27.3.

Then because the Circumferences BC, CK, KL, are equal to each other, the Angles BGC, CGK, KGL, will be alfo equal to one another; and fo 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 EHN; and if the Circumference BL be greater than the Circumference EN, the Angle BGL will be greater than the Angle EHN, and if lefs, lefs. Therefore here are four Magnitudes, viz. the two Circumferences BC, EF, and the two Angles BGC, EHF; and fince there are taken Equimultiples of the Circumference BC, and the Angle BGC; to wit, the Circumference BL, and the Angle BGL; as alfo Equimultiples of the Circumference EF, and the Angle EHF, viz. the Circumference EN, and the Angle EHN. And because it is proved if the Circumference BL 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 BC is to the Circumference EF; fo Def. 5. 5 is the Angle BGC to the Angle EHF; but as the Angle BGC is to the Angle EHF, fo is the Angle BAC to the Angle EDF; for the former are * double to the latter. Therefore as the Circumference BC is to the Circumference EF, fo is the Angle BGC to the Angle EHF; and fo the Angle BAC to the Angle EDF.

15. 5.

20.3.

+ 4. I.

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

I fay, moreover, that as the Circumference BC is to the Circumference EF, fo is the Sector GBC to the Sector HFE.

For join BC, CK, and affume the Points X, O, in the Circumferences BC, CK, and join BX, XC, CO, OK.

Then becaufe the two Sides BG, GC, are equal to the two Sides CG, GK, and they contain equal Angles, the Bafe BC fhall be † equal to the Bafe

CK;

*

CK; as likewise the Triangle GBC 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 ABC, is equal to the remaining Circumference which makes up the fame Circle, the Angle BXC is equal to the Angle COX; and fo the Segment BXC 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 * 24- 3other: Therefore the Segment BXC is equal to the Segment COK. But the Triangle BGC is also equal to the Triangle CGK; and fo the whole Sector BGC will be equal to the whole Sector CGK. By the fame Reason, the Sector GKL will be equal to the Sector GBC, or GCK; therefore the three Sectors BGC, CGK, KGL, are equal to one another; fo likewise are the Sectors HEF, HFM, HMN. Wherefore the Circumference LB is the fame Multiple of the Circumference B C, as the Sector GBL is of the Sector GBC. For the fame Reason, the Circumference NE is the fame Multiple of the Circumference EF, as the Sector HEN is of the Sector HEF; but if the Circumference BL be equal to the Circumference EN, then the Sector BGL will be equal to the Sector EHN; and if the Circumference BL exceeds the Circumference EN, then the Sector BGL will alfo exceed the Sector EHN, and if lefs, lefs. Therefore, fince there are four Magnitudes, to wit, the two Circumferences BC, EF, and the two Sectors GBC, EHF; and there are taken of the Circumference BL, and the Sector GBL, Equimultiples of the Circumference CB, and the Sector CGB; as also of the Circumference EN, 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 EN, the Sector BGL will alfo exceed the Sector EHN; and if equal, equal; if lefs, lefs. Therefore as the Circumference B C is to the Circumference EF, fo is the Sector GBC to the Sector HEF; which was to be demonftrated.

Coroll. 1.

Corell. 1. An Angle at the Center 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 BC to a Quadrant of the Circle: Wherefore if the Confequents be quadrupled, the Angle BAC fhall be to four Right Angles, as the Arc BC is to the whole Circumference.

2. The Arcs IL, BC, of unequal Circles, which fubtend equal Angles, whether at their Centers, or Circumferences, are fimilar; for IL is to the whole Circumference ILE, as the Angle IAL, is to four Right Angles; but as IAL, or BAC, is to four Right Angles, fo is the Arc BC to the whole Circumference BCF. 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 Semi-diameters AB, AC, cut off fimilar Arcs IL, BC, from concentric Circumferences.

The END of the SIXTH BOOK.

EUCLID's

EUCLID's

ELEMENTS.

воок XI.

I.

DEFINITION S.

A

Solid is that which has Length, Breadth,
and Thickness.

II. The Term of a Solid is a Superficies.
III. A Right Line is perpendicular to a
Plane, when it makes Right Angles with all the
Lines that touch it, and are drawn in the faid
Plane.

IV. A Plane is perpendicular to a Plane, when the
Right Lines in one Plane, drawn at Right An-
gles to the common Section of the two Planes,
are at Right Angles to the other Plane.
V. The Inclination of a Right Line to a Plane, is
the acute Angle contained under that Line, and
another Right one drawn in the Plane from that
End of the inclining Line, which is in the Plane
to the Point where a Right Line falls from the
other End of the inclining Line perpendicular
to the Plane.

VI. The

189

VI. The Inclination of a Plane to a Plane, is the acute Angle contained under the Right Lines drawn in both the Planes to the fame Point of their common Interfection, and making Right Angles with it.

VII. Planes are faid to be inclined fimilarly, when the faid Angles of Inclination are equal. VIII. Parallel Planes are fuch, which being produced never meet.

IX. Similar Jolid Figures are fuch that are contained under equal Numbers of fimilar Planes. X. Equal and fimilar folid Figures, are thofe that are contained under equal Numbers of fimilar and equal Planes.

XI. A folid Angle is the Inclination of more than two Right Lines that touch one another, and are not in the fame Superficies: Or, a folid Angle is that which is contained under more than two plane Angles which are not in the same Superficies, but being all at one Point.

XII. A Pyramid is a folid Figure comprehended under divers Planes fet upon one Plane, and put together at one Point.

XIII. A Prism is a folid Figure contained under Planes, whereof the two oppofite are equal, fimilar, and parallel, and the others Parallelograms. XIV. A Sphere is a folid Figure, made when the Diameter of a Semicircle, remaining at rest, the Semicircle is turned about till it returns to the fame Place from whence it began to move. XV. The Axis of a Sphere is that fixed Line, about which the Semicircle is turned.

XVI. The Center of a Sphere is the fame with that of the Semicircle. XVII. The Diameter of a Sphere, is a Right Line drawn thro' the Center, and terminated on either Side by the Superficies of the Sphere. XVIII. A Cone is a Figure defcribed when one of the Sides of a Right-angled Triangle, containing

the