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Take two ftraight lines DE, DF containing any angle EDF; and Book VI.

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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.

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PROP. XIII. PROB.

O find a mean proportional between two given
ftraight lines.

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

Place AB, BC in a straight line, and upon AC defcribe 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 a femicircle is a right angle 6, and becaufe in the right angled triangle ADC, DB is drawn from the right

angle perpendicular to the bafe, DB A

is a mean proportional between'

D

a. 11.

B

C

b. 31. 3

AB, BC the fegments of the bafe . therefore between the two c. Cor. 8.6

given ftraight lines AB, BC, a mean proportional DB is found. Which was to be done.

Book VI.

2.14 1.

b. 7. 5.

• c. 1. 6.

PROP. XIV. THEOR.

EQUAL 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 straight line; wherefore alfo FB, BG are in one ftraight line. the fides of the parallelograms AB, BC about the equal angles, are reciprocally proportional; that is, DB is to BE, as GB to BF.

A

Complete the parallelogram FE; and because the parallelogram AB is equal to BC, and that FE is another parallelogram, AB is to FE, as BC to FE♣. but as AB to FE, fo is the base DB to BE; and as BC to FE, fo is the bafe GB to BF; therefore as DB to BE, fo is GB to

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F

E

D

B

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c. 9. 5.

bout their equal angles are reciprocally proportional.

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.

Because as DB is to BE, fo GB to BF; and as DB to BE, fo is the parallelogram AB to the parallelogram FE; and as GB to BF, fo is parallelogram BC to parallelogram FE; therefore as AB to FE, fo BC to FE. wherefore the parallelogram AB is equal to the parallelogram BC. Therefore equal parallelograms, &c. Q. E. D.

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PROP. XV. THEOR.

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, are 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

D

Let the triangles be placed fo that their fides CA, AD be in one straight line; wherefore alfo EA and AB are in one straight line; and 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 triangle DAB b. but as triangle CAB to triangle BAD, fo is

the bafe CA to AD; and as triangle C

Book VI.

1. 14. 2)

b. 7. s

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c. 1. 6.

EAD to triangle DAB, fo is the bafe

EA to AB; as therefore CA to AD, fo is EA to AB4. wherefore d. ri. ¿ 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 to AB ; the triangle ABC is equal to the triangle ADE.

Having joined BD as before, becaufe as CA to AD, fo is EA to AB; and as CA to AD; fo is triangle BAC to triangle BAD ; and as EA to AB, fo triangle EAD to triangle BAD; therefore as triangle BAC to triangle BAD, fo is triangle EAD to triangle BAD; that is, the triangles BAC, EAD have the fame ratio to the triangle BAD. wherefore the triangle ABC is equal to the tri- . g. 5, angle ADE. Therefore equal triangles, &c. Q. E. D.

Book VI.

a. II. I.

b. 7. 5.

14. 6.

PROP. XVI. THEOR.

IF 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, Fis equal to the rectangle contained by CD, E.

2

From the points A, C draw AG, CH at right angles to AB, CD; and make AG equal to F, and CH equal to E, and complete the parallelograms BG, DH. because as AB to CD, so is E to F; and that E is equal to CH, and F to AG; AB is to CD, as CH to AG. therefore the fides of 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 to the parallelogram DH. and the parallelogram BG is contained by the ftraight lines T AB, F, because AG is equal

to F; and the parallelogram
DH is contained by CD and
E, because CH is equal to E.
therefore the rectangle con-
tained by the ftraight lines
AB, F is equal to that which
is contained by CD and E.

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G

H

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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, becaufe the rectangle contained by the ftraight 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, becaufe 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 equal parallelograms are reciprocally pro- Book VI. portional. wherefore as AB to CD, fo is CH to AG; and CH n is equal to E, and AG to F. as therefore AB is to CD, fo E to F. c. 14. 6, Wherefore if four, &c. Q. E, D,

IF

PROP. XVII. THEOR.

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 straight lines are proportionals.

Let the three ftraight 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 D equal to B; and because as A to B, fo B to C, and that B is equal to D; A is to B, as D to C. but if four ftraight lines a. 7. 8. be proportionals, the rec

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B is equal to D. therefore the rectangle contained by A, C is equal to the fquare of B.

And if the rectangle contained by A, C be equal to the fquare of B; A is to B, as B to C.

The fame conftruction being made, because the rectangle contained by A, C is equal to the fquare of B, and the fquare of B is equal to the rectangle contained by B, D, because B is equal to D; therefore the rectangle contained by A, C is equal to that contained by B, D. but if the rectangle contained by the extremes be equal to that contained by the means, the four ftraight lines are propor tionals b. therefore A is to B, as D to C; but B is equal to Di

b. 16. 4,

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