equiangular; therefore, as was fhewn in Prop. 67. the parallelogram KC, that is AC, is to EG, as MO to Q.

COR. 1. If two triangles ABC, DEF have two equal angles, 71. or two unequal, but given angles ABC, DEF, and if the ratios of the fides about thefe angles, viz. the ratios of AB to DE, and of BC to EF be given; the triangles fhall have a given ratio to one another.

B

GD H

CEF

Complete the parallelograms BG, EH; the ratio of BG to EH is given a; and therefore the triangles which are the halves of them have a given

ratio to one another.

a. 67.or 68. Dat.

b. 34. I. C. 15. 5.

COR. 2. If the bafes BC, EF of two triangles ABC, DEF have 72. a given ratio to one another, and if also the straight lines AG, DH which are drawn to the bases from the oppofite angles, either in equal angles, or unequal, but given, angles AGC, DHF have a given ratio to one another; the KA triangles fhall have a given ratio

to one another.

B G C

LD

EHF

Draw BK, EL parallel to AG, DH, and complete the parallelograms KC, LF. and because the angles AGC, DHF, or their equals, the angles KBC, LEF are either equal, or unequal, but given; and that the ratio of AG to DH, that is of KB to LE is given, as also the ratio of BC to EF; therefore the ratio of the a. 67.or 68. parallelogram KC to LF is given. wherefore alfo the ratio of the triangle ABC to DEF is given ".

I'

PROP. LXIX.

a parallelogram which has a given angle be applied to one fide of a rectilineal figure given in species; if the figure have a given ratio to the parallelogram, the parallelogram is given in fpecies.

Let ABCD be a rectilineal figure given in fpecies, and to one fide of it AB let the parallelogram ABEF having the given angle ABE be applied; if the figure ABCD has a given ratio to the parallelogram BF, the parallelogram BF is given in fpecies.

Thro' the point A draw AG parallel to BC, and thro' the point C draw CG parallel to AB, and produce GA, CB to the points H,

Dat.

b.}

61.

241. I. J15.5

c. 9. Dat. d. 35. 1.

e. 1. 6.

I.

a. 3. Def. K. because the angle ABC is given, and the ratio of AB to BC is given, the figure ABCD being given in fpecies; therefore the pa rallelogram BG is given a in species. and because upon the fame ftraight line AB the two rectilineal figures BD, BG given in species b. 53. Dat. are defcribed, the ratio of BD to BG is given ". and, by hypothefis, the fatio of 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 straight 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 témaining angle KBE is given. the angle EKB is alfo given, because it is equal to the angle ABK; therefore the triangle BKE is given in fpecies, and confequently the ratio 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, therefore the parallelogram BF is given a in fpecies.

D

A parallelogram fimilar A

to BF may be found thus;

take a ftraight line ÌM

N

M

B

P

HF KE

R

given in pofition and mag-
nitude; 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 53. 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,
thro' the point M draw OM parallel to LN, and complete the pa-
rallelogram 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 KBE is equal to MLO. and the angles BKE, LMO are equal, because the angle ABK is equal to NLM. therefore the triangles BKE,LMO are equiangular to one another, wherefore as BE to BK, fò is LO to LM. and because as the figure BF to BD, fo is the ftraight line LM to P; and as BD to BG, fo is P to Q ex aequali, as BF, that is a BH, to BG, fo is LM to Q. but BH is to BG, as KB to BC; as therefore KB to BC, so is

e

á

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, ás BE to BA, fo is LO to R, that is to LN. and the angles ABE, NLO are equal; therefore the parallelogram BF is fimilar to LS.

I

PROP. LXX.

62. 78.

F two straight lines have a given ratió to one another, See N. and upon one of them be described 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
described fimilar and fimilarly placed to FD. and because the ratio
of AB to CD is given, and upon them are defcribed the fimilar recti-
lineal figures AG, FD; the ratio of
AG to FD is given 2. and the ratio of
FD to AEB is given; therefore the A

ratio of AEB to AG is given; and the
angle ABG is given, because it is equal
to the angle FCD. because therefore M
the parallelogram AG which has a gi
ven angle ABG is applied to a fide AB

of the figure AEB given in species, and

E

F

B

a. 54. Dat. b. 9.

Data

GC

D

HKL

C

the ratio of AEB to AG is given, the parallelogram AG is given in c. 69. Dat. fpecies. but FD is fimilar to AG; therefore FD is given in fpecies.

A parallelogram fimilar to FD may be found thus; take a straight line H given in magnitude; and because the ratio of the figure AEB to FD is given, make the ratio of H to K the fame with it. alfo because the ratio of the straight line CD to AB is given, find by the 54. Dat. the ratio which the figure FD described 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 to L are given, the ratio of H to L is given .

81.

becaufe therefore as AEB to FD, so is H to K; and as FD to AG, fo is K to L; ex aequali, as AEB to AG, fo is H to L; therefore the ratio of AEB to AG is given. and the figure AEB is given in fpecies, and to its fide AB the parallelogram AG is applied in the given angle ABG, therefore by the 69. Dat. a parallelogram may be found fimilar to AG. let this be the parallelogram MN; MN alfo is fimilar to FD. for, by the construction, MN is fimilar to AG, and AG is fimilar to FD; therefore the parallelogram FD is fimilar to MN.

PROP. LXXI.

the extremes of three proportional ftraight lines have given ratios to the extremes of other three proportional straight lines; the means fhall also have a given ratio to one another. and if one extreme has a given ratio to one extreme, and the mean to the mean; likewife the other extreme fhall have to the other a given ratio.

Let A, B, C be three proportional ftraight lines, and D, E, F three other; and let the ratios of A to D, and of C to F be given. then the ratio of B to E is also given.

b

Because the ratio of A to D, as alfo of C to F is given, the a. 67. Dat. ratio of the rectangle A, C to the rectangle D, F is given a. but b. 17. 6. the fquare of B is equal to the rectangle A, C; and the square of E to the rectangle b D, F. therefore the ratio of the fquare c. 58. Dat. of B to the fquare of E is given; wherefore alfo the

ratio of the straight line B to E is given.

Next, let the ratio of A to D, and of B to E be
given; then the ratio of C to F is also given.
Because the ratio of B to E is given, the ratio of the

d. 54. Dat. square of B to the fquare of E is given . therefore the
ratio of the rectangle A, C to the rectangle D,F is given.
and the ratio of the fide A to the fide D is given; there-

e. 65. Dat. fore the ratio of the other fide C to the other F is given *.

ABC DEF

COR. And if the extremes of four proportionals have to the extremes of four other proportionals given ratios, and one of the means a given ratio to one of the means; the other mean fhall have a given ratio to the other mean. as may be fhewn in the fame manner as in the foregoing Propofition.

IF.

PROP. LXXII.

[F four ftraight lines be proportionals; as the first is to the straight line to which the fecond has a given ratio; fo is the third to the ftraight line to which the fourth has a given ratio.

Let A, B, C, D be four proportional straight lines, viz. as A to B, fo C to D; as A is to the straight line to which B has a given ratio, fo is C to the straight line to which D has a given ratio. Let E be the straight line to which B has a given ratio, and as B to E, fo make D to F. the ratio of B to E is given a, and therefore the ratio of D to F. and becaufe as A to B, fo is C to D; and as B to E, fo D to ABE F; therefore, ex aequali, as A to E, fo is C to F. and CDF E is the ftraight line to which B has a given ratio, and F that to, which D has a given ratio; therefore as A is to the straight line to which B has a given ratio, fo is C to that to which D has a given ratio.

PROP. LXXIII.

82.

a. Hyp.

83.

four straight lines be proportionals; as the first is to See N. the ftraight line to which the second has a given ratio, fo is the ftraight line to which the third has a given ratio to the fourth.

Let the straight line A be to B, as C to D; as A to the straight line to which B has a given ratio, fo is the straight

line to which C has a given ratio to D.

Let E be the straight line to which B has a given ra

tio, and as B to E, fo make F to C; because the ratio

of B to E is given, the ratio of C to F is given. and be- ABE
cause A is to B, as C to D; and as B to E, fo F to C; F C D
therefore, ex aequali in proportione perturbata, A is
to E, as F to D; that is A is to E to which B has a gi-
ven ratio, as F, to which C has a given ratio, is to D.

D d

a. 23.