A Treatise on Geometry and Its Application in the Arts

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Longman, Orme, Brown, Green, & Longmans, 1840 - Curves, Plane - 314 pages

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Contents

are equal the more distant the greater
26
25 Lines equally distant from the foot of the Perpendicular 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
32
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
37
CHAP IV
38
49 External Angle equal to the two remote Angles
39
50 External Angles of a Polygon
40
63 Isosceles Triangle
44
67 To bisect an Angle
45
71 Equilateral Triangle
46
OF CIRCLES Page 73 Centre and Radius
47
75 Circle a Symmetrical Figure
48
77 Beam Compasses
49
81 A Chord lies within a Circle
50
82 A Straight line cannot meet a Circle in more than two Points
51
87 Circles touching externally
52
92 Application to Wheels driven by Straps or Bands
54
93 Equal central Angles have equal Arcs
56
98 The Protractor
57
100 The Trisection of an Angle
59
102 The Multisection of an Angle
60
105 Central Angle double Circumferential Angle
61
106 Segment of a Circle
62
110 All Angles in same Segment are equal
63
113 In a lesser Segment obtuse in a greater acute
64
119 To find the Centre of a Circle
65
CHAP VI
67
128 A Trapezium
68
CHAP VII
75
CHAP VIII
85
197 Regular Octagon derived from the Square
91
Page
102
CHAP X
111
253 Triangles having an Angle in each equal and containing
118
263 To find a fourth Proportional numerically
124
272 Proportional Compasses
130
Application in Engraving and Printing
139
282 Reversing by a System of Squares
142
Reduction and reversing of Designs
143
CHAP XII
145
286 The Perpendicular from a given Point to a Plane
146
288 Lines from the given Point equally inclined to this Per pendicular are equal
147
Disadvantage of imperfect Forms
148
294 Angle under two Planes
149
295 Planes through a Perpendicular to a Plane are at right Angles to it
150
299 A straight Line parallel to a Plane
152
302 Three Points always in the same Plane
153
306 Points equidistant from a Plane are in a parallel Plane
154
309 A Plane through a given Line perpendicular to a given Plane
155
313 Parallel Lines between parallel Planes are equal
156
319 A right triangular Prism
158
337 Pyramids with equal Bases and equal Altitudes have
164
350 The Cube of the linear Unit is the Unit of Volume
170
372 Extensive Use of Cylinders in the Arts
176
379 Formation of cylindrical Surfaces by a circular Cutter
180
383 Formation of cylindrical Surfaces by the Flexure of plane Surfaces
181
385 The right circular Cylinder
182
389 Area of cylindrical Surface equal to the Rectangle under the Altitude and the Circumference of the Base
183
395 Volume of a Cylinder found by multiplying its Base by its Altitude
184
400 Position and Form of Lines determined by Projections
185
402 Planes produced in Agriculture and Gardening by cylin drical Rollers
186
404 One Cylinder rolling on another
187
408 Motion of Wheel Carriages on a Road
188
410 Method of drawing by a Steam Engine on Railways
189
413 Rapidity of the Process
191
416 Application to Paperstaining
192
CHAP XVI
194
428 Analogy between Cones and Pyramids
195
437 Area of Surface of right Cone equal to its Side multiplied by half the Circumference of its Base 1
196
441 Cone produced by the Lathe
198
444 Method of taking Likenesses in Profiles
199
446 The Structure of the Eye
200
448 Principles of Perspective
201
450 Application in architectural and mechanical drawing
202
451 Estimation of visual Magnitude
203
CHAP XVII
204
461 Parallels
205
466 Lesser Circles
206
472 Principle of Billiardplaying
207
477 Latitudes of Places
208
483 Surface of a Sphere between two Parallel Planes equal to Surface of circumscribed Cylinder between same Planes
209
485 Surface of Sphere equal to Surface of circumscribed Cy linder
210
515 Another Method
220
516 Solids of Revolution produced by Arcs revolving round their Chords and other Lines
221
517 Forms of Vases
222
519 Their Sections circular
223
522 Domes in Architecture
224
523 Art of Turning
225
CHAP XVIII
226
526 To construct the regular Tetraedron
227
Angles under its Faces equal
228
530 To construct the regular Octaedron
229
532 Angles under its Faces equal
230
538 Its Volume
231
540 Angles under its Faces equal
234
543 To determine the Angles under its Faces
235
545 Numerical Table of their Volumes and Surfaces
236
CHAP XIX
238
549 Produced by rolling a rightangled Triangle round a Cylinder
239
553 Form of the Threads
240
557 Ratio of the Velocity of Rotation to the Velocity of Pro gression
241
562 Micrometer Screws
242
563 Adjusting Screws
243
568 Buffers of Railway Carriages
244
571 Spiral Staircases
245
CHAP XX
246
579 Surfaces of Revolution
247
581 An Ellipse described by a Pencil and Cord
248
582 The Axes of an Ellipse
250
593 The Foci
251
596 To draw a Tangent at a Point in the Ellipse
252
597 Lines from the Foci equally inclined to the Tangent
253
602 Production of Echo
254
604 The Eccentricity of an Ellipse
255
607 Section of a Cylinder by a Plane
256
610 Circle on the Conjugate Axis as Diameter divides its Ordinates proportionally
257
611 An Ellipse the Projection of a Circle
258
616 Ellipse equal to a Circle whose Diameter is a mean Pro portional between its Axes
259
619 Conjugate Diameters
260
622 Area of such Parallelograms equal to Rectangle under the Axes
261
627 Rectangles under the Segments of intersecting Chords proportional to the Rectangles under the Segments of others parallel to them
262
629 These Rectangles proportional to the Squares of the parallel SemiDiameters
263
632 To find the Centre of an Ellipse
264
636 To find the Axes of a given Ellipse
265
640 Ellipse expressed algebraically
266
642 SemiDiameter a mean Proportional between the seg ments intercepted by an Ordinate and a Tangent from the Centre
268
646 The Directrices
269
648 The Parameter
270
651 Methods of Tracing an Ellipse by Points
271
652 Method by continued Motion with jointed Rules
272
653 Section of a Cone forming a Parabola
273
655 Focus of a Parabola
274
658 The Ellipse becomes a Parabola when its Axis becomes infinite
275
660 Diameters of a Parabola are parallel
276
663 Method of constructing a Parabola by Points
277
665 Diameter and Line to the Focus equally inclined to the Tangent
278
666 To draw a Tangent at a given Point in a Parabola
279
669 Tangent to a Parabola from a Point in its axis
280
671 Parabola described by continuous Motion
281
673 To draw a Diameter which shall be inclined at a given Angle to its Ordinates
282
675 The Section of a Cone producing an Hyperbola
283
678 Parallels to either Axis bisected by the other
284
680 Diameters bisected at Centre
285
681 Lines from the Foci equally inclined to the Tangent
286
683 Directrix
287
686 Square of the Ordinate proportional to Rectangle under Segments
288
687 Determination of the Position of the Asymptotes
289
689 Hyperbola described by continuous Motion
290
OF THE CURVATURE OF CURVES 690 The Curvature of a Circle uniform
292
692 The Circle measures the Curvature of all other Curves
294
694 The osculating Circle
295
698 The Normal of a Curve
296
703 The Involute of a Curve
297
708 Cases in which the Radius of Curvature becomes infi nite or vanishes
298
710 Point of Inflection or contrary Flexure
299
CHAP XXII
300
717 Its Base equal to Circumference of generating Circle
301
722 Tangent is parallel to corresponding Chord of generating Circle
303
728 Involute of the Cycloid
304
731 Area of Cycloid equal to three times that of the gene rating Circle
305
734 The Line of swiftest Descent is a Cycloid
306
736 The curtate and prolate Cycloids
307
THE CONCHOID 738 The Conchoid constructed by Points
308
739 Divided symmetrically by its Axis
309
743 To draw a Tangent to it
310
744 The inferior Conchoids
311
749 Inferior Conchoid nodated
312
TABLE OF CIRCUMFERENCES AND AREAS OF CIRCLES CORRE

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Common terms and phrases

altitude bisected called canvass centre chords circle circular circumference common base common vertex cone conical surface constructed corresponding sides curve cylinder cylindrical surface diagonals diameter directrix distance divided draw edges ellipse equal altitudes equal and parallel equal and similar equal bases equally distant equilateral triangle faces figure ABCDE fore fourth proportional generatrix geometrical gonal Hence increased or diminished line A B line of intersection linear unit lines be drawn magnitude manner meet motion moved multiplied number expressing parallel lines parallelogram parallelopiped pendicular perpendicular plane ABC plane surface plates polygon position produced pyramid whose base radius ratio rectangle regular polygon respectively equal revolution revolve right angle right-angled triangle rollers round segment similar figures similar triangles solid angle sphere spherical straight edge straight line surfaces of revolution threads three sides triangular prism truncated prism twice vertex vertices wheels

Popular passages

Page 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.
Appears in 289 books from 1723-2005
Page 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.
Appears in 134 books from 1796-1968
Page 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.
Appears in 95 books from 1812-1982
Page 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.
Appears in 291 books from 1714-2005
Page 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.
Appears in 64 books from 1762-1946
Page 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.
Appears in 51 books from 1828-1911
Page 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.
Appears in 33 books from 1828-1916
Page 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...
Appears in 328 books from 1812-2007
Page 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...
Appears in 20 books from 1840-2005
Page 40 - After remarking that the mathematician positively knows that the sum of the three angles of a triangle is equal to two right angles...
Appears in 131 books from 1820-2006

Bibliographic information

TitleA Treatise on Geometry and Its Application in the Arts
Cabinet cycl
Volume 71 of Cabinet cyclopaedia
Cabinet cyclopaedia: Natural philosophy
AuthorDionysius Lardner
PublisherLongman, Orme, Brown, Green, & Longmans, 1840
Original fromHarvard University
Digitized14 Aug 2007
Length314 pages
  
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