than one point. for, if it be possible, let the circle ACK touch the Book III. circle ABC in the points A, C, and join AC. therefore because the

two points A, C are in the circumference

K

b. 2. 3.

C

been fhewn that they cannot touch on the infide in more points than one. therefore one circle, &c. Q. E. D.

EQUAL ftraight lines in a circle are equally diftant

from the center; and those which are equally diftant from the center, are equal to one another.

Let the ftraight lines AB, CD in the circle ABDC be equal to one another; they are equally distant from the center.

Take E the center of the circle ABDC, and from it draw FF, EG perpendiculars to AB, CD. then because the straight line EF paffing thro' the center cuts the ftraight line AB, which does not pafs thro' the center, at right angles, it

also bisects it. wherefore AF is equal A
to FB, and AB double of AF. for the
fame reafon CD is double of CG. and
AB is equal to CD, therefore AFis equal
to CG. and because AE is equal to EC,
the fquare of AE is equal to the fquare
of EC. but the fquares of AF, FE are
equal to the fquare of AE, because the
angle AFE is a right angle; and for the

B

E

like reafon the fquares of EG, GC are equal to the fquare of EC. therefore the fquares of AF, FE are equal to the fquares of CG,

2. 3. 3.

b. 47. I.

Book III. GE, of which the fquare of AF is equal to the fquare of CG, be

cause AF is equal to CG; therefore the remaining fquare of FE is equal to the remaining fquare of EG, and the ftraight line FE is therefore equal to EG. but ftraight lines in a circle are faid to be equally diftant from the center, when

the perpendiculars drawn to them from A

.4. Def. 3 the center are equal. therefore AB, CD are equally distant from the center.

See N.

3. 20. T.

Next, if the straight lines AB, CD be equally diftant from the center, that is, if FE be equal to EG; AB is equal to CD. for, the fame conftruction be ing made, it may, as before, be demonftrated that AB is double of AF and

B

G

E

CD double of CG, and that the fquares of EF, FA are equal to the fquares of EG, GC; of which the fquare of FE is equal to the fquare of EG, because FE is equal to EG; therefore the remaining square of AF is equal to the remaining fquare of CG; and the ftraight line AF is therefore equal to CG. and AB is double of AF, and CD double of CG; wherefore AB is equal to CD. Therefore equal straight lines, &c. Q. E. D.

PROP. XV. THEO R.

HE diameter is the greatest ftraight line in a circle; and of all others, that which is nearer to the center is always greater than one more remote; and the greater is nearer to the center than the less.

Let ABCD be a circle, of which the diameter is AD, and center E; and let BC be nearer to the center than FG. AD is greater than any ftraight line BC which is not a

diameter, and BC greater than FG.

From the center draw EH, EK per-F

pendiculars to BC, FG, and join EB,
EC, EF; and becaufe AE is equal to (K
EB, and ED to EC, AD is equal to
EB, EC. but EB, EC, are greater than
BC, wherefore alfo AD is greater than
BC.

a

And because BC is nearer to the cen

AB

H

E

D

ter than FG, EH is less

than EK. but, as was demonftrated in Book III.

the preceding, 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 b. 5. Def. 3. which the fquare of EH is less than the fquare of EK, because EH is less 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 center than FG, that is, the fame construction being made, EH is lefs than EK. because BC is greater than FG, BH likewife is greater than FK. and the fquares of BH, HE are equal to the fquares of FK, KE, of which the fquare of BH is greater than the fquare of FK, because BH is greater than FK; therefore the fquare of EH is less than the fquare of EK, and the ftraight line EH less than EK. Wherefore the diameter, &c. Q. E. D.

PROP. XVI. THEOR.

THE 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 ftraight line can be drawn between that straight line and the circumference from the extremity, fo 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 fo 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 center of which is D, and the diameter AB; the straight 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 B draw DC to the point C where it

A

D

meets the circumference. and be

caufe DA is equal to DC, the angle

DAC is equal to the angle ACD;

but DAC is a right angle, therefore ACD is a right angle, and

2. 5. I.

Book III. the angles DAC, ACD are therefore equal to two right angles; which is impoffible . 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 must fall without the circle, as AE.

b. 17. 1.

C. 12. I.

d. 19. 1.

糖

6.2.3.

F

E

C

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 than a right angle, DA is greater than DG. but DA is equal to DH; therefore DH is greater than DG, the lefs than the greater, which is impoffible. therefore no straight line can be drawn from the point A between AE and the circumference, which does not cut the circle. or, which amounts to the fame thing, however great an acute angle a straight line makes with

B

A

D

the diameter at the point A, or however small an angle it makes with AE, the circumference paffes between that straight 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 less than any rectilineal angle.'

COR. From this it is manifeft that the ftraight 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. Also it is evident that there can be but one straight line which touches the circle in the fame point.'

To draw a ftraight line from a given point, either

without or in the circumference, which shall

touch a given circle.

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

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

circle.

a

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

point D draw b DF at right angles to EA, and join EBF, AB. b. 11. 1. AB touches the circle BCD.

Because E is the center 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 G
E common to the two triangles
AEB, FED; therefore the bafe
DF is equal to the bafe AB,
and the triangle FED to the
triangle AEB, and the other
angles to the other angles .

A

CE

therefore 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 center; but a straight line drawn from the extremity of

C. 4. K

a diameter, at right angles to it, touches the circled. therefore ABd. Cor. 16. 3,
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 center E, and DF at right angles
to DE; DF touches the circle d.

IF

PROP. XVIII. THEOR.

Fa ftraight line touches a circle, the ftraight line drawn from the center to the point of contact, shall be perpendicular to the line touching the circle.

Let the straight line DE touch the circle ABC in the point C, take the center F, and draw the ftraight 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 an acute angle; b. 17. 1. and to the greater angle the greatest fide is oppofite. therefore FCc. 19. 1.

c