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PROP. XXI. . Prop. I. B. 1 Apoll.
For, i possibie, et ACBae a traight linedrawn froni the verto A to the oin B, and whicli is no in the surisce of the oneri and Iet DEbe the straight line illi,hicli the coneis described and the circle EF the base andi DEie revolved in the circumference of EF, it illias through the poliati and the vertex N and thus Wo straight lines ACB AGB illhave the fame extremities: hic Mis absurd Theresere the traight line drawn rom thepolia Ario B, is no without the conical sur-facea there recit is in that furface.
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I a cone e cui by a plane passing through iis verteX, the section is a triangle.
Fig. 12. et thereae a cone hichias the sint Asor iis vertex, and the circle BDC se iis base, Iet ita cut through the potnt Aa an plane; and Iet the sections made in the surtaceae thelines AB, AC, and the section in the base thestraight 3. II. Elem. line BC ABC is a tri
For since the straight line drawn rom thepoint Aa B, is both in the culting plane andi 2I. l. the conica surface, it is the om-mon Sectio of the two , there re the sectionAB is a traight line se a like reason, the
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section AC is a straight linea and BC too is astraight line theresere the sectio ABC is atriangle.
PROP. XXIII. Prop. 4. B. I. ApolL
If the conical furface ori ei ther Sideo the vertex e cuti a plane Parallel to the circle hicli is tho
section of this plane illi the conical sursace is a circle havinciis centre in the aXis and the figure contained by this circle, and that par of the conica surisce hichis intercepte between it an ille
verteX, S A COIJ C. Let thereae a conica surface the VerteX of Fig. II.whic is A, Boebeing the circle in the circum- serence of whicli the strat glit line revolves whicli describes the sui face; et it e cui by
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an plane parallel to the circle BC, and let his plane mali in it a section DLE the line DL is the circumferen Ce of a Circle the Centre of hici, is in the axis Tahe the centre of
the circle BC, and et ita ea; oin AF AF,
Consequently is des. O. the axis, and meetsthe cui ting plane; et it me et it in neXt, letan plane pas through the fame AF and the Sectio made by this plane ill e 22. I. atriangle ABC. An die cause the potnis D, G, a re both in the cultin plane DLE, and in
the plane ABG, GDis 3. II. Eleria. a stra ight Jine Again, in the line L tali an potiatri; oin AH, and produc it AH then ill
21. . me et the circumferen ce BC; et it me et it in K, and oin GH, Κ and hecause the two parallel plane DLE, BC are ut by the plane ABC, their l6. 11. Elem. Common
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Elem. are also equat Aster the Same mannerit maybe demonstrated that an other traight Iines halever, rawn frona the poliat G to theline DLE, are equat Thecline DLE is, here- fore, the Circumference of a circle haviniit scenire, in the Xis. Co . The figure containe by the circle DLE, and that par of the conical furface whichis interceptedietween this circle and theloint A, is a Cone and the common sectionis thecuttin plane and the triangle passing throughthe axis, is a diameter of the circle DLE PROP. XXIV. Prop. 5. B. I. Apoll.9I a Calene Cone ut through theaXis by a plane at right angies tothe base, e ut also by another plane at right angi es to the trian-gle passing through the axis filiis ther plane ut ostri towardSthe verteX, a triangle similar to
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the triangle through the axis, both triangles ei nitione plane, ut subcontrarii Silva ted the Sectio made in the cone by this other plane is a circle.
pig. i4. et there e a Calene Cone, the vertex ofwhieli is the oin A, and the bas the circle BLC4 et ille ut through the axisi a plane perpendicular to the base, and Iet the sectionbe the triangle ABC; et it e also ut byanother plane at right angies to the triangle ABC and Iet this other plane Cut ossi to ard sthe vertex the triangle AG simila to the triangle ABC, but subcontrarit siluated; and Iet the sectio made in the sursace be theline GKH: his line is the circumserenc es a
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to the plane of the triangle ABG these per pendiculars, illi 38. 11. Elem. falli the common sections of the plane : accordingly, et themi HF LM: F, therei ore, is parallelto 6 11. Elem. LM 'nexi, through F drawDF paralleloo in the plane, there fore, whicli passes through FH DE is parallel to the l5. 11. Elem. base of the conoe and se this reason, the sectioni HE is 23. I. a circle, ofwhicli Dis a diameter the rectangle there- fore, containe by DF, FE is 35. 3. Elem.)equat to the quare of FH. An since Ereis parallela BC, the angle ADDis equa to the angi ABC and the angi AK is place dequat to the angi ABC there re the angle AK is also equat o ADE: and the angies atrare equat for the are opposite vertica an- gleso heresere the triangle DFG is simila tothe triangle KFE: there re a EF o FΚ, ois Ut FD; there fore the rectangle EFD sequat to the rectangle KFG. ut the rectan-gle EFD that is, the rectangi containe by DF, E has been prove tot equat O thesquare of FH therei ore the rectangi con-
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1ained by F, FG is equa to the Same quareo FH. I may, in like manner, e demon- inrated that the quare o an stra ight linewhateuer drawn froin the line HK, oras tohe perpendicular to GK, is equa to the rectan-gle Contained by the segment into hicli that strat glit line divides the sanae GK the sectionGH is, there re a circle havin GK se a diameter. A sectio of this in may be
Is a cone ut through the Yis by a plane, e cui lihewise by another plane, cuiliniit base in the di
rection o a traight in perpendicular to the base of the trian-gle passing through the aXis andis the common section of the triangle through the aYis, and of the plane culting the base of the one