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LIBRARY

OF THE

UNIVERSITY OF CALIFORNIA.

Class

ELEMENTARY SCIENCE APPLIED TO

SANITATION AND PLUMBERS' WORK

ELEMENTARY SCIENCE

APPLIED TO SANITATION

AND PLUMBERS' WORK

BY A. HERRING-SHAW

Associate, Royal Sanitary Institute ; Victoria University Sanitary Certificate ; Honours

Silver Medallist, City and Guilds of London Institute ; Assistant Lecturer in

Sanitary Engineering, Victoria University, JVLanchester ; Lecturer, etc.,

and Chief Assistant in the Municipal and Sanitary Engineering

Department, School of Technology, Manchester

SECOND EDITION

GUENEY AND JACKSON

10 PATERNOSTER ROW, LONDON, E.G.

1910

D. VAN NOSTRAND

YORK.

V

PEEFACE TO SECOND EDITION

IN presenting the Second Edition of Elementary Science applied to

Sanitation and Plumbers' Work, the author desires to express his

hearty thanks to all who were kind enough to offer criticisms and

express opinions anent the value of the book.

On mature consideration it has been thought advisable to include

a varied and extensive series of questions and answers in the mensura-

tion section, which will doubtless be useful to teachers and taught.

The calculations on heating and ventilation have been extended,

and several pages added to the mechanics section, in addition to

alterations and extensions in other parts of the book.

To the publishers the author's thanks are due for the skill and care

bestowed in the production of this edition, and, combined with the

extensions and alterations effected in the text, it is hoped the sphere

of usefulness of the work will be greatly extended.

A. HERRING-SHAW:

MUNICIPAL SCHOOL OF TECHNOLOGY,

MANCHESTER, 1910.

PEEFACE TO FIEST EDITION

THE object of the author in compiling the matter contained in the

following pages, is to produce a book on Elementary Science, fully

treated (in its fundamental principles, and application to Sanitation

and Plumbers' work) in such a manner as will be readily understood

by all students. The price has been fixed at a low figure, to bring the

vii

223099

viii PREFACE

book within the reach of all classes. It will be found useful to students

preparing for the Examinations of the City and Guilds of London

Institute, the Registered Plumbers' Company, and the Royal Sanitary

Institute of Great Britain. The examiners to these institutions, almost

without exception, complain of the lack of knowledge of Elementary

Science on the part of candidates who take the above examinations.

The author hopes that the contents will be of material assistance

in raising the standard of the work done by students of Sanitary

Engineering. Should a student find any difficulty in thoroughly

grasping any of the principles set forth in this book, the author will

be pleased to afford him assistance, if it be in his power to do so.

The sincere thanks of the author are due to Professor J. Radcliflfe,

M.Sc.(Tech.), and the members of the staff of the Municipal and

Sanitary Engineering Department, Municipal School of Technology,

Manchester, for valuable suggestions and assistance, and also to various

firms for the loan of electros.

A. HERRING-SHAW.

MUNICIPAL SCHOOL OF TECHNOLOGY,

MANCHESTER.

TABLE OF CONTENTS

ELEMENTARY SCIENCE

CHAP. PAGE

I. Definitions of Triangles, Circle, Quadrilaterals, etc. . 1 to 7

II. Problems of Triangles, Circle, Quadrilaterals, etc. . 8 14

III. Areas, Scale Drawing, Curves, the Ellipse . , , 15 21

IV. Solid Geometry, Plans, etc., Development of Surfaces of

Solids . 22 29

ARITHMETIC AND MENSURATION

V. Multiplication, Addition, etc., of Fractions and Decimals 30 to 39

VI. Involution and Evolution . . . . 40 44

VII. Ratio, Proportion, Percentage . . . 45 49

VIII. Measures (Tables), Duodecimals, Mensuration of Plane

Figures . . . . .... . 50 65

IX. Solid Figures, Capacities of Pipes, Cylinders, Tanks, etc. 66 73

X. Calculations on the Heating Capacities of Pipes and

Boilers, Discharge from Pumps, Ventilation Calcula-

tions/Flow of Air through Tubes, etc. . 74 85

PHYSICS

XI. Properties of Matter, Solids, Liquids, and Gases . 86 to 93

XII. Measurement of Volume, Specific Gravities, Balancing

Columns, Hydrostatics, Head of Water, Pressure due

to same, Capillary Attraction . . . . 94 107

XIII. Mechanics : Levers, Pulleys, Inclined Plane, Wedge,

Screw, Wheel and Axle . . . . 108 121

XIV. Work and Energy, Falling Bodies, Flow of Water through

Apertures and Pipes and over Weirs, Hydraulic Mean

Depth, Bursting Pressure of Pipes, the Hydraulic Ram 122 135

XV. Pneumatics, Barometers, Boyle's Law, Pressure Gauges,

the Syphon, Pumps . . . . . 136 148

XVI. Heat, Thermometers ; Effects of Heat upon Solids,

Liquids, and Gases ; Construction of Thermometers . 149 156

x CONTENTS

PHYSICS-Continued.

CHAP. PAGE

XVII. Expansion of Solids : Practical Application . . 157 to 162

XVIII. Expansion of Liquids and Gases, Maximum Density,

Absolute Temperatures for Gases, Normal Tempera-

ture and Pressure . . . . . 163 169

XIX. Specific Heat, Temperature of Fusion, Latent Heat of

Fusion, and Vaporisation ;. . . .170 180

XX. Transmission of Heat, Conduction, Convection, Eadia-

tion, Table of Conductors and Non - Conductors,

Natural and Mechanical Ventilation, Conduction and

Convection of Liquids and Gases, Humidity . . 181 198

CHEMISTKY

XXI. Elements and Compounds, Chemical and Physical

Changes, Metals and Non-metals, Symbols, Atomic

Weights, Mixtures and Compounds . . . 199 to 206

XXII. Composition of Air, Properties, etc., Oxidation (combus-

tion), Oxygen and Nitrogen, Carbon Dioxide . 207 214

XXIII. Hydrogen and Water, Composition and Preparation and

Properties the Voltameter, Solvency, Solubility of

Salts, Solubility of Gases, Natural Waters, Impurities

in same, Filtration of Water, Hard and Soft Waters 215 226

XXIV. Acids, Bases, and Salts : Preparation and Action upon

Metals of Hydrochloric, Nitric, and Sulphuric

Acids . . . . . . . 227 234

XXV. Carbon and Sulphur, with their Compounds : Charcoal,

CO 2 , CO, H 2 S,SO 2 ; Detection of Lead in Water . 235 ,,243

XXVI. The Common Metals, Iron, Lead, Zinc, Tin, Copper;

Ores of same, Extraction, Purification, Properties,

Uses ; White and Red Lead .... 244 252

XXVII. Alloys, Making of same ; Melting-points and Composi-

tion of various Alloys, including Hard and Soft

Solders, Amalgams, Fluxes ; their Uses . . 253 258

INDEX ..... 259

LIST OF TABLES

PAGE

Arithmetical Signs, etc. ... . . . .38

Measures . . . . . . . . 50-1

Constants for Areas of Polygons , . . . . .60

Coefficients for " Heating " Calculations . . . . .75

Cubic Space allowed for various purposes . . . . .82

Kates of Diffusion of various Gases . . . . .93

Metric Units ......... 95

Relation of English and Metric Units . . . . .96

Specific Gravities . . . . . ' . . .103

Tensile Strengths of Metals . . . . . . .132

Barometric Levels and Weather . . . . . ] 37

Coefficients of Linear, Surface, and Cubical Expansion . . .158

Scale of Absolute Temperatures . . . . . .168

Specific Heats of Solids and Liquids . . . . 1 74

Temperatures of Fusion and of Vaporisation . . . 1 76

Latent Heats . .- . . . . . .176

Boiling-points of Liquids . . . . . . . 1 76

Conductors of Heat ..... .183

Non-conductors of Heat . . . . . . .183

Dew-points at Temperatures 10 to 100 Fahr. . . . .192

Glaisher's Factors . . . . . . . .193

Common Elements ........ 201-2

Composition of Pure Air ..... . 214

Dissolved Impurities in Natural Waters ..... 222

Strengths and other particulars of Metals ..... 252

Alloys of Lead and Tin . .' . . . . .254

Very fusible Alloys ........ 255

Alloys of Antimony . . . . . . .255

Hard Solders . .256

Composition of Brasses . ... . . . . 256

Composition of Bronzes . . ..',. . . . 257

Gold Alloys . | . . . . . . .257

Silver Alloys . . . . . . . .258

Fluxes . 258

LIST OF ILLUSTRATIONS AND FIGURES

Definitions PAGE

1 Definition of a Point (Geometry) ..... 2

2 a Line . . . . ; .2

3 a Line . . . . . ' . . 2

4 a Curved Line . .' . . . . 2

5 a Horizontal Line . * ' . ' . 2

6 ,, a Vertical Line . . * . . .2

7 an Oblique Line . . . . . . . 3

8 Parallel Line . . , . . .3

9 an Angle . . . . . . 3

10 a Right Angle . . . . . . 3

11 an Obtuse Angle . . -. . .4

12 an Acute Angle ...... 4

13 a Circle ....... 4

14 Centre of Circle . , . . . . 5

15 Diameter ...... 5

16 Radius ... . . . . .5

17 a Segment ...... 5

18 an Arc ....... 5

19 a Sector . . . . . 5

20 ,. a Triangle ...... 5

21 an Equilateral Triangle . . . . . 5

22 an Isosceles Triangle * * * - .5

23 a Scalene Triangle i 5

24 a Right-angled Triangle .... 5

25 an Obtuse-angled triangle .... 5

26 an Acute-angled Triangle . . . . 5

27 a Quadrilateral Figure . . . i . 6

28 a Parallelogram . . . . . 6

29 a Square . . < . -'".-'. . 6

30 an Oblong . . . . . . 6

31 a Rhombus . . . . . . 6

32 a Rhomboid ^ . 6

33 a Trapezoid . . . . . . 6

34 a Trapezium . . . . . 6

35 a Polygon . . . . . , .', 6

LIST OF ILLUSTRATIONS AND FIGURES xiii

Problems PAOE

1 To Bisect a Given Line ...... 8

2a Draw a Line Parallel to Given Line . . . .8

26 Draw a Line Parallel through Given Point / . . . 8

3 Bisect a Given Angle . . . .' . . 9

4 Make an Angle equal to a Given Angle . . . . 9

5 Construction of Angle containing a Given Number of Degrees . 9

6 Division of a Line into Equal Parts . . . . . 10

7 Construction of Equilateral Triangle . . . . .10

8 Isosceles Triangle . . . . .10

9 Right-angled Triangle . . . .11

10 Triangle, three sides given . . . .11

11 Square. . . . , .11

12 Square, Diagonal given .. . . .11

13 a Rhombus ...... 12

14 To Find Centre of a Circle . . . . . .12

15 Describe Circle passing through three Points . . .12

16 Describe Circle about a Triangle . . . . .12

17 Describe a Tangent to a Circle . . . . .12

18 Construction of Regular Polygons . . . . .13

19 Regular Hexagon . . . . .14

20 To Inscribe any Regular Polygon within a Circle . . . 14

21 A, B, C, and D Construction of Ellipse ;. 19-21

22 Projection of a Cube Figure . . . . .23

23 Plan and Elevation of a Pyramid . . . . .24

24 Cone 24

1 The Protractor ... ... 2

2 Areas of Parallelograms . . . . . .16

3 Triangles . . .' . . ..16

4 Triangles . . . . . . 16

5 Three sides of Triangles . . . . .16

6 Construction of Scale, | full size . . ., . .17

7 ,, Scale, oV full size . . . . 17

8 Scale, 1 inch == 5 yards . . . . . 17

9 a Diagonal Scale . . . . ..17

10 a Diagonal Scale, showing tenths . . .17

11 a Diagonal Scale, showing tenths and hundredths 17

12 a Diagonal Scale, showing yards, feet, and inches 17

13 Ellipse from Cones . . . . .19

14 Illustrations of Plan and Elevation . . ; . , . . 24

15 Development of Surface of a Skew-cut Cylinder . . .25

16 Curved Surface of a Skew-cut Cone . . 26

xiv LIST OF ILLUSTRATIONS AND FIGURES

Figures Continued. PAGE

1 7 Development of Sides of Frustum of Pyramid . . .27

18 Surface of a Sphere . . . ... 27

19 an Ogee Turret . .' -: . . '. . 28

20 Plans and Sections of Buildings ..... 79

21 To Demonstrate Weight of Air . . . .87

22 Elasticity . . . . % . ' 89

23 that Liquids find their own Level . . ; 90

24^

j- that Liquids communicate Pressure . 91

26 that Liquids communicate Pressure equally in all

directions . . . . .91

27 The Bramah Press . . ... . .92

28 Measurement of Volume of a Solid . . . .94

29 Chemical Balance . . . . . . .95

30 Specific Gravity Bottle . . . . . .97

31 Spring Balance . . . . .98

32 Balance . . . . . .99

33 by Flotation. . . . .100

34 The Hydrometer . . . . . . .101

35 U-Tube 102

36 Hare's Apparatus . . . . . .103

37 Capillarity of Mercury and of Water . . , .. . 106

38 and its Prevention on Roofs . . . . 107

39 The Lever . . . ... . . .109

40 Three Orders of Levers . . . . . .109

41 Application of Levers to Problems . . . . .110

42 Levers to Apparatus . . . . .111

43 Levers to Apparatus . . . . .112

44 Levers to Apparatus . . . . 1. 112

45 Fixed and Movable Pulleys . . . . _'. '.113

A.R \

V Group of First System of Pulleys . ' . . .114

48 ^

> Second System of Pulleys . , . . , v 115

[The Inclined Plane .... . H6-7

51J

52 The Wedge . ", . . . . . 118

53 A, B, C, the Screw . . . . , . .119

54 The Screw "Lifting Jack" ... . . . .120

55 The Wheel and Axle . . . . . .121

56 Falling Bodies . ... . . . .123

57 Vena contracta . . . . . . . .126

58 Discharging Mouthpiece . . . . . 127

LIST OF ILLUSTRATIONS AND FIGURES xv

Figures Continued. PAGE

59 Conoidal Mouthpiece ..... .127

60 Weir .127

61 The Hydraulic Ram .... .133

62 Construction of a Barometer Tube . . ' : '. . 137

63 Clock-face Barometer . . . . . . .138

64 Construction of Barometer . . . . 138

65 Fortin's Barometer . . . . . . .139

66 Aneroid Barometer . . . . . . .139

67 "Boyle's Law" Tube . , , . . . .140

68 Balancing Columns . . . . . . ,141

69 with closed end . . . . . 141

70 The Syphon . . . . . . . 142

71 (emptying a Cylinder) - . . . . 142

72 W.-C. Flushing Tank . . . . 143

73 Field's Flush Tank '".-"., 143

74 Suction Pump ........ 144

75 Lift Pump . . -. . . . 145

76 Force Pump . . . ... . . .145

77 Ram Pump . . - : . .146

78 Double-acting Pump . . . . .. . .146

79 Centrifugal Pump . . . . . . 147

80 Expansion of Solids . . . " . . . .150

81 Liquids . . , . . . . . 150

82 Gases . . . . .150

83 Construction of a Thermometer . . . . . 151

j- Fixing Boiling-point of Thermometer . . . .152

85J

86 Freezing-point of Thermometer '. . . .152

87 Comparison of three Thermometer Scales . . . .153

88 Graph of Centigrade and Fahrenheit Scales . . . .154

89 Maximum Thermometer . . . . . .156

90 Minimum Thermometer . . . . . .156

91 Linear Expansion of Compound Bar . . **'- . .157

92 Force of Contraction of Bar when Cooling . . . .159

93 Superficial Expansion . . . .' . . .160

94 Cubical Expansion . . . . . ... 160

95 A B Expansion Loop and Joint . . . . .161

96 Comparison of Expansion of Three Liquids . . . .164

97 Maximum Density of Water . . .. . . ..165

98 Diagram showing Expansion of Water . .. . .166

99 Expansion of Gases . . . . . . .,167

100 Specific Heat of Four Metals . . . . . ^172

101 Apparatus for finding Melting-point of Metals and Alloys . 175

xvi LIST OF ILLUSTRATIONS AND FIGURES

Figures Continued. PAGE

102 Apparatus for finding Latent Heat of Vaporisation of Water . 178

103 Conductivity of Substance .- . . . ... 181

104 Cooling Effect of Wire Gauze on a Flame . . . .182

105 Miner's Safety Lamp . . . . . ' . .182

106 Apparatus for determining Relative Conductivity of Metals . 183

107 Convection Currents set up by Heat. . . . .184

108 set up by Ice . . . . . 185

109 through Pipes . . \ . . .185

110 in Heating Apparatus . . . .186

111 Cylinder System of Hot-water Supply . . . . . .188

112 Convection in Gases . . . . . . .189

113 The Hygrometer ....... 191

114 \

_,_ I Ventilation of Buildings . . . 195-6

1 lOA I

116 J

117 Distilling Apparatus .......

118 Apparatus Demonstrating Indestructibility of Matter

119 Action of Phosphorus on Water .....

120 Preparation of Oxygen ......

121 Nitrogen ......

122 Composition of Air by Weight .....

123 Action of Sodium on Water . .

124 The Voltameter .......

1 25 Preparation of Hydrogen from Steam ....

126 Kipp Apparatus .......

127 Synthesis of Water .......

128 The Eudiometer Tube .......

129 Air in Water . . . . . .-..(..

130 Pasteur Filter . . . . . .-.'.

131 Berkefeld Filter . . . . <

132 Preparation of HC1 . . . . ' .

133 Solubility of HC1 . . .

134 Preparation of H 2 SO 4 . . .

135 Reverberatory Furnace ......

136 Preparation of Lead Carbonate .....

ELEMENTARY SCIENCE APPLIED TO

SANITATION AND PLUMBERS' WORK

ELEMENTARY PRACTICAL GEOMETRY

CHAPTER I

DEFINITIONS OF TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

THE following has been carefully considered and selected to give the

necessary information in a manner that will be easily understood, with-

out previous instruction in the subject, as far as directly concerns

Plumbers' Work.

INSTRUMENTS

As there is a great benefit derived from the working of the various

problems, and later, in the making of both working and finished

drawings, it is perhaps advisable to give a list and a description of what

is requisite as regards drawing instruments.

A good drawing board, 30 ins. by 22 ins., of yellow pine framed

with hard wood, and with its surface perfectly level and the corners

true right angles.

A "J"-square of suitable length, bound with a bevelled hard wood

edge. Two set squares, having angles of 45 and 60 respectively

the first 6 ins. Ions: and the latter 10 ins. long.

The compasses should have firmly fastened needle points. Buy

just the instruments you require, i.e., a 4 J-in. half-set with knee joints and

ink and pencil points, and a ruling pen for inking-in lines. It is prefer-

able to use a pricker for marking off distances, etc. This can be bought,

or a good one can be made by breaking a sewing needle and forcing

the blunt end into the wood of a common penholder, leaving the point

projecting half an inch. The foregoing instruments may be carried

in a roll of wash leather.

i

TRIANGLES, CIRCLE, -QUADRILATERALS, ETC.

SCALES

These are indispensable. A set of paper scales can be obtained for

a few pence.

The pencils should be of good quality, and for construction a hard

HH or 3H is the best ; for lining-in, an H pencil may be used. When

sharpened, finish to a fine chisel edge with glasspaper.

A protractor (Fig. 1) is an instrument used for measuring angles

already drawn, or for drawing an angle

of a given degree (see Definitions). It

consists either of a circular or semi-cir-

cular disc, made in metal or celluloid. It

may also be used as a flat rule. The

semi-circular celluloid type is a convenient

form; it is very thin, has easily read

degrees, and can be used with a fair

amount of accuracy. Though perhaps not necessary at first, it is

almost impossible to work without one in the later stages.

DRAWING PAPER

For ordinary pencil work smooth cartridge paper is the most suit-

able ; when the drawings have to be inked - in, a better quality is

desirable, such as Whatman's cartridge papers.

DEFINITIONS

Axioms and Explanations of Terms, etc.

Def. 1. A Point denotes position only. It is shown by a dot, or a dot

enclosed in a small circle.

Lines

Def. 2. A Line has length and position, but neither breadth nor thick-

ness. . It is indicated by letters placed at its extremities as A, B.

Various methods of drawing lines are used, such as thick, thin,

dotted, and chain lines.

Def. 3. A Straight line is the shortest distance between two given

points ; it is also called a Right line.

Def. 4. A Curved line is nowhere straight. There are different kinds

of curved lines, such as (a) a simple curve, and (6) a compound

curve.

Def. 5. A Horizontal line is a level line, similar to the surface of still

water.

Def. 6. A Vertical or Perpendicular line is perfectly upright, like a

plumb line.

LINES

3

Def. 7. An Oblique line is neither horizontal nor vertical.

Def. 8. Parallel lines are the same distance apart, and cannot meet,

however far they may be produced.

0DEF I

DEF. 2 and 3

DEF 5

DEF 6

DEF. 9 ^d 12

DEF. 10

DEF II

Angles

Def. 9. An Angle is the inclination of two straight lines to each other

which meet together ; when two straight lines meet at a point

AOB, the "corner" they form is termed "an angle."

Def, 10. When a line EO meets another CD at a point O, so that the

adjacent angles on each side are equal, then that line is said to

be perpendicular, and the angles formed are right angles.

A Right angle is divided into 90 degrees for the purposes of

measuring, etc.

4 TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

In Def. 10, if we continue the line EO below CD, then we have

four right angles. Thus it will be seen (if a circle is described with

centre O) that a circle contains four right angles, or 360.

Def. 11. An Obtuse angle is greater than a right angle.

Def. 12. An Acute angle is less than a right angle.

The Circle

Def. 13. A Circle is a plane figure contained by one line called the

" Circumference," and is such, that all lines drawn from a certain

point within the figure to the circumference are equal to one

another.

TRIANGLES 5

Def. 14. This point is called the "Centre" (Euc. I., Def. 15, 16).

Def. 15. A line drawn through the centre and terminating at each

end in the circumference, is the Diameter of the circle, and

divides the circle into two equal parts, i.e., semi-circles.

Def. 16. The Radius of a circle is the distance from the centre to

any point in the circumference, and a line drawn at right angles

to this line touching the circumference is called a Tangent to

the circle.

Def. 17. A Segment of a circle is the figure contained by a straight

line and the portion of the circumference which it cuts off.

Def. 18. An Arc is a portion of the circumference of a circle.

Def. 19. A Sector is any portion enclosed by two radii and an arc,

i.e., J circle = a Quadrant, J = a Sextant, and -J circle = an

Octant.

Triangles

Any figure formed by straight lines is termed rectilineal.

Def. 20. Trilateral figures, or triangles, are those which are formed by

three straight lines.

Triangles are named (1st) from the comparative lengths of their sides to

each other :

Def. 21. An Equilateral triangle has 3 equal sides,

Def. 22. An Isosceles 2 of its sides equal,

Def. 23. A Scalene 3 unequal sides ;

and (2nd) from the magnitude of their angles :

Def. 24. A Right-angled triangle has one of its angles a right angle,

i.e., 90. The side opposite the right angle is called the " Hypo-

tenuse." In Def. 24 AB is the hypotenuse.

Def. 25. An Obtuse-angled triangle has one obtuse angle.

Def. 26. An Acute-angled triangle has three acute angles (see Def. 22).

In all triangles the sum of the three angles always equals two

right angles, or 180. Thus, if two angles of a triangle are given,

these together, subtracted from 180, will give the third, or in the case

of an isosceles triangle, if the angle at the apex or top be given, the

result of this substracted from 180 and afterwards divided by 2 would

give the angles at the base :

180 - Angle at apex (say 40) H AO / , i . i \

1 ^ = 70 (each angle at base),

for if the three sides of a triangle are equal, the angles are equal ; and

if the two opposite sides are equal then the two opposite angles are

also equal.

TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

Quadrilaterals

Def. 27. A Quadrilateral figure or Quadrangle is bounded by four

straight lines, the four angles equalling four right angles.

Def. 28. A Parallelogram is a quadrilateral figure in which the opposite

sides are parallel. The Square, Rectangle or Oblong, the Rhombus,

and the Rhomboid are parallelograms.

Def. 29. A Square has all its sides equal and its angles right angles.

Def. 30. A Rectangle or Oblong has its opposite sides equal and its

angles right angles.

Def. 31. A Rhombus has all its sides equal, but its angles are not right

angles.

Def. 32. A Rhomboid has its opposite sides equal, but its angles are

not right angles.

Def. 33. A Trapezoid has only two sides parallel.

Def. 34. A Trapezium (A) has none of its sides parallel, but may have

two of its sides equal, as (B). It is then termed a Trapezium or

kite.

Polygons

Def. 35. A and B. A Polygon is a plane figure bounded by more than

four straight lines.

If the sides are equal it is termed a regular Polygon.

If the sides are unequal it is termed an irregular Polygon.

POLYGONS

Polygons are named according to the number of their

sides, viz. :

A Pentagon has five sides.

A Hexagon has six sides.

A Heptagon has seven sides.

An Octagon has eight sides.

A Nonagon has nine sides.

A Decagon has ten sides.

An Undecagon has eleven sides.

A Duodecagon has twelve sides.

CHAPTER II

PROBLEMS OF TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

LINES AND ANGLES

Problem 1. To bisect a given line AB or Arc CD.

With A as centre and a radius greater than half the line

AB, describe the arc EF.

With B as centre and same radius, describe GH, intersect-

ing EF at 1 and 2.

Join 1 and 2, cutting AB at O.

Then AO = OB, and the line AB is bisected, and the line 1-2

is perpendicular to AB.

Treat the arc CD by the same method as shown, and the

line 1-2 is perpendicular to the tangent of the arc CD.

FX

\

PROB

\H

PROB. 2

OF THE

UNIVERSITY OF CALIFORNIA.

Class

ELEMENTARY SCIENCE APPLIED TO

SANITATION AND PLUMBERS' WORK

ELEMENTARY SCIENCE

APPLIED TO SANITATION

AND PLUMBERS' WORK

BY A. HERRING-SHAW

Associate, Royal Sanitary Institute ; Victoria University Sanitary Certificate ; Honours

Silver Medallist, City and Guilds of London Institute ; Assistant Lecturer in

Sanitary Engineering, Victoria University, JVLanchester ; Lecturer, etc.,

and Chief Assistant in the Municipal and Sanitary Engineering

Department, School of Technology, Manchester

SECOND EDITION

GUENEY AND JACKSON

10 PATERNOSTER ROW, LONDON, E.G.

1910

D. VAN NOSTRAND

YORK.

V

PEEFACE TO SECOND EDITION

IN presenting the Second Edition of Elementary Science applied to

Sanitation and Plumbers' Work, the author desires to express his

hearty thanks to all who were kind enough to offer criticisms and

express opinions anent the value of the book.

On mature consideration it has been thought advisable to include

a varied and extensive series of questions and answers in the mensura-

tion section, which will doubtless be useful to teachers and taught.

The calculations on heating and ventilation have been extended,

and several pages added to the mechanics section, in addition to

alterations and extensions in other parts of the book.

To the publishers the author's thanks are due for the skill and care

bestowed in the production of this edition, and, combined with the

extensions and alterations effected in the text, it is hoped the sphere

of usefulness of the work will be greatly extended.

A. HERRING-SHAW:

MUNICIPAL SCHOOL OF TECHNOLOGY,

MANCHESTER, 1910.

PEEFACE TO FIEST EDITION

THE object of the author in compiling the matter contained in the

following pages, is to produce a book on Elementary Science, fully

treated (in its fundamental principles, and application to Sanitation

and Plumbers' work) in such a manner as will be readily understood

by all students. The price has been fixed at a low figure, to bring the

vii

223099

viii PREFACE

book within the reach of all classes. It will be found useful to students

preparing for the Examinations of the City and Guilds of London

Institute, the Registered Plumbers' Company, and the Royal Sanitary

Institute of Great Britain. The examiners to these institutions, almost

without exception, complain of the lack of knowledge of Elementary

Science on the part of candidates who take the above examinations.

The author hopes that the contents will be of material assistance

in raising the standard of the work done by students of Sanitary

Engineering. Should a student find any difficulty in thoroughly

grasping any of the principles set forth in this book, the author will

be pleased to afford him assistance, if it be in his power to do so.

The sincere thanks of the author are due to Professor J. Radcliflfe,

M.Sc.(Tech.), and the members of the staff of the Municipal and

Sanitary Engineering Department, Municipal School of Technology,

Manchester, for valuable suggestions and assistance, and also to various

firms for the loan of electros.

A. HERRING-SHAW.

MUNICIPAL SCHOOL OF TECHNOLOGY,

MANCHESTER.

TABLE OF CONTENTS

ELEMENTARY SCIENCE

CHAP. PAGE

I. Definitions of Triangles, Circle, Quadrilaterals, etc. . 1 to 7

II. Problems of Triangles, Circle, Quadrilaterals, etc. . 8 14

III. Areas, Scale Drawing, Curves, the Ellipse . , , 15 21

IV. Solid Geometry, Plans, etc., Development of Surfaces of

Solids . 22 29

ARITHMETIC AND MENSURATION

V. Multiplication, Addition, etc., of Fractions and Decimals 30 to 39

VI. Involution and Evolution . . . . 40 44

VII. Ratio, Proportion, Percentage . . . 45 49

VIII. Measures (Tables), Duodecimals, Mensuration of Plane

Figures . . . . .... . 50 65

IX. Solid Figures, Capacities of Pipes, Cylinders, Tanks, etc. 66 73

X. Calculations on the Heating Capacities of Pipes and

Boilers, Discharge from Pumps, Ventilation Calcula-

tions/Flow of Air through Tubes, etc. . 74 85

PHYSICS

XI. Properties of Matter, Solids, Liquids, and Gases . 86 to 93

XII. Measurement of Volume, Specific Gravities, Balancing

Columns, Hydrostatics, Head of Water, Pressure due

to same, Capillary Attraction . . . . 94 107

XIII. Mechanics : Levers, Pulleys, Inclined Plane, Wedge,

Screw, Wheel and Axle . . . . 108 121

XIV. Work and Energy, Falling Bodies, Flow of Water through

Apertures and Pipes and over Weirs, Hydraulic Mean

Depth, Bursting Pressure of Pipes, the Hydraulic Ram 122 135

XV. Pneumatics, Barometers, Boyle's Law, Pressure Gauges,

the Syphon, Pumps . . . . . 136 148

XVI. Heat, Thermometers ; Effects of Heat upon Solids,

Liquids, and Gases ; Construction of Thermometers . 149 156

x CONTENTS

PHYSICS-Continued.

CHAP. PAGE

XVII. Expansion of Solids : Practical Application . . 157 to 162

XVIII. Expansion of Liquids and Gases, Maximum Density,

Absolute Temperatures for Gases, Normal Tempera-

ture and Pressure . . . . . 163 169

XIX. Specific Heat, Temperature of Fusion, Latent Heat of

Fusion, and Vaporisation ;. . . .170 180

XX. Transmission of Heat, Conduction, Convection, Eadia-

tion, Table of Conductors and Non - Conductors,

Natural and Mechanical Ventilation, Conduction and

Convection of Liquids and Gases, Humidity . . 181 198

CHEMISTKY

XXI. Elements and Compounds, Chemical and Physical

Changes, Metals and Non-metals, Symbols, Atomic

Weights, Mixtures and Compounds . . . 199 to 206

XXII. Composition of Air, Properties, etc., Oxidation (combus-

tion), Oxygen and Nitrogen, Carbon Dioxide . 207 214

XXIII. Hydrogen and Water, Composition and Preparation and

Properties the Voltameter, Solvency, Solubility of

Salts, Solubility of Gases, Natural Waters, Impurities

in same, Filtration of Water, Hard and Soft Waters 215 226

XXIV. Acids, Bases, and Salts : Preparation and Action upon

Metals of Hydrochloric, Nitric, and Sulphuric

Acids . . . . . . . 227 234

XXV. Carbon and Sulphur, with their Compounds : Charcoal,

CO 2 , CO, H 2 S,SO 2 ; Detection of Lead in Water . 235 ,,243

XXVI. The Common Metals, Iron, Lead, Zinc, Tin, Copper;

Ores of same, Extraction, Purification, Properties,

Uses ; White and Red Lead .... 244 252

XXVII. Alloys, Making of same ; Melting-points and Composi-

tion of various Alloys, including Hard and Soft

Solders, Amalgams, Fluxes ; their Uses . . 253 258

INDEX ..... 259

LIST OF TABLES

PAGE

Arithmetical Signs, etc. ... . . . .38

Measures . . . . . . . . 50-1

Constants for Areas of Polygons , . . . . .60

Coefficients for " Heating " Calculations . . . . .75

Cubic Space allowed for various purposes . . . . .82

Kates of Diffusion of various Gases . . . . .93

Metric Units ......... 95

Relation of English and Metric Units . . . . .96

Specific Gravities . . . . . ' . . .103

Tensile Strengths of Metals . . . . . . .132

Barometric Levels and Weather . . . . . ] 37

Coefficients of Linear, Surface, and Cubical Expansion . . .158

Scale of Absolute Temperatures . . . . . .168

Specific Heats of Solids and Liquids . . . . 1 74

Temperatures of Fusion and of Vaporisation . . . 1 76

Latent Heats . .- . . . . . .176

Boiling-points of Liquids . . . . . . . 1 76

Conductors of Heat ..... .183

Non-conductors of Heat . . . . . . .183

Dew-points at Temperatures 10 to 100 Fahr. . . . .192

Glaisher's Factors . . . . . . . .193

Common Elements ........ 201-2

Composition of Pure Air ..... . 214

Dissolved Impurities in Natural Waters ..... 222

Strengths and other particulars of Metals ..... 252

Alloys of Lead and Tin . .' . . . . .254

Very fusible Alloys ........ 255

Alloys of Antimony . . . . . . .255

Hard Solders . .256

Composition of Brasses . ... . . . . 256

Composition of Bronzes . . ..',. . . . 257

Gold Alloys . | . . . . . . .257

Silver Alloys . . . . . . . .258

Fluxes . 258

LIST OF ILLUSTRATIONS AND FIGURES

Definitions PAGE

1 Definition of a Point (Geometry) ..... 2

2 a Line . . . . ; .2

3 a Line . . . . . ' . . 2

4 a Curved Line . .' . . . . 2

5 a Horizontal Line . * ' . ' . 2

6 ,, a Vertical Line . . * . . .2

7 an Oblique Line . . . . . . . 3

8 Parallel Line . . , . . .3

9 an Angle . . . . . . 3

10 a Right Angle . . . . . . 3

11 an Obtuse Angle . . -. . .4

12 an Acute Angle ...... 4

13 a Circle ....... 4

14 Centre of Circle . , . . . . 5

15 Diameter ...... 5

16 Radius ... . . . . .5

17 a Segment ...... 5

18 an Arc ....... 5

19 a Sector . . . . . 5

20 ,. a Triangle ...... 5

21 an Equilateral Triangle . . . . . 5

22 an Isosceles Triangle * * * - .5

23 a Scalene Triangle i 5

24 a Right-angled Triangle .... 5

25 an Obtuse-angled triangle .... 5

26 an Acute-angled Triangle . . . . 5

27 a Quadrilateral Figure . . . i . 6

28 a Parallelogram . . . . . 6

29 a Square . . < . -'".-'. . 6

30 an Oblong . . . . . . 6

31 a Rhombus . . . . . . 6

32 a Rhomboid ^ . 6

33 a Trapezoid . . . . . . 6

34 a Trapezium . . . . . 6

35 a Polygon . . . . . , .', 6

LIST OF ILLUSTRATIONS AND FIGURES xiii

Problems PAOE

1 To Bisect a Given Line ...... 8

2a Draw a Line Parallel to Given Line . . . .8

26 Draw a Line Parallel through Given Point / . . . 8

3 Bisect a Given Angle . . . .' . . 9

4 Make an Angle equal to a Given Angle . . . . 9

5 Construction of Angle containing a Given Number of Degrees . 9

6 Division of a Line into Equal Parts . . . . . 10

7 Construction of Equilateral Triangle . . . . .10

8 Isosceles Triangle . . . . .10

9 Right-angled Triangle . . . .11

10 Triangle, three sides given . . . .11

11 Square. . . . , .11

12 Square, Diagonal given .. . . .11

13 a Rhombus ...... 12

14 To Find Centre of a Circle . . . . . .12

15 Describe Circle passing through three Points . . .12

16 Describe Circle about a Triangle . . . . .12

17 Describe a Tangent to a Circle . . . . .12

18 Construction of Regular Polygons . . . . .13

19 Regular Hexagon . . . . .14

20 To Inscribe any Regular Polygon within a Circle . . . 14

21 A, B, C, and D Construction of Ellipse ;. 19-21

22 Projection of a Cube Figure . . . . .23

23 Plan and Elevation of a Pyramid . . . . .24

24 Cone 24

1 The Protractor ... ... 2

2 Areas of Parallelograms . . . . . .16

3 Triangles . . .' . . ..16

4 Triangles . . . . . . 16

5 Three sides of Triangles . . . . .16

6 Construction of Scale, | full size . . ., . .17

7 ,, Scale, oV full size . . . . 17

8 Scale, 1 inch == 5 yards . . . . . 17

9 a Diagonal Scale . . . . ..17

10 a Diagonal Scale, showing tenths . . .17

11 a Diagonal Scale, showing tenths and hundredths 17

12 a Diagonal Scale, showing yards, feet, and inches 17

13 Ellipse from Cones . . . . .19

14 Illustrations of Plan and Elevation . . ; . , . . 24

15 Development of Surface of a Skew-cut Cylinder . . .25

16 Curved Surface of a Skew-cut Cone . . 26

xiv LIST OF ILLUSTRATIONS AND FIGURES

Figures Continued. PAGE

1 7 Development of Sides of Frustum of Pyramid . . .27

18 Surface of a Sphere . . . ... 27

19 an Ogee Turret . .' -: . . '. . 28

20 Plans and Sections of Buildings ..... 79

21 To Demonstrate Weight of Air . . . .87

22 Elasticity . . . . % . ' 89

23 that Liquids find their own Level . . ; 90

24^

j- that Liquids communicate Pressure . 91

26 that Liquids communicate Pressure equally in all

directions . . . . .91

27 The Bramah Press . . ... . .92

28 Measurement of Volume of a Solid . . . .94

29 Chemical Balance . . . . . . .95

30 Specific Gravity Bottle . . . . . .97

31 Spring Balance . . . . .98

32 Balance . . . . . .99

33 by Flotation. . . . .100

34 The Hydrometer . . . . . . .101

35 U-Tube 102

36 Hare's Apparatus . . . . . .103

37 Capillarity of Mercury and of Water . . , .. . 106

38 and its Prevention on Roofs . . . . 107

39 The Lever . . . ... . . .109

40 Three Orders of Levers . . . . . .109

41 Application of Levers to Problems . . . . .110

42 Levers to Apparatus . . . . .111

43 Levers to Apparatus . . . . .112

44 Levers to Apparatus . . . . 1. 112

45 Fixed and Movable Pulleys . . . . _'. '.113

A.R \

V Group of First System of Pulleys . ' . . .114

48 ^

> Second System of Pulleys . , . . , v 115

[The Inclined Plane .... . H6-7

51J

52 The Wedge . ", . . . . . 118

53 A, B, C, the Screw . . . . , . .119

54 The Screw "Lifting Jack" ... . . . .120

55 The Wheel and Axle . . . . . .121

56 Falling Bodies . ... . . . .123

57 Vena contracta . . . . . . . .126

58 Discharging Mouthpiece . . . . . 127

LIST OF ILLUSTRATIONS AND FIGURES xv

Figures Continued. PAGE

59 Conoidal Mouthpiece ..... .127

60 Weir .127

61 The Hydraulic Ram .... .133

62 Construction of a Barometer Tube . . ' : '. . 137

63 Clock-face Barometer . . . . . . .138

64 Construction of Barometer . . . . 138

65 Fortin's Barometer . . . . . . .139

66 Aneroid Barometer . . . . . . .139

67 "Boyle's Law" Tube . , , . . . .140

68 Balancing Columns . . . . . . ,141

69 with closed end . . . . . 141

70 The Syphon . . . . . . . 142

71 (emptying a Cylinder) - . . . . 142

72 W.-C. Flushing Tank . . . . 143

73 Field's Flush Tank '".-"., 143

74 Suction Pump ........ 144

75 Lift Pump . . -. . . . 145

76 Force Pump . . . ... . . .145

77 Ram Pump . . - : . .146

78 Double-acting Pump . . . . .. . .146

79 Centrifugal Pump . . . . . . 147

80 Expansion of Solids . . . " . . . .150

81 Liquids . . , . . . . . 150

82 Gases . . . . .150

83 Construction of a Thermometer . . . . . 151

j- Fixing Boiling-point of Thermometer . . . .152

85J

86 Freezing-point of Thermometer '. . . .152

87 Comparison of three Thermometer Scales . . . .153

88 Graph of Centigrade and Fahrenheit Scales . . . .154

89 Maximum Thermometer . . . . . .156

90 Minimum Thermometer . . . . . .156

91 Linear Expansion of Compound Bar . . **'- . .157

92 Force of Contraction of Bar when Cooling . . . .159

93 Superficial Expansion . . . .' . . .160

94 Cubical Expansion . . . . . ... 160

95 A B Expansion Loop and Joint . . . . .161

96 Comparison of Expansion of Three Liquids . . . .164

97 Maximum Density of Water . . .. . . ..165

98 Diagram showing Expansion of Water . .. . .166

99 Expansion of Gases . . . . . . .,167

100 Specific Heat of Four Metals . . . . . ^172

101 Apparatus for finding Melting-point of Metals and Alloys . 175

xvi LIST OF ILLUSTRATIONS AND FIGURES

Figures Continued. PAGE

102 Apparatus for finding Latent Heat of Vaporisation of Water . 178

103 Conductivity of Substance .- . . . ... 181

104 Cooling Effect of Wire Gauze on a Flame . . . .182

105 Miner's Safety Lamp . . . . . ' . .182

106 Apparatus for determining Relative Conductivity of Metals . 183

107 Convection Currents set up by Heat. . . . .184

108 set up by Ice . . . . . 185

109 through Pipes . . \ . . .185

110 in Heating Apparatus . . . .186

111 Cylinder System of Hot-water Supply . . . . . .188

112 Convection in Gases . . . . . . .189

113 The Hygrometer ....... 191

114 \

_,_ I Ventilation of Buildings . . . 195-6

1 lOA I

116 J

117 Distilling Apparatus .......

118 Apparatus Demonstrating Indestructibility of Matter

119 Action of Phosphorus on Water .....

120 Preparation of Oxygen ......

121 Nitrogen ......

122 Composition of Air by Weight .....

123 Action of Sodium on Water . .

124 The Voltameter .......

1 25 Preparation of Hydrogen from Steam ....

126 Kipp Apparatus .......

127 Synthesis of Water .......

128 The Eudiometer Tube .......

129 Air in Water . . . . . .-..(..

130 Pasteur Filter . . . . . .-.'.

131 Berkefeld Filter . . . . <

132 Preparation of HC1 . . . . ' .

133 Solubility of HC1 . . .

134 Preparation of H 2 SO 4 . . .

135 Reverberatory Furnace ......

136 Preparation of Lead Carbonate .....

ELEMENTARY SCIENCE APPLIED TO

SANITATION AND PLUMBERS' WORK

ELEMENTARY PRACTICAL GEOMETRY

CHAPTER I

DEFINITIONS OF TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

THE following has been carefully considered and selected to give the

necessary information in a manner that will be easily understood, with-

out previous instruction in the subject, as far as directly concerns

Plumbers' Work.

INSTRUMENTS

As there is a great benefit derived from the working of the various

problems, and later, in the making of both working and finished

drawings, it is perhaps advisable to give a list and a description of what

is requisite as regards drawing instruments.

A good drawing board, 30 ins. by 22 ins., of yellow pine framed

with hard wood, and with its surface perfectly level and the corners

true right angles.

A "J"-square of suitable length, bound with a bevelled hard wood

edge. Two set squares, having angles of 45 and 60 respectively

the first 6 ins. Ions: and the latter 10 ins. long.

The compasses should have firmly fastened needle points. Buy

just the instruments you require, i.e., a 4 J-in. half-set with knee joints and

ink and pencil points, and a ruling pen for inking-in lines. It is prefer-

able to use a pricker for marking off distances, etc. This can be bought,

or a good one can be made by breaking a sewing needle and forcing

the blunt end into the wood of a common penholder, leaving the point

projecting half an inch. The foregoing instruments may be carried

in a roll of wash leather.

i

TRIANGLES, CIRCLE, -QUADRILATERALS, ETC.

SCALES

These are indispensable. A set of paper scales can be obtained for

a few pence.

The pencils should be of good quality, and for construction a hard

HH or 3H is the best ; for lining-in, an H pencil may be used. When

sharpened, finish to a fine chisel edge with glasspaper.

A protractor (Fig. 1) is an instrument used for measuring angles

already drawn, or for drawing an angle

of a given degree (see Definitions). It

consists either of a circular or semi-cir-

cular disc, made in metal or celluloid. It

may also be used as a flat rule. The

semi-circular celluloid type is a convenient

form; it is very thin, has easily read

degrees, and can be used with a fair

amount of accuracy. Though perhaps not necessary at first, it is

almost impossible to work without one in the later stages.

DRAWING PAPER

For ordinary pencil work smooth cartridge paper is the most suit-

able ; when the drawings have to be inked - in, a better quality is

desirable, such as Whatman's cartridge papers.

DEFINITIONS

Axioms and Explanations of Terms, etc.

Def. 1. A Point denotes position only. It is shown by a dot, or a dot

enclosed in a small circle.

Lines

Def. 2. A Line has length and position, but neither breadth nor thick-

ness. . It is indicated by letters placed at its extremities as A, B.

Various methods of drawing lines are used, such as thick, thin,

dotted, and chain lines.

Def. 3. A Straight line is the shortest distance between two given

points ; it is also called a Right line.

Def. 4. A Curved line is nowhere straight. There are different kinds

of curved lines, such as (a) a simple curve, and (6) a compound

curve.

Def. 5. A Horizontal line is a level line, similar to the surface of still

water.

Def. 6. A Vertical or Perpendicular line is perfectly upright, like a

plumb line.

LINES

3

Def. 7. An Oblique line is neither horizontal nor vertical.

Def. 8. Parallel lines are the same distance apart, and cannot meet,

however far they may be produced.

0DEF I

DEF. 2 and 3

DEF 5

DEF 6

DEF. 9 ^d 12

DEF. 10

DEF II

Angles

Def. 9. An Angle is the inclination of two straight lines to each other

which meet together ; when two straight lines meet at a point

AOB, the "corner" they form is termed "an angle."

Def, 10. When a line EO meets another CD at a point O, so that the

adjacent angles on each side are equal, then that line is said to

be perpendicular, and the angles formed are right angles.

A Right angle is divided into 90 degrees for the purposes of

measuring, etc.

4 TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

In Def. 10, if we continue the line EO below CD, then we have

four right angles. Thus it will be seen (if a circle is described with

centre O) that a circle contains four right angles, or 360.

Def. 11. An Obtuse angle is greater than a right angle.

Def. 12. An Acute angle is less than a right angle.

The Circle

Def. 13. A Circle is a plane figure contained by one line called the

" Circumference," and is such, that all lines drawn from a certain

point within the figure to the circumference are equal to one

another.

TRIANGLES 5

Def. 14. This point is called the "Centre" (Euc. I., Def. 15, 16).

Def. 15. A line drawn through the centre and terminating at each

end in the circumference, is the Diameter of the circle, and

divides the circle into two equal parts, i.e., semi-circles.

Def. 16. The Radius of a circle is the distance from the centre to

any point in the circumference, and a line drawn at right angles

to this line touching the circumference is called a Tangent to

the circle.

Def. 17. A Segment of a circle is the figure contained by a straight

line and the portion of the circumference which it cuts off.

Def. 18. An Arc is a portion of the circumference of a circle.

Def. 19. A Sector is any portion enclosed by two radii and an arc,

i.e., J circle = a Quadrant, J = a Sextant, and -J circle = an

Octant.

Triangles

Any figure formed by straight lines is termed rectilineal.

Def. 20. Trilateral figures, or triangles, are those which are formed by

three straight lines.

Triangles are named (1st) from the comparative lengths of their sides to

each other :

Def. 21. An Equilateral triangle has 3 equal sides,

Def. 22. An Isosceles 2 of its sides equal,

Def. 23. A Scalene 3 unequal sides ;

and (2nd) from the magnitude of their angles :

Def. 24. A Right-angled triangle has one of its angles a right angle,

i.e., 90. The side opposite the right angle is called the " Hypo-

tenuse." In Def. 24 AB is the hypotenuse.

Def. 25. An Obtuse-angled triangle has one obtuse angle.

Def. 26. An Acute-angled triangle has three acute angles (see Def. 22).

In all triangles the sum of the three angles always equals two

right angles, or 180. Thus, if two angles of a triangle are given,

these together, subtracted from 180, will give the third, or in the case

of an isosceles triangle, if the angle at the apex or top be given, the

result of this substracted from 180 and afterwards divided by 2 would

give the angles at the base :

180 - Angle at apex (say 40) H AO / , i . i \

1 ^ = 70 (each angle at base),

for if the three sides of a triangle are equal, the angles are equal ; and

if the two opposite sides are equal then the two opposite angles are

also equal.

TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

Quadrilaterals

Def. 27. A Quadrilateral figure or Quadrangle is bounded by four

straight lines, the four angles equalling four right angles.

Def. 28. A Parallelogram is a quadrilateral figure in which the opposite

sides are parallel. The Square, Rectangle or Oblong, the Rhombus,

and the Rhomboid are parallelograms.

Def. 29. A Square has all its sides equal and its angles right angles.

Def. 30. A Rectangle or Oblong has its opposite sides equal and its

angles right angles.

Def. 31. A Rhombus has all its sides equal, but its angles are not right

angles.

Def. 32. A Rhomboid has its opposite sides equal, but its angles are

not right angles.

Def. 33. A Trapezoid has only two sides parallel.

Def. 34. A Trapezium (A) has none of its sides parallel, but may have

two of its sides equal, as (B). It is then termed a Trapezium or

kite.

Polygons

Def. 35. A and B. A Polygon is a plane figure bounded by more than

four straight lines.

If the sides are equal it is termed a regular Polygon.

If the sides are unequal it is termed an irregular Polygon.

POLYGONS

Polygons are named according to the number of their

sides, viz. :

A Pentagon has five sides.

A Hexagon has six sides.

A Heptagon has seven sides.

An Octagon has eight sides.

A Nonagon has nine sides.

A Decagon has ten sides.

An Undecagon has eleven sides.

A Duodecagon has twelve sides.

CHAPTER II

PROBLEMS OF TRIANGLES, CIRCLE, QUADRILATERALS, ETC.

LINES AND ANGLES

Problem 1. To bisect a given line AB or Arc CD.

With A as centre and a radius greater than half the line

AB, describe the arc EF.

With B as centre and same radius, describe GH, intersect-

ing EF at 1 and 2.

Join 1 and 2, cutting AB at O.

Then AO = OB, and the line AB is bisected, and the line 1-2

is perpendicular to AB.

Treat the arc CD by the same method as shown, and the

line 1-2 is perpendicular to the tangent of the arc CD.

FX

\

PROB

\H

PROB. 2

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