There is a cone whose height is 27 feet, and whose base is 7 feet in diameter : what are its contents ? Place the square of 7 (49) opposite 1: then opposite A is the area of the base. of 27 is 9. Place 9 opposite 1: then opposite the area (38:6) is the answer, 3464 solid feet. TO FIND THE SOLID CONTENTS OF A FRUSTRUM OF A PYRAMID. Rule.-To the product of one end by the other, add the sum of the squares of each end. Place this opposite 144. Then opposite of the length, is the ans ver. Example.- What are the contents of a stick of timber whose larger end is 12, whose smaller end is 8 inches, and whose length is 30 feet ? The product of one end by the other is 96, the square of 12 is 144, the square of 8 is 64. These, all added, make 96 144 64 304. Place this opposite 144. then opposite 10 (f of the length) is the answer, 214 feet. TO FIND THE SOLID CONTENTS OF A FRUSTRUM OF A CONE. RULE.-Multiply each diameter by itself separately, multiply one diameter by the other, add these three products together. Now place the length opposite 382 : then opposite the products thus added, is the answer. To find the Circumference of a Circle from its Diameter, or its Diameter from its Circumfer ence. a Rule.-Place letter c, (found on the circular, opposite fig. 1: then the figures on the fixed part are diameters, and those on the circle are circumferences. Opposite each diameter is its circumference. Example. What is the circumference of a circle whose diameter is 9 inches? Place c opposite fig. 1: then opposite 9 is 28.2, (28 inches and 2 tenths,) the answer. To find the Area of a Circle. RULE.—Place the square of the diameter opposite 1: then opposite the letter A is the area. a Example.—What is the area of a circular garden whose diameter is 11 rods? Place 121 (the square of 11) opposite 1: then opposite letter A is 95.03 rods, the answer. To find the side of a Square equal in area to any given Circle. Rule.-Place '886, found on the circular, opposite fig. 1: then opposite any diameter of a circle upon the fixed part, is the side of a square equal in area, on the circular. Example.—What is the side of a square equal in area to a circle 4 feet in diameter ? Place '886 opposite fig. 1: then opposite 4 is 3-55 feet, the answer. To find the side of the greatest Square that can be inscribed in any given Circle. RULE.—Place '707, found on the circular, opposite fig. 1: then opposite any diameter of a circle (found on the fixed part,) is the side of its in scribed square. : Example.—What is the side of an inscribed square equal in area to a circle 45 rods in diameter ? Place '707 opposite fig. 1: then opposite 45, on the fixed part, is 31.8 rods, the answer. To find the length of one side of the greatest Cube that can be taken from a Globe of a given diameter. RULE.—Place 577, found on the circular, opposite fig. 1: then opposite any diameter, on the fixed part, is the length of one side of the greatest cube. Example. What is the length of the side of the greatest cube that can be taken from a globe 82 inches in diameter ? Place 577 (the gauge point for the side of an inscribed cube) opposite fig. 1: then opposite 82, on the fixed part, is 47:3 (471) inches, the answer. To find the length of the side of the greatest equi lateral triangle that can be inscribed in a given circle. Rule.—Place 87, found on the circular, opposite fig. 1: then opposite any diameter on the fixed part, is the length of the side of an inscribed triangle And opposite the length of the side of any triangle on the circular, is the diameter required to inscribe it in. Example.- What is the length of one side of the greatest equilateral triangle that can be inscribed in a circle 62 inches in diameter ? Place 87 opposite fig. 1: then opposite 62, on the fixed part, is 54 inches, the answer. What is the least diameter of a circle in which a triangle may be inscribed whose side is 6.5 inches (61)? Place 87 opposite fig. 1: then opposite 6-5, on the circular, is 7.48 (7108) inches, the answer. ) 4 To find the length of the side of the greatest figure that can be inscribed in a given circle. 66 66 66 Rule for a 5. 437. Octagon 8 3.83 Nonagon 9 oi 337 Decagon 10 31 Undecagon 11 282 Dodecagon 12 opposite fig. 1: then opposite any given diameter on the fixed part, is the length of the side of the greatest figure that can be inscribed in it. Example 1.- What is the length of one side of the greatest pentagon, or five-sided figure, that can be inscribed in a circle whose diameter is 51 inches ? Place 589 opposite 1: then opposite 51, on the fixed part, is 30 inches, the answer. 26 a Example 2.- What is the length of one side of the greatest nonagon (nine-sided figure) that can be inscribed in a circle 82 feet in diameter ? Place 337 opposite fig. 1: then opposite 82, found on the fixed part, is 27:6 (27) feet, the answer. Example 3.-What is the least diameter of a circle |