PAGE To measure an Irregular Body another Way, more exactly

152 To find the side of a Cube equal to any given Solid

153

The CONIC SECTIONS

154

159

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

OF A CONE
To find the Foci of any Ellipfis
To delineate an Ellipfis
To find the Circumference of an Ellipfis
To find the Area of an Ellipfis
To find the Area of a Segment of an Ellipsis
To find the Focus of a Parabola
To delineate a Parabola
To find the Length of an Arch of a Parabola
To find the Area of a Parabola
To find the Area of a Frustum 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

159 160 161 162 163 164 165

166

a

a

167 163 169 170

172 • 173

a

a

The CUBATURE; OR, MENSURATION OF.
SOLIDS ARISING FROM The Sections

174 OF A CONE To find the Solidity of a Spheroid

174 To find the Solidity of the Segment of a Spheroid 176 To find the Solidity of the Middle Zone of a Spheroid 177 To find the Solidity of a Parabolic Conoid

178 To find the Solidity of a Frustum of a Parabolic Conoid

179 To find the Solidity of a Parabolic Spindle

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

Spindle
To find the Solidity of an Hyperbolic Conoid

182 To find the Solidity of the Frustum of an Hyperbolic Conoid

183

181

T

Ε Ν Τ S.
To gauge, or find the Contents of Houshold Uten-

lils, such as Tuns, Tubs, Coppers, Calks, &c.

PAGE

1840

The FIVE REGULAR BODIES

188

188

a

Of the Tetraëdron

To find the Solidity of a Tetraëdron

1897

Of the Hexaëdron

190

To find the Solidity of an Hexaëdron

191

Of the Octaëdron

192

To find the Solidity of an Oxtaëdron

193

Of the Dodecaëdron

194

To find the Solidity of a Dodecaëdron

195

Of the Icofaëdron

196

To find the Solidity of an Icofaëdron

197

To find the Solid Contents of the Five Regular Bodies

another Way

To find the Superficial Contents of the Five Regular

Bodies

199

To find the Length of the Sides of the Five Regular

Solids inscribed in a Sphere of any given Dinnen-

fions

198

200

a

ADDITIONAL PROBLEMS

203

To continue a Right Line to a greater Length than

can be drawn by a Ruler at one Oreration 203

To find the Length of any Arch of a Circle

204

To divide a given Line into an infinite Number of

Parts

205

To shew that an Angle, as well as a Line, may,

be con-

tinually 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 203

To find the Area of an Oblique plain Triangle, with-

out falling a Perpendicular

209

The Shepherd's Problem

To divide the Area of a Circle into any Number of

equal Parts by concentric Circles

210

a

1

220

221 222

PAGE To find the Area of any Space of Archimedes' Spiral 212 To find the Area of a Cycloid

213 To find the Area of a Segment, or Part of a Sector of a Circle

214 to describe a Parabola, by having only the Base and Height given

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

217 To delineate an Hyperbola, the Transverse and Conjugate Diameters being given

219 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 separately
shall produce more than the whole Tree

223 To cut a Tree so that the Part next the greater End may measure the most possible

224 To determine, geometrically, the Point in a given

Right Line, from which the Sum of the Distances
of two Objects shall be the least poffible

225 The Nature of Cube Numbers exemplified in measuring Stacks of Hay

226 To find the Difference of the Areas of Isoperimetrical Figures

227 To find the Side of a Cubic Block of Gold, which

being coined into Guineas, would pay off the
National Debt

229 To find what Annuity would pay off the National

Debt of 250 Millions in 30 Years, at 4 per Cent.
Compound Interest

230 Of Magic Squares

231 To Square the Circle

233 To raise the Earth according to the Proposal of the

great Geometrician Archimedes of Syracuse 238

Plato, a celebrated Greek Philofopher, who flourished about 350 Years before Chrif, was used, in his Leca tures, to illustrate and demonstrate to his Pupils the Truth of his Propofitions by Geometry; and Euclid, who lived about fourscore Years after him, being educated in PLATO' School, is said to have compiled his whole System of Geo metrical Elements only in Reference to Applications of tha Kind. But now, the Utility of Geometry extends to every Art and Science in Human Life.

E RR A T U M. Page 73, line 7, after the Period, read, “ With the fame Extent, a one Foot in b, make a Mark at c."

THE YOUNG

Geometrician's Companion.

DEC M A L

AR I TH ME T I C.

HIS is a particular Kind of Arithmetic, which

T ;

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

Definitions.

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

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