The Solutions of the Geometrical Problems: Consisting Chiefly of Examples in Plane Co-ordinate Geometry, Proposed at St. John's College, Cambridge, from Dec. 1830 to Dec. 1846. With an Appendix, Containing Several General Properties of Curves of the Second Order, and the Determination of the Magnitude and Position of the Axes of the Conic Section Represented by the General Equation of the Second Degree |
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... curve ; and the position of a point in the conic section in any proposed direction has been determined by its points of intersection with a given straight line . In a subject which for ages has exercised the skill PREFACE . vii.
... curve ; and the position of a point in the conic section in any proposed direction has been determined by its points of intersection with a given straight line . In a subject which for ages has exercised the skill PREFACE . vii.
Page 18
... direction CB . Produce CB to D , take CD 2cc and upon CD C c ' describe a circle ; from any point E in its circumference draw AE , EC , EB , then will AEC BEC . = △ 4. Let AB ( fig . 21 ) be the given base ; draw AC per- pendicular to ...
... direction CB . Produce CB to D , take CD 2cc and upon CD C c ' describe a circle ; from any point E in its circumference draw AE , EC , EB , then will AEC BEC . = △ 4. Let AB ( fig . 21 ) be the given base ; draw AC per- pendicular to ...
Page 44
... direction , the loci of the intersections of each ellipse with its original position , are two straight lines whose equations are A Ꮎ y = x tan 2 و Ꮎ and y ∞ cot 2 7. Two given unequal circles touch each other externally ; shew that ...
... direction , the loci of the intersections of each ellipse with its original position , are two straight lines whose equations are A Ꮎ y = x tan 2 و Ꮎ and y ∞ cot 2 7. Two given unequal circles touch each other externally ; shew that ...
Page 55
... direction with equal angular velocities ; shew that the curve traced upon the first by a pencil P fixed perpendicular to the plane of the second is a circle : or if the planes revolve in GEOMETRICAL PROBLEMS . NO . VIII . DEC . 1837 ...
... direction with equal angular velocities ; shew that the curve traced upon the first by a pencil P fixed perpendicular to the plane of the second is a circle : or if the planes revolve in GEOMETRICAL PROBLEMS . NO . VIII . DEC . 1837 ...
Page 56
... directions the equation to the curve is a2 - 2ar cos 0 + y2 200 ( a2 = 2 ) 2 where A is the origin , BP = a , AB = c , and the prime radius is the line originally in the position AB . 15 . ABCD is any quadrilateral . Bisect AC , BD in E ...
... directions the equation to the curve is a2 - 2ar cos 0 + y2 200 ( a2 = 2 ) 2 where A is the origin , BP = a , AB = c , and the prime radius is the line originally in the position AB . 15 . ABCD is any quadrilateral . Bisect AC , BD in E ...
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Common terms and phrases
2c sin² a₁ a²² ABCD angular points asymptotes axes axis axis-major ay² b₁ bisected C₁ chord of contact co-ordinates conic section conjugate conjugate hyperbola constant cos² cosw curve diagonals diameter draw ellipse equal Euclid find the locus fixed point given points given straight lines hence hk g hyperbola inclined inscribed joining the points latus rectum Let A fig line joining m₁ meet middle point n₂ pair of tangents parabola parallel parallelogram pass through three perpendicular plane points of contact points of intersection polar equation polygon position quadrilateral figure radius remaining sides respectively right angles shew Similarly sin w sin² ST JOHN'S COLLEGE t₁ t₂ tangents be drawn tangents drawn three sides touch vertex y₁
Popular passages
Page 54 - If two triangles which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel to one another ; the remaining sides shall be in a straight line. Let ABC, DCE be two triangles which have the two sides BA, AC proportional to the two CD, DE, viz.
Page 117 - Similar triangles are to one another in the duplicate ratio of their homologous sides.
Page 117 - IF from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other touches it ; the rectangle contained by the whole line which cuts the circle, and the part of it without the circle,. shall be equal to the square of the line which touches it.
Page 96 - The rectangle contained by the diagonals of a quadrilateral ,figure inscribed in a circle, is equal to both the rectangles contained by i'ts opposite sides.
Page 16 - MAGNITUDES which have the same ratio to the same magnitude are equal to one another ; and those to which the same magnitude has the same ratio are equal to one another.
Page 28 - To divide a given straight line into two parts, so that the rectangle contained by the whole and one part may be equal to the square on the other part*.
Page 28 - THEOREM. lf the first has to the second the same ratio which the third has to the fourth, but the third to the fourth, a greater ratio than the fifth has to the sixth ; the first shall also have to the second a greater ratio than the fifth, has to the sixth.
Page 10 - ... not in the same plane with the first two ; the first two and the other two shall contain equal angles.
Page 87 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.