Then because the Circumferences BC, CK, KL, are equal to each other; the Angles BGC, CGK, K GL, will be alfo equal to one another; and fo* 27. 3. the Circumference BL is the fame Multiple of the Circumference BC, as the Angle BGL is of the Angle BGC. For the fame Reason, the Circumference NE is the fame Multiple of the Circumference EF, as the Angle EHN is of the Angle EHF; but if the Circumference BL be equal to the Circumference EN, then the Angle BGL fhall be equal to the Angle EH N; and if the Circumference B L be greater than the Circumference E N, the Angle B G L will be greater than the Angle EH N; and if lefs, lefs. Therefore, here are four Magnitudes, viz. the two Circumferences B C, E F, and the two Angles BGC, EHF; and fince there are taken Equimultiples of the Circumference BC, and the Angle BGC, to wit, the Circumference B L, and the Angle BGL; as alfo Equimultiples of the Circumference E F, and the Angle EHF, viz. the Circumference E N, and the Angle EHN; and because it is proved, if the Circumference B L exceeds the Circumference EN, the Angle BGL will likewife exceed the Angle EHN; and if equal, equal; if lefs, lefs; it fhall be as the Circumference B C is to the Circumference EF, fo is the Angle BGC to the Angle E HF; † Def. 5. 5. but as the Angle BGC is to the Angle EHF, fo is

# 20.3°

the Angle BAC to the Angle EDF, for the for- 1 15.5 mer are double to the latter: Therefore, as the Circumference BC is to the Circumference E F, fo is the Angle BGC to the Angle EHF; and to the Angle B A C to the Angle ED F.

Wherefore, in equal Circles, the Angles have the fame Proportion as the Circumferences they land on, whether they be at the Centres, or at the Circumferences.

I fay, moreover, that as the Circumference BC is to the Circumference E F, fo is the Sector G B C to the Sector HF E.

For, join B C, C K, and affume the Points X, O, in the Circumferences B C, CK; and join BX, X C, CO, O K.

Then, because the two Sides B G, GC, are equal to the two Sides C G, GK, and they contain equal

Angles,

+4.1.

24. 3.

Angles, the Bafe BC fhall be + equal to the Bafe CK, as likewife the Triangle G B C to the Triangle GCK. And, because the Circumference BC is equal to the Circumference CK, and the Circumference remaining, which makes up the whole Circle A B C, is equal to the remaining Circumference, which makes up the fame Circle A B C, the Angle B XC is equal to the Angle COK; and fo the Segment B XC is fimilar to the Segment COK; and they are upon equal Right Lines BC, CK; but fimilar Segments of Circles, that ftand upon equal Right Lines, are equal to each other: Therefore the Segment B XC is equal to the Segment COK. But the Triangle BGC is alfo equal to the Triangle CGK, and fo the whole Sector B G C will be equal to the whole Sector CG K. By the fame Reafon, the Sector GKL will be equal to the Sector GBC, or GCK; therefore, the three Sectors B G C, CGK, KG L, are equal to one another; fo likewife are the Sectors HEF, HFM, H MN. Wherefore the Circumference L B is the fame Multiple of the Circumference B C, as the Sector G BL is of the Sector G B C. For the fame Réafan, the Circumference NE is the fame Multiple of the Circumference E F, as the Sector HEN is of the Sector HEF; but if the Circumference B L be equal to the Circumference EN, then the Sector B G L will be equal to the Sector EH N; and if the Circumference B L exceeds the Circumference E N, then the Sector B G L will also exceed the Sector EHN; and if lefs, lefs. Therefore, fince there are four Magnitudes, to wit, the two Circumferences BC, E F, and the two Sectors GBC, EHF; and there are taken the Circumference B L, and the Sector G BL, Equimultiples of the Circumference C B, and the Sector GBC; as also the Circumference E N, and the Sector HEN Equimultiples of the Circumference EF, and the Sector HEF; and because it is proved, that if the Circumference BL exceeds the Circumference E N, the Sector BGL will alfo exceed the Sector EHN; and, if + Def. 5. 5, equal, equal; if lefs, lefs; therefore, † as the Circumference BC is to the Circumference E F, fo is the Sector GBC to the Sector HEF; which was to be demonftrated.

Coroll,

Coroll. 1. An Angle at the Centre of a Circle is to four Right Angles, as an Arc on which it ftands is to the whole Circumference; for as the Angle BAC is to a Right Angle, fo is the Arc B C to a Quadrant of the Circle: Wherefore, if the Confequents be quadrupled, the Angle B A C fhall be to four Right Angles as the Arc BC is to the who le Circumference.

2. The Arcs I L, BC, of unequal Circles, which fubtend equal Angles, whether at their Centres, or Circumferences, are fimilar; for I L, is to the whole Circumference IL E, as the Angle I A L is to four Right Angles; but as I A L, or BA C, is to four Right Angles, fo is the Arc B C to the whole Circumference B C F. Therefore, as IL is to the whole Circumference ILE, fo is BC to the whole Circumference B CF; and fo the Arcs IL, BC, are fimilar.

3. Two Semidiameters, A B, A C, cut off fimilar Arcs IL, BC, from concentric Circumferences,

The END of the SIXTH BOOK.

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