A Treatise on Analytic Geometry: Especially as Applied to the Properties of Conics; Including the Modern Methods of Abridged Notation

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Wilson, Hinkle & Company, 1869 - Geometry, Analytic - 574 pages
 

Contents

THE NORMAL
327
Constructions by means of it
329
ratio
330
THE CURVE REFERRED TO ANY TWO CONJUGATES
335
Equation to Diameter Conjugate to that through any given point 375
336
CONJUGATE PROPERTIES OF TANGENT
339
Rectangle under Subtangent and Abscissa
344
AREA OF THE ELLIPSE
357
29
361
Focal Center bisects the Axes Corresponding inter
363
Verification of Figure of Curve by means of its equation
367
Analogy of Hyperbola to Ellipse with respect to Circle on Trans
368
Condition that a Right Line touch an Hyperbola Eccentric Angle
383
its length in terms of the Abscissa
384
Equation to the Normal referred to the Axes
386
Construction of Tangent by means
388
x
396
THE CURVE REFERRED TO ANY TWO CONJUGATES
399
its length
404
Polar of Centerof any point on Axis of xon Transverse Axis
410
Equation to Planar Bisector of angle between any two Planes
411
An Ge 3
415
Derivation of the name
416
Right Lines joining two Fixed Points on Curve
422
EXAMPLES ON THE HYPERBOLA
428
Two Right Lines intersecting a particular case of the Hyperbola
437
Equation to Locus of middle points Parallel Chords
439
NOTE
444
41
447
Equation to Parabola referred to any Diameter and Vertl Tangent
455
THE POLE AND THE POLAR
459
AREA OF THE PARABOLA
468
THE CURVES IN SYSTEM AS SUCCESSIVE PHASES OF
480
The Polar Relation
486
Development of the Center
495
Development of the Asymptotes in general symbols
501
Fundamental Anharmonic Property of Conics
507
CHAPTER FIRST
513
Distance between any Two Points in Space
520
General Form of Equation to any Plane
526
solved as Common Sections of Surfaces
537
Criterion of the Form of any Surface furnished by its Sections
543
Classification of QUADRICS or Surfaces of Second Order
553

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Page 254 - Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion...
Page 476 - A conic section is by definition the locus of a point whose distance from a fixed point is in a constant ratio to its distance from a fixed straight line.
Page 238 - ... proved that, if a point and a straight line be such that the straight line joining the centre of a circle to the point is at right angles to the line, and the rectangle contained by the distances of the point and the line from the centre is equal to the square on the radius of the circle, the point is the pole of the line, and the line the polar of the point with respect to the circle. In the diagram O is the centre of the circle : H is a point in OP such that the rectangle OP, OH is equal to...
Page 192 - To find the locus of the centre of a circle which passes through a given point and touches a given straight line.
Page 221 - The straight lines joining the vertices of a triangle to the middle points of the opposite sides meet in a point* which is for each line the point of trisection further from the vertex.
Page 250 - The perpendiculars erected at the middle points of the sides of a triangle meet in a point equidistant from the three vertices.
Page 249 - The lines drawn from the angles of a triangle to the middle points of the opposite sides meet in a point.
Page 169 - The two fixed points, F' and F, are called the foci of the curve ; and the variable distances with a constant difference, F'P and FP, are termed its focal radii. The portion A' A of the right line drawn through the foci, is called the transverse axis.
Page 319 - The perpendicular from the vertex to the base of an isosceles triangle bisects the base. 2. The perpendicular from the vertex to the base of an isosceles triangle bisects the vertical angle.
Page 222 - What is the locus of a point, the sum of the squares of the perpendiculars from which on the sides of a triangle is constant?

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