To find the square of a greater number than is contained in the table. RULE 1.—If the number required to be squared exceed, by 2, 3, 4, or any other number of times, any number contained in the table, let the square affixed to the number in the table be multiplied by the square of 2, 3, or 4, &c., and the product will be the answer sought. EXAMPLE.-Required the square of 2595. 2595 is three times greater than 865; and the square of 865, as per table, is 748225. Then, 748225 × 32 = 6734025, Ans. RULE 2.-If the number required to be squared be an odd number, and do not exceed twice the amount of any number contained in the table, find the two numbers nearest to each other, which, added together, make that sum; then, the sum of the squares of these two numbers, as per table, multiplied by 2, will exceed the square required by 1. EXAMPLE.-Required the square of 1865. = × To find the cube of a greater number than is contained in the table. RULE.-Proceed, as in squares, to find how many times the number required to be cubed exceeds the number contained in the table. Multiply the cube of that nnmber by the cube of as many times as the number sought exceeds the number in the table, and the product will be the answer required. EXAMPLE.-Required the cube of 3984. 3984 is 4 times greater than 996; and the cube of 996, as per table, is 988047936. Then 988047936 × 43 = 63235067904, Ans. To find the square or cube root of a higher number than is in the table. RULE. Refer to the table, and seek in the column of squares or cubes, the number nearest to that number whose root is sought, and the number from which that square or cube is derived will be the answer required, when decimals are not of importance. EXAMPLE.-Required the square root of 542869. In the table of squares, the nearest number is 543169; and the number from which that square has been obtained is 737. Therefore, ✔542869 = 737 nearly, Ans. To find more nearly the cube root of a higher number than is in the table. RULE.-Ascertain, by the table, the nearest cube number to the number given, and call it the assumed cube. Multiply the assumed cube and the given number, respectively, by 2; to the product of the assumed cube add the given number, and to the product of the given number add the assumed cube. Then, by proportion, as the sum of the assumed cube is to the sum of the given number, so is the root of the assumed cube to the root of the given number. EXAMPLE.-Required the cube root of 412568555. Per table, the nearest number is 411830784; and its cube root is 744. Therefore, 411830784 x 2 + 412568555 = 1236230123. And, 412568555 x2+411830784-1236967894. Hence, as 1236230123 1236967894 :: 744: 744.369, very nearly, Ans. To find the square or cube root of a number containing decimals. Subtract the square root or cube root of the integer of the given number from the root of the next higher number, and multiply the difference by the decimal part. The product, added to the root of the integer of the given number, will be the answer required. = EXAMPLE.-Required the square root of 321-62. ✓ 321 17.9164729, and✓ 32217.9443584; the difference (·0278855) × ·62 + 17·9164729 = 17.9337619, Ans. THE CONIC SECTIONS. THE plane figures formed by the cutting of a cone by a plane, are five in number, viz: The Triangle, the Circle, the Ellipse, the Hyperbola, and the Parabola. The methods of finding their linear and superficial admeasurement have been already described; the several directions in which the section of the cone is to be made, in order to produce them, are as follows: The Triangle is formed by cutting the cone through the vertex and any part of the base. The Circle, by cutting the cone through the sides, parallel to the base. The Ellipse, by a cut passing obliquely, or at an angle with the base, through both sides of the cone. The Hyperbola, by cutting through one side and the base parallel to the axis, or at a greater angle with the base than that made by the opposite side. The opposite Hyperbola is formed by continuing the cutting plane through an opposite and equal cone, produced by continuing the sides of the first cone through its vertex. The Parabola, by cutting through one side and the base of the cone in a direction parallel to the opposite side, or making an equal angle with the base. The Ellipse has two vertices, being the points in the curve at the extremities of the longest diameter; the Hyperbola has one vertex, or, rather, the opposite Hyperbolas one each; the Parabola has one only. The Transverse Axis is the line uniting the two vertices. The Conjugate Axis is a line drawn through the centre of the transverse axis, and at right angles to it. A Diameter is a right line drawn through the centre, in any direction, and terminated at each end by the curve. A Conjugate Diameter is a line drawn through the centre of any diameter, parallel to the tangent of the curve at the extremity of such diameter. An Ordinate to a Diameter is a line between the diameter and the curve, parallel to its conjugate. The part of the diameter cut off by an ordinate and terminated by its vertex, is called the Abscissa. The Parameter, or latus rectum, is a line drawn through the focus, at right angles to the transverse axis, and terminated by the curve. The parameter of a diameter, in the ellipse and hyperbola, is a third proportional to the diameter and its conjugate; in the parabola, it is a third proportional to one abscissa and its ordinate. The Focus is that point in the transverse axis where the ordinate is equal to half the parameter. |