PAGE To measure an Irregular Body another Way, more exactly To find the Side of a Cube equal to any given The CONIC SECTIONS 152 153 154 The QUADRATURE; OR, MENSURATION OF 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 To delineate an Hyperbola To find the Length of an Arch of an Hyperbola The CUBATURE; OR, MENSURATION OF To find the Solidity of a Spheroid To find the Solidity of the Segment of a Spheroid To find the Solidity of a Fruftum of a Parabolic To find the Solidity of a Parabolic Spindle To find the Solidity of an Hyperbolic Conoid To find the Solidity of the Fruftum of an Hyperbolic 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 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 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 a Circle To find the Length of the Transverse and Conjugate To delineate an Hyperbola, the Tranfverfe and Con- 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 To determine, geometrically, the Point in a given 213 214 215 217 219 220 221 222 223 224 225 226 To find the Difference of the Areas of Ifoperimetrical 227 To find the Side of a Cubic Block of Gold, which To find what Annuity would pay off the National 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. |