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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.
By the foregoing proportions, therefore, the focus of either curve may be found.
The Ellipse has two foci; as have likewise the opposite Hyperbolas; but the Parabola has one only. The Ellipse has its several parts lying within the circumference of the curve; the axis and centre of the Hyperbola lie on the outside, in consequence of the axis being drawn between the vertices of the two opposite Hyperbolas. The axis of the Parabola is of infinite length, because the axis can only touch one point or vertex in the curve.
To find the area of a four-sided figure, whether it be a square, fig. 1, a parallelogram, fig. 2, a rhombus, fig. 3, or a rhomboid, fig. 4.
RULE.-Multiply the length, a b, or c d, by the breadth or perpendicular height; the product will be the area.
To find the area of a triangle, whether it be isosceles, fig. 5, scalene, fig. 6, or right-angled, fig. 7. RULE.-Multiply the length, a b, of one of the sides, by the perpendicular, cd, falling upon it; half the product will be the area.
To find the length of one side of a right-angled triangle, when the lengths of the other two sides are given.
RULE 1.-To find the hypothenuse, a e, fig. 7, add together the squares of the two legs, a b and b c, and extract the square root of that sum.
RULE 2.-To find one of the legs, subtract the square of the leg, of which the length is known, from the square of the hypothenuse, and the square root of the difference will be the answer.
OF REGULAR POLYGONS.
To find the Area of a regular Polygon.
RULE.-Multiply the length of a perpendicular, drawn from the centre to one of the sides (or the