Sidebilder
PDF
ePub

lefs bthan EK: But, as was demonftrated in the preceding, Book III. BC is double of BH, and FG double of FK, and the fquares of EH, HB are equal to the fquares of EK, KF, of which the b5. Def. 3. fquare of EH is lefs than the fquare of EK, because EH is lefs than EK; therefore the fquare of BH is greater than the fquare of FK, and the straight line BH greater than FK; and therefore BC is greater than FG.

Next, Let BC be greater than FG; BC is nearer to the cen- . tre than FG, that is, the fame construction being made, EH is lefs than EK: Because BC is greater than FG, BH likewise is greater than KF: And the fquares of BH, HE are equal to the squares of FK, KE, of which the square of BH is greater than the fquare of FK, becaufe BH is greater than FK; therefore the fquare of EH is less than the fquare of EK, and the straight line EH lefs than EK. Wherefore the diameter, &c. Q. E. D.

PROP. XVI, THE OR.

THE ftraight

HE ftraight line drawn at right angles to the dia- see N. meter of a circle, from the extremity of it, falls without the circle; and no straight line can be drawn between that straight line and the circumference from the extremity, so as not to cut the circle; or which is the fame thing, no ftraight line can make fo great an acute angle with the diameter at its extremity, or so small an angle with the straight line which is at right angles to it, as not to cut the circle,

Let ABC be a circle, the centre of which is D, and the diameter AB: the ftraight line drawn at right angles to AB from its extremity A, fhall fall without the circle.

For, if it does not, let it fall, if

poffible, within the circle, as AC, and draw DC to the point C where

it meets the circumference: And

C

[blocks in formation]

therefore ACD is a right angle, and the angles DAC, ACD are

therefore equal to two right angles; which is impoflible b: b 17. I F

Therefore

Book III. Therefore the straight line drawn from A at right angles to BA does not fall within the circle: In the fame manner, it may be demonstrated that it does not fall upon the circumference; therefore it muft fall without the circle, as AE.

C 12. I.

d 19. 1.

€ 2. 3.

And between the straight line AE and the circumference no ftraight line can be drawn from the point A which does not cut the circle: For, if poffible, let FA be between them, and from the point D draw DG perpendicular to FA, and let it meet the circumference in H And because AGD is a right angle, and DAG lefs b than a right angle: DA is greater than DG: But DA is equal to DH; therefore DH is greater than DG,

:

the less than the greater, which is
impoffible: Therefore no ftraight
line can be drawn from the point
A between AE and the circumfe-
rence, which does not cut the cir-
clę, or, which amounts to the fame B
thing, however great an acute angle
a ftraight line makes with the dia-
meter at the point A, or however
fmall an angle it makes with AE,

d

F

C

E

[blocks in formation]

the circumference paffes between that ftraight line and the perpendicular AE. And this is all that is to be understood, when, in the Greek text and translations from it, the angle of ⚫ the femicircle is faid to be greater than any acute rectilineal angle, and the remaining angle lefs than any rectilineal an• gle.'

[ocr errors]

COR. From this it is manifest that the straight line which is drawn at right angles to the diameter of a circle from the extremity of it, touches the circle; and that it touches it only in one point, because, if it did meet the circle in two, it would fall within it e. Alfo it is evident that there can be but one ftraight line which touches the circle in the fame point,'

[ocr errors]

T

PROP, XVII. PRO B,

'O draw a straight line from a given point, either without or in the circumference, which hall touch

a given circle.

First, let A be a given point without the given circle BCD;

it

it is required to draw a straight line from A which fhall touch Book III. the circle.

Find the centre E of the circle, and join AE; and from a 1. 3. the centre E, at the distance EA, describe the circle AFG;

from the point D draw b DF at right angles to EA, and join b 11. I EBF, AB. AB touches the circle BCD.

Because E is the centre

of the circles BCD, AFG, EA is equal to EF: And ED to EB; therefore the two fides AE, EB are equal to the two FE, ED, and they contain the angle at E common to the two triangles AEB, FED; therefore the base DF is equal to the bafe AB, and the triangle FED to the trian

[blocks in formation]

gle AEB, and the other angles to the other angles : There- c 4. I. fore the angle EBA is equal to the angle EDF: But EDF is a right angle, wherefore EBA is a right angle: And EB is drawn from the centre: But a ftraight line drawn from the extremity of a diameter, at right angles to it, touches the circle d: There- d Cor.16. fore AB touches the circle; and it is drawn from the given point A. Which was to be done.

But, if the given point be in the circumference of the circle, as the point D, draw DE to the centre E, and DF at right angles to DE; DF touches the circle d

IF

PROP. XVIII. THEOR.

F a ftraight line touches a circle, the ftraight line drawn from the centre to the point of contact, fhall be perpendicular to the line touching the circle.

Let the ftraight line DE touch the circle ABC in the point ; take the centre F, and draw the straight line FC: FC is perpendicular to DE.

For, if it be not, from the point F draw FBG perpendicular to DE; and because FGC is a right angle, GCF is b an acute b 17. 1 angle; and to the greater angle the greatest c fide is oppofite: c 19. Le

F 2.

Therefore

Book III. Therefore FC is greater than FG; but FC is equal to FB; therefore FB is greater than FG, the lefs than the greater, which is impoffible: Wherefore FG is not perpendicular to DE: In the fame manner it may be fhewn, that no other is perpendicular to it befides FC, that is, FC is perpendicular to DE. Therefore, if a ftraight line, &c. Q.E. D.

a 18.3.

I

[blocks in formation]

PROP. XIX. THEOR.

F a straight line touches a circle, and from the point of contact a ftraight line be drawn at right angles to the touching line, the centre of the circle fhall be in that line,

Let the ftraight line DE touch the circle ABC in C, and from C let CA be drawn at right angles to DE; the centre of the circle is in CA.

For, if not, let F be the centre, if poffible, and join CF :
Because DE touches the circle ABC,

and FC is drawn from the centre to
the point of contact, FC is perpendi.
cular a to DE; therefore FCE is a
right angle: But ACE is also a right
angle; therefore the angle FCE is
equal to the angle ACE, the lefs to B
the greater, which is impoffible :
Wherefore F is not the centre of the

F

See N.

circle ABC: In the fame manner, it D

C

E

may be fhewn, that no other point

which is not in CA, is the centre; that is, the centre is in CA. Therefore, if a straight line, &c. Q. E. D.

PROP. XX. THEOR.

THE angle at the centre of a circle is double of the angle at the circumference, upon the fame base, that is, upon the fame part of the circumference,

Let

~

Let ABC be a circle, and BEC an angle at the centre, and Book III. BAC an angle at the circumference, which have the fame circumference BC for their base; the angle

BEC is double of the angle BAC.

a

A

First, let E the centre of the circle be within the angle BAC, and join AË, and produce it to F: Because EA is equal to EB, the angle EAB is equal to the angle EBA; therefore the angles EAB, EBA are double of the angle EAB; but the angle BEF is equal to the angles EAB, EBA; therefore alfo the angle BEF is double of the angle EAB: For the fame reason, the angle FEC is double of the angle EAC : Therefore the whole angle BEC is double of the whole angle BAC.

Again, let E the centre of the circle be without the angle BDC,and join DE and produce it to G. It may be demonftrated, as in the firft cafe, that the angle GEC is double of the angle GDC, and that GEB a part of the first is double of GDB a part of the other; therefore the remaining angle BEC is double of the remaining angle BDC, Therefore the angle at the centre, &c. Q. E. D.

B

B

PROP. XXI. THEOR.

F

a 5. 1.

Cb 32. I

A

D

E

HE angles in the fame fegment of a circle are e-
qual to one another.

TH

[merged small][merged small][merged small][ocr errors][merged small]

See N.

F 3

the

« ForrigeFortsett »