Sidebilder
PDF
ePub

PROP. XXI. THEOR.

RECTILINEAL figures which are fimilar to the same

rectilineal figure, are alfo fimilar to one another.

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

Book VI.

their fides about the equal angles proportionals. Again, because Ba. 1. Def. 6. is fimilar to C, they are cquiangular, and have

their fides about the e

qual angles proportionals. therefore the fi

A

gures A, B are each of

B

them equiangular to C, and have the fides about the equal angles of each of them and of C proportionals. Wherefore the rectilineal figures A and B are equiangularb, and have their fides about the equal b. 1. Ax. 1. angles proportionals. Therefore A is fimilar to B. Q. E, D,

C. II. S.

PROP. XXII. THEOR.

F four straight lines be proportionals, the fimilar rectilineal figures fimilarly defcribed upon them shall also be proportionals. and if the fimilar rectilineal figures fimilarly described upon four straight lines be proportionals, those straight lines fhall be proportionals.

Let the four straight lines AB, CD, EF, GH be proportionals, viz. 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.

a

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

b. 11. 5.

C. 22. 5.

to

Book VI. to X, fo EF to O. but as AB to X, fo is d the rectilineal KAB to the rectilineal LCD, and as EF to O, fo is the rectilineal MF to the d. 2. Cor. rectilineal NH. therefore as KAB to LCD, fob is MF to NH.

20.6.

b. 11. 5.

C. 12. 6.

f. 18. 6.

And if the rectilineal KAB be to LCD, as MF to NH; the ftraight line AB is to CD, as EF to GH.

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

[blocks in formation]

g. 9.5.

See N.

gures 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 ftraight lines, &c. Q. E. D.

g

[blocks in formation]

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 AC to the parallelogram CF, is the fame with the ratio which is compounded of the ratios of their fides.

Let

Let BC, CG be placed in a straight line, therefore DC and CE are Book VI. alfo in a ftraight line; and complete the parallelogram DG, and, taking any straight line K, make b as BC to CG, fo K to L; and as a. 14. 1. DC to CE, fo make b L to M. therefore the ratios of K to L, and b. 12.6. 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 is faid to be compounded of the ratios of K to L, and L to M. wherefore alfo c. A. Def. 5. K has to M, the ratio compounded

A

D H

G

B

C

d. f. 6.

e. 11. 5.

KLM

É F

of the ratios of the fides. and be-
caufe as BC to CG, fo is the pa-
rallelogram AC to the parallelo-
gram CH4; but as BC to CG, fo
is K to L; therefore K is to L,
as the parallelogram AC to the pa-
rallelogram CH. again, because as
DC to CE, fo is the parallelogram
CH to the parallelogram CF; but
as DC to CE, fo is L to M; where-
fore 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 paral-
lelogram AC to the parallelogram CH; and as L to M, so the pa-
fallelogram CH to the parallelogram CF; ex aequali f, K is to M, as
the parallelogram 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 compounded of the ratios of the fides. Wherefore equiangular
parallelograms, &c. Q. E. D.

f. 22. 5.

[blocks in formation]

THE

HE parallelograms about the diameter of any paral- See N.
lelogram, are fimilar to the whole, and to one ano-
ther.

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.

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

angle

c. 4. 6.

Book VI. angle ABC is equal to the angle AEF. and each of the angles n BCD, EFG is equal to the oppofite angle DAB b, and therefore are b. 34. 1. equal to one another; wherefore the parallelograms ABCD, AEFG are equiangular. and because the angle ABC is equal to the angle AEF, and the angle BAC common to the two triangles BAC, EAF, they are equiangular to one another; therefore as AB to BC, fo is AE to EF. and because the oppofite fides of parallelograms are equal to one another b, AB is d 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 proportionals; and they

d. 7. 5.

G

A

E

B

F

H

D K

C

e. 1. Def. 6. are therefore fimilar to one another. for the fame reason, the parallelogram ABCD is similar 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 f. 21. 6. fimilar to one anotherf, therefore the parallelogram GE is fimilar to KH. Wherefore the parallelograms, &c. Q. E. D.

See N.

a. Cor.45.1.

b.

PROP. XXV. PROB.

To defcribe a rectilineal figure which shall be fimilar

to one, and equal to another given rectilineal figure.

Let ABC be the given rectilineal figure, to which the figure to be described 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.

a

Upon the straight line BC describe the parallelogram BE equal to the figure ABC; alfo upon CE defcribe the parallelogram CM S 19.1. equal to D, and having the angle FCE equal to the angle CBL. 214.1. therefore BC and CF are in a straight line b, as alfo LE and EM. c. 13. 6. between BC and CF find a mean proportional GH, and upon GH d. 18.6. described 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 firft is to the third, e. 2. Cor. fo is the figure upon the firft to the fimilar and fimilarly described

20. 6.

figure

figure upon the fecond; therefore as BC to CF, fo is the rectilineal Book VI. figure ABC to KGH. but as BC to CF, fo is f the parallelogram

BE to the parallelogram EF. therefore as the rectilineal figure ABC f. 1.6. is to KGH, fo is the parallelogram BE to the parallelogram EF8. g. 11. 5. and the rectilineal figure ABC is equal to the parallelogram BE;

[blocks in formation]

therefore the rectilineal figure KGH is equal to the parallelogram h. 14. 5. 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 has been described fimilar to the figure ABC, and equal to D. Which was to be done.

IF

PROP. XXVI. THEOR.

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 are

[blocks in formation]

bout the fame diameter, they are fimilar to one another. where- a. 24 6. fore as DA to AB, fo is GA to AK. but because ABCD and b. 1. Def.6. AEFG are fimilar parallelograms, as DA is to AB fo is GA to AE.

there.

« ForrigeFortsett »