REC

PROP. XXI. THEOR.

ECTILINEAL figures which are fimilar to the
fame rectilineal figure, are alfo fimilar to one

another.

Book VI,

Again, because
Again, because a.1. Def. 6,

Let each of the rectilineal figures A, B be fimilar to the rectilineal figure C. the figure A is fimilar to the figure B. Because A is fimilar to C, they are equiangular, and also have their fides about the equal angles proportionals. B is fimilar to C, they are equiangular, and have their fides about the equal angles proportionals. therefore the figures A, B are each of

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them equiangular to C, and have the fides about the equal angles of each of them and of C proportionals. Wherefore the recti

lincal figures A and B are equiangular, and have their fides about b. 1. Ax. 1. the equal angles proportionals. Therefore A is fimilar to B. c. 11. 5. Q. E. D.

PROP. XXII. THEOR,

IF four ftraight lines be proportionals, the fimilar rectilineal figures fimilarly defcribed upon them fhall also be proportionals. and if the similar rectilineal figures fimilarly defcribed upon four ftraight lines be proportionals, those straight lines fhall be proportionals.

Let the four straight lines AB, CD, EF, GH be proportionals, yiz. AB to CD, as EF to GH, and upon AB, CD let the similar rectilineal figures KAB, LCD be fimilarly defcribed; and upon EF, GH the fimilar rectilineal figures MF, NH, in like manner. the rectilineal figure KAB is to LCD, as MF to NH.

To AB, CD take a third proportional X; and to EF, GH a a. 11. 6. third proportional O. and becanfe AB is to CD, as EF to GH, b. 11. 5. therefore CD is to X, as GH to O; wherefore ex aequali, as AB c. 22. 5.

Book VI. to X, fo EF to O. but as AB to X, fo is the rectilineal KAB to the rectilineal LCD, and as EF to O, fo is the rectilineal MF to d. 2. Cor. the rectilineal NH. therefore as KAB to LCD, fob is MF to NH. And if the rectilineal KAB be to LCD, as MF to NH; the ftraight line AB is to CD, as EF to GH.

20.6.

b. 11. 5.

C. 12. 6.

f. 18. 6,

Make as AB to CD, fo EF to PR, and upon PR defcribe ! the rectilineal figure SR fimilar and fimilarly fituated to either of

6.9.5.

See N.

E

F G H

PR

the figures MF, NH. then because as AB to CD, fo EF to PR, and that upon AB, CD are defcribed the fimilar and fimilarly fituated rectilineals KAB, LCD, and upon EF, PR, in like manner, the fimilar rectilineals MF, SR; KAB is to LCD, as MF to SR; but, by the Hypothefis, KAB is to LCD, as MF to NH; and therefore the rectilineal MF having the fame ratio to each of the two NH, SR, these are equal to one another. they are alfo fimilar, and fimilarly fituated; therefore GH is equal to PR. and because as AB to CD, fo is EF to PR, and that PR is equal to GH; AB is to CD, as EF to GH. If therefore four straight lines, &c. Q. E. D.

PROP. XXIII. THEOR.

EQUIANGULAR parallelograms have to one

another the ratio which is compounded of the

ratios of their fides,

Let AC, CF be equiangular parallelograms, having the angle BCD equal to the angle ECG. the ratio of the parallelogram AÇ to the parallelogram CF, is the fame with the ratio which is com pounded of the ratios of their fides.

Let BC, CG be placed in a ftraight line, therefore DC and CE Book VI. are alfo in a straight line; and complete the parallelogram DG,

and, taking any straight line K, make bas BC to CG, fo K to L; a. 14 1. and as DC to CE, fo make b L to M. therefore the ratios of K to b. 12. 6. L, and L to M are the fame with the ratios of the fides, viz. of BC to CG, and DC to CE. But the ratio of K to M is that which

A

D

H

G

C

d. 1.6.

is faid to be compounded of the ratios of K to L, and L to M. c. A. Def. 5. wherefore alfo K has to M, the ratio compounded of the ratios of the fides. and because as BC to CG, fo is the parallelogram AC to the parallelogram CH4; but as BC to CG, fo is K to L; therefore K is

B

gram CF; but as DC to CE, fo is L to M; wherefore L is to M, as the parallelogram CH to the parallelogram CF. therefore fince it has been proved that as K to L, fo is the parallelogram AC to the parallelogram CH; and as L to M, fo the parallelogram CH to the parallelogram CF; ex aequali f, K is to M, as the parallelo- f. 22. 5v gram AC to the parallelogram CF. but K has to M the ratio which is compounded of the ratios of the fides; therefore, alfo the parallelogram AC has to the parallelogram CF the ratio which is copounded of the ratios of the fides. Wherefore equiangular parallelograms, &c. QE. D.

TH

PROP. XXIV. THEOR.

HE parallelograms about the diameter of any pa- see N. rallelogram, are fimilar to the whole, and to one

another.

Let ABCD be a parallelogram, of which the diameter is AC; and EG, HK the parallelograms about the diameter. the parallelograms EG, HK are fimilar both to the whole parallelogram ABCD, and to one another.

2

Because DC, GF are parallels, the angle ADC is equal to the a. 29. 1. angle AGF. for the fame reafon, becaufe BC, EF are parallels, the

b. 34. I.

Book VI. angle ABC is equal to the angle AEF. and each of the angies BCD, EFG is equal to the oppofite angle DAB, and therefore are equal to one another; wherefore the parallelograms ABCD, AEFG are equiangular. and becaufe the angle ABC is equal to the angle AEF, and the angle BAC common to the two triangles BAC, EAF,

c. 4. 6.

d. 7. 5.

they are equiangular to one another ; A E

c

therefore as AB to BC, fo is AE to
EF. and because the oppofite fides of
parallelograms are equal to one ano-
ther, AB is to AD, as AE to AG;
and DC to CB, as GF to FE; and alfo
CD to DA, as FG to GA. therefore
the fides of the parallelograms ABCD,
AEFG about the equal angles are pro-

D K

B

e. 1. Def.6. portionals; and they are therefore fimilar to one another . for the fame reafon, the parallelogram ABCD is fimilar to the parallelogram FHCK. Wherefore each of the parallelograms GE, KH is fimilar to DB. but rectilineal figures which are fimilar to the fame rectilineal figure, are alfo fimilar to one another f, therefore the parallelogram GE is fimilar to KH. Wherefore the parallelo grams, &c. Q. E. D.

f. 21. 6.

O defcribe a rectilineal figure which fhall be fimilar to one, and equal to another given rectilineal figure.

Let ABC be the given rectilineal figure, to which the figure to be defcribed is required to be fimilar, and D that to which it must be equal. It is required to defcribe a rectilineal figure fimilar to' ABC and equal to D.

C

a

Upon the ftraight line BC defcribe the parallelogram BE equal to the figure APC; alfo upon CE defcribe the parallelogram CM equal to D, and having the angle FCE equal to the angle CBL. therefore BC and CF are in a straight line b, as alfo LE and EM. between BC and CF find a mean proportional GH, and upon GH describe the rectilineal figure KGH fimilar and fimilarly fituated to the figure ABC. and because BC is to GH, as GH to' CF, and if three ftraight lines be proportionals, as the first is to the third, fo is the figure upon the first to the fimilar and fimilarly

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defcribed figure upon the fecond; therefore as BC to CF, fo is the Book VI. rectilineal figure ABC to KGH. but as BC to CF, fo is f the parallelogram BE to the parallelogram EF. therefore as the rectilineal f. 1. 6. figure ABC is to KGH, fo is the parallelogram BE to the parallelogram EF 8. and the rectilineal figure ABC is equal to the pa- g. 11. 5.

rallelogram BE; therefore the rectilineal figure KGH is equal bh. 14 5. to the parallelogram EF. but EF is equal to the figure D, wherefore alfo KGH is equal to D; and it is fimilar to ABC. Therefore the rectilineal figure KGH nas been defcribed fimilar to the figure ABC, and equal to D. Which was to be done.

IF

PROP. XXVI. THEOR.

F two fimilar parallelograms have a common angle, and be fimilarly fituated; they are about the fame diameter.

Let the parallelograms ABCD, AEFG be fimilar and fimilarly fituated, and have the angle DAB common.

ABCD and AEFG

A

B

about the fame diameter, they are fimilar to one another where- a. 24. 6.

fore as DA to AB, fo is GA to AK. but becaufe ABCD and b.1. Def.6. AEFG are fimilar parallelograms, as DA is to AB fo is GA to AE.