3.C.1.Ax. 3, I. And arc BC= arc CK,

4 27, III, Def. 11, III.

5 D. 2. 24, III.

6 D. 2.

7 Sim.

8 Sim.

9 Conc.

10 Sim.

11

12 Hyp.

13 D. 9.

14 D. 10.

15 D. 11.

16 Def. 5, V. 17 Rec.

.. rem. arc BLC =rem. arc CLK;

ско.

and

ZO, and seg. BXC is sim. to seg.

BC= CK,

=

..seg. BXC = seg. CKO. And ▲ BGC = A CGK, .. the whole sect. BGC So sectors KGL = BGC = and sectors EHF = FHM = .. the mult. which are BL is of arc BC, that same mult. sect. BGL is of sect BGC;

the whole sect. CGK.
CGK,
MHN;

and the mult. which arc EN is of arc EF, that mult. sect. EHN is of sect EHF;

and if arc BL => or < arc EN, the sect. BGL => or < sect. EHN.

Now there are 4 Ms, the arcs, BC, EF, and the sects. BGC, EHF;

and of arc BC and sect. BGC, arc BL and sect. BGL are equims.;

and of arc EF and sect. EHF, arc EN and sect. EHN are equims.;

and if arc BL > = or < arc EN, the sect. BGL> or < sect. EHN;

.. arc BC: arc EF = sect. BGC: sect, EHF. . In equal circles, angles, &c.

COR. 1. The sectors are to each other as their angles.

Q. E. D.

For, if arc BC: arc EF = / BGC: EHF; and arc BC: arc EF = sect. BGC sect. EHF, then 11, V. sect. BGC: sect EHF = ▲ BGC: ▲ EHF.

:

COR. II. Similar sectors of the same or equal circles are equal.

COR. III. An angle at the centre of a circle is to four rt. angles as the arc on which it stands to the circumference of the circle.

Ꮓ

For, the at the centre: one rt. = arc subtending central arc subtending a rt. or quadrant;

ce.

Then, 4, V. at cen. : 4 rt. s = arc of central ▲ : whole

COR. IV. In different circles the arcs of equal angles at the centres or circumferences are similar.

COR. V. Hence, similar segments are contained by similar arcs and vice versa.

SCH. 1. If the arcs and sector had all been in one circle the proof would have been the same,-for H would have coincided with G, and D,E,F,M,N would have been points in the circumference of circle ABL.

2. The second part of the Proposition was added by THEON, the Ptolemaist, and father of the renowned but unfortunate HYPATIA, of Alexandria, in the time of the elder Theodosius;—it is given in the Commentary on PTOLEMY's Almagest.

3. That the angles at the circumferences are as the arcs on which they stand, follows also as a Corollary from Prop. 20, bk. III.

Prop. 33. a. In the same or in equal circles AGB, CHD angles, whether at the centres, as Ls AEB, CFD, or at the circumferences, Ls AGB, CHD, have the same ratio as the arcs, AB, CD. on which they stand.

E & C. 1 Def. 1, V. | Let AM or CN be a com. meas. of arcs AB, CD;

And. s AEB, CFD are mults. of /s G & H by 2, .. G: Harc AMB : arc CND

Q. E. D.

COR. Since the Zs and arcs are proportional when commensurable, ..they are also proportional when incommensurable. See Sch. 3-5 & Use 1, 16, VI.

PROP. 33 b. In the same circle, ABK, or in equal circles, the sectors BGC, GCK, that stand on equal arcs, BC, CK, are equal.

PROP 33 c. In the same circle ABK, as in fig. to 33 b. or in equal circles, ABG, CHD, as in fig. to 33 a, sectors AEB, CFD, have the same ratio as the arcs, AMB, CND, on which they stand.

E. & C. 1 Fig. 33, a.

2

3 Pst. 1, I.

D. 1 33, b. VI.
2 C. 2.

3 7 a V.

4 11, V.

Let are AM arc CN, & be a com. measure of arcs
AB, CD;

& let arc AB = 4 arc AM, & arc CD 3 arc CN,
then join EM and FN.

COR. Since the arcs and sectors are thus proportional when commensurable, ..they must also be proportional when incommensurable.

4. From this 33rd Proposition it results, that the angle at the centre of a circle is said to be measured by the arc on which it stands.

USE and APPL. 1. If arcs ACB, AEB, of different circles have a common chord, AB, the lines, AC, AD, diverging from one of its extremities, A, will cut the arcs proportionally, i. e. BF: FE = BD: DC.

2. The arcs, A, A' of unequal circles are in a ratio compounded of their central angles a, a' and their radii, R, R'.

C.

2

With rad. = R desc. an a',
and let the subtending arc be m.

D. 1 H. & C. 2 . arcs A and m have an eq. R,
2 C. & H. and . m and A' have eq. central
3 Cor. 5, 20, VI... m : A', =R:R'

4 D. 1 & 2.

A: maa';

s;

3. Central angles a, a', are in a ratio compounded of the direct ratio of their arcs A, A', and the inverse ratio of their radii R, R'.

Let each of the equal ratios be compounded with R′ : R;

2

OBS. "And herewith," remarks Captain Thomas Rudd, Chiefe Engineer to Charles I., "is the first six Books of EUCLIDE ended. There be hereafter added certain Propositions, which although they be not EUCLIDES, yet because they are both witty and usefull, I thought it good not to omit." RUDD'S Euclides Elements, A.D. 1651, p. 253.

SUBSIDIARY PROPOSITIONS.

PROP. B.-THEOR.

If an angle of a triangle be bisected by a st. line which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the st. line which bisects the angle.

CON. 5, IV. To desc. a about a given A. Pst 2, 1, I.

contained

contained by the

DEM. 21, III. The s in the same seg. of a O are equal.
3, II. If a st. line be divided into any two pts, the
by the whole and one of the pts is eq. to the
two parts together with the square of the aforesaid part.
35, III. If two st. lines cut one another within a the rect. contained
by the segs. of one is eq. to the rect. contained by the segs. of the
other.