The Geometry of Conics, Part 1 |
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Page 12
... produced in- definitely has become a Tangent . Thus in fig . p . 14 suppose to move along the curve up to Q. Then ultimately QQ ' becomes the tangent at Q. There are various ways in which a chord may be made to assume a position of ...
... produced in- definitely has become a Tangent . Thus in fig . p . 14 suppose to move along the curve up to Q. Then ultimately QQ ' becomes the tangent at Q. There are various ways in which a chord may be made to assume a position of ...
Page 18
... produced . Then by Euc . I. 32 , < PSO - SPP + SpP = 2SpP . Similarly , QSO = < = 2 SqQ . Hence : [ 27 . ( i ) If P , Q lie on opposite sides of the axis , then by addition , < PSQ = 2 ( SpP + SqQ ) = 2 ( SpP + pqT ) = 2PTQ . [ Euc . I ...
... produced . Then by Euc . I. 32 , < PSO - SPP + SpP = 2SpP . Similarly , QSO = < = 2 SqQ . Hence : [ 27 . ( i ) If P , Q lie on opposite sides of the axis , then by addition , < PSQ = 2 ( SpP + SqQ ) = 2 ( SpP + pqT ) = 2PTQ . [ Euc . I ...
Page 19
... produce TS to t . Then < PSt = 1⁄2 PSQ [ 29 . = PTQ . [ 30 . Hence by Euc . I. 32 , < STP + SPT = STQ + STP . Therefore < SPT = STQ . And △ TSP = TSQ . [ 29 . Therefore the triangles SPT , STQ are similar . 32. PROP . XIII . If SY be ...
... produce TS to t . Then < PSt = 1⁄2 PSQ [ 29 . = PTQ . [ 30 . Hence by Euc . I. 32 , < STP + SPT = STQ + STP . Therefore < SPT = STQ . And △ TSP = TSQ . [ 29 . Therefore the triangles SPT , STQ are similar . 32. PROP . XIII . If SY be ...
Page 24
... produced , such that SA ' is to A'X as the eccentricity . 43. The ellipse lies wholly on the same side of the directrix with the focus . For imagine two of its points O , P to lie on opposite sides . Let OP cut the directrix in R. Then ...
... produced , such that SA ' is to A'X as the eccentricity . 43. The ellipse lies wholly on the same side of the directrix with the focus . For imagine two of its points O , P to lie on opposite sides . Let OP cut the directrix in R. Then ...
Page 28
... produced * . This proposition illustrates a remark made above ; for it is immaterial whether the eccentricity Sp : pm be less than unity or wholly unrestricted . In other words , the theorem is proved as simply for all conics at once as ...
... produced * . This proposition illustrates a remark made above ; for it is immaterial whether the eccentricity Sp : pm be less than unity or wholly unrestricted . In other words , the theorem is proved as simply for all conics at once as ...
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Common terms and phrases
abscissa auxiliary circle bisects the angle Bishop of Carlisle CA² Cambridge CB² CD² centre CN² concyclic Conic Sections conjugate diameters conjugate radii constant ratio CP² CS² CV² draw drawn ellipse equally inclined Fellow of Trinity fixed point focal chord focal perpendicular focal radii focus Geometry Hence intersect J. W. DONALDSON late Fellow latus rectum locus meet the asymptotes meet the curve meet the directrix meets the axis middle point minor axis normal parabola parallel chords parallelogram Pembroke College PN² point Q principal ordinate PROP QV² radius rectangular hyperbola right angles right-angled triangles Second Edition Shew shewn sides SP² St John's College straight line joining subtends a right supplementary angles tangent Third Edition Treatise Trinity College vertex
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