to EG, fo is H to L: And the figures A, EG are fimilar, and M is a mean proportional between H and L; therefore, as was fhewn in the preceding propofition, CD is to EF as H to M.

PROP. LX.

Fa rectilineal figure be given in fpecies and magnitude, the fides of it fhall be given in magnitude.

Let the rectilineal figure A be given in fpecies and magnitude; its tides are given in magnitude.

Take a ftraight line BC given in pofition and magnitude;

and upon BC defcribe the figure D fimilar, and fimilarly a 18. 6. placed, to the figure A, and let EF be the fide of

M

K

b 56. dat.

gure D given in fpecies is defcribed, D is given b in magnitude, and the figure A is given in magnitude, therefore the ratio of A to D is given: And the figure A is fimilar to D; therefore the ratio of the fide EF to the homologous fide BC is given ; and BC is given, wherefore d EF is given: And the ratio of EF to EG is given e, therefore EG is given. And, c 3. def. in the fame manner, each of the other fides of the figure A can be fhewn to be given.

f

PROBLEM.

c 58. dat.

d 2. dat.

To defcribe a rectilineal figure A fimilar to a given figure D, and equal to another given figure H. It is prop. 25. b. 6. Elem. Because each of the figures D H is given, their ratio is gi ven, which may be found by making fupon the given ftraight fcor. 45. line BC the parallelogram BK equal to D, and upon its fide CK making the parallelogram KL equal to H in the angle KCL equal to the angle MBC; therefore the ratio of D to H, that is, of BK to KL, is the fame with the ratio of BC to CL: And because the figures D, A are fimilar, and that the ratio of D to A, or H, is the fame with the ratio of BC to CL; by the 58th dat. the ratio of the homologous fides BC, EF is the fame with the ratio of BC to the mean proportional between BC and CL. Find EF the mean proportional; then EF is the

fide

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 becaule D is to A, as g BC to CL, that is as the figure BK to KL; and h 14. 5. that D is equal to BK, therefore A h is equal to KL, that is, to H.

82. Cor.

20. 6.

57See N.

a 1. def.

I'

PROP. LXI.

F 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

C

D

Take a straight line EF given in pofition and magnitude; and becaufe the parallelogram AD is given in magnitude, a rectilineal figure equal to it can be found a. And a parallelogram equal to this b Cor. 45. figure can be applied b to the given ftraight line EF in an angle equal to E the given angle BAC. Let this be the parallelogram EFHG having the angle FEG equal to the angle BAC. And because the parallelo: grams AD, EH are equal, and have the angles at A and E equal; the fides about them are recipro cally proportional c; therefore as AB to EF, fo is EG to AC; and AB, EF, EG are given, therefore alfo AC is given ¿, Whence the way of finding AC is manifeft.

C 14. 6. d 12. 6.

H.

See N.

G

PROP. LXII.

H

IF a parallelogram has a given angle, the rectangle con-
tained 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.
From the point A draw AE perpendi. B
cular to BC; because the angle ABC is
given, as also the angle AEB; the triangle

a 43. dat. ABE is given a in fpecies; therefore the
ratio of BA to AE is given. But as BA

A

E

D

F

b 1. 6.

to AE, fo is b the rectangle AB. BC to the GK
rectangle AE, BC; therefore the ratio of

H

the

the rectangle AB, BC to AE, BC, that is c, 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 F in 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 paral lelogram AC; and AC has a given ratio to its half the triangle d ABC; therefore the rectangle AB, BC has a given e ra- d. 14. 1. tio to the triangle ABC.

And the ratio of the rectangle to the triangle is found thus; make the triangle FGK as was fhewn in the proposition; 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 fhewn, 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 æquali, 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 fecond, fo is the other fide of the fecond to the ftraight line to which the other fide of the firft has the fame ratio which the first parallelogram has to the fecond. 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 ftraight 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 fecond is given.

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

ther

c. 9. dat.

ther fide of the firft has the fame ratio which AC has to DF.

Produce the ftraight line AB, and make as BC to EF, so 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 a. 14. 6. reciprocally proportional; wherefore a the parallelogram BH is equal to DF ; and AB is to BG, as the parallelogram D AC is to BH, that is, to DF; as there fore BC is to EF, so is DE to BG which is the straight line to which AB has the E fame ratio that AC has to DF.

74. 73.

See N.

3.

A

F

And if the ratio of the parallelogram AC to DF be given, then the ratio of the straight 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.

PROP. LXIV.

IF two parallelograms have unequal, but given angles, and if as a fide of the firft to a fide of the fecond, fo the other fide of the second be made to a certain straight line; 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 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 thall 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.

At the point B of the straight line BC make the angle CBK equal to the angle EFG, and complete the parallelogram KBCL. And because the ratio of AC to EG is given, and that a. 35. AC is equal a to the parallelogram KC, therefore the ratio of KC to EG is given; and KC, EG are equiangular; there b. 63. dat. fore as BC to FG, fo is b 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 line M; therefore KB has a given ratio to M; and the ratio

of

of AB to BK is given, because the triangle ABK is given in c 43. dat. fpecies; therefore the ratio of AB to M is given d.

ΚΑ

B

pa

d 9. dat.

L D

C

b 63. dati

H

M F

G

And if the ratio of AB to M be given, the ratio of the rallelogram AC to EG is given; for fince the ratio of KB to BA is given, as alfo the ratio of AB to M, the ratio of KB to M is given d; and because the parallelograms KC, EG are equiangular, as BC to FG, fo is b EF to the ftraight line to which KB has the fame ratio which the parallelogram KC has to EG; but as BC to FG, fo is EF to M; therefore KB is to M, as the parallelogram KC is to EG; and the ratio of KB to M is given, therefore the ratio of the parallelogram KC, that is, of AC to EG, is given.

E

COR. And i two triangles ABC, EFG have two equal angles, or two unequal, but given, angles ABC, EFG, and if as BC a fide of the first to FG a fide of the fecond, fo the other fide of the fecond EF be made to a straight line M; if the ratio of the triangles be given, the ratio of the other fide of the first to the ftraight line M is given.

75-1

Complete the parallelograms, ABCD, EFGH; and because, the ratio of the triangle ABC to the triangle EFG is given, the ratio of the parallelogram AC to EG is given, because the pa-e 15. 56 rallelograms are double f of the triangles; and becaufe BC is to f 41. 1. FG, as EF to M, the ratio of AB to M is given by the 63d dat. if the angles ABC, EFG are equal; but if they be unequal, but given angles, the ratio of AB to M is given by this propofi

tion.

And if the ratio of AB to M be given, the ratio of the parallelogram AC to EG is given by the fame propofitions: and therefore the ratio of the triangle ABC to EFG is given.

PROP. LXV.

IF two equiangular parallelograms have a given ratio to one another, and if one fide has to one fide a given ratio; the other fide fhall alfo have to the other fide a given ratio.

Let the two equiangular parallelograms AB, CD have a given ratio to one another, and let the fide EB have a given ratio to the fide FD; the other fide AE has alfo a given ratio to the other fide CF.