A Treatise on Geometry and Its Application in the Arts

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Longman, Orme, Brown, Green, & Longmans, 1840 - 314 σελίδες
 

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are equally inclined to it Lines more remote the greater their inclination
27
CHAP III
29
31 Every Line perpendicular to one of two Parallels is per pendicular to the other
31
34 Parallels are equidistant
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37 Systems of rectangular Parallels
33
38 The T Square
34
39 Rolling Parallel Ruler Machine for ruling Paper Spinning Frame
35
42 Alternate Angles equal
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CHAP IV
38
49 External Angle equal to the two remote Angles
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50 External Angles of a Polygon
40
63 Isosceles Triangle
44
67 To bisect an Angle
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71 Equilateral Triangle
46
CHAP W of CIRCLES 73 Centre and Radius
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75 Circle a Symmetrical Figure
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77 Beam Compasses
49
81 A Chord lies within a Circle
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82 A Straight line cannot meet a Circle in more than two Points
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87 Circles touching externally
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92 Application to Wheels driven by Straps or Bands
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93 Equal central Angles have equal Arcs
56
98 The Protractor
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15 Supplement of an Angle
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102 The Multisection of an Angle
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106 Segment of a Circle
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110 All Angles in same Segment are equal
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113 In a lesser Segment obtuse in a greater acute
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119 To find the Centre of a Circle
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CHAP VI
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128 A Trapezium
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CHAP VII
75
CHAP VIII
85
197 Regular Octagon derived from the Square
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Page
102
CHAP X
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253 Page Triangles having an Angle in each equal and containing 254 Sides proportional are similar 118 Perpendiculars proportional to Sides in s...
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255 Areas of similarTriangles proportional to the Squares of 256 the corresponding Sides 119 Similar Figures resolved into similar Triangles
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259 Areas of similar Figures as the Squares of their corre sponding sides
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260 Areas of Circles as the Squares of their Diameters 122 Circles on the Sides of a rightangled Triangle equal to 262 the Circle on the Hypothenu...
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265 The Square of the Mean is equal to the Rectangle under 266 the Extremes 124 Rectangles under the Segments of intersecting Chords 267 in ...
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269 The Square of the Tangent is equal to the Rectangle under Angle in the alternate Segment
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271 To find a fourth Proportional geometrically
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272 Proportional Compasses
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273 To find a third Proportional geometrically
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276 To find a Line whose Square is equal to the Area of a given Figure
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CHAP XI
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277 To construct a Figure equal and similar to another and 278 similarly placed 133 To construct a Figure similar to another on a different 279 s...
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280 every kind 136 Construction of the lateral Reversion of a Geometrical Figure
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Application in Engraving and Printing
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282 Reversing by a System of Squares
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Reduction and reversing of Designs
143
CHAP XII
145
286 The Perpendicular from a given Point to a Plane
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288 Lines from the given Point equally inclined to this Per pendicular are equal
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Disadvantage of imperfect Forms
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294 Angle under two Planes
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295 Planes through a Perpendicular to a Plane are at right Angles to it
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299 A straight Line parallel to a Plane
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302 Three Points always in the same Plane
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306 Points equidistant from a Plane are in a parallel Plane
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309 A Plane through a given Line perpendicular to a given Plane
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312 Lines between parallel Planes equally inclined to them are equal
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CHAP XIV
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346 The Volumes of Prismsor Pyramids with equal Bases
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348 Volume of a truncated triangular Prism equal to
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CHAP XV
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379 Formation of cylindrical Surfaces by a circular Cutter
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382 Manufacture of Candles
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385 The right circular Cylinder
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389 Area of cylindrical Surface equal to the Rectangle under the Altitude and the Circumference of the Base
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395 Volume of a Cylinder found by multiplying its Base by its Altitude
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400 Position and Form of Lines determined by Projections
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402 Planes produced in Agriculture and Gardening by cylin drical Rollers
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404 One Cylinder rolling on another
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408 Motion of Wheel Carriages on a Road
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448 Principles of Perspective
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450 Application in architectural and mechanical drawing
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451 Estimation of visual Magnitude
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CHAP XVII
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461 Parallels
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468 Lesser Circles equally distant from Centre are equal
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472 Principle of Billiardplaying
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500
213
their Chords and other Lines
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524 Definition of a regular Solid
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539 To construct a regular Icosaedron
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CHAP XIX
238
567 Spring Steelyard
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568 Buffers of Railway Carriages
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571 Spiral Staircases
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CHAP XX
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579 Surfaces of Revolution
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581 An Ellipse described by a Pencil and Cord
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582 The Axes of an Ellipse
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593 The Foci
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596 To draw a Tangent at a Point in the Ellipse
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597 Lines from the Foci equally inclined to the Tangent
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604 The Eccentricity of an Ellipse
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607 Section of a Cylinder by a Plane
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610 Circle on the Conjugate Axis as Diameter divides its Ordinates proportionally
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611 An Ellipse the Projection of a Circle
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616 Ellipse equal to a Circle whose Diameter is a mean Pro portional between its Axes
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619 Conjugate Diameters
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Area of such Parallelograms equal to Rectangle under the Axes
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Rectangles under the Segments of intersecting Chords proportional to the Rectangles under the Segments of others parallel to them
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629 These Rectangles proportional to the Squares of the parallel SemiDiameters
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632 To find the Centre of an Ellipse
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636 To find the Axes of a given Ellipse
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640 Ellipse expressed algebraically
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642 SemiDiameter a mean Proportional between the seg ments intercepted by an Ordinate and a Tangent from the Centre
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646 The Directrices
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648 The Parameter
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651 Methods of Tracing an Ellipse by Points
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652 Method by continued Motion with jointed Rules
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653 Section of a Cone forming a Parabola
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655 Focus of a Parabola
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658 The Ellipse becomes a Parabola when its Axis becomes infinite
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660 Diameters of a Parabola are parallel
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663 Method of constructing a Parabola by Points
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665 Diameter and Line to the Focus equally inclined to the Tangent
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666 To draw a Tangent at a given Point in a Parabola
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669 Tangent to a Parabola from a Point in its axis
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671 Parabola described by continuous Motion
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673 To draw a Diameter which shall be inclined at a given Angle to its Ordinates
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675 The Section of a Cone producing an Hyperbola
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678 Parallels to either Axis bisected by the other
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680 Diameters bisected at Centre
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681 Lines from the Foci equally inclined to the Tangent
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683 Directrix
287
686 Square of the Ordinate proportional to Rectangle under Segments
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687 Determination of the Position of the Asymptotes
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689 Hyperbola described by continuous Motion
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CHAP xxi
292
692 The Circle measures the Curvature of all other Curves
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694 The osculating Circle
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698 The Normal of a Curve
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703 The Involute of a Curve
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708 Cases in which the Radius of Curvature becomes infi nite or vanishes
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710 Point of Inflection or contrary Flexure
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CHAP XXII
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717 Its Base equal to Circumference of generating Circle
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722 Tangent is parallel to corresponding Chord of generating Circle
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727 Radius of Curvature of the Cycloid
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731 Area of Cycloid equal to three times that of the gene rating Circle
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734 The Line of swiftest Descent is a Cycloid
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736 The curtate and prolate Cycloids
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The Conchoid constructed by Points
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Divided symmetrically by its Axis
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To draw a Tangent to it
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744 The inferior Conchoids
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749 Inferior Conchoid nodated
312

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Σελίδα 84 - All the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Σελίδα 44 - If two triangles have the three sides of the one equal to the three sides of the other, each to each, the triangles are congruent.
Σελίδα 124 - Four quantities are in proportion when the ratio of the first to the second is equal to the ratio of the third to the fourth.
Σελίδα 83 - Therefore all the interior angles of the figure, together with four right angles, are equal to twice as many right angles as the figure has sides.
Σελίδα 40 - EBF, there are two angles in the one equal to two angles in the other, each to each ; and the side EF, which is opposite to one of the equal angles in each, is common to both ; therefore the other sides are equal ; (i.
Σελίδα 40 - If one angle of a triangle be equal to the sum of the other two, the greatest side is double of the distance of its middle point from the opposite angle.
Σελίδα 169 - The volume of the frustum of any pyramid is equal to the sum of the volumes of three pyramids whose common altitude is the altitude of the frustum, and whose bases are the lower base, the upper base, and the mean proportional between the bases of the frustum.
Σελίδα 46 - Euclid's, and show by construction that its truth was known to us ; to demonstrate, for example, that the angles at the base of an isosceles triangle are equal...
Σελίδα 212 - ... solid is, as before, a solid inscribed in a larger sphere ; and, since the perpendicular on any side of the revolving polygon is equal to the radius of the inner sphere, the proposition is identical with Prop. 26. COR The solid circumscribed about the smaller sphere is greater than four times the cone whose base is a great circle of the sphere and whose height is equal to the radius of the sphere. For, since the surface of the solid is greater than four times the great circle of the inner sphere...
Σελίδα 40 - After remarking that the mathematician positively knows that the sum of the three angles of a triangle is equal to two right angles...

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