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CONTENTS.

URES....

Page
CONIC SECTIONS.

Page

NoTATION.

11

Construction of Figures ...........

48

EXPLANATIONS OF CHARACTERS... 12 Definitions ....

54

To construct a Parabola...........

54

U. S. WEIGHTS AND MEASURES. To construct a Hyperbola..

54

55

Measures of Length.

13

Ellipse

Parabolas ...............

56

Measures of Surface.

14

57

Measures of Capacity

Hyperbolas

14

Measures of Solidity.

15

15

Measures of Weight

MENSURATION OF SURFACES.

Miscellaneous

16 Triangles, Trapeziums, and Trape-

Measures of Value

16

zoids

59

Mint Value of Foreign Coins..

16 Regular Polygons ...

60

Regular Bodies, Irregular Figures,

FOREIGN WEIGHTS AND MEASURES. and Circles.

61

Arcs of a Circle....

62

Measures of Length:

17

Sectors, Segments, and Zones ...... 63

Measures of Surface

18 Ungulas and Ellipses ....

64

Measures of Capacity.

18 Parabolas and Hyperbolas

65

Measures of Solidity

19 Cylindrical Rings and Cycloids.... 66

Measures of Weight...

20 Cylinders, Cones, Pyramids, Spheres,

SCRIPTURE AND ANCIENT MEAS- and Circular Spindles......

67

21

By Mathematical Formule.

Table for finding the Distances of Lines of Circle, Ellipse, and Para-

Objects at sea...

21 bola..

67

Reduction of Longitude

22 Areas of Quadrilaterals, Circle, Cyl-

inder, Spherical Zone or Segment,

VULGAR FRACTIONS

23 Circular Spindle, Spherical Tri-

DECIMAL FRACTIONS

26 angle, or any Surface of Revolu-

tion...

68

Duodecimals.........

30

Capillary Tube....

69

RULE OF THREE.

31 Useful Factors....

69

70

Compound Proportion

32 Examples in Mensuration.........

Involution ............

32

Evolution...

32 AREAS OF THE SEGMENTS OF A

Arithmetical Progression

34 CIRCLE

72

Geometrical Progression

35

LENGTHS OF CIRCULAR ARCS...... 75

Permutation....

36

Combination

36

To find the Length of an Elliptic

Curve.......

77

Position

36

Fellowship

37 LENGTHS OP SEMI-ELLIPTIC ARCS.. 78

Alligation

38

AREAS OF THE ZONES OF A CIRCLE. 80

Compound Interest ......

38

Discount ........

39

MENSURATION OF SOLIDS.

Equation of Payments ..

39

Annuities ....

40

Of Cubes and Parallelopipedons.... 81

Perpetuities ...

41

" Regular Bodies...

81

Chronological Problems....

41

Cylinders, Prisms, and Ungulas 81

To find the Moon's Age

42

Cones and Pyramids ...

82

Table of Epacts, Dominical Letters,

Wedges and Prismoids.

83

&c. ...

42

Spheres:

83

Promiscuous Questions.

43

Spheroids

84

Circular Spindles ..

85

GEOMETRY.

Elliptic Spindles

85

Definitions ....

46 Parabolic Conoids and Spindles. 86
....... 168

Page

HYDROSTATICS.

Of Hyperboloids and Hyperbolic Co-

Page

noids ....

86 Of Pressure...

167

Cylindrical Rings.

87 Construction of Banks............ 168

By Mathematical Formule.........

87 Flood Gates....

Cask Gauging...

88 Pipes ....

......... 168

Examples in Mensuration.

89 Hydrostatic Press.....

......... 169

AREAS OF CIRCLES .....

91 HYDRAULICS AND HYDRODYNAMICS.

CIRCUMFERENCES OF CIRCLES..... 95 Of Sluices

170

SQUARES, CÚBES, AND ROOTS...... 99

Of Vertical Apertures or Slits..... 170

Of Streams and Jets ....

171

To find the Square of a Nurnber Velocity of Streams....

171

above 1000.

116 To find the Velocity of Water run-

To find the Cube or Square Root of ning through Pipes....

172

a higher. Number than is contain- Waves

172

ed in the Table...

118 Table showing the Head necessary

To find the Cube of a Number above

to overcome the Friction of Water

1000

118 in Horizontal Pipes...

173

To find the Sixth Root of a Number 118 General Rules :.

174

To find the Cube or Square Root of Table of the Rise of Watar in Rivers 175

a Number consisting of Integers

WATER WHEELS....

176

and Decimals....

119

SIDES OF EQUAL SQUARES..

120

To find the Power of a Stream..... 176

Barker's Mill..

178

PLANE TRIGONOMETRY....

123

To find the Centre of Gyration of a

Oblique-angled Triangles.

124 Water Wheel....

179

Notes

179

NATURAL Sines, COSINES, AND TAN-

GENTS

...

125

PNEUMATICS.

Sines and Secants ................. 127

Weight, Elasticity, and Rarity of

MECHANICAL POWERS.

Air....

180

Measurement of Heights by means

Lever.

128

of the Barometer.

181

Wheel and Axle..........

130

Velocity and Force of Wind........ 181

Inclined Plane

131

Wedge...

133

STATICS...

Screw

133

Pulley.

135 Pressure of Earth against Walls .. 182

DYNAMICS........

....... 183

CENTRES OF GRAVITY.

PENDULUMS .........

185

Surfaces....

Solids.

138 CENTRE OF GYRATION............. 186

GRAVITATION.......

139 CENTRES OF PERCUSSION AND Os-

CILLATION..........

188

Promiscuous Examples...

142

Gravities of Bodies ............

143 CENTRAL FORCES.................

190

Specific Gravities ....

143 Fly Wheels ............

192

Proof of Spirituous Liquors....... 144 Governors....

192

TABLE OF SPECIFIC GRAVITIES..... 145

GUNNERY .............

193

STRENGTH OF MATERIALS. FRICTION........

194

Cohesion:

148

Transverse Strength.

149

HEAT.

Deflerion..

156

Communication of Caloric....... 196

Journals of Shafts

158

Radiation of Caloric...

197

Gudgeon's and Shafts

160 Specific Caloric.........

197

200

Teeth Or WHEELS.............. 161 | Evaporation...

Congelation and Liquefaction 200

Velocity of Wheels...........

162

Distillation

200

Strength of Wheels.

163

Miscellaneous ...

201

General Explanations concerning

Melting Points of Alloys.

202

Wheels .....

164

GUNPOWDER..

203

HORSE POWER.

165

Dimensions of Powder Barrels.... 203

ANIMAL STRENGTH.

LIGHT.....

204

Men....

.............. 204

.......... 166 TONNAGE

165

Horses ...............

Page

Page

PILING OF BALLS AND SHELLS..... 206 Compositions.

240

Weights and Dimensions of Balls

Sizes of Nuts

..... 240

and Shells ..........

207

Screws.

241

241

WINDING ENGINES.........

Strength of Copper.

208

Digging

241

FRAUDULENT BALANCES.

209 Coal Gasi..

241

MEASURING OF TIMBER.

242

Composition Sheathing Nails 242

STEAM ........

Cement.....

...... 211

... 242

Brown Mortar.

242

Steam acting Expansively....

...... 214

Bricks, Laths, &c.........

243

Hay...

243

STEAM-ENGINE.

Hills in an Acre of Ground.

243

243

Condensing Engines...

215 Displacement of Vessels.

Boilers.

217

Weight of Lead Pipe..

244

Tin

244

Non-condensing Engines.......

218

Boilers ..

Relative

218

Prices of American

General Rules .........

345

219

Wrought Iron.

Saturation in Marine Boilers...... 220

Power required to Punch Iron and

Smoke Pipes, or Chimneys ......

220

245

Copper Plates...

Belts.

220

IRON.

Power of Engines...

221

To find the Volume the Steam of a Weight of Square Rolled Iron...... 246

Cubic Foot of Water occupies · 222 Weight of Round Rolled Iron...... 247

To find the Power of an Engine ne- Weight of Flat Rolled Iron.... 248

cessary to raise Water.

Weight of a Square Foot of Cast

To find the Velocity necessary to

and Wrought Iron, Copper, and

Discharge a Given Quantity of

Lead

251

Water..

223 CAST IRON........

251

COMBUSTION.

252

Weight of Cast Iron Pipes ..

Weight of a Square Foot....

254

Fuel

224

Weight of Cast Iron and Lead Balls 255

Analysis of Fuels..

225 Weight of Copper Rods or Bolts.... 256

Anthracite.

226 Weight of Riveted Copper Pipes .... 256

Charcoal.................

226

Coke.............

257

LEAD.

257

WATER.....iiii...

...... 227

BRASS.......

257

MOTION OF BODIES IN FLUIDS ..... 229

CABLES AND ANCHORS....

232

258

AIR....................

CABLES......

259

DIMENSIONS AND WEIGHT OF GUNS,

SHOT, AND SHELLS, U. S. ARMY. 233 Tables of Hemp Cables and Ropes .. 259

Weight and Dimensions of Leaden

To ascertain the Strength of Cables. 259

Balls

To ascertain the Weight of Manilla

234

259

Expansion of Shot

Ropes and Hawsers.....

234

Weight and Dimensions of Shot.... 234

To ascertain the Strength of Ropes. 260

To ascertain the Weight of Cable-laid

DIMENSIONS AND WEIGHT OF Guns,

Ropes.

260

SHOT, AND SHELLS, U.S. Navy.. 236

To ascertain the Weight of Tarred

Penetration of Shot and Shells...... 237 Ropes and Cables

260

Penetration of Shells .............. 238

BLOWING ENGINES.....

261

MISCELLANEOUS.

Dimensions of a Furnace, Engine,

&c. ........

261

Recapitulation of Weights of Vari-

ous Substances

239

MISCELLANEOUS NOTES.

Weights of a Cubic Foot of Various

Substances

...... 239 On Materials.

262

Slating..

...... 239 Solders, Cements, and Paints ....... 263

Capacity of Cisterns..

240 | Paints, Lackers, and Staining..

NOTATION.

=C.

1=1.
2 = II.

As often as a character is repeated, 3=III.

so many times is its value repeated. 4 = IV. 55V.

A less character before a greater 6 = VI.

diminishes its value, as IV=I from 13VII.

V, or 1 subtracted from 5=4.
8=VIII.
9=IX.

A less character after a greater in10 =X.

creases its value, as XI = X+I, or 20=XX.

10+1=11.
30 =XXX.
40 XL.
50 = L.
60 = LX.
70 LXX.
80 = LXXX.
90 =XC.
100
500
:D, or 10.

For every annexed, this be-
{
comes 10 times as many.

For every C and , placed one at 1,000 M, or CIO. each end, it becomes 10 times as

many. 2,000 = MM. 6,000 =V, or 15. 6,000 = VI. 10,000 = X, or CCIII. 50,000 L, or 1.3.

A bar, thus, over any

number, increases it 1000 60,000LX.

times. 100,000 =, or CCCI. 1,000,000 =M, or CCCCI300. 2,000,000 =MM. EXAMPLES.-1840, MDCCCXL.

18560, XVIIIDLX.

EXPLANATIONS OF THE CHARACTERS USED IN THE FOLLOWING

TABLES AND CALCULATIONS. Equal to, as, 12 inches 1 foot, or 8x8 = 16 X4. + Plus, or more, signifies addition; as, 4+6+5 = 15.

- Minus, or less, signifies subtraction; as, 15–5=10. X Multiplied by, or into, signifies multiplication; as, 8X9=72. Divided by, signifies division; as, 72-7-9=8. : Is to

? :: so is

to 16. : to ✓ Prefixed to any number signifies that the square root of that

number is required ; as, ✓16=4; that is, 4X4=16. y Signifies that the cube root of that number is required; as,

364 = 4; that is, 4X4X4= 64. added to a number signifies that that number is to be squared ;

thus, 42, means that 4 is to be multiplied by 4. 3 added to a number signifies that that number is to be cubed ;

thus, 4*, is =4X4X4=64. The power, or number of times a number is to be multiplied by itself is shown by the num

ber added; as, 2 3 4 5, &c. The bar signifies that the numbers are to be taken together ;

as, 8–2+6= 12, or 3X5+3= 24. Docimal point, signifies when prefixed to a number, that that

number has a unit (1) for its denominator; as, .l is jó, .155

is 15.5., &c. is Signifies difference, and is placed between two quantities when

it is not evident which of them is the greater. • Degrees,' minutes, " seconds," thirds. < Signifies angle. a—, a-, -3, &c., denote inverse powers of a, and are equal

1 1
to
al' a?' a3'

&c. 7 Is put between two quantities to express that the former is

greater than the latter; as, a 7 b, reads a greater than b. L Signifies the reverse; as, a lb, reads a less than b. () Parentheses are used to show that all the figures within them

are to be operated upon as if they were only one ; thus,

(3+2)x5=25. p is used to express the ratio of the circumference of a circle to

its diameter 3.1415926, &c. A AA" A" signifies A, A prime, A second, A third, &c. axd, a.d, or ad, signifies that a is to be multiplied by d.

Note. The degrees of temperature used in this work are those of Fahrenheit.

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