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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.

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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 e d, by the breadth or perpendicular height; the product will be the area.

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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 c, 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

radius of its inscribed circle) by the length of one side, and this product again by the number of sides; and half the product will be the area of the polygon.

[For a table of the areas of regular polygons, see pages 67, 68.1

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To find the Area of a Trapezium, fig. 8.

RULE 1.-Draw a diagonal line, a c, to divide the trapezium into two triangles; find the areas of these triangles separately, and add them together.

RULE 2. Divide the trapezium into two triangles, by the diagonal a c, and let two perpendiculars, bf, and de, fall on the diagonal from the opposite angles; then, the sum of these perpendiculars multiplied by the diagonal, and divided by 2, will be the area of the trapezium.

To find the Area of a Trapezoid, fig. 9.

RULE 1.-Multiply the sum of the two parallel sides, a h, d c, by a p, the perpendicular distance

between them, and half the product will be the

area.

RULE 2.-Draw a diagonal, a c, to divide the trapezoid into two triangles; find the areas of those triangles separately, and add them together.

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To find the Area of an Irregular Polygon, a b c d e f g, fig. 10.

RULE.-Draw diagonals to divide the figure into trapeziums and triangles; find the area of each separately, by either of the rules before given for that purpose; and the sum of the whole will be the area of the figure.

To find the Area of a Long Irregular Figure, dcab, fig. 11.

RULE.-Take the breadth in several places, and at equal distances from each other; add all the breadths together, and divide the sum by this number, for the mean breadth; then multiply the mean

breadth by the length of the figure, and the product

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To find the Circumference of a Circle when the Diameter is given; or the Diameter when the Circumference is given.

RULE 1.—Multiply the diameter by 3·1416, and the product will be the circumference; or divide the circumference by 3.1416, and the quotient will be the diameter.

RULE 2.-As 7 is to 22, so is the diameter to the circumference. As 22 is to 7, so is the circumference to the diameter.

RULE 3-AS 113 is to 355, so is the diameter to the circumference. As 355 is to 113, so is the circumference to the diameter.

To find the Area of a Circle.

RULE 1.-Multiply the square of the diameter by 7854; or the square of the circumference by

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