Algebraic Geometry Santa Cruz 1995: Summer Research by David R. Morrison, Janos Kolla Summer Research Institute on

By David R. Morrison, Janos Kolla Summer Research Institute on Algebraic Geometry

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Extra info for Algebraic Geometry Santa Cruz 1995: Summer Research Institute on Algebraic Geometry, July 9-29, 1995, University of California, Santa Cruz (Proceedings of Symposia in Pure Mathematics) (Pt. 2)

Sample text

Xlii Summary of V 8, V 9 &. V 10 V 8 The basic theorem on minima in the parabola. See Fig. 8. For a point E on the axis [where Er > l/lR), the minimum is found by marking off half the latus rectum, EZ, towards the vertex and erecting a perpendicular from Z, ZH, to cut the section in H. g. e. the difference between the squares on the two lines equals the square on the intercept on the axis between the perpendiculars from the two points. Using the basic property of the parabola [I 11) and the "geometrical algebra" familiar from early propositions of Euclid's Book II [see pp.

5) on p. xxxii). II 27 Two tangents to an ellipse or circle will be parallel if the line joining the points of tangency passes through the center of the section; otherwise they will meet on the side of the center on which that line lies. Used for the ellipse in V 71. II 28 The line bisecting two parallel chords in a conic section is a diameter to that section. This follows from II 5-6 and the definition of a diameter. Used in VI 19, VI 20 & VI 25. II 30 If two tangents are drawn from a point to a conic section, the dia- meter through that point (where the tangents intersect) will bisect the line joining the points of tangency.

284-309, where it is treated from a modern viewpoint. Here I give only a brief analysis of the content and interconnections of the individual propositions. 342). Book V; summary of V 1-3 xxxix the book may be stated as follows. For a given conic section and a given point P: (1) to construct all minimum and/or maximum straight lines from P to the conic; (2) to determine how the distance between P and a variable point X on the conic changes as PX moves away from the position of minimum or maximum.

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