Book VI. B, and DH equal to C; and having joined GH, draw EF

b 2. 6.

E

31. 1. parallel a to it through the point E: and because GH is parallel to EF, one of the fides of the triangle DEF, DG is to GE, as DH to HF b; but DG is equal to A, GE to B, and DH to C; therefore, as A is to B, fo is C to HF. Wherefore to the three given straight lines, A, B, C a fourth proportional HF is found. Which was to be done.

a II. I.

O find a mean proportional between two given ftraight lines.

Let AB, BC be the two given straight lines; it is required to find a mean proportional between them.

Place AB, BC in a straight line, and upon AC describe the femicircle ADC, and from

the point B draw a BD at
right angles to AC, and join
AD, DC.

Because the angle ADC
in à femicircle is a right

b 31. 3. angle b, and because in the right angled triangle ADC,

DB is drawn from the right

angle perpendicular to the bafe, DB is a mean proportional

e Cor. 8. 6. between AB, BC the fegments of the bafe c; therefore be

tween

tween the two given straight lines AB, BC, a mean proportional DB is found. Which was to be done.

Book VI.

QUAL parallelograms which have one angle of the one equal to one angle of the other, have their fides about the equal angles reciprocally proportional And parallelograms that have one angle of the one equal to one angle of the other, and their fides about the equal angles reciprocally proportional, are equal to one another.

Let AB, BC be equal parallelograms, which have the angles at B equal, and let the fides DB, BE be placed in the fame ftraight line; wherefore alfo FB, BG are in one straight linea the fides of the parallelograms AB, BC about the e- a 14. 1. qual angles, are reciprocally proportional; that is, DB is to BE, as GB to BF.

Complete the parallelogram FE; and because the parallelogram AB is equal to BC,

and that FE is another pa- A rallelogram, AB is to FE,

as BC to FE b; but as AB.

to FE, fo is the base DB to BE c; and, as BC to FE, fo is the base GB to BF; therefore, as DB to BE, fo is GB to BF d, Wherefore, the fides of the parallelograms AB, BC a

F

E

b 7.5.

D

B

c I. 6.

bout their equal angles are reciprocally proportional.

G

But, let the fides about the equal angles be reciprocally proportional, viz. as DB to BE, fo GB to BF; the parallelogram AB is equal to the parallelogram BC.

Becaufe, as DB to BE, fo is GB to BF; and as DB to BE, fo is the parallelogram AB to the parallelogram FE; and as GB to BF, fo is the parallelogram BC to the parallelogram FE; therefore as AB to FE, fo BC to FEd; wherefore the parallelogram

d 11. 5.

Book VI. parallelogram AB is equal e to the parallelogram BC. There fore equal parallelograms, &c. Q. E. D.

e g. 5.

a 14. 1.

EQUAL triangles which have one angle of the one equal to one angle of the other, have their fides about the equal angles reciprocally proportional: And triangles which have one angle in the one equal to one angle in the other, and their fides about the equal angles reciprocally proportional, equal to one another..

Let ABC, ADE be equal triangles, which have the angle BAC equal to the angle DAE, the fides about the equal angles of the triangles are reciprocally proportional; that is, CA is to AD, as EA to AB.

B

Let the triangles be placed fo that their fides CA, AD be in one ftraight line; wherefore alfo EA and AB are in one ftraight line a; join BD. Because the triangle ABC is equal to the triangle ADE,and that ABD is another triangle, therefore as the triangle CAB is to the triangle BAD, fo is triangle EAD to tri

b 7.5. angleDAB b: but as triangle CAB to triangle BAD, fo is the bale CA to AD c;

C 1.6.

and as triangle EAD

to triangle DAB, fo

C

A

D

E

is the bafe EA to AB ;, as therefore CA to AD, so is EA

d II. 5. to AB d; wherefore the fides of the triangles ABC, ADE about the equal angles are reciprocally proportional.

But let the fides of the triangles ABC, ADE about the equal angles be reciprocally proportional, viz. CA to AD, as

EA

EA to AB; the triangle ABC is equal to the triangle Book VI.

ADE.

Having joined BD as before; because, as CA to AD, fo is

EA to AB; and as CA to AD, so is triangle ABC to tri

ngle BADc; and as EA to AB, fo is triangle EAD to 1.6. riangle BAD; therefore d as triangle BAČ to triangle d 11. 5. BAD, fo is triangle EAD to triangle BAD; that is, the riangles BAC, EAD have the fame ratio to the triangle 3AD: wherefore the triangle BAC is equal e to the triangle 9. 5. ADE. Therefore equal triangles, &c. Q. E. D.

PROP. XVI. THE OR.

F four ftraight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means: And if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four ftraight lines are proportionals.

Let the four ftraight lines, AB, CD, E, F be proportionals, viz. as AB to CD, fo E to F; the rectangle contained by AB, F is equal to the rectangle contained by CD, E.

From the points A, C draw a AG, CH at right angles to a 11. 1. AB, CD; and make AG equal to F, and CH equal to E, and complete the parallelograms BG, DH: because, as AB to CD, fo is E to F; and that E is equal to CH, and F to AG; AB is b to CD, as CH to AG: therefore the fides of b 7. 5. the parallelograms BG, DH about the equal angles are reciprocally proportional; but parallelograms which have their fides about equal angles reciprocally proportional, are equal to

one another; therefore the parallelogram BG is equal toc 14. 6.

Book VI the parallelogram DH and the parallelogram BG is contained by the ftraight E

lines AB, F; because AG
is equal to F; and the pa-
rallelogram DH is contain-
ed by CD and E;' because
CH is equal to E: there-
fore the rectangle contain-
ed by the straight lines AB,
Fis equal to that which is
contained by CD and E.

And if the rectangle contained by the ftraight lines AB, F be equal to that which is contained by CD, E; thefe four lines are proportionals, viz. AB is to CD, as E to F.

The fame conftruction being made, because the rectangle contained by the straight lines AB, F is equal to that which is contained by CD, E, and that the rectangle BG is contained by AB, F, because AG is equal to F; and the rectangle DH by CD, E, because CH is equal to E; therefore the parallelogram BG is equal to the parallelogram DH; and they are equiangular: but the fides about the equal angles of 14. 6. equal parallelograms are reciprocally proportional d; wherefore, as AB to CD, fo is CH to AG; and CH is equal to E, and AG to F: as therefore AB is to CD, fo E to F. Wherefore, if four, &c. Q. E. D.

PRO P. XVII. THE OR.

F three ftraight lines be proportionals, the rectangle contained by the extremes is equal to the fquare of the mean: And if the rectangle contained by the extremes be equal to the fquare of the mean, the three ftraight lines are proportionals.

Let the three straight lines A, B, C be proportionals, viz. as A to B, fo B to C; the rectangle contained by A, C is equal to the fquare of B.

Take