Elemento of Geometry, Theoretical and Practical: Containing a Full Explanation of the Construction and Use of Tables, and a New System of Surveying

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Pratt, Woodford & Company, 1848 - Mathematics - 324 pages
 

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Contents

Multiplying or dividing dividend or divisor
34
A fraction an expression of division
35
Division by a fractionreciprocal
36
Fractions cleared of subdenominators
37
21 Scholia Signs common factors c
38
Elimination
39
SECTION II ExponentsProportionVariation 1 Reciprocal powers
40
Root of a powerextension
41
and evolutionequal multiples
44
Inverse or reciprocal proportion
46
Continued proportiongeometrical progression
47
Variation definednotation
48
Inversion compositioninvolutionmultiples comparison com bined comparison
49
Analysis of Equations 1 A single equation resolvable into several distinct equations Ex amples
50
Quadraticsrule for solvingsum and product of roots
51
Classes of biquadratics solvable as quadratics
52
Rule for putting problems into equation
53
Discussion of the two values of the unknown quantity
55
Equation between constants and variables
57
CoŽfficients equated
59
BOOK SECOND PLANE GEOMETRY DEPENDING ON THE RIGHT LINE Section I Comparison of Angles 1 Definitions Geometrysolids surfaces ...
63
Corollariesscholiam on parallels
64
Sum of adjacent angles constant method
65
Corollariesright angles sum of adjacent angles lines forming one and the same straight line vertical angles sum of angles round a point
66
ApplicationsRightangle Surveyors Cross
67
Parallels method reversion and superposition
68
Corollariesthe converse conditions determined by the equality of alternate angles secant perpendicular lines parallel to the same angles having parallel...
69
Sum of external angles of polygon
70
Exercises
71
Polygons when equal how proved
72
Proportional Lines
78
Line bisecting the vertical angle of a triangle
84
Comparison of Plane Figures
90
BOOK THIRD
101
Inscribed anglesconsequences tangent
104
Exercises
112
Tangent and consequences
118
Ellipse hyperbola and parabola how related
125
PAGE
128
BOOK FIRST
131
Derivative of y fx Axa + B 2b + Coco + and the converse
137
Logarithms rules of operation
145
Development of y fx when ay x computations
146
Interpolation
153
Exercises
158
Exponential theorem or development of y a
159
General Laws relating to the Development of Functions depending on a single Variable 1 Ratio of the increment of a continuous function to that of it...
162
Derivative of a polynomial and multiple function
163
Use of intermediate and converse functions
164
Derivative of a functional product
165
Power of a function derivative
166
Fraction of functions derivative
167
Expansion of a function Maclaurins theorem
168
Exercises
169
BOOK SECOND PLANE TRIGONOMETRY SECTION I Trigonometrical Analysis 1 Construction and definitionscomplementary arcs sine c
170
Sum of the squares of the sine and cosine the sine an increasing the cosine a decreasing function
171
Secant and tangent
172
Sine and cosine derivatives of
173
Sine and cosine developed in terms of arc
174
Trigonometrical equations solved
191
Resolution of Triangles and Mensuration of Heights and Distances 1 Projection of one line upon another
192
Fundamental relation between the sides and angles of a triangle
193
Sum of squares of two sidescorollary
194
Relation of sides and opposite angles
195
Quadrature of the Circle the Ellipse and Parabola 1 Circular sector how measured area of circle
201
Ratio of circumference to diameter arcs of similar sectors areas of circles circles and their like parts how related
202
Incremental vanishing arc of continuous curve
203
Segmental arca derivative of
204
Area of parabola
205
Proximate area of continuous curve
206
Exercises
207
BOOK THIRD SURVEYING SECTION I Description and Use of Instruments 1 The chainlength division how used field notes
208
The surveyors crossconstruction and use
210
Vernier or nonius
211
Theodolitehow adjusted used
212
Variation of needle
214
Leveling
215
Plotting 1 Graphical problemsperpendiculars parallels c
216
Problems of constructionxa +6 c
218
Graduation of the circlechords
220
To plot a field Diagonal scale sector 992 200
224
Computation of Areas 1 Last side and diagonals of polygonal fields
225
Corollaries Similar figures and proportional lines the Pantograph
227
Exercises
229
Area of polygon in terms of the sides and their inclinations
231
Corollariestriangle parallelogram similar polygons equilateral
232
BOOK FIRST
245
Surface of revolution derivative
251
Consequences ellipsoidal frustrum ellipsoid sphere Para
257
Spherical triangle measured consequences
262
cosa cosb cosc + sind sinc cosA consequences
264
Sides how related to opposite angles consequences
267
Elimination first second thirdconsequences
268
Napiers analogies
272
Napiers Rules modified
273
Cases in spherical trigonometry
275
Exercises
276
Projections of the Sphere 1 Orthographic projection consequences
280
How made
282
Gnomonic projection
283
A conic section
284
How made
286
Conical projection how made
288
Exercises
290
BOOK THIRD NAVIGATION SECTION I Problem of the Course 1 Difference of latitude
291
Difference of longitude consequences
292
Parallel sailing
294
Problem of the Place
295
Latitude by meridian altitude
296
Time consequences
298
Longitude by lunar distance
302
Description and Use of Instruments 1 CourseMariners compass
304
Zenith distancesextant adjustments use depression of horizon re fraction parallax semidiameter
305
II 312
312
TABLESLogarithms of numbers
316
Logarithmic Sines
320
Logarithmic Tangents
322

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Page 13 - If two triangles have two sides of the one equal to two sides of the...
Page 264 - ... greater than the angle contained by the two sides equal to them, of the other ; the base of that which has the greater angle, shall be greater than the base of the other. Let ABC, DEF be two triangles, which have the two sides AB, AC, equal to...
Page 217 - Through a given point to draw a line parallel to a given straight line.
Page 195 - As the sine of the angle opposite the given side, is to the sine of the angle opposite the required side ; so is the given side to the required side.
Page 197 - To find the other side: — • as the sum of the two given sides is to their difference, so is the tangent of half the sum of their opposite angles to the tangent of half their difference...
Page 170 - The sine of an arc is the perpendicular let fall from one extremity of the arc on the diameter which passes through the other extremity.
Page 45 - In a series of equal ratios, any antecedent is to its consequent, as the sum of all the antecedents is to the sum of all the consequents. Let a: 6 = c: d = e :/. Then, by Art.
Page 83 - ... by four times the square of the line joining the middle points of the diagonals.
Page 304 - N. by E. NNE NE by N. NE NE by E. ENE E. by N. East E. by S. ESE SE by E. SE SE by S.
Page 177 - The sum of the sines of two arcs is to their difference, as the tangent of half the sum of those arcs is to the tangent of half their difference.

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