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IN equal Circles, the Angles as well those at the Center as those at the Cir∣cumference, as also the Sectors are in the same Ratio as the Arks, which serve to them as Base.
If the Circles ANC, DOF, be equal; there shall be the same Ratio of the Angle ABC to the Angle DEF, as of the Ark AC to the Ark DF. Let the Ark AG, GH, HC, he equal Arks, and consequently Aliquot parts of the Ark AC, and let be divided
the Ark DF, into so many equal to AG, as may be found therein, and let the Lines EI, EK, and the rest be drawn.
Demonstration. All the Angles ABG, GBH, HBC, DEI, IEK, and the rest, are equal (by the 37th. of the third;) so then AG an Aliquot part of AC, is found in the Ark DF, as many times as the Angle ABG an Aliquot part of the Angle ABC, is found in DEF; there is therefore the same Ratio of the Ark AC to the Ark DF, as of the Angle ABC to the Angle DEF. And because N and O are the halfs of the Angles ABC, DEF, they shall be in the same Ratio as are those Angles; there is therefore the same Ratio of the Angle N to the Angle O, as of the Ark AC to the Ark DF.
It is the same with Sectors; for if the Lines AG, GH, HC, DI, IK, and the rest were drawn, they would be equal (by the 28th. of the Third;) and each Sector would be divided into a Triangle and a Segment. The Triangles would be equal (by the 8th. of the first;) and the little Segments would be also equal (by the 24th. of the third;) thence all those little
Sectors would be equal: and so as many as the Ark BF containeth of Aliquot parts of the Ark of AC, so many the Sector DKF would contain Aliquot parts of the Sector AGC. There is therefore the same Reason of Ark to Ark, as of Sector to Sector.