than a right angle. in the fame manner, because ED is lefs than 1. 17. 1. DG or DF, the angle DFE is less than a right angle. and because the triangles ABC, DEF have the angle ABC equal to the angle DEF, and the fides about the angles BAC, EDF proportionals, and each of the other angles ACB, DFE less than a right angle; the triangles ABC, DEF are fimilar. and DEF is given in fpe- m. 7. 6. cies, wherefore the triangle ABC is also given in species.

Thirdly, Let the given ratio be the ratio of a greater to a less, that is, let the fide AB adjacent to the given angle be greater than AC. and, as in the last case, take a straight line DE given in pofition and magnitude, and make the angle DEF equal to the given angle ABC; therefore EF is given in pofition. alfo draw DG perpendicular to EF; therefore if the ratio of BA to AC be the

E

A

C. 32. Dat

G F

b. 43. Dat.

fame with the ratio of ED to the perpen- B dicular DG, the triangles ABC, DEG are fimilar m, because the angles ABC, DEG are equal, and DGE is a right angle. therefore the angle ACB is a right angle, and the triangle ABC is given in fpecies. But if, in this last case, the given ratio of BA to AC be not the fame with the ratio of ED to DG, that is, with the ratio of BA to the perpendicular AM drawn from A to BC; the ratio of BA to AC must be less than the ratio of BA to AM, because AC is 0. 8. 5 greater than AM. make as BA to AC, fo ED to DH; therefore the ratio of ED to DH is less than the ratio of (BA to AM, that is than the ratio of) ED to DG; and confequently DH is greater P than DG; and because BA is greater than AC, ED is B e than DH. from the center D, at greater the distance DH, defcribe the circle KHF

A

P. 10. 5.

C

D

e. A. 5.

which neceffarily meets the ftraight line EF

in two points, becaufe DH is greater than E K DG, and lefs than DE, let the circle meet

EF in the points F, K which are given, as was fhewn in the preceding cafe; and, DF, DK being joined, the triangles DEF, DEK are given in fpecies, as was there fhewn. from the center A, at the diftance AC describe a circle meeting BC again in L. and if the

m. 7. 6.

angle ACB be less than a right angle, ALB must be greater than a
right angle; and on the contrary. in the same manner, if the angle
DFE be lefs than a right angle, DKE must be greater than one;
and on the contrary. let each of the angles
ACB, DFE be either less or greater than a
right angle; and because in the triangles
ABC, DEF the angles ABC, DÉF are e-
qual, and the fides BA, AC, and ED, DF
about two of the other angles proportionals, BL
the triangle ABC is fimilar to the triangle
DEF. in the fame manner, the triangle
ABL is fimilar to DEK. and the triangles

A

M

D

DEF, DEK are given in fpecies, therefore E K
alfo the triangles ABC, ABL are given in

H

a. 9. 1. b. 3. 6.

45.

fpecies. and from this it is evident, that, in this third cafe, there are always two triangles of a different fpecies to which the things mentioned as given in the Propofition can agree.

IF

PROP. XLVIII.

a triangle has one angle given, and if both the fides together about that angle have a given ratio to the remaining fide; the triangle is given in fpecies.

Let the triangle ABC have the angle BAC given, and let the fides BA, AC together about that angle have a given ratio to BC; the triangle ABC is given in species.

Bifect a the angle BAC by the straight line AD; therefore the angle BAD is given. and because as BA to AC, so is " BD to DC, by permutation, as AB to BD, fo is AC to CD; and as BA and AC together to BC, fo

c

c. 12. 5. is AB to BD. but the ratio of BA and AC

together to BC is given, wherefore the ratio

of AB to BD is given; and the angle BAD

d. 47. Dat. is given, therefore 4 the triangle ABD is gi-B

d

A

D

ven in fpecies. and the angle ABD is therefore given; the angle e. 43. Dat. BAC is alfo given, wherefore the triangle ABC is given in fpecies *.

A triangle which fhall have the things that are mentioned in the Propofition to be given, can be found in the following manner. let EFG be the given angle, and let the ratio of H to K be the

given ratio which the two fides about the angle EFG muft have to the third fide of the triangle. therefore because two sides of a triangle are greater than the third fide, the ratio of H to K must be the ratio of a greater to a less. bifect a the angle EFG by the a. 9. I. ftraight line FL, and by the 47th Propofition find a triangle of which EFL is one of the angles, and in which the ratio of the fides about the angle oppofite to FL is the fame with the ratio of H to K; to do which, take FE given in pofition and magnitude, and draw EL perpendicular to FL. then, if the ratio of H to K be the fame with the ratio of FE to EL, produce EL and let it meet FG in P; the triangle FEP is that which was to be found. for it has the given angle EFG, and because this angle is bifected by FL, the fides EF, FP together are to EP, asb FE to EL, that is as H to K.

But if the ratio of H to K be not the fame with the ratio of FEE to EL, it must be less than it, as

H

F

K

b. 3. 6.

G

P

N

f

was fhewn in Prop. 47. and in this cafe there are two triangles each of which has the given angle EFL, and the ratio of the fides about the angle oppofite to FL the fame with the ratio of H to K. by Prop. 47. find thefe triangles EFM, EFN each of which has the angle EFL for one of its angles, and the ratio of the fide FE to EM or EN the fame with the ratio of H to K; and let the angle EMF be greater, and ENF lefs than a right angle. and becaufe H is greater than K, EF is greater than EN, and therefore the angle EFN, that is the angle NFG, is lefs than the angle f. 18. I ENF. to each of these add the angles NEF, EFN; therefore the angles NEF, EFG are lefs than the angles NEF, EFN, FNE, that is than two right angles; therefore the ftraight lines EN, FG must meet together when produced; let them meet in O, and produce EM to G. each of the triangles EFG, EFO has the things mentioned to be given in the Propofition. for each of them has the given angle EFG, and because this angle is bifected by the straight line FMN, the fides EF, FG together have to EG the third fide the ratio of FE to EM, that is of H to K. in like manner, the fides EF, FO together have to EO the ratio which H has to K.

46.

I

PROP. XLIX.

F a triangle has one angle given, and if the fides about another angle, both together, have a given ratio to the third fide; the triangle is given in species.

Let the triangle ABC have one angle ABC given, and let the two fides BA, AC about another angle BAC have a given ratio to BC; the triangle ABC is given in species.

Suppose the angle BAC to be bifected by the straight line AD; BA and AC together are to BC, as AB to BD, as was fhewn in the preceding Propofition. but the ratio of BA and AC together to BC is given, therefore alfo the ratio of AB to BD is given. and ■. 44. Dat. the angle ABD is given, wherefore the triangle ABD is given in fpecies; and confequently the angle BAD, and its double the angle BAC are given; and the angle ABC is given. therefore

b. 43. Dat. the triangle ABC is given in fpecies b.

H

A

B

D

C

E

K

M

L

G

A triangle which shall have the things
mentioned in the Propofition to be given,
may be thus found. Let EFG be the gi-
ven angle, and the ratio of H to K the
given ratio; and by Prop. 44. find the
triangle EFL which has the angle EFG
for one of its angles, and the ratio of the F
fides EF, FL about this angle the fame

with the ratio of H to K; and make the angle LEM equal to the
angle FEL. and because the ratio of H to K is the ratio which
two fides of a triangle have to the third, H must be greater than
K; and becaufe EF is to FL, as H to K, therefore EF is greater
than FL, and the angle FEL, that is LEM is therefore less than
the angle ELF. wherefore the angles LFE, FEM are less than two
right angles, as was fhewn in the foregoing Propofition, and the
ftraight lines FL, EM must meet if produced; let them meet in
G. EFG is the triangle which was to be found; for EFG is one
of its angles, and because the angle FEG is bifected by EL, the
two fides FE, EG together have to the third fide FG the ratio of
EF to FL, that is the given ratio of H to K.

PROP. L.

IF
F from the vertex of a triangle given in fpecies, a
straight line be drawn to the base in a given angle;
it fhall have a given ratio to the base.

From the vertex A of the triangle ABC which is given in fpecies, let AD be drawn to the base BC in a given angle ADB; the ratio of AD to BC is given.

Because the triangle ABC is given in fpecies, the angle ABD is given, and the angle ADB is given; therefore the triangle ABD is given a in fpecies; wherefore the ratio of AD to AB is given. and the ratio B

b

D

A

C

of AB to BC is given; and therefore the ratio of AD to BC

is given.

76.

a. 43. Dat.

b.
9. Dat.

R

PROP. LI.

ECTILINEAL figures given in fpecies, are divided
into triangles which are given in fpecies.

47.

Let the rectilineal figure ABCDE be given in fpecies; ABCDE may be divided into triangles given in fpecies.

Join BE, BD, and because ABCDE is given in fpecies, the

angle BAE is given a, and the ratio of BA

a. 3. Def.

A

b. 44, Dat.

E

to AE is given ; wherefore the triangle BAE is given in fpecies, and the angle AEB is therefore given 2. but the whole B angle AED is given, and therefore the remaining angle BED is given. and the ratio of AE to EB is given, as alfo the

C

ratio of AE to ED; therefore the ratio of BE to ED is given c. c. 9. Dat. and the angle BEI) is given, wherefore the triangle BED is given in fpecies in the fame manner the triangle BDC is given in

fpecies. therefore rectilineal figures which are given in species are divided into triangles given in species.