PAGE

To measure an Irregular Body another Way, more

exactly

To find the Side of a Cube equal to any given
Solid

The CONIC SECTIONS

152

153

154

The QUADRATURE; OR, MENSURATION OF
SURFACES ARISING FROM THE SECTIONS

159

OF A CONE

To find the Foci of any Ellipfis

159

To delineate an Ellipfis

160

To find the Circumference of an Ellipfis

161

To find the Area of an Ellipfis

162

To find the Area of a Segment of an Ellipfis

163

To find the Focus of a Parabola

164

To delineate a Parabola

165

To find the Length of an Arch of a Parabola

166

167

168

169

170

172

To find the Area of a Parabola

To find the Area of a Fruftum of a Parabola
Of an Hyberbola

To delineate an Hyperbola

To find the Length of an Arch of an Hyperbola
To find the Area of an Hyperbola

The CUBATURE; OR, MENSURATION OF
SOLIDS ARISING FROM THE SECTIONS
OF A CONE

To find the Solidity of a Spheroid

To find the Solidity of the Segment of a Spheroid
To find the Solidity of the Middle Zone of a Spheroid 177
To find the Solidity of a Parabolic Conoid

To find the Solidity of a Fruftum of a Parabolic
Conoid

To find the Solidity of a Parabolic Spindle
To find the Solidity of the middle Zone of a Parabolic
Spindle

To find the Solidity of an Hyperbolic Conoid

To find the Solidity of the Fruftum of an Hyperbolic
Conoid

178

179

180

181

182

183

PAGE

To gauge, or find the Contents of Houfhold Utenfils, fuch as Tuns, Tubs, Coppers, Cafks, &c. 184

197

To find the Solidity of an Icofaëdron

To find the Solid Contents of the Five Regular Bodies

another Way

To find the Superficial Contents of the Five Regular

Bodies

To find the Length of the Sides of the Five Regular Solids infcribed in a Sphere of any given Dimenfions

ADDITIONAL PROBLEMS

To continue a Right Line to a greater Length than
can be drawn by a Ruler at one Operation
To find the Length of any Arch of a Circle
To divide a given Line into an infinite Number of

198

199

200

203

203

204

Parts

205

To fhew that an Angle, as well as a Line, may be continually diminished, and yet never be reduced to nothing

206

To reduce a Parallelogram to a Square equivalent in
Area to it

207

To increase the Surface of a Geometrical Parallelogram 208 To find the Area of an Oblique plain Triangle, with

out falling a Perpendicular

The Shepherd's Problem

To divide the Area of a Circle into any Number of equal Parts by concentric Circles

209

210

211

PAGE

To find the Area of any Space of Archimedes' Spiral 212
To find the Area of a Cycloid
To find the Area of a Segment, or Part of a Sector of

a Circle
To-describe a Parabola, by having only the Base and
Height given

To find the Length of the Transverse and Conjugate
Axis of an Hyperbola

To delineate an Hyperbola, the Tranfverfe and Con-
jugate Diameters being given

To find the Solidity of a Circular, Elliptical, Parabo-lical, or Hyperbolical Spindle

To find the Solidity of a Fruftum, or Segment of an

Elliptical, Parabolical, or Hyperbolical Spindle

To find the Solidity of a Wedge

To cut a Tree so that the two Parts measured feparately
fhall produce more than the whole Tree
To cut a Tree so that the Part next the greater End
may measure the most poffible

To determine, geometrically, the Point in a given
Right Line, from which the Sum of the Distances
of two Objects fhall be the leaft poffible
The Nature of Cube Numbers exemplified in mea-
furing Stacks of Hay

213

214

215

217

219

220

221

222

223

224

225

226

To find the Difference of the Areas of Ifoperimetrical
Figures

227

To find the Side of a Cubic Block of Gold, which
being coined into Guineas, would pay off the
National Debt

To find what Annuity would pay off the National
Debt of 250 Millions in 30 Years, at 4 per Cent.
Compound Intereft

Of Magic Squares

To Square the Circle

To raise the Earth according to the Propofal of the great Geometrician Archimedes of Syracufe

229

230

231

233

238

PLATO, a celebrated Greek Philofopher, who flourifhed about 350 Years before Chrift, was ufed, in his Lectures, to illuftrate and demonftrate to his Pupils the Truth of his Propofitions by Geometry; and EUCLID, who lived about fourscore Years after him, being educated in PLATO'S School, is faid to have compiled his whole Syftem of Geo metrical Elements only in Reference to Applications of that Kind. But now, the Utility of Geometry extends to every Art and Science in Human Life.

ERRATUM.

Page 73, line 7, after the Period, read, "With the fame Extent, and

one Foot in b, make a Mark at c.

THE YOUNG

Geometrician's Companion.

DE CIMA L.

ARITHMETIC.

HIS is a particular Kind of Arithmetic, which

T enables us to treat Fractions as whole Numbers;

and it is of the greatest Use in all Parts of Mathematical Learning. It receives its Name from Decem (Latin för Ten), because it always fuppofes the Unit or Integer, let it be what it will, whether Pound, 1 Mile, 1 Gallon, to be divided into ten equal Parts, and each of those into 10 more, and fo on, as far as we please.

Definitions.

A Fraction is a Number expreffing fome Part or Parts of an Unit or Integer: So the Half, a Third, or Tenth Part of any Thing are Fractions.

Every Fraction confifts of two Numbers, the Numerator, and the Denominator. The Denominator fhews into how many Parts the Unit or Integer is divided; and the Numerator is the Number expreffing how many of those Parts are intended by the Fraction.