PAGE To measure an Irregular Body another Way, more exactly To find the side of a Cube equal to any given Solid 152 153 The CONIC SECTIONS 154 } 164 The QUADRATURE; OR, MENSURATION OF SURFACES ARISING FROM THE Sections 159 OF A CONE 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 To delineate a Parabola 165 To find the Length of an Arch of a Parabola 166 To find the Area of a Parabola 167 To find the Area of a Frustum of a Parabola 168 Of an Hyberbola 169 To delineate an Hyperbola 170 To find the Length of an Arch of an Hyperbola 172 To find the Area of an Hyperbola • 173 The CUBATURE; OR, MENSURATION OF SOLIDS ARISING FROM THE SECTIONS 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 182 To find the Solidity of the Frustumn of an Hyperbolic Conoid 183 174 181 PAGES To find the Solidity of a Tetraedron To find the Solidity of an Hexaëdron To find the Solidity of an Oxtaëdron To find the Solidity of a Dodecaëdron To find the Solidity of an Icofaedron To find the Solid Contents of the Five Regular Bodies To find the Superficial Contents of the Five Regular To find the Length of the Sides of the Five Regular To continue a Right Line to a greater Length than can be drawn by a Ruler at one Operation 203 To find the Length of any Arch of a Circle To divide a given Line into an infinite Number of To shew that an Angle, as well as a Line, may be con- tinually diminished, and yet never be reduced to To reduce a Parallelogram to a Square equivalent in To increase the Surface of a Geometrical Parallelogram 208 equal Parts by concentric Circles 2 IL 219 220 221 222 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 Con jugate Diameters being given To find the Solidity of a Circular, Elliptical, Parabo lical, or Hyperbolical Spindle Elliptical, Parabolical, or Hyperbolical Spindle 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 Diftances 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 229 To find what Annuity would pay off the National Debt of 250 Millions in 30 Years, at 4 per Cent. 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 Alourished about 350 Years before Christ, was used, in his Lectures, to illustrate and demonstrate to his Pupils the Truth of his Propositions 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 everi Art and Science in Human Life. E R R 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 co' THE YOUNG Geometrician's Companion. D E CI M A L· ARI TH ME T I C. T HIS is a particular Kind of Arithmetic, which 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 for Ten), because it always supposes the Unit or Integer, let it be what it will, whether i Pound, 1 Mile, i 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 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 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. |