a 3. def.

the points H, K; because the angle ABC is given", and the ratio of B to BC is given, the figure ABCD being given in fpecies; therefore, the parallelogram BG is given in fpecies. And because upon the fame ftraight line AB the two rectilineal figures BD, BG given in fpecies are defcribed, the ratio of b 53. dat. BD to BG is given b; and, by hypothefis, the ratio of

c 9. dat.

d 35. I. e 1. 6.

BD to the parallelogram BF is given; wherefore the ratio of
BF, that is, of the parallelogram BH, to BG is given, and
therefore the ratio of the ftraight line KB to BC is given;
and the ratio of BC to BA is given, wherefore the ratio of
KB to BA is given: And because the angle ABC is given, the
adjacent angle ABK is given; and the angle ABE is given,
therefore the remaining angle KBE is given. The angle EKB
is alfo given, becaufe it is equal to the angle ABK; therefore
the triangle BKE is given in fpecies, and confequently the ra
tio of EB to BK is given; and the ratio of KB to BA is given,
wherefore the ratio of EB
to BA is given; and the
angle ABE is given, there-
fore the parallelogram
BF is given in fpecies.
A parallelogram fimi-

lar to BF may be found
thus: Take a ftraight line

D

N

G

C

S

Ο

M

B

LM given in pofition and HF KEP

R

magnitude; and becaufe the angles ABK, ABE are given, make the angle NLM equal to ABK, and the angle NLO equal to ABE. And because the ratio of BF to BD is given, make the ratio of LM to P the fame with it; and because the ratio of the figure BD to BG is given, find this ratio by the 53d dat. and make the ratio of P to Q the fame. Alfo, because the ratio of CB to BA is given, make the ratio of Q to R the fame; and take LN equal to R; through the point M draw OM parallel to LN, and complete the parallelogram NLOS; then this is fimilar to the parallelogram BF.

Because the angle ABK is equal to NLM, and the angle ABE to NLO the angle KEE is equal to MĻO; and the angles BKE, LMO are equal, because the angle ABK is equal to NM; therefore, the triangles BKE, LMO are equiangular to one another; wherefore as BE to BK, fo is LO to LM; and because as the figure BF to BD, fo is the straight line LM to P; and as BD to BG, fo is P to Q; ex aequali, as BF, that is BH, to BG, fo is LM to Q: But BH is to

BG,

BG, as KB to BC; as therefore KB to BC, fo is LM to Q; and because BE is to BK as LO to LM; and as BK to BC, fo is LM to Q: And as BC to BA, fo Q was made to R ; therefore, ex aequali, as BE to BA, fo is LO to R, that is to LN; and the angles ABE, NLO are equal; therefore the paralle logram BF is fimilar to LS.

PROP.

LXX.

62.78.

IF two straight lines have a given ratio to one another, See N. and upon one of them be defcribed a rectilineal figure given in fpecies, and upon the other a parallelogram having a given angle; if the figure have a given ratio to the parallelogram, the parallelogram is given in fpecies.

Let the two straight lines AB, CD have a given ratio to one another, and upon AB let the figure AEB given in fpecies be defcribed, and upon CD the parallelogram DF having the given angle FCD; if the ratio of AEB to DF be given, the parallelogram DF is given in fpecies.

Upon the ftraight line AB, conceive the parallelogram AG to be defcribed fimilar, and fimilarly placed to FD; and because the ratio of AB to CD is given, and upon them are described the fimilar rectilineal figures AG, FD; the ratio of AG to FD is given; and the ratio of FD to ALB A is given; therefore the ratio of

F

B

a 54. dat.

b 9. dat.

AEB to AG is given; and the angle

GC

D

ABG is given, because it is equal to

the angle FCD; because therefore M

the parallelogram AG which has a

N

given angle ABG is applied to a fide

AB of the figure AEB given in fpe

HKL

cies, and the ratio of AEB to AG is given, the parallelogram AG is given in fpecies; but FD is fimilar to AG; therefore e 69. date FD is given in fpecies.

A parallelogram fimilar to FD may be found thus: Take a ftraight line H given in magnitude; and because the ratio of the higure AEB to FD is given, make the ratio of H to K the fame with it: Alfo, because the ratio of the ftraight line CD to. AB is given, find by the 54th dat. the ratio which the figure FD defcribed upon CD has to the figure AG defcribed upon AB fimilar to FD; and make the ratio of K to L the fame with this ratio: And because the ratios of H to K, and of K

82. Cor.

20. 6. h 14. 5.

57.

See N.

a 1. def.

fide of the figure to be defcribed, homologous to BC the fide of D, and the figure itself can be defcribed by the 18th prop. b. 6. which, by the conftruction, is fimilar to D; and because D is to A, as BC to CL, that is as the figure BK to KL; and that D is equal to BK, therefore A is equal to KL, that is, to H.

PROP. LXI.

IF a parallelogram given in magnitude has one of its fides and one of its angles given in magnitude, the other fide alfo is given.

Let the parallelogram ABDC given in magnitude, have the fide AB and the angle BAC given in magnitude, the other fide AC is given.

A

B

Take a straight line EF given in pofition and magnitude; and because the parallelogram AD is given in magnitude, a rectilineal figure equal to it can be found". And a parallelogram equal to this b Cor. 45. figure can be applied to the given ftraight line EF in an angle equal to E the given angle BAC. Let this be

I.

€ 14. 6. d 12. 6.

H.

See N.

the parallelogram EFHG having
the angle FEG equal to the angle
BAC. And becaufe the parallelo-
grams AD, EH are equal, and have

H

D

the angles at A and E equal; the fides about them are reciprocally proportional; therefore as AB to EF, fo is EG to AC; and AB, EF, EG are given, therefore alfo AC is given 4. Whence the way of finding AC is manifeft.

PROP. LXII.

IF a parallelogram has a given angle, the rectangle contained by the fides about that angle has a given ratio to the parallelogram.

Let the parallelogram ABCD have the given angle ABC, the rectangle AB, BC has a given ratio to the parallelogram AC.

a

From the point A draw AE perpendicular to BC; because the angle ABC is given, as alfo the angle AEB, the triangle 43. dat. ABE is given in fpecies; therefore the ratio of BA to AE is given. But as BA to AE, fo is the rectangle AB, BC to the rectangle AE, BC; therefore the ratio of

b x. 6.

B

A

D

the rectangle AB, BC to AE, BC, that is, to the parallelo- c 35. 1. gram AC, is given.

And it is evident how the ratio of the rectangle to the pa rallelogram may be found, by making the angle FGH equal to the given angle ABC, and drawing, from any point Fin one of its fides, FK perpendicular to the other GH; for GF is to FK, as BA to AE, that is, as the rectangle AB, BC, to the parallelogram AC.

COR. And if a triangle ABC has a given angle ABC, the rectangle AB, BC contained by the fides about that angle, fhall have a given ratio to the triangle ABC.

66.

Complete the parallelogram ABCD; therefore, by this propofition, the rectangle AB, BC has a given ratio to the parallelogram AC; and AC has a given ratio to its half the triangle ABC; therefore the rectangle AB, BC has a given ra- d 14. 1. tio to the triangle ABC.

с

And the ratio of the rectangle to the triangle is found thus: Make the triangle FGK, as was shown in the propofition; the ratio of GF to the half of the perpendicular FK is the fame with the ratio of the rectangle AB, BC to the triangle ABC. Becaufe, as was shown, GF is to FK, as AB, BC to the paralle logram AC; and FK is to its half, as AC is to its half, which is the triangle ABC; therefore, ex aequali, GF is to the half of FK, as AB, BC rectangle is to the triangle ABC.

PROP. LXIII.

IF two parallelograms be equiangular, as a fide of the first to a fide of the second, fo is the other fide of the second to the straight line to which the other fide of the firft has the fame ratio which the first parallelogram has to the second. And confequently, if the ratio of the first parallelogram to the fecond be given, the ratio of the other fide of the first to that straight line is given; and if the ratio of the other fide of the first to that straight line be given, the ratio of the first parallelogram to the second is given.

Let AC, DF be two equiangular parallelograms, as BC, a fide of the firft, is to EF, a fide of the fecond, fo is DE, the other fide of the fecond, to the straight line to which AB, the o

ther

e 9. dat.

56.

1

a 14. 6.

74.73.

See N.

a 35. I.

A

ther fide of the first has the fame ratio which AC has to DF.
Produce the ftraight line AB, and make as BC to EF, fo
DE to BG, and complete the parallelo-
gram BGHC; therefore, because BC, or
GH, is to EF, as DE to BG, the fides
about the equal angles BGH, DEF are
reciprocally proportional; wherefore
the parallelogram BH is equal to DF;
and AB is to BG, as the parallelogram
AC is to BH, that is, to DF; as there-
fore BC is to EF, fo is DE to BG, which
is the ftraight line to which AB has the
fame ratio that AC has to DF.

G

D

And if the ratio of the parallelogram AC to DF be given, then the ratio of the ftraight line AB to BG is given; and if the ratio of AB to the ftraight line BG be given, the ratio of the parallelogram AC to DF is given.

IF

PROP. LXIV.

two parallelograms have unequal, but given angles, and if as a fide of the first to a fide of the fecond, fo the other fide of the fecond be made to a certain ftraight line; if the ratio of the firft parallelogram to the fecond be given, the ratio of the other fide of the first to that ftraight line fhall be given. And if the ratio of the other fide of the first to that ftraight line be given, the ratio of the first parallelogram to the fecond fhall be given.

Let ABCD, EFGH be two parallelograms which have the unequal, but given, angles ABC, EFG; and as BC to FG, fo make EF to the ftraight line M. If the ratio of the parallelogram AC to EG be given, the ratio of AB to M is given.

a

At the point B of the straight line BC make the angle CBK equal to the angle EFG, and complete the parallelogram KBCL. And becaule the ratio of AC to EG is given, and that AC is equal to the parallelogram KC, therefore the ratio of KC to EG is given; and KC, EG are equiangular; thereb 63. dat, fore as BC to FG, fo is EF to the ftraight line to which KB has a given ratio, viz. the fame which the parallelogram KC has to EG: But as BC to FG, fo is EF to the ftraight Jine M; therefore KB has a given ratio to M; and the ratio

of