g 2 Cor. 20.6.

h 14. 5.

57.

See N.

a 1. def.

b Cor. 45.

1.

c 14 6.

d 12. 6.

H. See N.

side of the figure to be described, homologous to BC the side of D, and the figure itself can be described by the 18th prop. book 6, which, by the construction, is similar to D; and because D is to A, as 8 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, H. PROP. LXI.

IF a parallelogram given in magnitude has one of its sides and one of its angles given in magnitude, the other side also is given.

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

b

D

E

F

B

Take a straight line EF given in position and magnitude; and because the parallelogram AD is given in magnitude, a rectilineal figure equal to it can be found a. And a parallelogram equal to this figure can be applied to the given straight line EF in an angle equal to the given angle BAC. Let this be the parallelogram EFHG, having the angle FEG equal to the angle BAC. And because the parallelograms AD, EH are equal, and have the angles at A and E equal; the sides about them are reciprocally proportionale; therefore as AB to EF, so is EG to AC; and AB, EF, EG are given, therefore also AC is given. Whence the way of finding AC is manifest.

PROP. LXII.

G

H

IF a parallelogram has a given angle, the rectangle contained by the sides 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 cular to BC; because the angle ABC is given, as also the angle AEB, the triangle a 43 dat. ABE is given in species: therefore the ratio of BA to AE is given. But as BA to AE, so is the rectangle AB, BC to the rectangle AE, BC; therefore the ratio of

b 1. 6.

A

BE

F

D

the rectangle AB, BC to AE, BC, that is, to the parallelogram c 35, 1. AC is given.

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

COR. And if a triangle ABC has a given angle ABC, the 66, rectangle AB, BC contained by the sides about that angle, shall have a given ratio to the triangle ABC.

Complete the parallelogram ABCD; therefore, by this proposition, the rectangle AB, BC has a given ratio to the parallelogram AC; and AC has a given ratio to its half the triangle 4 d 41, 1. ABC; therefore, the rectangle AB, BC has a given ratio to the e 9. dat. triangle ABC.

e

And the ratio of the rectangle to the triangle is found thus: make the triangle FGK, as was shown in the proposition; the ratio of GF to the half of the perpendicular FK is the same with the ratio of the rectangle AB, BC to the triangle ABC. Because, as was shown, GF is to FK, as AB, BC to the parallelogram 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.

IF two parallelograms be equiangular, as a side of the first to a side of the second, so is the other side. of the second to the straight line to which the other side of the first has the same ratio which the first parallelogram has to the second.. And consequently, if the ratio of the first parallelogram to the second be given, the ratio of the other side of the first to that straight line is given; and if the ratio of the other side 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 side of the first, is to EF, a side of the second, so is DE, the other side of the second, to the straight line to which AB, the

56.

a 14. 6.

74, 73.

See N.

a 35.1.

B

other side of the first, has the same ratio which AC has to DF.
Produce the straight line AB, and make as BC to EF, so
DE to BG, and complete the parallelo-
A
gram BGHC; therefore because B or
GH is to EF, as DE to BG, the sides
about the equal angles BGH, DEF are
reciprocally proportional; wherefore a
the parallelogram BH is equal to DF;
and AB is to BG, as the parallelogramTM
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 same ratio that AC has to DF.

G

D

E

F

C

H

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 ra tio of AB to the straight 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 side of the first to a side of the second, so the other side of the second be made to a certain straight line; if the ratio of the first parallelogram to the second be given, the ratio of the other side of the first to that straight line shall be given. And if the ratio of the other side of the first to that straight line be given, the ratio of the first parallelogram to the second shall be given.

Let ABCD, EFCH be two parallelograms which have the unequal, but given angles ABC, EFG; and as BC to FG, so make EF to the straight line M. If the ratio of the parallelo. gram 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 AC to EG is given, and that AC is equal a to the parallelogram KC, therefore the ratio of KC to EG is given; and KC, EG are équiangular; thereb 63. dat.fore as BC to FG, so is b EF to the straight line to which KB has a given ratio, viz. the same which the parallelogram KC has to EG; but as BC to FG, so is EF to the straight line M; therefore KB has a given ratio to M; and the ratio

*

of AB to BK is given, because the triangle ABK is given in species; therefore the ratio of AB to M is given a.

c 43. dat.

And if the ratio of AB to M be given, the ratio of the pa- d 9. dat. rallelogram AC to EG is given; for since the ratio of KB to

BA is given, as also the ratio of AB to M, the ratio of KB to M is given; and because the parallelograms KC, EG are /equiangular, as BC to FG, so is

EF to

KAL D

B

M

E

b 63. dat. C

H

G

the straight line to which KB has the
same ratio which the parallelogram KC
has to EG; but as BC to FG, so is EF to
M; therefore KB is to M, as the parallel-
ogram 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.

COR. And if two triangles ABC, EFG, have two equal angles, 75, or two unequal, but given, angles ABC, EFG, and if as BC a side of the first to FG a side of the second, so the other side of the second EF be made to a straight fine M; if the ratio of the triangles be given, the ratio of the other side of the first to the. straight line M is given.

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 parallel- e 15. 5. ograms are double f of the triangles; and because BC is to FG, f 41, 1. 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 proposition.

And if the ratio of AB to M be given, the ratio of the parallelogram AC to EG is given by the same proposition; 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 side have to one side a given ratio; the other side shall also have to the other side a given ratio.

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

68.

Because the two equiangular parallelograms AB, CD have a given ratio to one another; as EB, a side of the first, is to FD, a 63. dat. a side of the second, so is a FC, the other side of the second, to the straight line to which AE, the other side of the first, has the same given ratio which the first parallelogram AB has to the other CD. Let this straight line be EG; therefore the ratio of AE to EG is given; and EB is to FD, as FC to EG, therefore the ratio of FC to EG is given, because the ratio of EB to FD is given; and because the ratio of AE to EG, as also the ratio of FC to EG is given; the ratio of AE to CF is given þ.

b 9. dat.

A

E

BT

G'

H.K L

.

The ratio of AE to CF may be found thus: take a straight line H given in magnitude; and because the ratio of the paral lelogram AB to CD is given, make the ratio of H to K the same with it. And because the ratio of FD to EB is given, make the ratio of K to L the same: the ratio of AE to CF is the same with the ratio of H to L. Make as EB to FD, so FC to EG, therefore, by inversion, as FD to EB, so is EG to FC; and as AE to EG, so is a (the parallelogram AB to CD, and so is) H to K; but as EG to FC, so is (FD to EB, and so is) K to L; therefore, ex æquali, as AE to FC, so is H to L.

69.

PROP. LXVI.

IF two parallelograms have unequal, but given angles, and a given ratio to one another; if one side have to one side a given ratio, the other side has also a given ratio to the other side.

Let the two parallelograms ABCD, EFGH which have the given unequal angles ABC, EFG, have a given ratio to one another, and let the ratio of BC to FG be given; the ratio also of AB to EF is given.

At the point B of the straight line BC make the angle CBK equal to the given angle EFG, and complete the parallelo-gram BKLC; and because each of the angles BAK, AKB, is a 43. dat. given, the triangle ABK is given in species; therefore the ratio of AB to BK is given; and because, by the hypothesis,