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

25 Lines equally distant from the foot of the Perpendicular
25
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
58
15 Supplement of an Angle 20
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
112 Angle in a Semicircle right 6S 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
134 Resolved into equal Triangles
69
138 Parallel Rulers
70
143 Rectangle
71
151 Symmetrical Trapezium
73
CHAP VII
75
162 Construction of an Equilateral Triangle
76
165 Tangents from same Point equal
77
167 Triangle with given Angles circumscribed about a Circle
78
172173 Properties of Rectangle inscribed in a Circle
79
179 Circumscribed Parallelogram
80
183 Sum of internal Angles
81
186 Sum of external Angles equal four right Angles
83
CHAP VIII
85
193 Magnitude of Angles
89
196 Regular Hexagon
90
197 Regular Octagon deri from the Square
91
CHAP IX
93
206 Area of a Rectangle
94
211 Ratio of two Magnitudes
98
219 Areas of all rectilinear Figures found by resolving them into Triangles
99
223 Area of a Circle equal to half the Product of its Radius and its Circumference
100
224229 Numerical expressions for the Area of a Circle
101
Page
102
232235 Squares on the sides of a rightangled Triangle equal to the Square of the Hypothenuse
103
236 To find a Line whose Square is equal to several given Squares
107
238239 Given two sides of a rightangled Triangle to find the Third
108
CHAP X
111
246 Similar Triangles
112
247E Sides of a Triangle divided proportionally by a Parallel to the Base
113
249 Equiangular Triangles similar
115
268 The Angle under the Chord and Tangent is equal to
126
CHAP XI
133
281 Examples of copying by systems of Squares
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
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
312 Lines between parallel Planes equally inclined to them are equal
156
CHAP XIV
162
348 Volume of a truncated triangular Prism equal to
169
CHAP XV
175
379 Formation of cylindrical Surfaces by a circular Cutter
180
382 Manufacture of Candles
181
385 The right circular Cylinder 182
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
196
441 Cone produced by the Lathe
198
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
linder
210
great Circle
211
495 Volume of a Sphere equal to the Volume of a Cone whose Base is equal to the Surface of the Sphere and whose Altitude is equal to its Radius
212
500 The entire Volumes and Surfaces of a Sphere circum scribed Cylinder and circumscribed equilateral Cone are in the continued Ratio of Two to ...
213
501 The celestial Sphere
215
502 A Sphere contains within a given Surface the greatest possible Volume
216
505 A spherical Sector
217
512 Developable Surfaces
218
514 Methods of lining or coating a Spherical Surface
219
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
539 To construct a regular Icosaedron
232
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 O
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
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
the Axes
261
625 Squares of the Ordinates to any Diameter proportional 627 Rectangles under the Segments of intersecting Chords proportional to the Rectangle...
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
278 To construct a Figure similar to another on a different
278
666 To draw a Tangent at a given Point in a Parabola
279
669 Tangent to a Parabola from a Point in 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
CHAP XXI
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
727 Radius of Curvature 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 CORRB

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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.
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.
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.
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.
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.
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.
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.
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...
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...
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...

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