.

g 5. 6.

1

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

PROP

PROP. XLVI.

43.

F the fides of a right angled triangle about one of the acute angles have a given ratio to one another; the triangle is given in fpecies.

Let the fides 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 fpecies.

Take a straight line DE given in pofition and magnitude; and because the ratio of AB to BC is given, make as AB to BC, fo 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, fo is DE to EF; and AB is lefs b than BC, therefore b 19. 1. DE is less than EF. From the pointD draw DG at right angles c A. 5. to DE, and from the centre E at the distance EF, def

A

cribe a circle which fhall

D

meet DG in two points; let B

F

C

G be either of them, and

join EG; therefore the cir

E

G

cumference of the circle is

e 32. dat

f 28. dat.

given din pofition; and the ftraight line DG is given e in po- d 6. def. fition, because it is drawn to the given point D in DE given in pofition, in a given angle; therefore f the point G is given; and the points D, E are given, wherefore DE, EG, GD are given g in magnitude, and the triangle DEG in fpecies h. g 29. dat. And because the triangles ABC, DEG have the angle BACh 42. dat. equal to the angle EDG, and the fides about the angles ABC, DEG proportionals, and each of the other angles BCA, EGD lefs than a right angle; the triangle ABC is equiangular i and i 7. 6. fimilar to the triangle DEG: But DEG is given in fpecies; therefore the triangle ABC is given in fpecies: And in the fame 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.

PROP.

44.

See N.

PROP. XLVII.

Fritangle hiven, of if the fides about another Fa triangle has one, of its angles which is not a angle have a given ratio to one another; the triangle is given in fpecies.

Let the triangle ABC have one of its angles ABC a given, but not a right angle, and let the fides BA, AC about another angle BAC have a given ratio to one another; the triangle ABC is given in fpecies.

First, Let the given ratio be the ratio of equality, that is, let the fides BA, AC, and confequently the angles ABC, ACB, be equal; and because the angle ABC is given, the angle ACB, and also the remaining a angle BAC is given; therefore the triangle b. 43. dat. ABC is given b in fpecies; and it is evident

a 32. 1.

A

that in this cafe the given angle ABC must be acute.

C

Next, Let the given ratio be the ratio of a lefs to a greater, that is, let the fide AB adjacent to the given angle be lefs than the fide AC: Takea ftraight line DE given in pofition and magnitude, and make the angle DEF equal to the given f. 32. dat. angle ABC; therefore EF is given in pofition; and because the ratio of BA to AC is given,

< A. 5.

as BA to AC, fo make ED to DG; and becaufe the ratio of ED to DG is given, and ED is given, the ftraight line DG is d 2. dat. given d, and BA is less than AC, therefore ED is lefs e than DG. From the centre D, at the distance DG defcribe the circle GF meeting EF in F, and join DF; and because the circle is given fin pofition, as also the straight line EF, the g 28. dat. point F is given g; and the points D, E are given; where

f 6. def.

And be

h 29. dat. fore the ftraight lines DE, EF, FD are given b in mag. i 42. dat. nitude, and the triangle DEF in fpecies i. k 18. 1. caufe BA is lefs than AC, the angle ACB is lefs k 1.7.1. than the angle ABC, and therefore ACB is lefs than

a