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... the ⚫s A & B each on the Oce.

Again. DF <DA < rad. CA.. F is within the O.

And

from G or H in AB produced, CG, or CH > rad. CA;

the s G & H are without the O.

Hence the st. line HF meets the Oce only in two points.

Any point being assumed, &c.

USE.-1. Practically the centre of a is found, by bisecting any chord, A B with a perp. CE, terminated in the Oce; and CE being bisd. in F, F is the centre.

2. The First Prop. bk. III. is applicable to all cases in which the centre of any circular object, as of the hori- A zontal section of a tree, may be required. A circular disk of metal, a wheel, a flower-bed, any object possessing the circular form will have its centre found in the same way.

Q. E. D.

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PROP. 2.-THEOR.

If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.

CON. 1, III. Pst. 1 & 2.

DEM. Def. 15, I.

5, I. The angles at the base of an isosc. A are equal to each other; and if the equal sides be produced, the s on the other side of the base shall be equal.

16, I. If one side of a ▲ be produced the ext. ▲ is greater than either of the int. opp. angles.

19, 1. The gr. of every ▲ is subtended by the gr. side, or has the gr. side opposite to it.

E1 Hyp. Let A B C be O, and A B any two ⚫s in the Oce; 2 Conc. the st. line from A to B within the O.

SUP.-If not within, and it is possible, let it be without, as

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5 Def. 15, I. ad imp. Now DB=DF .. DF > DE; an

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COR. 1.-A st. line, A B, cannot cut the Oce of a points than two; for, every st. line joining any two points in the Oce falls within the O, neither co-inciding with any other points in the Oce, nor meeting it except in the two given points.

COR. 2.-A st. line which touches a circle meets it only in one point.

COR. 3.-A circle is concave towards its centre.

SCH.-Instead of the ex absurdo demonstration of this Proposition, a direct method of proof, founded on Axiom A, bk. III. was given by COMMANDINE, who lived between A D. 1509 and 1575; he applied himself to mathematics at Verona, and in 1572, at Pesaro, published Euclid's Elements in fifteen books, in Latin.

C. 1

2

Assum.

1, III. Pst. I.

In the given line AB, take any E;
find D cen. of the O; and join DA,
DE, DB.

D. 1 Def. 15, I. 5, I. in ADAB, DA=DB;

to B,

DAB=LDBA;

in ▲ AED, AE is prod.

..ext. DEB > int. and opp. A
DAE;

but DAE=/ DBE;

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and

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.. ▲ DEB > < DBE.

But DEB>DBE..DB > DE;

C

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So is every

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.. the line AB, joining A and B falls within the O.

Q. E. D.

USE.-On this proposition are grounded those which show, that a circle touches a st. line in only one point; for if the st. line touched two points of the Oce the st. line would be drawn from one point of the Oce to the other, and consequently would fall within the circle, contrary to the very definition of such a line, that it does not cut the circumference. THEODOSIUS of Tripolis, a mathematician who lived some time after the reign of Trajan, compiled a work on the Properties of the Sphere and of the circles described on its surface, an edition of which was published at Oxford in 1675: he used Prop. 2. bk. III. to demonstrate that a Globe resting on a plane surface cannot touch the plane in any but a single point; otherwise the plane would enter the globe.

PROP. 3.-THEOR.

If a straight line drawn through the centre of a circle bisect a straight line in it which does not pass through the centre, it shall cut it at right angles; and conversely, if it cuts it at right angles, it shall bisect it.

CON. 1, III. Pst. 1.

DEM. Def. 15, I. 8, I. Def. 10, I. 5. I.

26, I. If two As have two /s of the one equal to two s of the other, each to each, and one side equal to one side, viz., either the sides adj. to the equals in each, or the sides opp. to them, then shall the other sides be equal, each to each, and also the third of the one to the third of the other.

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II. Also CD cutting AB at rt. Ls, bis. AB, i.e. AF=FB.

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COR. 1.-A st. line E F, bisecting any chord, A B, at rt. Ls, passes through the centre E of the circle.

COR. 2.-All chords, A B, G H par. to the tang. C K, at either extremity of the diam. CD, are bisected by the diam.

COR. 3.-The st. line which bisects AB, the com. chord of two circles, ACB, ALB, at rt. ≤s, passes through E & D, the centres of both circles.

COR. 4.-When in a circle there are several chords, AB, GH par. to each other the locus of their points of bisection is in that diam. CD, which is at rt. ▲s to them; and if the line, CD, which bisects one chord be a perp. that line bisects all the par. chords at rt. ▲s.

SCH.-This Prop. might be more briefly given; "If a diam. cut any other chord at rt.s, it shall bisect it; and conversely, if a diam. bisect any other chord, it shall cut it at rt.

s."

USE AND APP.-The use of this Prop. extends to the various cases, in which arcs, or circles, have to be drawn, and their centres ascertained; as,—

1. Given a circle, ABC, to find its centre, as in the last figure.

S. 1, Pst. 1. 10, I. 11, I., Draw any ch. A B; bis. it in F by the perp. EF ; prod. E to ⚫s C, D, forming diam. CD; bis. CD in E; and E is the centre.

2 Pst. 2.

3 10. I.

D. 1 C. 1. Cor. 3, III.. 2 C. 2., Def. 15, I. | ·.·

CD bis. ABL.. DC through the centre ;
CD is a diam... its mid. E cen. of

Q. E. F.

II. Given an arc ABD, to find the cen. of the of which it is an arc.

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N.B. With rad. C A, or CD, the circle, AB DE, may be completed.

III. Through three points, A, B, D, not in a st. line, to draw a circle.

S. 1 Pst. 1. 10, I. 2 11. I.

3 Cor. 3, III. Pst. 3.

Join A B, BD, and bis. them in F, G;

at F & G raise perps. intersecting in G;
C is the centre of through A, B, D:

and with rad. A C, or B C the circle
may be drawn.

N.B. The Dem. is given in Prop. 9, III.

K

E

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